ENHANCED ADIC FORMALISM, BIDUALITY, AND PERVERSE
FOR HIGHER ARTIN STACKS
t -STRUCTURES
YIFENGLIUANDWEIZHEZHENG
Inthissequelof[17,18],wedeveloptheadicformalismandextendpreviousresultstoadic
Abstract.
complexes. Inaddition,weintroduceperverse
t -structuresonArtinstacksforgeneralperversity,based
onGabber’sworkonschemes. OurresultsgeneralizeresultsofLaszloandOlssononadicformalismand
∞ -categoriesinthesenseofLurie,byenhancing
middleperversity. Wecontinuetoworkintheworldof
∞ -categories.
allthederivedcategories,functors,andnaturaltransformationstothelevelof
Contents
Introduction
0.1. Results
0.2. Refinement of Deligne’s construction
0.3. Acknowledgments
1. The adic formalism
1.1. Adic objects
1.2. The natural transformations α R and α R
1.3. Evaluation functors
1.4. Adic complexes as limits
1.5. Enhanced six operations and the usual t -structure
1.6. Adic dualizing complexes
2. Perverset -structures
2.1. Perversity evaluations
2.2. Perverset -structures
2.3. Adic perverse t -structures
3. The m-adic formalism and constructibility
3.1. The m-adic formalism
3.2. Finiteness conditions and the usual t -structure
3.3. Constructible adic complexes
3.4. Constructible adic perverse t -structures
3.5. Compatibility with Laszlo–Olsson
4. Descent properties
4.1. Hyperdescent
4.2. Proper descent
4.3. Flat descent
References
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35
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37
Introduction
This is the subsequent article of [17,18] in which we develop a theory of Grothendieck’s six operations
for lisse-étale sheaves on Artin stacks and prove all expected properties including the Base Change theorem. Now we develop the adic formalism and extend all the previous results to adic sheaves/complexes.
Date :February25,2013.
2010 MathematicsSubjectClassification.
14F05,14A20,18D05,18G30.
1
2
YIFENG LIU AND WEIZHE ZHENG
This extends all the previous theories including SGA 5 [2], Deligne [6], Ekedahl [7], Behrend [3] and
Laszlo–Olsson [14]. In addition, we introduce perverset -structures on Artin stacks for general perversity,
based on the work of Gabber [9], which generalizes the construction in [15]. The existing theories all have
one common limitation, that is, the constructibility assumption.
We feel this unnecessary and in fact
insu_cient in certain context, for example, the ` -adic cohomology of Artin stacks which are just locally of
finite type. In this article, we provide the adic formalism for general lisse-étale sheaves, which coincides
the old theories after restricting to constructible ones.
Like our preceding article, the approach we are taking is di_erent from all the previous theories. We
work in the world of ∞ -categories in the sense of Lurie [19,20], by enhancing all the derived categories,
∞ -categories. At this level, we may use some new
functors, and natural transforms to the level of
∞machineries among which the most important ones are gluing objects, Adjoint Functor Theorem,
categorical descent, all in [19,20], and some other techniques developed in [18]. In particular, we obtain
several other special descent properties for the derived category of lisse-étale sheaves.
0.1. Results. In this section, we will state our constructions and results only in the classical setting
of Artin stacks on the level of usual derived categories (which are homotopy categories of the derived
∞ -categories), among other simplification. We will provide the precise references of the complete results
in later chapters, for higher Deligne–Mumford stacks and higher Artin stacks, stated on the level of
stable ∞ -categories. Our statements follow right after those in [18, ø.1]. In particular, we will use the
definitions and notations from the section there.
Let X be an Artin stack and λ be a ringed diagram, that is, a functor from a partially ordered set to the
category of unital commutative rings. Recall that D cart ( X lis-´et ,λ ) is the full subcategory of D( X lis-´et ,λ )
spanned by complexes whose cohomology sheaves are all Cartesian. We define in §1.4 a strictly full
subcategory D(X ,λ ) adic of D cart ( X lis-´et ,λ ) consisting of adiccomplexes, possessing the property that the
inclusion D( X ,λ ) adic → D cart ( X lis-´et ,λ ) admits a right adjoint functor R X . Let f : Y → X be a morphism
of Artin stacks. We then define operations:
_
f : D( X ,λ ) adic → D( Y,λ ) adic ,
kk
X
f _ : D( Y,λ ) adic → D( X ,λ ) adic ;
: D( X ,λ ) adic × D( X ,λ ) adic → D( X ,λ ) adic ,
op
Hom X : D( X ,λ ) adic × D( X ,λ ) adic → D( X ,λ ) adic .
_
The pairs ( f ,f _ ) and ( k K , Hom ( K , k )) for every K 2 D( X ,λ ) adic are pairs of adjoint functors.
We fix a nonempty set L of rational primes. If X is L-coprime, f : Y → X is locally of finite type, and
λ is L-torsion ringed diagram, then we have another pair of adjoint functors:
f ! : D( Y,λ ) adic → D( X ,λ ) adic ,
f ! : D( X ,λ ) adic → D( Y,λ ) adic .
There operations satisfy the similar properties as in the non-adic version (see Proposition 1.5.1 and 1.5.2).
In §1.6, we introduce the adic dualizing complex and prove the biduality property, which will be used
later to prove the compatibility between our theory and Laszlo–Olsson’s [13,14] under their restrictions.
The adic formalism introduced above doesnot assume the constructibility at the first place. In other
words, we are free to talk about adic complexes for any sheaves. In particular, in terms of Grothendieck’s
fonctions-faisceaux dictionary, we make sense of divergent integrals on stacks over finite fields, those
appear for example in [8].
In §2, we define the perverset -structure, in both non-adic and adic setting, for general “perversity” for
(higher) Artin stacks, while in all previous theory only middle perversity is considered [15]. We define a
perversitysmooth(resp.étale)evaluation p on an Artin (resp. a Deligne–Mumford) stack X (Definition
2.1.8) to be an assignment to each atlas u : X → X a weak perversity function pu on X in the sense of
X is a scheme, a perversity
Gabber [9], satisfying certain compatibility condition. In particular, when
étale evaluation is same as a weak perversity function.
Theorem 0.1.1 ((Adic) perverse t -structure, §§2.2, 2.3). Let X bean L-coprime Artinstack equipped
withaperversitysmoothevaluation p and λ bean L-torsionringeddiagram.
A) There is a unique up to equivalence
t -structure ( pD ≤ 0 ( X ,λ ) , p D ≥ 0 ( X ,λ )) on D( X ,λ ) =
u : X → X,
D cart ( X lis - ´et,λ ) , called the
perverse t -structure , such that for every atlas
ENHANCED ADIC FORMALISM FOR ARTIN STACKS
3
u_ p D ≤ 0 ( X ,λ ) = pu D ≤ 0 ( X,λ ) and u_ p D ≥ 0 ( X ,λ ) = pu D ≥ 0 ( X,λ ) , where the corresponding
t -structureonthescheme X isdefinedbyGabber [9].
B) If f : Y → X isasmoothmorphism,then f _ isperverse t -exactwithrespecttocompatibleperversity
smoothevaluations p on X and q on Y.
p ≤0
p
C) Wehaveaboveresultsintheadicsetting,where
D ( X ,λ ) adic = D ≤ 0 ( X ,λ ) ∩ D( X ,λ ) adic .
D) Moreover,theclassicaldescriptionoftheperverse t -structureviacohomologyonstalksagainholds
(Propositions2.2.7and2.3.2).
In particular, when p = 0, we recover the usual t -structure in the non-adic case and obtain the similar
usual t -structure in the adic case. When p is the middle perversity evaluation, we generalize the classical
notion of middle perverse t -structure for schemes, in both non-adic and adic cases.
In §3, we introduce a special case of the adic formalism, namely, them-adic formalism on which there
is a good notion of constructibility. Such formalism is enough for most applications.
Let _ be a ring
and m _ Λ be a principal ideal, satisfying the conditions in De_nition 3.1.1. The typical example is that
Λ is a 1-dimensional valuation ring and m is a proper ideal. The pair (_ , m) corresponds to a ringed
diagram _ • with the underlying category N = { 0 → 1 → 2 →∙∙∙} and _ n = Λ / mn +1 . We call this
setup as the m-adic formalism. An Artin stack X is locally Λ / m-bounded (see Definition 3.2.5) if there is
an atlas X → X such that X has finite (étale) cohomological dimension with respect to the coe_cient
ring _ / m. For such Artin stack X , we have refined description of the usual t -structure on D( X , Λ•) adic
_
and the functor R X . In particular, we show that f : D( X , Λ•) adic → D( Y, Λ •) adic is t -exact (with respect
to usual t -structures) for an arbitrary morphism f : Y → X between locally _ / m-bounded Artin stacks
(Proposition 3.2.4).
Now we fix a pair (_ , m) as above such that _ is Noetherian and _ / m is L-torsion. Let S be either
a quasi-excellent scheme or a regular scheme of dimension ≤ 1, which is L-coprime and locally _ / mbounded, and only consider Artin stacks which are locally of finite type over S. In this setup, we defined
the intersection D cons ( X lis-´et , Λ •) ∩ D( X , Λ•) adic of constructible complexes and adic complexes as the
category of constructible adic complexes. We denote this category by D cons ( X , Λ •) adic which is a full
subcategory of D( X , Λ •) adic . In §3.3 1, we show that the usual t -structure on D( X , Λ •) adic restricts to
the one on Dcons ( X , Λ •) adic . Moreover, the six operations mentioned previously restrict to the following
refined ones:
_
f : D cons ( X , Λ•) adic → D cons ( Y, Λ•) adic ,
kk
X
!
f : D cons ( X , Λ •) adic → D cons ( Y, Λ•) adic ;
•)
•)
•)
: D (cons
( X , Λ •) adic × D (cons
( X , Λ•) adic → D (cons
( X , Λ •) adic ,
op
•)
(+)
Hom X : D (cons
( X , Λ •) adic × D (+)
cons ( X , Λ •) adic → D cons ( X , Λ •) adic .
If f is quasi-compactandquasi-separated, then we have
(+)
f _ : D (+)
cons ( Y, Λ•) adic → D cons ( X , Λ•) adic ,
•)
•)
f ! : D (cons
( Y, Λ •) adic → D (cons
( X , Λ•) adic .
In §3.4, we show that under certain conditions on (_ , m) and the perversity smooth evaluation p, the
adic perverse t -structure restricts to the one on D cons ( X , Λ •) adic . In §3.5, we show that our theory of
constructible adic formalism coincides with Laszlo–Olsson [15] under their assumptions. In particular,
when p is the middle perversity smooth evaluation (that is, the middle perversity function in the case of
schemes), the corresponding (adic) perverset -structure coincides with the one defined by Laszlo–Olsson
[15], under their further restrictions on (_ , m) and X .
In §4, we prove several additional ∞ -categorical descent properties of derived∞ -categories and their
adic version we have constructed. In particular, we have the following theorem, which is the incarnation
on the level of usual derived categories of the main result in §4.1.
Theorem 0.1.2 (Proposition 4.1.9) . Let f : Y → X bemorphismofArtinstacksand
y : Y+0 → Y bea
+
smoothsurjectivemorphism. Let Y• beasmoothhypercoveringof Y withthemorphism yn : Y+n → Y+• 1 =
Y. Put f n = f ◦ yn : Y+n → X .
1We actually denote the (
explanation.
∞ -)category of constructible adic complexes by
D cons ( X , Λ •) in Ÿ3.3.
See §0.2 for an
4
YIFENG LIU AND WEIZHE ZHENG
A) Foreverycomplex K in D ≥ 0 ( Y,λ ) (resp. D ≥ 0 ( Y,λ ) adic ), wehaveaconvergentspectralsequence
p,q
E1 = H q( f py_
_
pK
) ) Hp+ q f _ K
p,q
( resp. E1 = H q( f yp_
_
K
p
) ) H p+ q f _ K ) .
B) If X is L-coprime; λ is L-torsion, and f is locally of finite type, then for every complex K in
D ≤ 0 ( Y,λ ) (resp. D ≤ 0 ( Y,λ ) adic ), wehaveaconvergentspectralsequence
q
!
p+ q
˜ p,q
E
f !K
1 = H ( f • p ! y• p K ) ) H
q
˜ p,q
( resp. E
1 =H( f
y! K
• p! • p
) ) H p+ q f ! K ) .
Finally, we would like to emphasize that all conventions and notations from [18], especially those in
ø.5 there, will be continually adopted in the current article, unless otherwise specified.
0.2. Refinement of Deligne’s construction. Let _ be the ring of integers of a finite extension of Q` ,
m be the maximal ideal of _. Let X be a Noetherian separated algebraic space over Z [1/` ]. Deligne
[6, 1.1.2] defines a category D• ( X, Λ •) adic (which he denotes by D• ( X, Λ) ) as the 2-limit of D • ( X, Λ n),
L
for n 2 N, where the transition functors are the derived tensor k Λ n Λ m for m → n , in the 2-category
of categories. Moreover, he defines a subcategory Dbcons ( X, Λ •) as the 2-limit of D bcons , ftd ( X, Λn), where
b
ftd stand for _nite Tor-dimension. If Hom( K,L
n
n ) is finite for all K,L 2 D cons ( X, Λ •) (this is the case if
X is of finite type over a finite field or an algebraically closed field), then D bcons ( X, Λ •) is a triangulated
category.
We provide a refinement of Deligne’s construction via limit in (part of) §§1.2, 1.3 and 1.4. Indeed, if
we consider the enhancementD ( X ,λ ) adic , which is a presentable stable ∞ -category, of D(X ,λ ) adic , then
D ( X ,λ ) adic is naturally the ∞ -categorical limit of the diagram ξ → D ( X , Λ( ξ )) , where the transition
functors are the ( ∞ -categorical) derived tensor k Λ( ξ 0) Λ( ξ ) for ξ → ξ 0. Here, λ = (Ξ , Λ) is an arbitrary
ringed diagram and X is an Artin stack (see Corollary 1.4.3).
0.3. Acknowledgments. We thank Ofer Gabber, Luc Illusie, Aise Johan de Jong, Joël Riou, Shenghao
Sun, and Xinwen Zhu for useful conversations. Part of this work was done during a visit of the first
author to the Morningside Center of Mathematics, Chinese Academy of Sciences, in Beijing. He thanks
the Center for its hospitality. The second author was partially supported by China’s Recruitment Program
of Global Experts, as well as Hua Loo-Keng Key Laboratory of Mathematics and National Center for
Mathematics and Interdisciplinary Sciences, Chinese Academy of Sciences.
1. The adic formalism
In this chapter, we provide the adic formalism. In §1.1, we work in the general nonsense and define the
abstract “adic objects” in a system assigning each coe_cient a (diagram of) ∞ -category. We then define
the adic category to be the full subcategory spanned by adic objects. In §1.2, we study a natural and
fundamental relation between the adic category and certain limit category constructing from the same
system, refining a consideration of Deligne. In §1.3, we record some simple but important properties of
the system arising from algebraic geometry, namely, those ∞ -categories that are derived categories of
(lisse-)étale sheaves on schemes or Artin stacks. In this geometric case, we may identify the adic category
and the limit category mentioned previously, which is proved in §1.4. In §1.5, we construct the enhanced
adic operation maps and study the usual t -structure on the adic categories. §1.6, we introduce the adic
dualizing complex and prove the biduality property, which will be used later to prove the compatibility
between our theory and Laszlo–Olsson’s [13,14] under their restrictions.
1.1. Adic objects. Let R _ Ring be a full subcategory. We letRind R _ Rind denote the full subcategory
spanned by those ringed diagrams (_, Λ) such that _: _ op → Ring factors through R. In particular, if R
is the subcategory spanned by torsion (resp.L-torsion) rings, then Rind R is Rind tor (resp. Rind L- tor ).
Let C: N( Rind R) op → Pr Lst be a functor. We denote by C(Ξ , Λ) the image of (_ , Λ) under C. Fix an
object (_ , Λ) of Rind R. For every morphism ' : ξ → ξ 0 in _, there is a commutative diagram in Rind R of
ENHANCED ADIC FORMALISM FOR ARTIN STACKS
5
the form
iξ
(Ξ , Λ) o
o
(Ξ /ξ , Λ /ξ )
=
pξ
/( { ξ } , Λ( ξ ))
/
'˜
i'
_
iξ0
_
pξ 0
_ 0
/( { ξ 0} , Λ(
(Ξ ,_Λ) o
(Ξ /ξ 0 ,_Λ /ξ 0 )
_ ξ )) ,
o
/
which induces the following diagram in Pr Lst by applying C:
_
_
iξ
C(Ξ , Λ)
O
= O
/C(Ξ /ξ , Λ /ξ ) o
O
/
o
Oi _'
_
C( { ξ } , Λ( ξ ))
O
O'˜ _
_
iξ0
pξ 0
/C(Ξ /ξ 0 , Λ /ξ 0 ) o
C( { ξ 0} , Λ( ξ 0)) ,
/
o
(resp. pξ 0_ ) be a right adjoint of p_ξ (resp. p_ξ 0 ) and α ' : ˜'p_
C(Ξ , Λ)
where k _ = C( k ). Let pξ _
natural transformation.
pξ
ξ 0_
→ pξi_
_
'
be the
Definition 1.1.1 (Adic object) . An object K of C(Ξ , Λ) is adic (with respect to C) if the natural morphism
_
_
'p
˜
_
ξ 0_ i ξ 0 K
α ' ( i ξ 0K )
kkkkkk→ pξi_ i _'
_
ξ 0K
is an equivalence for every morphism ' : ξ → ξ 0 in _. The target of
It is clear that adic objects are stable under equivalence.
α ' ( i _ξ 0 K ) is equivalent to pξi_ _ξ K .
Let C: N( Rind R) op → Pr Lst be a functor and (_, Λ) be an object of Rind R. We let C(Ξ , Λ) adic _ C(Ξ , Λ)
denote the (strictly) full subcategory spanned by adic objects. It is clearly a stable ∞ -category and the
inclusion is an exact functor. We emphasize that the full subcategory C(Ξ , Λ) adic depends on the functor
C: N( Rind R) op → Pr Lst , not just on C(Ξ , Λ) . If C0 is another such functor, then ( C × C0)(Ξ , Λ) adic is
equivalent to C(Ξ , Λ) adic × C0(Ξ , Λ) adic .
Definition 1.1.2. We introduce the following.
A) A functor C: N( Rind R) op → Cat ∞ is topological if
(a) C factors through Pr Lst .
(b) For every object (_, Λ) of Rind R such that _ admits a final object, C(Ξ , Λ) adic is presentable.
(c) For every morphism (•,γ ) : (Ξ 0 , Λ 0) → (Ξ 1 , Λ 1) of Rind R that carries the final object ξ 0 of
Ξ0 to the final object ξ 1 of _ 1, the following diagram
_
C(Ξ 1 , Λ 1) o
o
pξ 1
C( { ξ 1 } , Λ 1( ξ 1 ))
_
p 0
_
_
0 o ξ
C(Ξ _, Λ )
C( { ξ 0 } , _Λ 0( ξ 0 ))
o
is right adjointable. Moreover, its right adjoint is in Pr Lst .
B) A morphism δ : C0 → C1 in Fun(N( Rind R) op , Cat ∞ ) is topological if
(a) Both C0 and C1 are topological.
(b) For every object (_ , Λ) of Rind R with the final object ξ , the following diagram
0
0_
C0 (Ξ , Λ) o
o
pξ
_
_
p 1ξ
C1 (Ξ_, Λ) o
o
is right adjointable.
C0 ( { ξ } , Λ( ξ ))
_
C1 ( { ξ } _, Λ( ξ ))
6
YIFENG LIU AND WEIZHE ZHENG
top
top
C) We denote by Fun (N( Rind R) op , Cat ∞ ) (resp. Fun (N( Rind R) op , Pr Lst )) the subcategory
op
op
of Fun(N( Rind R) , Cat ∞ ) (resp. Fun(N( Rind R) , Pr Lst )) spanned by topological functors
and topological morphisms.
In other words, an n -cell of Fun(N( Rind R) op , Cat ∞ ) (resp.
top
top
op
L
Fun(N( Rind R) , Pr st )) belongs to Fun (N( Rind R) op , Cat ∞ ) (resp. Fun (N( Rind R) op , Pr Lst ) )
if all its vertices are topological functors and edges are topological morphisms.
Lemma 1.1.3. We have
A) Let C: N( Rind R) op → Cat ∞ beatopologicalfunctor. Thenforeverymorphism (•,γ ) : (Ξ 0 , Λ 0) →
(Ξ 1 , Λ 1) in Rind R ,thefunctor C(•,γ ) : C(Ξ 1 , Λ 1) → C(Ξ 0 , Λ 0) preservesadicobjects,thatis,it
carriesadicobjectstoadicobjects.
B) Let δ : C0 → C1 beatopologicalmorphism. Thenforeveryobject
(Ξ , Λ) of Rind R , thefunctor
0
1
→
C
C
δ(Ξ , Λ) : (Ξ , Λ)
(Ξ , Λ) preservesadicobjects.
Proof. We prove B), and A) is similar.
Let ' : ξ → ξ 0 be a morphism in _.
diagram with all solid squares commutative (up to obvious homotopies):
i 0ξ
C0 (Ξ , Λ)
O
O
δ(Ξ , Λ)
y
C (Ξ , Λ)y
O
O
_
δ/ξ
1_
iξ
1
/C1/ξ
/ O
O
=
~
~
p 0ξ _
/C0/ξ
/ O
O
0_
i'
=
i 0ξ 0
i'
_
i 1ξ 0
y
C1 (Ξ , Λ)y
p 0ξ 0_
/C0/ξ 0
/
1_
δ(Ξ , Λ)
δ/ξ
~
/C1/ξ 0 ~
/
δ( ξ )
1
_
C0 (Ξ , Λ)
/C0 ( ξ )
/ O
O
{
/C ( ξ ) {
/ O
O
1
pξ _
Consider the following
'˜
0
1_
'˜
0_
/C0 ( ξ 0)
/
0
δ( ξ )
p 1ξ 0_
{
/C1 ( ξ 0) {,
/
0
1
where we have written δ/ξ : C/ξ → C/ξ instead of δ(Ξ /ξ , Λ /ξ ) : C0 (Ξ /ξ , Λ /ξ ) → C1 (Ξ /ξ , Λ /ξ ) and
δ( ξ ) : C0 ( ξ ) → C1 ( ξ ) instead of δ( { ξ } , Λ( ξ )) : C0 ( { ξ } , Λ( ξ )) → C1 ( { ξ } , Λ( ξ )) for short. By definition,
we need to show that for every adic object K of C0 (Ξ , Λ) , the natural morphism
1
'˜p1
_ 1
1_
ξ 0_ i ξ 0 δ(Ξ , Λ) K
1_
α ' ( i ξ 0 δ(Ξ , Λ) K )
kkkkkkkkkkk→ p1ξ _ i 1ξ _ δ(Ξ , Λ) K
is an equivalence. But we have the following commutative diagram in hC1 ( ξ )
δ( ξ ) '˜p0_
0
0_
ξ 0_ i ξ 0 K
' '˜δ1_ ( ξ 0) p0ξ 0_ i 0ξ 0_ K
_
δ( ξ )( α 0' ( i 0ξ 0 K ))
/'˜p1_
/
1
ξ 0_ δ/ξ
0
i 0ξ _0 K ' '˜p1_
1
1_
ξ 0_ i ξ 0 δ(Ξ , Λ) K
_
α 1' ( i 1ξ 0 δ(Ξ , Λ) K )
_
δ( ξ ) p0ξ__ i 0φ_ i 0ξ 0_ K ' δ( ξ ) p0ξ _ i 0ξ _ K
_
/p1ξ _ δ/ξi 0ξ _ K ' p1ξ _ i 1ξ _ δ(Ξ , Λ)
_ K ,
/
where the horizontal arrows are equivalences since δ is topological. Therefore, α '1 ( i 1ξ 0_ δ(Ξ , Λ) K ) is an
equivalence sinceα 0' ( i 0ξ 0_ K ) is.
_
top
Lemma 1.1.4. The ∞ -category Fun (N( Rind R) op , Pr Lst ) admitssmalllimitsandthoselimitsarepreservedundertheinclusion Funtop (N( Rind R) op , Pr Lst ) _ Fun(N( Rind R) op , Pr Lst ) .
Proof. We will frequently use [20, 6.2.3.18] in this article.
First, we show that for a limit diagram C: K / → Fun(N( Rind R) op , Pr Lst ) such that C| K factors through
Funtop (N( Rind R) op , Pr Lst ) , then the limit functor C∙
is topological. To check the requirement Ac), we
pick a morphism (• ,γ ) : (Ξ 0 , Λ 0) → (Ξ 1 , Λ 1) of Rind R that sends the final object ξ 0 of _ 0 to the final
ENHANCED ADIC FORMALISM FOR ARTIN STACKS
object ξ 1 of _ 1. By assumption, the functor ( C | K ) | (Δ 1 × Δ 1)
(Ξ 0 , Λ 0)
pξ 0
op
7
, where _ 1 × Δ 1 is the diagram
/( { ξ 0 } , Λ0( ξ 0 ))
/
(• ,γ )
_
(Ξ 1 ,_Λ 1)
_
/( { ξ 1 } , Λ_1( ξ 1 )) ,
/
K × Δ 1 → FunRAd (Δ 1 , Cat ∞ ) . By [20, 6.2.3.18] and the fact that
can be viewed as a functor
Pr Lst _ Cat ∞ preserves small limits, the limit functor C∙
satisfies (c), and the limit diagram factors
→ Ck = C( k ) satisthrough Pr Lst . In particular, for every object k of K , the natural morphism C∙
∙
fies condition Bb). To check that C
satisfies condition Ab), we only need to show that the diagram
D : K / → Fun(N( Rind op
C
)
,
at
)
that
carries
k to (_ , Λ) 7→Ck (Ξ , Λ) adic is a limit diagram. Let C∞ (Ξ , Λ) 0
∞
R
k
C∞ (Ξ , Λ) and contains
be the limit limk K C (Ξ , Λ) adic , which is naturally a strictly full subcategory of
∞
∞
C (Ξ , Λ) adic by [18, 3.1.5] and the proof of Lemma 1.1.3 B). We need to prove
C (Ξ , Λ) 0 _ C∞ (Ξ , Λ) adic .
0
By definition, we may assume _ admits a final object ξ and need to show that for every other object
ξ 2 Ξ, the diagram
pξ 1
_
C∞ (Ξ /ξ , Λ /ξ ) 0 o
O
o
_
i' O
pξ
C∞ ( { ξ } , Λ( ξ ))
O
O'˜ _
_
pξ 0
C∞ (Ξ , Λ) 0 o
C∞ ( { ξ 0} , Λ( ξ 0))
o
induced by the map φ : ξ → ξ 0 is right adjointable. This follows from the fact that, for every
k 2 K,
∞
∞
0
k
k
k
k
k
k
the corresponding diagram with C ( ) replaced by C ( ) and C ( ) replaced by C ( ) adic is right
adjointable.
Second, we show that for an arbitrary diagram C: K / → Funtop (N( Rind R) op , Pr Lst ), the induced mor→ lim( C | K ) in Fun(N( Rind op
phism C∙
R ) , Cat ∞ ) is topological. To check condition Bb), we pick an
k
object (_ , Λ) of Rind R with a final object ξ of _. Then this again follows from [20, 6.2.3.18].
_
Lemma 1.1.5. Let D beafullsubcategoryofan ∞ -category C and f : D → C betheinclusion. Thenthe
pullbackof f inthecategory SetΔ byanyfunctor g : C0 → C withsourcein Cat ∞ isapullbackin Cat ∞ .
Proof. This follows immediately from [18, 3.1.5] applied to the pullback of id C by g.
_
Proposition 1.1.6. Let C: N( Rind R) op → Pr Lst be a topological functor. For every object (Ξ , Λ) , the
inclusion C(Ξ , Λ) adic → C(Ξ , Λ) isamorphismin Pr Lst .
By the proposition, the inclusion C(Ξ , Λ) adic → C(Ξ , Λ) admits a right adjoint R : C(Ξ , Λ) → C(Ξ , Λ) adic
which we call the colocalizationfunctor .
Proof. By definition, the inclusion C(Ξ , Λ) adic _ C(Ξ , Λ) fits into the following diagram
C(Ξ , Λ) adic
_
C(Ξ_, Λ)
/Q ξ 2 Ob(Λ) C(Ξ /ξ , Λ/ξ ) adic
/
Q ξ 2 Ob(Λ)
_
iξ
_
/Q ξ 2 Ob(Λ) C_(Ξ /ξ , Λ/ξ ) ,
/
which is a pullback diagram in Cat ∞ by the above lemma. Since pξ _ commutes with small colimits by
condition Ac) of Definition 1.1.2, C(Ξ /ξ , Λ/ξ ) adic admits small colimits and the inclusion into C(Ξ /ξ , Λ /ξ )
preserves such colimits. By condition Ab) of Definition 1.1.2, the source is presentable (and stable), so
the above inclusion is a morphism in Pr Lst . Therefore, the right vertical arrow is a morphism in Pr Lst since
Λ is small. Moreover, the functor Q ξ 2 Ob(Λ) i _ξ preserves small colimits since eachi _ξ does and _ is small.
Therefore, the inclusion C(Ξ , Λ) adic → C(Ξ , Λ) is a morphism in Pr Lst , because the inclusionPr Lst _ Cat ∞
_
preserves small limits.
8
YIFENG LIU AND WEIZHE ZHENG
Corollary 1.1.7. Let C: N( Rind R) op → Pr Lst beatopologicalfunctorand (Ξ , Λ) beanobjectof Rind R .
Let F : C(Ξ , Λ) → C(Ξ , Λ) beafunctoradmittingrightadjointssuchthattheset
F ( C(Ξ , Λ) adic ) iscontained in C(Ξ , Λ) adic . Then the induced functor F | C(Ξ , Λ) adic : C(Ξ , Λ) adic → C(Ξ , Λ) adic admits right
adjointsaswell.
Proof. By the above proposition, C(Ξ , Λ) adic is presentable (without assuming that _ admits the final
object). Therefore, by Adjoint Functor Theorem, we only need to show that F | C(Ξ , Λ) adic preserves
C(Ξ , Λ) adic _ C(Ξ , Λ) preserves small
small colimits. This then follows from the fact that the inclusion
colimits, and the assumptions on F .
_
To conclude this section, we introduce another formalism, which makes use of a limit construction
instead of adic objects. This can be seen as a refinement of Deligne’s construction [6, 1.1.2].
Notation 1.1.8. Let C: N( Rind R) op → Pr Lst be a topological functor. For every objectλ = (Ξ , Λ) of Rind R,
there is a tautological functor λ _ : Ξ → Rind R sending ξ to ( _, Λ( ξ )) . Composing with C, we obtain a
diagram C ◦ N( λ _ ) op : N(Ξ) op → N( Rind R) op → Pr Lst . Let C(Ξ , Λ) be the limit limk C ◦ N( λ _ ) op , viewed as
an object of Pr Lst . We will construct a natural functor C(Ξ , Λ) adic → C(Ξ , Λ) in §1.2 (in a coherent way)
and show in §1.4 that it is an equivalence if C comes from the derived ∞ -categories of (higher) stacks.
α R and α R .
1.2. The natural transformations
PrL
P rL
Notation 1.2.1. We denote by Funtop (N( Rind R) op , Mon P f st ( Cat ∞ )) _ Fun(N( Rind R) op , Mon P f st ( Cat ∞ ))
top
the subcategory spanned by cells that factor through Fun
(N( Rind R) op , Cat ∞ ) . Here we view
L
Pr
Fun(N( Rind R) op , Mon P f st ( Cat ∞ )) as a subcategory of Fun(Pf , Fun(N( Rind R) op , Cat ∞ )) .
We construct two endofunctors
A.1)
A dicR , A dicR : Funtop (N( Rind R)
op
P rL
, Mon P f st ( Cat ∞ )) → Funtop (N( Rind R)
op
PrL
, Mon P f st ( Cat ∞ )) ,
and a natural transformation α R : A dicR → A dicR. If we represent objects of the functor category by
their compositions with the restriction functor G ζ in [18, 1.5.8 D)], then A dicR (resp. A dicR) sends an
•
C 0f
_
•
object C0 × C kkkkkk→ C0 to
•
C 0f
_
•
C0adic × Cadic kkkkkk→ C0adic
0
•
C 0f
_
•
0
(resp. C × C kkkkkk→ C ) .
Here, Cadic (resp. C) takes the value Cadic (Ξ , Λ) (resp. C(Ξ , Λ) ) at (_ , Λ) , which justifies the notation.
Rind R → Cat 1 sending (_ , Λ) to _.
By definition, there is a tautological functor
Applying
→
R
Rind univ
Grothendieck’s construction, we obtain an op-fibration π: Rind univ
ind
.
The
objects
of
R
R
R
are pairs ((_ , Λ) ,ξ ) where (_ , Λ) is an object of Rind R and ξ is an object of _. Moreover, there are
univ
functors puniv : Rind R × [1] → Rind R sending ((_ , Λ) ,ξ ) × [1] to pξ : (Ξ /ξ , Λ /ξ ) → ( { ξ } , Λ( ξ )) , and
univ
i univ : Rind R _R ind R Rind R → Rind R sending the morphism ((_ , Λ) ,ξ ) → (Ξ , Λ) to i ξ : (Ξ /ξ , Λ /ξ ) →
(Ξ , Λ) .
Let
PrL
Cuniv : N( Rind R) op → Fun(Funtop (N( Rind R) op , Mon P f st ( Cat ∞ )) × Pf , Cat ∞ )
Cuniv and carrying
be the universal functor, and letCuniv
adic be the functor with the same source and target as
P r Lst
L
(Ξ , Λ) to ( C,X ) 7→(G X ◦ C) adic (Ξ , Λ). Here GX : Mon P f ( Cat ∞ ) → Pr st is the evaluation functor at X
defined in [18, 1.5.8 D)]. By the definition of adic objects, we can apply partial right adjunction to
univ op
Cuniv
) with respect to direction 2 to obtain a functor
adic ◦ N( p
N( Rind univ
) op × Δ 1 → Fun(Funtop (N( Rind R)
R
op
PrL
, Mon P f st ( Cat ∞ )) × Pf , Cat ∞ ) .
univ op
Composing this with Cuniv
) , we obtain a functor
adic ◦ N( i
F : N( Rind R)
op
top
_N( R ind R ) op N( Rind univ
) op → Fun(Fun (N( Rind R)
R
op
PrL
, Mon P f st ( Cat ∞ )) × Pf , Cat ∞ ) ,
univ op
whose restriction to N( Rind R) op is (equivalent to) Cuniv
) is (equivalent
adic and restriction to N( Rind R
univ
univ
op
to) Cadic ◦ N( t ) , where t : Rind R → Rind R is the functor carrying ((_ , Λ) ,ξ ) to ( { ξ } , Λ( ξ )) . Let F 0 be
univ
univ
the right Kan extension of F | N( Rind R ) op along the inclusion N(Rind R ) op _ N( Rind R) op _N( R ind R ) op
ENHANCED ADIC FORMALISM FOR ARTIN STACKS
9
univ
N( Rind R ) op . Then the restriction of F → F0 to N( Rind R) op provides the desired natural transformation
α R : A dicR → A dicR. To see this, the only nontrivial point is the following: By construction and the
definition of right Kan extensions, for every object λ of Rind R, F0 ( λ ) is the limit of the diagram
N( Rind R)
op
λ/
univ
× N( R ind R ) op N( Rind univ
) op → N( Rind R ) op
R
F
k→Fun(Funtop (N( Rind R)
PrL
op
, Mon P f st ( Cat ∞ )) × Pf , Cat ∞ ) ,
op
, Mon P f st ( Cat ∞ )) × Pf , Cat ∞ ) ,
which is equivalent to the limit of the diagram
{ λ }×
N( R ind R ) op
univ
N( Rind R
univ
) op → N( Rind R ) op
F
k→Fun(Funtop (N( Rind R)
PrL
by [19, 4.1.2.10, 4.1.2.15].
The latter diagram, restricted to each fixed object ( C,X ) of
P r Lst
top
op
Fun (N( Rind R) , Mon P f ( Cat ∞ )) × Pf , is nothing but G X ◦ C ◦ N( λ _ ) op .
Notation 1.2.2. We denote by Funtop (N( Rind R) op , Pr Lst , cl )) the subcategory of Fun(N( Rind R)
spanned by cells that factor through Fun top (N( Rind R) op , Cat ∞ ) after viewing Fun(N( Rind R)
as a subcategory of Fun(N(F in _) , Fun(N( Rind R) op , Cat ∞ )) .
, Pr Lst , cl )
op
, Pr Lst , cl )
op
We construct two endofunctors
A dicR , A dicR : Funtop (N( Rind R)
A.2)
op
, Pr Lst , cl ) → Funtop (N( Rind R)
op
, Pr Lst , cl ) ,
and a natural transformation α R : A dicR → A dicR , such that A dicR (resp. A dicR ) sends an objectC to
Cadic (resp. C ), where aG ◦ Cadic _ (Ξ , Λ) = (G ◦ C ) (Ξ , Λ) adic (resp. aG ◦ C _ (Ξ , Λ) = (G ◦ C ) (Ξ , Λ) ).
Here G: Pr Lst , cl → Pr Lst is the forgetful functor defined in [18, 1.5.3]. The construction of α R is exactly
the same as that of α R : A dicR → A dicR.
Remark 1.2.3. We have the following properties of compatibility.
A) If R0 _ R, then the following diagram
Funtop (N( Rind R)
op
PrL
P rL
/Funtop (N( Rind R 0 ) op , Mon P f st ( Cat ∞ ))
/
, Mon P f st ( Cat ∞ ))
A dic R 0
A dic R
_
PrL
top
op
Fun (N( Rind R) _, Mon P f st ( Cat ∞ ))
_
L
_ , Mon PP rf st ( Cat ∞ ))
/Funtop (N( Rind R 0 ) op
/
commutes up to homotopy. The same holds for A dicR and α R .
B) The map pf [18, A.1)] induces a map
Fun(N( Rind R)
op
, pf): Fun((Δ )1
op
, Funtop (N( Rind R)
op
, Pr Lst , cl )) → Funtop (N( Rind R)
op
PrL
, Mon P f st ( Cat ∞ )) .
Moreover, by construction, the following diagram
Fun((Δ 1)
op
Fun((Δ )
top
, Fun
1 op
(N( Rind R)
op
, Pr Lst , cl ))
op
Fun(N( R ind R ) , pf)
/Funtop (N( Rind R)
/
op
PrL
, Mon P f st ( Cat ∞ ))
A dic R
, A dic R )
_
_
Fun(N( R ind R ) , pf)
PrL
L
op
/Funtop (N( Rind R) op_, Mon P f st ( Cat ∞ ))
Fun((Δ 1) op , Funtop (N(
_ Rind R) , Pr st , cl ))
/
commutes up to homotopy. We have the same phenomena forA dicR and α R .
op
The following lemma is similar to Lemma 1.1.4 and will be used in §1.3.
Lemma 1.2.4. We have
A) The ∞ -category Funtop (N( Rind R)
top
servedundertheinclusion Fun
op
PrL
, Mon P f st ( Cat ∞ )) admitssmalllimitsandthoselimitsarepre-
(N( Rind R)
op
PrL
, Mon P f st ( Cat ∞ )) _ Fun(N( Rind R)
op
PrL
, Mon P f st ( Cat ∞ )) .
10
YIFENG LIU AND WEIZHE ZHENG
top
B) The ∞ -category Fun (N( Rind R) op , Pr Lst , cl )) admitssmalllimitsandthoselimitsarepreserved
top
undertheinclusion Fun (N( Rind R) op , Pr Lst , cl )) _ Fun(N( Rind R) op , Pr Lst , cl )) .
Proof. Both follow from Lemma 1.1.4, (the dual of) [19, 5.1.2.3], and [20, 3.2.2.5].
1.3. Evaluation functors.
Chp Ar
L
Chp DM
C hp Ar
_
Recall that we have maps
_
Ar
EO: δ2, { 2} Fun(Δ 1 , ChpL ) cart
F 0,A
_
DM
EO: δ2, { 2} Fun(Δ 1 , Chp ) cart
F 0,A
→
→
PrL
→ Fun(N( Rind L-tor ) op , Mon P f st ( Cat ∞ ));
PrL
→ Fun(N( Rind tor ) op , Mon P f st ( Cat ∞ ));
_
EO : ( ChpAr ) op → Fun(N( Rind) op , Pr Lst , cl ) .
pξ
For an object (_ , Λ) of Rind admitting the _nal object ξ 2 Ξ, we have both the projection (Ξ , Λ) k→
sξ
_
( { ξ } , Λ( ξ )) and the inclusion ( { ξ } , Λ( ξ )) k→ (Ξ , Λ). Now if C = Chp Ar EO ( X ), where X is a higher Artin
_
stack, the induced functors pξ,s
_ ξ : C(Ξ , Λ) → C( { ξ } , Λ( ξ )) are canonically and coherently equivalent in
a sense that we now explain.
Let Rind fin _ Rind be the subcategory spanned by objects (Ξ , Λ) admitting final objects and morphisms preserving final objects. We have natural functors Rind fin → Fun([1], Rind) sending (Ξ , Λ) to
pξ
sξ
(Ξ , Λ) k→ ( { ξ } , Λ( ξ )) and ( { ξ } , Λ( ξ )) k→ (Ξ , Λ) , respectively, whereξ is the final object of _.
_
Apply the functor Chp Ar EO and the forgetful functor Pr Lst , cl → Cat ∞ . On the one hand, we obtain a
functor
p
_
Ar
fin
EO : ( Chp ) op × N( Rind ) op → Fun(Δ 1 , Cat ∞ )
Chp Ar
_
pξ
sending (X, (Ξ , Λ)) to D ( X, ( { ξ } , Λ( ξ ))) k→ D ( X, (Ξ , Λ)) . On the other hand, we have
_
s
Chp Ar EO
: ( ChpAr ) op × N( Rind fin ) op → Fun(Δ 1 , Cat ∞ )
_
sξ
sending (X, (Ξ , Λ)) to D ( X, (Ξ , Λ)) k→ D ( X, ( { ξ } , Λ( ξ ))) . We view these functors as
Ar
fin
( Chp ) op × N( Rind ) op × N( F in _) → Fun(Δ 1 , Cat ∞ ) .
p
_
_
RAd
s
factorsthrough Fun
(Δ 1 , Cat ∞ ) and Chp Ar EO factorsthrough
Chp Ar EO
p
_
_
1
, Cat ∞ ) . Moreover, Chp Ar EO is equivalent to _ ◦ Chp Ars EO , where _ is the equivalence
1
, Cat ∞ ) → FunRAd (Δ 1 , Cat ∞ ) in [20, 6.2.3.18 B)].
Lemma 1.3.1. The functor
FunLAd (Δ
LAd
Fun
(Δ
In particular, p_ξ is a left adjoint of s_ξ .
qc . sep op
Proof. When k = k 2, we have an equivalence between the two functors (Sch
) × ( Rind) op → A b
_
sending (X, (Ξ , Λ)) to exactandmonoidal functors pξ,s
_ ξ : Mod( X, (Ξ , Λ)) → Mod( X, Λ( ξ )) , respectively.
In this case the lemma follows from the construction in [18, 2.2].
p
_
In general, we first show by induction on k that the functor Chp k -Ar EO factors through
p
_
RAd
k that Chp k -Ar EO
Fun
(Δ 1 , Cat ∞ ). This holds by descent.
Next we prove by induction on
_
s
and _ ◦ Chp k -Ar EO are equivalent. This holds by [18, 4.1.1] since smooth surjective morphisms are of
p
_
_
s
_
both Chp Ar EO and Chp Ar EO -descent.
Remark 1.3.2. In general, if (_ , Λ) is an object of
sξ
Rind and ξ 2 Ξ, we have successive inclusions
iξ
eξ : ( { ξ } , Λ( ξ )) k→ (Ξ /ξ , Λ/ξ ) k→ (Ξ , Λ) which induce the evaluationfunctor(at
_
eξ :
ξ)
D ( X, (Λ , Ξ)) → D ( X, Λ( ξ ))
for a higher Artin stack X . By the above lemma,e_ξ and pξ _ ◦ i _ξ are canonically and coherently equivalent.
For brevity, we sometimes also write K ξ for e_ξ K for an object K 2 D ( X, (Ξ , Λ)).
The functor
Y e_ξ : D ( X, (Λ , Ξ)) → Y D ( X, Λ( ξ))
ξ2 Ξ
ξ2 Ξ
ENHANCED ADIC FORMALISM FOR ARTIN STACKS
11
qc . sep
is conservative. This is obvious whenX is in Sch
. The general case follows, because simplicial limits
of conservative functors are conservative.
_
qc . sep
Lemma 1.3.3. Let X beaschemein Sch
, C = Chp Ar EO ( X ) : N( Rind) op → Pr Lst . Let (Ξ , Λ) bean
p_ξ : C( { ξ } , Λ( ξ )) →
objectof Rind suchthat Λ admitsafinalobject ξ . Thentheimageofthenaturalmap
C(Ξ , Λ) is contained in C(Ξ , Λ) adic . Moreover, the induced map p_ξ : C( { ξ } , Λ( ξ )) → C(Ξ , Λ) adic is an
equivalenceof ∞ -categories. Inparticular, C(Ξ , Λ) adic ispresentable.
Proof. By the definition of adic objects, to show the first assertion, it su_ces to show that the unit
transformation id C( { ξ } , Λ( ξ )) → pξp_ _ξ is an equivalence. This follows from condition Bb) of Defintion 1.1.2
applied to pξ , which is a special case of Lemma 1.3.1. For the second assertion, we only need to show
_
that for every adic complex K 2 D (Ξ , Λ) adic , the adjunction map pp
ξ ξ _ K → K is an equivalence. Since
_
_ _
_
Q ξ 02 Ξ eξ 0 is conservative, this is equivalent to showing that b: eξ 0 pp
ξ ξ _ K → eξ 0 K is an equivalence for
0
0
every object ξ 2 Ξ. Let φ be the map ξ → ξ. Since K is adic, the composite
˜_
φp
a
˜_
k→pξ 0_ p_ξ 0 φp
ξ_K
_ _
' pξ 0_ ipp
φ ξ
ξ_K
ξ_K
b
k→pξ 0_ i _φ K
is an equivalence. Moreover, we have shown thata is an equivalence. Therefore,b is an equivalence. _
Proposition 1.3.4. We have
A) The map
EO ) defined above factors through the subcategory
Chp
L
P
PrL
r
top
st
Fun (N( Rind L- tor ) op , Mon P f ( Cat ∞ )) (resp. Fun (N( Rind tor ) op , Mon P f st ( Cat ∞ )) ).
_
Themap Chp Ar EO factorsthroughthesubcategory Funtop (N( Rind) op , Pr Lst , cl ) .
Chp Ar
L
EO
(resp.
DM
top
B)
Proof. By the construction of the above maps by descent and Lemma 1.2.4, we only need to show for
PrL
_
top
A) that Sch qc . sep EO factors though Fun (N( Rind tor ) op , Mon P f st ( Cat ∞ )) , and for B) that Sch qc . sep EO
factors through Funtop (N( Rind) op , Pr Lst , cl ) . We check the conditions in Definition 1.1.2: Aa) is automatic;
_
Ab) follows from Lemma 1.3.3; Ac) and Bb) follow from Lemma 1.3.1 and Ba) is then automatic.
Remark 1.3.5. Composing T
[18, B.1)] and the functor
N( PTopos) × N( Rind) → N( RingedPTopos)
induced by [18, C.4)], carrying ( T, (Λ , Ξ)) to ( T Ξ, Λ), we obtain a functor
PT opos EO
_
: N( PTopos)op → Fun(N( Rind) op , Pr Lst , cl ) .
The arguments of this section show that PT opos EO
1.4. Adic complexes as limits.
_
factors through Fun top (N( Rind) op , Pr Lst , cl ) .
We put A dic = A dicR ing , A dic = A dicR ing , and α
= α R ing .
12
YIFENG LIU AND WEIZHE ZHENG
Definition 1.4.1 (Enhanced adic operation map). Define the following four groups of data. Each contains
two functors and a natural transformation between them.
_
adic
EO
Chp Ar
L
= A dicR ing L-tor ◦
Chp Ar
L
Ar
0
EO: δ2, { 2} Fun(Δ 1 , ChpL ) cart
F ,A
EO
Chp Ar
L
α
EO
Chp Ar
L
= A dicR ing L-tor ◦
Chp Ar
L
cart
0
EO: δ2, { 2} Fun(Δ 1 , ChpAr
L ) F ,A
_
C hp DM
EO: δ2, { 2} Fun(Δ 1 , ChpDM ) cart
F 0,A
Chp DM EO
α
Chp DM EO
_
adic
Chp Ar EO
_
Chp Ar EO
_
α
Chp Ar EO
_
adic
PT opos EO
= A dicR ing tor ◦
C hp DM
EO: δ2, { 2} Fun(Δ 1 , Chp
_
α
PT opos EO
→
→ Funtop (N( Rind L-tor ) op , Mon P f st ( Cat ∞ )) ,
_
= A dicR ing tor ◦
_
→ Funtop (N( Rind L-tor ) op , Mon P f st ( Cat ∞ )) ,
PrL
= α R ing L- tor ◦ ( Chp Ar
EO × id Δ 1 );
L
adic
Chp DM EO
PT opos EO
PrL
→
_
DM cart
) F 0,A
PrL
→
→ Funtop (N( Rind tor ) op , Mon P f st ( Cat ∞ )) ,
→
→ Funtop (N( Rind tor ) op , Mon P f st ( Cat ∞ )) ,
PrL
= α R ing tor ◦ ( Chp DM EO × id Δ 1 );
= A dic ◦
Chp Ar
_
EO : ( ChpAr ) op → Funtop (N( Rind) op , Pr Lst , cl ) ,
= A dic ◦
Chp Ar
_
Ar
top
EO : ( Chp ) op → Fun (N( Rind) op , Pr Lst , cl ) ,
_
= α ◦ ( Chp Ar EO × id Δ 1 );
= A dic ◦
Chp Ar
_
EO : ( PTopos)op → Funtop (N( Rind) op , Pr Lst , cl ) ,
= A dic ◦
Chp Ar
_
top
L
EO : ( PTopos)op → Fun (N( Rind) op , Pr st , cl ) ,
_
= α ◦ ( PT opos EO × id Δ 1 ) .
By construction and Remark 1.2.3, the six functors in first three groups satisfy (P4).
For objects X of PTopos (resp. ChpAr ) and λ = (Ξ , Λ) of Rind, we write D ( X,λ ) adic for
_
_
adic
adic
PT opos EO ( X,λ ) (resp. Chp Ar EO ( X,λ ) ), which is a full (symmetric monoidal) subcategory of
D ( X,λ ) . Similarly, we write D ( X,λ ) for Chp Ar EO_ ( X,λ ) (resp. PT opos EO_ ( X,λ ) ). Recall that _ is
by definition a partially ordered set.
α
_
Proposition 1.4.2. The natural transformation
is a natural equivalence. In other words,
PT opos EO
foreveryobject T of PTopos andeveryobject λ = (Ξ , Λ) of Rind ,thesymmetricmonoidalfunctor
_
α
PT opos EO
( T,λ ) : D ( T,λ ) adic → D ( T,λ )
' lim D ( T, Λ( ξ ))
k
op
Ξ
isanequivalence.
Ξ
Proof. By definition D ( T,λ ) adic is a full subcategory of D ( T,λ ) = D (Mod( T´et
, Λ)). We analyze the
Ξ
D
construction of the functor α : D ( T , Λ) adic → lim
(
Λ(
))
.
First,
we
have
a
functor _ 1 × N(Ξ) →
T,
ξ
Ξ op
k
Pr Lst sending _ 1 × ( ' : ξ → ξ 0) to the square
D ( T Ξ /ξ , Λ /ξ )
i'
pξ _
'˜ _
_
_
D ( T Ξ /ξ _0 , Λ /ξ 0 )
/D ( T, Λ( ξ ))
/
_ 0
/D ( T, Λ(
_ ξ )) .
/
This corresponds to a projectively fibrant simplicial functor F : C[N( D )] → Set+Δ , where D = [1] × Ξ.
D
◦
Let φD : C[N( D )] → D be the canonical equivalence of simplicial categories and put F 0 = (Fibr
+
+
+
+ Ξ
0
0
St φop ◦ Un N( D ) op ) F : D → SetΔ . We write F in the form F : [1] → ( SetΔ ) . Applying the marked
D
p ξ 0_
+
unstraightening functor Un φ for the weak equivalence of simplicial categories φ : C[N(Ξ) op ] → Ξop , we
+
obtain a morphism ~α : F1 → F2 of Cartesian fibrations in the category ( SetΔ ) / N(Ξ) op . Moreover, by
[19, 5.2.2.5], both F1 and F 2 are coCartesian fibrations as well, but ~α does not send coCartesian edges
to coCartesian ones in general. By a similar argument, we have a map
op
D ( T Ξ, Λ) → Map coCart
,F 1 ) := Map [N(Ξ)
N(Ξ) op (N(Ξ)
op
((N(Ξ) op]) , ( F,1 E)) ,
ENHANCED ADIC FORMALISM FOR ARTIN STACKS
13
where E is the set of coCartesian edges of F1 .
Composing with the obvious inclusion and
Map N(Ξ) op (N(Ξ) op , α˜ ) , we obtain a map
α 0 : D ( T Ξ, Λ) → MapN(Ξ) op (N(Ξ) op ,F 2 ). We have the
op
D
equivalence MapcoCart
,F 2 ) ' lim
N(Ξ) op (N(Ξ)
k Ξ op ( T, Λ( ξ )) , and the following pullback diagram
α
D ( T Ξ, Λ) adic
_
D ( T Ξ_, Λ)
op
/MapcoCart
,F 2 )
N(Ξ) op (N(Ξ)
/
_ op
/Map N(Ξ) op (N(Ξ)
,F 2 ) ,
_
/
by the definition of adic objects, where vertical arrows are inclusions. We also note that α 0 commutes
with small colimits by [19, 5.1.2.2].
ξ of _, we have an exact evaluation functor
e_ξ : Mod( T Ξ, Λ) →
Recall that for every object
Mod( T, Λ( ξ )) (on the level of Abelian categories) that admits a (right exact) left adjoint
eξ ! : Mod( T, Λ( ξ )) → Mod( T Ξ, Λ). We define a truncation functor t ≤ ξ : Mod( T Ξ, Λ) → Mod( T Ξ, Λ) such
that for a sheaf F• 2 Mod( T Ξ, Λ),
α
0
( t ≤ ξ F• ) ξ 0 =
( Fξ 0
0
if ξ 0 ≤ ξ,
otherwise,
which is exact and admits a right adjoint. Let Δ / Ξ be the category of simplices of N(_) of dimension
≤ 1. Then all n -cells of N(Δ / Ξ ) are degenerate forn ≥ 2. Define a functor
op
0
β : N( Δ / Ξ ) → Fun(Map N(Ξ)
sending a typical subcategory ξ → ( ξ → ξ 0)
op
(N(Ξ) op ,F 2 ) , D ( T Ξ, Λ))
ξ 0 of Δ / Ξ to
/L eξ 0! ◦ _ξ 0,
L eξ ! ◦ _ξ o
t ≤ ξ ◦ L eξ 0! ◦ _ξ 0
o
/
ξ . The functor
where _ξ : Map N(Ξ) op (N(Ξ) op ,F 2 ) → D ( T, Λ( ξ )) is the restriction to the fiber at
0
0
Ξ
.
Ξ
Ξ
Fun(α, D ( T , Λ)) ◦ β extends to a functor N( Δ / Ξ ) → Fun( D ( T , Λ) , D ( T , Λ)) carrying ( ξ → ( ξ →
ξ 0)
ξ 0) / to
L eξ ! ◦ _ξ ◦ α 0 o
o
t ≤ ξ ◦ L eξ 0! ◦ _ξ 0 ◦ α 0
/L eξ 0! ◦ _ξ 0 ◦ α 0
/
)id_, u
) _u
0
◦ α 0) ' (lim β 0) ◦ α 0 → id. One checks that the restriction
which induces a natural transformation lim(
β
k→
k→
0
op
Ξ
op (N(Ξ)
|MapcoCart
β = lim
β
,F
2 ) takes values inD ( T , Λ) adic . We prove that β ◦ α → id is an equivalence.
N(Ξ)
k→
op
Pick an object K of D ( T Ξ, Λ) adic . We need to show that the diagram βK. : N( Δ / Ξ ) . → D ( T Ξ, Λ) ,
depicted as
/L eξ 0! K ξ 0
L eξ ! K ξ o
t ≤ ξ L eξ 0! K ξ 0
o
/
&_ w
K
& _, w
is a colimit diagram. We only need to check this after applying e_ξ 0 for every ξ0 2 Ξ, since e_ξ 0 commutes
with colimits. The composed functor e_ξ 0 ◦ βK. has value (equivalent to) K ξ 0 on the cone point, vertices
{ ξ } , ( ξ → ξ 0) of Δ / Ξ for ξ ≥ ξ0 and 0 otherwise, with all morphisms being either identities onK ξ 0 or 0, or
op
e_ξ 0 ◦ βK. | N( Δ / Ξ )) ' K ξ 0
the zero morphism 0→ K ξ 0 . It is clear that e_ξ 0 ◦ βK. induces an equivalencek→
lim(
in D ( T, Λ( ξ0 )) .
For the other direction, the functor Fun N(Ξ) op (N(Ξ) op ,F 2 ) ,α 0) ◦ β 0 also extends to a functor
N( Δ / Ξ ) . → Fun(Map N(Ξ)
op
(N(Ξ) op ,F 2 ) , Map N(Ξ)
op
(N(Ξ) op ,F 2 ))
14
YIFENG LIU AND WEIZHE ZHENG
carrying ( ξ → ( ξ → ξ 0)
ξ 0) / to
α 0 ◦ L eξ ! ◦ _ξ o
o
α 0 ◦ t ≤ ξ ◦ L eξ 0! ◦ _ξ 0
/α 0 ◦ L eξ 0! ◦ _ξ 0
/
)id_, u
) _u
0
◦ β 0) ' α 0 ◦ (lim β 0) → id, where the equivalence of two
α
which induces a natural transformation lim(
k→
k→
op
functors is due to the fact that α 0 commutes with colimits. Restricting to Map coCart
,F 2 ) , one
N(Ξ) op (N(Ξ)
obtains a map α ◦ β → id which is an equivalence by an argument similar to the above. Therefore, α is
an equivalence and the proposition follows.
_
Corollary 1.4.3. The three natural transformations
ral equivalences. In particular, for every object
symmetricmonoidalfunctor
_
α
Chp Ar EO
α
α
EO , Chp DM EO ,
C hp Ar
L
Ar
and
_
α
Chp Ar EO
areallnatu-
and every object λ = (Ξ , Λ) of Rind , the
X of Chp
( X,λ ) : D ( X,λ ) adic → D ( X,λ )
' lim D ( X, Λ( ξ ))
k
op
Ξ
isanequivalence.
adic
_
Proof. It su_ces to show the second assertion. Since smooth surjective morphisms are of both
Chp Ar EO _
qc . sep
and Chp Ar EO -descent, we may assume thatX is an object of Sch
, that is, a disjoint union of quasi_
compact and separated schemes. In this case, it su_ces to apply Proposition 1.4.2 toT = X ´et.
Remark 1.4.4. In the special case where _: _ op → Ring is a constant functor with value Λ (by abuse of
notation), we have an equivalence
A.3)
_
D ( X,λ ) adic k→D ( X,λ )
'
Y
D ( X, Λ)
π0 (Ξ)
given by the product functor Q π0 (Ξ) e_ξ i , where we have arbitrarily chosen an objectξi in each connected
component _i of _. The resulting functor is independent of such choices up to equivalence.
Assume _ is connected for simplicity. Let π : (Ξ , Λ) → ( _, Λ) be the projection. Then π_ : D ( X, Λ) →
D ( X,λ ) adic is an equivalence since its composition with (1.3) is the identity.
In particular, the right
adjoint functor (between the underlying ∞ -categories) R π_ | D ( X,λ ) adic : D ( X,λ ) adic → D ( X, Λ) is an
equivalence as well. The special case _ = N was proved in [14, 2.2.5] assumingthefinitenesscondition
oncohomologicaldimension .
1.5. Enhanced six operations and the usual t -structure. Recall that [20, 1.4.4.12], for a presentable
stable ∞ -category D , a t -structure is accessibleif the full subcategory D ≤ 0 is presentable. For a topos
X 2 PTopos or a schemeX 2 Schqc . sep , the usual t -structure on D ( X,λ ) is accessible by [20, 1.3.5.21].
For a higher Artin stack X , the usual t -structure on D ( X,λ ) is accessible by construction [18, 4.3.7] (Part
B) of (P6)).
Let D ≤ n ( X,λ ) adic = D ≤ n ( X,λ ) ∩ D ( X,λ ) adic and D ≥ n ( X,λ ) adic = D ≤ n • 1 ( X,λ ) ?adic . Recall by Proposition 1.1.6 that the inclusion D ( X,λ ) adic _ D ( X,λ ) admits a right adjoint, that is, the colocalization
functor R X : D ( X,λ ) → D ( X,λ ) adic . By Lemma 1.1.5 and [19, 5.5.3.12], D ≤ n ( X,λ ) adic is presentable.
The inclusion D ≤ n ( X,λ ) adic _ D ( X,λ ) preserves all small colimits and D ≤ n ( X,λ ) adic is closed under
extension. By [20, 1.4.4.11 A)], the pair ( D ≤ n ( X,λ ) adic , D ≥ n ( X,λ ) adic ) define an accessiblet -structure,
called the usual t -structure, on D ( X,λ ) adic .
Now we define six operations for adic complexes and study their behavior under the above
t -structure.
It is clear that kk
preserves the subcategoryD ( X,λ ) adic . Therefore, we have the induced (derived)
tensor product
3L: kk
= kk X : D ( X,λ ) adic × D ( X,λ ) adic → D ( X,λ ) adic ,
that is left t -exact with respect to the above
t -structure. Moving the first factor of the source
D ( X,λ ) adic × D ( X,λ ) adic to the target side, we can write the functor kk
in the form D ( X,λ ) adic →
FunL( D ( X,λ ) adic , D ( X,λ ) adic ) , because the tensor product on D ( X,λ ) adic is closed. Taking opposites
ENHANCED ADIC FORMALISM FOR ARTIN STACKS
15
op
and applying [19, 5.2.6.2], we obtain a functor D ( X,λ ) adic → FunR( D ( X,λ ) adic , D ( X,λ ) adic ) , which can
be written as
op
3R: Hom ( k , k ) = Hom X ( k , k ) : D ( X,λ ) adic × D ( X,λ ) adic → D ( X,λ ) adic .
Moreover, we have
op
Hom X ( k , k ) = R X ◦ Hom X ( k , k ) | D ( X,λ ) adic × D ( X,λ ) adic .
For every morphism π: λ 0 → λ of Rind, π_ preserves adic objects by 1.1.3 A), so that we have
_
_
0
π = π | D ( X,λ ) : D ( X,λ ) adic → D ( X,λ ) adic ,
which admits a right adjoint
0
0
π_ = R X ◦ π_ | D ( X,λ ) : D ( X,λ ) adic → D ( X,λ ) adic .
Let f : Y → X be a morphism of higher Artin stacks. Then f
have the induced functor
_
1L: f = f _ | D ( X,λ ) adic : D ( X,λ ) adic → D ( Y,λ ) adic ,
_
preserves adic objects. Therefore, we
which admits a right adjoint, denoted by
1R: f _ ' R X ◦ f _ | D ( Y,λ ) adic : D ( Y,λ ) adic → D ( X,λ ) adic .
_
It follows from the definition and the corresponding properties of f _ and f _ , that f is left t -exact, and
f _ is right t -exact.
ChpAr
If f is a morphism, locally of finite type, of higher Artin stacks in
L (resp. of higher Deligne–
λ is an object of
Mumford stacks, resp. locally quasi-finite of higher Deligne–Mumford stacks) and
Rind L- tor (resp. Rind tor , resp. Rind), then f ! preserves adic objects. Therefore, we have the induced
functor
2L: f ! = f ! | D ( Y,λ ) adic : D ( Y,λ ) adic → D ( X,λ ) adic ,
which admits a right adjoint, denoted by
!
2R: f ' R Y ◦ f ! | D ( X,λ ) adic : D ( X,λ ) adic → D ( Y,λ ) adic .
It follows from the definition and the corresponding properties of f ! and f ! [18, 6.1.9], that f ! [2d] (resp.
!
f [k 2d]) is left (resp. right) t -exact if d = dim + ( f ) < ∞ .
The t -structure on D ( X,λ ) naturally induces a t -structure on D ( X,λ ). More precisely, for every integer
≤n
n , an object K of D ( X,λ ) = limk Ξ op D ( X, Λ( ξ )) is in D ( X,λ ) if and only if K ξ is in D ≤ n ( X, Λ( ξ ))
for all objects ξ of _. Here, K ξ is the image of K under the natural functor lim k Ξ op D ( X, Λ( ξ )) →
≥
≤ •
D ( X, Λ( ξ )) . Let D n ( X,λ ) = D n 1 ( X,λ ) ? . We have induced functors and properties similar to those
_
α
for D ( X,λ ) adic . The equivalenceChp Ar EO ( X,λ ) is compatible with the two t -structures on D ( X,λ ) adic
and D ( X,λ ), as well as the various induced functors.
In what follows, we will identify D ( X,λ ) adic and D ( X,λ ) (as stable ∞ -categories with t -structures)
and the corresponding functors. We will use the underlined version in all notations. In particular, we
will view D ( X,λ ) as a (strictly) full subcategory of D ( X,λ ).
Proposition 1.5.1. The six operations and the usual t -structureintheaboveadicsettinghaveallthe
propertiesstatedinPropositions [18, 6.1.1, 6.1.2, 6.1.3, 6.1.4, 6.1.7, 6.1.8]andCorollary [18, 6.1.6].
Proof. (The analogues of) Propositions [18, 6.1.1, 6.1.2] are encoded in
Chp Ar
L
EO '
adic
EO
Chp Ar
L
and
adic
Chp DM EO.
EO '
Propositions [18, 6.1.3, 6.1.4] follow from the previous two propositions. Proposition [18, 6.1.7] follows by restricting to the full subcategory spanned by adic objects, sincef _ preserves
adic objects in this case. Proposition [18, 6.1.8] follows from property (P4) of Chp Ar
EO and Chp DM EO.
L
Corollary [18, 6.1.6] follows from Proposition [18, 6.1.1] and 1.5.2 B).
_
Chp DM
With slight modification as follows, Proposition [18, 6.1.5] also holds in the adic setting by restriction
to the full subcategory of adic objects.
16
YIFENG LIU AND WEIZHE ZHENG
Proposition 1.5.2 (Poincaré duality) . Let f : Y → X beaflat(resp.flatandlocallyquasi-finite)morAr
DM
phismof ChpL (resp. Chp ), locallyoffinitepresentation. Let λ beanobjectof Rind L- tor (resp. Rind ).
Then
_
A) Thereisanaturaltransformation
uf : f ! ◦ f hdi→ id X foreveryinteger d ≥ dim + ( f ) .
_
B) If f ismoreoversmooth,theinducednaturaltransformation
uf : f ! ◦ f hdim f i→ id X isacounit
_
!
transformation, so that the induced map f hdim f i→ f is a natural equivalence of functors
D ( X ,λ ) → D ( Y,λ ) .
_
Remark 1.5.3. By Poincaré duality, f is also right t -exact if f is a smooth morphism of higher Artin
Ar
stacks in ChpL and λ is an object of Rind L- tor or if f is an étale morphism of higher Deligne–Mumford
stacks. We will show in §3.2 that this holds for more general morphismsf under a finiteness assumption.
Lemma 1.5.4. Let
(Ξ , Λ) be an object of Rind and ξ be an object of Ξ, K
D ( X, (Ξ , Λ)) . Thefollowingdiagram
D ( X, (Ξ , Λ)) o
o
_
•
K
be an adic object of
D ( X, (Ξ , Λ))
_
eξ
eξ
_
eξ K
•
_
_
D ( X, _Λ( ξ )) o
D ( X, _Λ( ξ ))
o
eξ ! ( L
isrightadjointableanditstransposeisleftadjointable. Inotherwords,thenaturalmorphisms
e_ξ K ) → ( eξ ! L ) K and e_ξ Hom ( K , L 0) → Hom ( e_ξ K ,e _ξ L 0) are equivalences for objects L of
D ( X, Λ( ξ )) and L 0 of D ( X, (Ξ , Λ)) .
Proof. By [18, 6.1.7], we may assume that ξ is the final object of _. In this case, e_ξ can be identified
_ _
with π_ , where π: (Ξ , Λ) → ( { ξ } , Λ( ξ )) . Since K is adic, the morphism πe
ξ K → K is an equivalence.
A left adjoint of the transpose of the above diagram is then given by the diagram
D ( X, (Ξ , Λ)) o
o
•
_
π
•
K
_
D ( X, (Ξ
_ , Λ)) o
o
D ( X, Λ( ξ ))
π
_
_
eξ K
_
D ( X, Λ(
_ ξ )) .
_
Ar
1.6. Adic dualizing complexes. Let X be an object of Chp and λ = (Ξ , Λ) be an object of Rind. Let
O be an objects inD ( X,λ ) (resp. D ( X,λ ) ). By adjunction of the pair of functors k K and Hom ( K , k )
(resp. k K and Hom ( K , k ) ), we have a natural transformation δO : id → hHom (h Hom ( k , O) , O)
(resp. δO : id → hHom (h Hom ( k , O ) , O )) between endofunctors of hD ( X ,λ ) (resp. hD ( X ,λ )), which is
called the bidualitytransformation 2.
In the following of this section, we fix an L-coprime base schemeS that is a disjoint union of schemes
which are excellent, quasi-compact, and admit a global dimension function for which we fix one. Let
Rind L- dual be the full subcategory of Rind L- tor spanned by ringed diagrams _: _op → Ring such that Λ( ξ )
is an ( L-torsion) Gorenstein ring of dimension 0 for every object ξ of _.
Definition 1.6.1 (Potential dualizing complex) . Let λ = (Ξ , Λ) be an object of Rind L-dual . For an object
qc . sep
f : X → S of ChpAr
, we say a complexO in D ( X,λ ) is a pinned/potentialdualizing
lft / S with X in Sch
complex (on X ) if
A) O is adic, and
B) for every object ξ of _, Oξ = e_ξ O 2 D ( X, Λ( ξ )) is a pinned/potential dualizing complex.
˜ O : id → Hom ( Hom ( 2 , O ) , O ) (resp. δ
˜ O : id →
δO can be enhanced to a natural transformation
δ
˜
Hom ( Hom ( 2 , O) , O )) between endofunctors of D ( X ,λ ) (resp. D ( X ,λ )), that is, h δO = δO . We omit the detail since
wedonotneedsuchenhancementintheproofbelow.
2In fact,
ENHANCED ADIC FORMALISM FOR ARTIN STACKS
17
Ar
For a general objectf : X → S of Chplft / S, we say a complexO in D ( X,λ ) is a pinned/potentialdualizing
qc . sep
complex if for every atlas u : X 0 → X with X 0 in Sch
, u! O is a pinned/potentialdualizingcomplex
on X 0 .
Proposition 1.6.2. Let f : X → S beanobjectof ChpAr
lft / S and λ beasabove. Thefullsubcategoryof
D ( X,λ ) spannedbyallpinned/potentialdualizingcomplexesisequivalenttothenerveofanordinarycat_ π0 ( X ) . Moreover,pinned/potential
egoryconsistingofonlyoneobject O with Hom(O , O ) = _ lim
Λ(Ξ)
k ξ2 Ξ
dualizingcomplexesareconstructibleandcompatibleunderextensionofscalars.
In the proof, we will use the following observation that is essentially [19, A.3.2.27]. LetC: K / → Cat ∞
be a functor which is a limit diagram. Let X,Y be two objects in the limit ∞ -category C∙
and write
X,Y
( X,Y ) is naturally the homotopy
k
k the natural images in Ck for every vertex k of K . Then Map C∙
limit (in the ∞ -category H of spaces) of a diagramK → H sending k to Map Ck ( X,Y
k
k ).
qc . sep
Proof. When _ = _ is a singleton and X is in Sch
, the proposition is proved in [10] (see [18, 6.3.3
A)]). We also note that if OS is a pinned dualizing complex onS, then f ! OS is a pinned dualizing complex
Ar
k - Ar
on X . First, we prove by induction on k that for an object f : X → S of Chplft / S with X in Chp
,
0
0
O
O
O
O
A) For any two pinned dualizing complexes and , Map D ( X, Λ) ( , ) is discrete;
B) There is a unique distinguished equivalenceo : O → O 0 such that for every atlas u : X 0 → X with
!
X 0 in Schqc . sep , uo
is the one preserving pinning.
It is clear that once the equivalence o in B) exists, it is compatible under f ! for every smooth morphism
•
!
f . Choose an atlas u : Y → X (with Y in Chp( k 1) - Ar ). Since u is of universal Chp Ar
EO -descent,
L
both A) and B) follow from the induction hypothesis, the above observation, and the fact that limit
of k -truncated spaces is k -truncated (following [19, 5.5.6.7]). Second, we show that MapD ( X, Λ) ( O , O ) '
π0 MapD ( X, Λ) ( O , O ) is isomorphic to _ π0 ( X ) . Without lost of generality, we assume that X is connected.
Choose an atlas u = ` I ui : ` I Yi → X with Yi in Schqc . sep and is connected. We have the following
commutative diagram
α
/π0 Map D ( X, Λ) ( O , O )
Λ
/
β
Λ
_
π0 Map D ( Y,i_Λ) ( u!i O ,u !i O ) .
/L I
/
Since u is conservative,β is injective. Since _ → π0 Map D ( Y,i Λ) ( u!i O ,u !i O ) is an isomorphism for everyi 2
)
I , α is injective. If we write elements of L I π0 Map D ( Y,i Λ) ( u!i O ,u !i O ) in the coordinate form ( ...,λ,...i
with respect to the basis consisting of distinguished equivalences, then the image uof! must belong to the
diagonal sinceX is connected. Therefore,α is an isomorphism. The fact that pinned dualizing complexes
are constructible and compatible under extension of scalars follows from the case of schemes.
Now we consider the general coe_cient λ = (Ξ , Λ) . First, we construct a pinned dualizing complex
OS,λ on the base scheme
S. Let Δ / Ξ be the category of simplices of _. Then alln -simplices of N(Δ / Ξ ) are
degenerate forn ≥ 2. For every objectξ of _, denote OS,ξ the pinned dualizing complex inD ( S, Λ( ξ )) , and
recall the functors eξ ! and t ≤ ξ in the proof of Proposition 1.4.3. Define a functor δ : N( Δ / Ξ ) → D ( S,λ )
sending a typical subcategory ξ
( ξ ≤ ξ 0) → ξ 0 of Δ / Ξ to
!
/L eξ 0! OS,ξ 0 ,
L eξ ! OS,ξ o
L eξ ! Eξ ≤ ξ 0 OS,ξ 0 ' t ≤ ξ L eξ 0! OS,ξ 0
o
/ _
where the left arrow is given by the distinguished equivalence Eξ ≤ ξ 0 OS,ξ 0 k→ OS,ξ . It is easy to see that
OS,λ = limk δ, viewed as an element inD ( S,λ ) , satisfies the two requirements in Definition 1.6.1, hence is
Ar
a pinned dualizing complex. For an object f : X → S of Chplft / S, let Of,λ = f ! O . Then it is a pinned
dualizing complex on X . The rest of the proposition follows from the fact that Of,λ is adic, Proposition
1.4.3, the observation before the proof, and the same assertion when _ is a singleton.
_
In what follows, we write
similarly for D , D.
D = D X = Hom ( k , ΩX,λ ), D = D
X
= h D X = h Hom ( k , ΩX,λ ) , and
18
YIFENG LIU AND WEIZHE ZHENG
Proposition 1.6.3. Let K 2 D ( X,λ ) suchthat δΩ X,
Then δΩ X,λ ( K ) isanequivalenceaswell.
Λ( ξ )
( e_ξ K ) isanequivalenceforeveryobject
ξ of Ξ.
Proof. We need to show that the natural morphism K → D D K is an isomorphism (in the homotopy
category D ( X,λ )). By definition,
D D K ' hHom ( K , hHom ( K , ΩX,λ ))
' hHom ( K , hR X hHom ( K , ΩX,λ ))
' hR X hHom ( K , hHom ( K , ΩX,λ )) .
It su_ces to show that δΩ X,λ ( K ) : K → hHom ( K , hHom ( K , ΩX,λ )) is an equivalence. In fact, since
K is adic, we have
_
_
_
_
eξ hHom ( K , hHom ( K , ΩX,λ )) ' hHom ( eξ K , hHom ( eξ K ,e ξ ΩX,λ ))
' hHom ( e_ξ K , hHom ( e_ξ K , ΩX, Λ( ξ ) ))
for every object ξ 2 Ξ by Lemma 1.5.4, which is equivalent to e_ξ K by the assumption.
_
2. Perverse t -structures
In §2.1, we define the general notion of perversity, which we call
perversitysmooth/étaleevaluation for
higher Artin/Deligne–Mumford stacks. Then we define the perverset -structure for a perversity evaluation
on an Artin stack in §2.2 using descent. In 2.3, we define the adic perverset -structure. In both cases, we
provide the description of such t -structures in terms of cohomology on stalks as in the classical situation.
2.1. Perversity evaluations.
qc . sep
Definition 2.1.1. Let X be a scheme inSch
.
A) Following [9, ù], a weakperversityfunction on X is a function p : |X |→ Z [{ + ∞} such that
for every n 2 Z , the set { x 2 |X || p( x ) ≥ n } is ind-constructible.
B) An admissible perversity function on X is a weak perversity function p such that for every
x 2 |X |, there is an open dense subset U _{ x } satisfying that for every x 0 2 U , p( x 0) ≤
0
p( x ) + 2 codim(x,x
).
C) A codimensionperversityfunction on X is a function p : |X |→ Z [{ + ∞} such that for every
immediate étale specialization x 0 of x , p( x 0) = p( x ) + 1.
Remark 2.1.2.
A) A weak perversity function on a locally Noetherian scheme is locally bounded from below.
B) An admissible perversity function on a scheme that is locally Noetherian and of finite dimension
is locally bounded from above.
C) A codimension perversity function on a scheme is not necessarily a weak perversity function.
D) A codimension perversity function that is also a weak perversity function is an admissible perversity function. If X is locally Noetherian, a codimension perversity function is a weak perversity
function and hence an admissible perversity function.
E) A codimension perversity function is the opposite of a dimension function in the sense of [24,
2.1.8]. If X is locally Noetherian and admits a dimension function, then it is universally catenary
by [24, 2.2.6]. In this case, immediate étale specializations coincide with immediate Zariski
specializations [24, 2.1.4].
F) If p is a weak (resp. admissible, resp. codimension) perversity function on X and d : |X |→
Z [{ + ∞} is a locally constant function, then p + d is a weak (resp. admissible, resp. codimension)
perversity function on X .
Definition 2.1.3. A function q : N → Z or Z → Z is called admissible if q and 2 k q are both increasing,
where 2( x ) = 2 x and similarly for 0 and 1, which will be used below.
qc . sep
Let f : Y → X be a smooth morphism of schemes in Sch
, p : |X |→ Z [{ + ∞} be a function,
_
_
q : N → Z be a function. We define the pullback f qp : |Y |→ Z [{ + ∞} by ( f qp
)( y ) = p( f ( y)) k
_
q(tr .deg[k ( y ) : k ( f ( y))]) for every point y 2 |Y |. In particular, f 0 = p ◦ f .
ENHANCED ADIC FORMALISM FOR ARTIN STACKS
19
Lemma 2.1.4. Let f : Y → X beamorphism(resp.étalemorphism,resp.étalemorphism)ofschemes
_
in Schqc . sep . If p isaweak(resp.admissible,resp.codimension)perversityfunctionon
X ,then f 0p
isa
weak(resp.admissible,resp.codimension)perversityfunctionon
Y.
_
Proof. We have f 0p
= p ◦ f . If p is a weak perversity function, then
_
{ y 2 |Y || f 0p
( y) ≥ n} = f
•1
( { x 2 |X || p( x ) ≥ n } )
0
y,y
is ind-constructible by [1, IV 1.9.5 (vi)]. The other two cases follow from the trivial fact that codim(
)=
0
0
codim(f ( y ) ,f ( y)) for every specialization y of y on Y .
_
Schqc . sep ,locallyoffinite
Lemma 2.1.5. Let f : Y → X beamorphismoflocallyNoetherianschemesin
type.
_
X , q : N → Z beanincreasingfunction. Then
f qp
isa
A) Let p beaweakperversityfunctionon
weakperversityfunctionon Y .
_
B) Let p beanadmissibleperversityfunctionon X , q : N → Z beanadmissiblefunction. Then f qp
isanadmissibleperversityfunctionon
Y.
_
C) Let p beacodimensionperversityfunctionon X . Then f 1p
isacodimensionperversityfunction
on Y .
Proof. A) For a locally closed subsetZ of a schemeX , we endow it with the reduced induced subscheme
structure. For every point y 2 |Y |, let Uy _{ y } be a nonempty open subset such that the induced
morphism f y : { y }→{ f ( y ) } is flat. Such an open subset exists by [1, IV 6.9.1]. For y 0 2 Uy , we have
0
0
0
0
δ( y,y ) := tr .deg[k ( y ) : k ( f ( y))] k tr .deg[k ( y ) : k ( f ( y ))] = codim( y,U
y
× f ( y ) { f ( y 0) } ) ≥ 0
by [1, 14.3.13] becausef y is universally open [1, IV 2.4.6]. Therefore, for n 2 Z ,
_
•
{ y 2 |Y || f qp
( y ) ≥ n } = [ f 1 { x 2 |X || p( x ) ≥ n + q(tr .deg[k ( y ) : k ( f ( y ))]) }∩ U,y
y 2| Y |
_
is a union of ind-constructible subsets, and hence is itself ind-constructible. In other words,f p
is a weak
perversity function.
B) Let y 2 |Y | be a point, x = f ( y) , Ux _{ x } be a dense open subset such that p( x 0) ≤ p( x ) +
0
_
_
0
2 codim( x,x
). We prove that for y 0 2 Uy ∩ f • 1 ( Ux ), f qp
( y 0) ≤ f qp
( y ) + 2 codim(y,y
). We may
0
0
_
_
assume that p( x ) 2 Z . Let x = f ( y ) . We have f qp ( y ) = p( x ) k q(tr .deg[k ( y ) : k ( x )]) and f qp
( y0) =
0
0
0
p( x ) k q(tr .deg[k ( y ) : k ( x )]). Moreover, by [1, IV 6.1.2],
0
0
0
δ( y,y ) = codim( y,y ) k codim(x,x ) .
Therefore,
_
0
_
0
0
0
f qp ( y ) k f qp ( y) = p( x ) k p( x ) + q(tr .deg[k ( y ) : k ( x )]) k q(tr .deg[k ( y ) : k ( x )])
0
0
0
≤ 2 codim(x,x
) + 2δ( y,y ) = 2codim( y,y ) .
_
In other words, f p
is an admissible perversity function on Y .
C) This is essentially proved in [24, 2.5.2].
_
Definition 2.1.6 (Pointed schematic neighborhood). Let X be a higher Artin (resp. Deligne–Mumford)
stack. A pointedsmooth(resp.étale)schematicneighborhood of X is a triple ( X,u,x
0
0
0 ) where u 0 : X 0 →
X is a smooth (resp. an étale) morphism withX 0 in Schqc . sep , and x 0 2 |X 0 | is a scheme-theoretical point.
A morphism v : ( X,u,x
1
1
1 ) → ( X,u,x
0
0
0 ) of pointed smooth (resp. étale) schematic neighborhoods is a
smooth (resp. an étale) morphism v : X 1 → X 0 such that there is a triangle
B.1)
v
X1
u1
/X 0
/
u0
~
X ~
and v( x 1 ) = x 0 . We say (X,u,x
1
1
1 ) dominates ( X,u,x
0
0
0 ) if there is such a morphism. The category of
pointed smooth (resp. étale) schematic neighborhoods ofX is denoted by Vosm ( X ) (resp. Vo´et( X )).
20
YIFENG LIU AND WEIZHE ZHENG
v : ( X,u,x
Lemma 2.1.7. Let X beahigherArtinstack,
1
1
1 ) → ( X,u,x
0
0
0 ) beamorphismofpointed
smoothschematicneighborhoodsof X . Thenthecodimensionof x 1 inthebasechangescheme X 1,x 0 =
X 1 ×←{
v.
X 0 x 0 } dependsonlyonthesourceandthetargetof
( X,u,x
1
1
We will let δ( X,u,x
0
0
( X,u,x
2
2
2)
1)
0)
( X,u,x
2
2
denote this codimension. It is clear that δ( X,u,x
0
0
dominates (X,u,x
1
1
1 ).
Moreover, if v is étale,
( X,u,x
1
1
δ( X,u,x
0
0
1)
0)
2)
0)
( X,u,x
2
2
= δ( X,u,x
1
1
2)
1)
( X,u,x
1
1
+ δ( X,u,x
0
0
1)
0)
if
= 0.
Proof. Note that codim( x,X
1
1 ,x 0 ) = dim x 1 ( v) k tr .deg[k ( x 1 ) : k ( x 0 )] . It is clear that dim x 1 v = dim x 1 u1 k
dim x 0 u0 does not depend onv. We will show that tr .deg[k ( x 1 ) : k ( x 0 )] does not depend onv either. Let
f : Y → X be an atlas of X with Y a scheme inSchqc . sep . Let
v
Y1
0
0
u1
be the base change of (2.1), f 0 : Y0 → X 0
qc . sep
scheme inSch
, and let
/Y0
/
u
0
~ 0
Y ~
and f 1 : Y1 → X 1 . Let w0 : Y00 → Y0 be an atlas with Y00 a
Y10
w1
v
00
/Y00
/
w0
_ v0
_
/Y_0
Y_1
/
be the base change. Then v00 is a smooth morphism of schemes in Schqc . sep . Since f 0 ◦ w0 : Y00 → X 0
0
is smooth and surjective, the base change scheme Y0,x 0 = Y00 ×←{
X 0 x 0 } is nonempty and smooth over
the residue field k ( x 0 ) of x 0 . Similarly, we have a nonempty schemeY10,x 1 , smooth over k ( x 1 ). Choose a
generic point y10 of Y10,x 1 . Then its image y00 in Y00,x 0 is a generic point. Let y be the image of y00 in Y .
Then tr .deg[k ( x 1 ) : k ( x 0 )] = tr .deg[k ( y10 ) : k ( y)] k tr .deg[k ( y00) : k ( y)] , which does not depend on v.
_
For a higher Artin (resp. Deligne–Mumford) stack X and a function p : Ob(Vo sm ( X )) → Z [{ + ∞}
(resp. p : Ob(Vo ´et( X )) → Z [{ + ∞} ), we have, by restriction, the function pu 0 : |X 0 |→ Z [{ + ∞} for
qc . sep
every smooth (resp. étale) morphism u0 : X 0 → X with X 0 in Sch
. If f : Y → X is a smooth (resp.
an étale) morphism of higher Artin (resp. Deligne–Mumford) stacks, then composition with f induces a
functor f : Vosm ( Y ) → Vosm ( X ) (resp. f : Vo´et( Y ) → Vo´et( X ) ), and we let f _ p = p ◦ f .
Definition 2.1.8 ((admissible/codimension) perversity evaluations). Let X be a higher Artin stack. A
smoothevaluation on X is a function p : Ob(Vo sm ( X )) → Z [{ + ∞} such that once (X,u,x
1
1
1 ) dominates
( X,u,x
1
1
1)
≤
≤
( X,u,x
.
p( X,u,x
p( X,u,x
0
0
0 ), p( X,u,x
0
0
0)
1
1
1)
0
0
0 ) + 2δ( X,u,x
0
0
0)
A perversitysmoothevaluation (resp. admissibleperversitysmoothevaluation , codimensionperversity
sm
smooth evaluation) on X is an evaluation p such that for every ( X,u,x
0
0
0 ) 2 Ob(Vo ( X )) , pu 0 is a
weak perversity function (resp. admissible perversity function, codimension perversity function) on X 0 .
Similarly, we define étale evaluations and (admissible/codimension) perversity étale evaluations on a
´et
higher Deligne–Mumford stack X using Vo ( X ).
qc . sep
Remark 2.1.9. If X is a scheme inSch
, the map from the set of étale evaluations on X to the set of
functions |X |→ Z [{ + ∞} , carrying p to pid X , is bijective. Under this bijection, the notions of (weak)
perversity, admissible perversity, and codimension perversity coincide. If f : Y → X is a morphism of
schemes inSchqc . sep , then f _ for étale evaluations coincide with f 0_ for functions.
Example 2.1.10.
A) Let X be a higher Artin (resp. Deligne–Mumford) stack.
Any constant smooth (resp. étale)
evaluation is an admissible perversity smooth (resp. étale) evaluation.
B) Let f : Y → X be a morphism of higher Deligne–Mumford stacks, locally of finite type, p be
an étale evaluation on X , q : N → Z be a function. We define an étale evaluation f q_ p on Y as
ENHANCED ADIC FORMALISM FOR ARTIN STACKS
´et
follows. For any object ( Y,v,y
0 0 0 ) of Vo ( Y ), there exists a morphism (Y,v,y
1 1
´et
in Vo ( Y ) such that there exists a diagram
Y1
f0
v1
21
1)
→ ( Y,v,y
0 0
0)
/Y
/
f
_
X_0
_
u0
/X,_
/
qc . sep
where X 0 is in Sch
and u0 is étale.
We put f q_ p( Y,v,y
0 0 0 ) = p( X,u,f
0
0 0 ( y1 )) k
q(tr .deg[k ( y1 ) : k ( f 0 ( y1 ))]). This clearly does not depend on choices. If p is a perversity étale
evaluation, then f 0_ p is a perversity étale evaluation by Lemma 2.1.4.
C) Let f : Y → X be a morphism of higher Artin stacks with X being a higher Deligne–Mumford
stack, locally of finite type, p be an étale evaluation on X . Let q : Z → Z be an admissible
_
function. We define a smooth evaluation f q_ p on Y by ( f q_ p)( Y,v,y
0 0 0 ) = ( v0 ◦ f ) q0 ( y0 ) for every
sm
0
0
object ( Y,v,y
0 0 0 ) of Vo ( Y ) , where q : N → Z is the function q ( n ) = q( n k dim y 0 v0 ) . If p is a
perversity étale evaluation, then f 0_ p is a perversity smooth evaluation. If X is locally Noetherian
and p is a perversity (resp. admissible perversity, resp. codimension perversity) étale evaluation,
then f q_ p (resp. f q_ p, resp. f 1_ p) is a perversity (resp. admissible perversity, resp. codimension)
smooth evaluation by Lemma 2.1.5.
In particular, if X is an object of Schqc . sep and p : |X |→ Z [{ + ∞} is a function, then we have
_
a smooth evaluation f qp
on Y .
2.2. Perverse t -structures. Let C be a stable ∞ -category equipped with a t -structure. We say that C
is weaklyleftcomplete (resp. weaklyrightcomplete ) if C≤∙
= T n C≤• n (resp. C© = T n C≥ n ) consists
i
of zero objects. The family (H ) i 2 Z is conservative if and only if C is weakly left complete and weakly
right complete (cf. [4, 1.3.7]). The following generalizes [20, 1.2.1.19].
Lemma 2.2.1. Let C beastable ∞ -categoryequippedwitha t -structure. Considerthefollowingconditions
A) The ∞ -category C isleftcomplete(seethedefinitionpreceding [20, 1.2.1.17]).
B) The ∞ -category C isweaklyleftcomplete.
C admits countableproducts and thereexists a such thatcountable
Then A)implies B). Moreover, if
productsofobjectsof C≤ 0 belongto C≤ a ,thenB)impliesA).
ˆ consists of zero
Proof. The first assertion is obvious since the image of C≤∙
under the functor C → C
objects.
To show the second assertion, it su_ces to replace
f ( n k 1) by f ( n k a k 1) in the proof of [20, 1.2.1.19].
_
Let X be a schemeX in Schqc . sep , p : |X |→ Z [{ + ∞} be a function, λ = (Ξ , Λ) be an object of Rind.
p
p
Following Gabber [9, ú], we define full subcategories D ≤ 0 ( X,λ ) , D ≥ 0 ( X,λ ) _ D ( X,λ ) as follows. For
K in D ( X,λ ),
• K belongs to p D ≤ 0 ( X,λ ) if and only if ij_x x_ K 2 D ≤ p( x ) ( x,λ ) for all x 2 X ;
• K belongs to p D ≥ 0 ( X,λ ) if and only if K 2 D (+) ( X,λ ) and ij!x x_ K 2 D ≥ p( x ) ( x,λ ) for all x 2 X .
Here x is a geometric point above x , i x : x → X ( x ) , j x : X ( x ) → X . We will omit j x_ from the notations
when no confusion arises. For an immersion i : Z → X in Schqc . sep , z 2 Z , we have an equivalence
i !z ' ii!z ! of functors D (+) ( X,λ ) → D + ( z,λ ).
p
p
Gabber showed in [9] that if p is a weak perversity function and _ = _, then ( D ≤ 0 ( X,λ ) , D ≥ 0 ( X,λ ))
is a t -structure on D ( X,λ ). This generalizes easily to the case of general _ as follows. By [20, 1.4.4.11],
p
p
p
there exists a t -structure ( D ≤ 0 ( X,λ ) , D 0) on D ( X,λ ). For K 2 D ≤ 0 ( X,λ ) , L 2 D ≥ 0 ( X,λ ) , we have
≥1
0
a_ Hom ( K , L [1]) 2 D ( λ ), so that Hom(K , L [1]) = H (Ξ ,a _ Hom ( K , L [1])) = 0, where a : X ´et →_
p
is the morphism of topoi. Thus D ≥ 0 ( X,λ ) _ D 0. For every ξ 2 Ξ, the functor L eξ ! : D ( X, Λ( ξ )) →
D ( X,λ ) is left t -exact for the t -structures ( p D ≤ 0 ( X, Λ( ξ )) , p D ≥ 0 ( X, Λ( ξ ))) and ( p D ≤ 0 ( X,λ ) , D 0) . It
p
follows that e_ξ is right t -exact for the same t -structures. Therefore D 0 _ D ≥ 0 ( X,λ ) .
22
YIFENG LIU AND WEIZHE ZHENG
p
p
Thus ( D ≤ 0 ( X,λ ) , D ≥ 0 ( X,λ )) is a t -structure on D ( X,λ ) . By definition, this t -structure is accessible,
and, if p takes values in Z , weakly left complete. By [9, 3.1], this t -structure is weakly right complete,
thus right complete.
qc . sep
D ( Y,λ ) carries
By definition, _for any morphism f : Y → X of schemes in Sch
, f _ : D ( X,λ ) →
_
p ≤0
fp
p
fp
≤0
_
≥0
0
0
D ( X,λ ) to
D ( Y,λ ). Moreover, if f is étale, then f carries D ( X,λ ) to
D ≥ 0 ( Y,λ ). We
will prove an analogue of this for smooth morphisms in Proposition 2.2.5.
qc . sep
Lemma 2.2.2. Let f : Y → X beasmoothmorphismin SchL
, λ beanobjectof Rind L- tor , y bea
_
!
!
pointof Y , x = f ( y ) . Thenthereisanequivalence i y ◦ f ' g ◦ i !x hdi offunctors D (+) ( X,λ ) → D + ( y,λ ) ,
where g : y → x , d = tr .deg[k ( y ) : k ( x )] .
Proof. Consider the diagram with Cartesian squares
iy
y
/V
/
g
j
0
ix
/Yx
/
/Y{ x }
/
f { x}
fx
i
0
/Y
/
f
_
&x_ i x /{ x_} i /X_
_
&_
/
/
where V is a regular integral subscheme ofYx such that the image of y in V is a generic point. We have
a sequence of equivalences of functors
_
!
_
_
0
0
_
0
i !y ◦ f ! ' i y ◦ j ! ◦ i x ◦ i ◦ f ! ' i y ◦ j ! ◦ i x ◦ f {! x } ◦ i !
' i _y ◦ j ! ◦ f x! ◦ i _x ◦ i !
'
'
_
iy ◦
i _y ◦
_
' g ◦
(f x ◦ j
(f x ◦ j
) ◦ i !x
_
) ◦ i !x hdi
by Poincaré duality
!
by Poincaré duality
i !x hdi .
_
. sep
Schqc
,
L
Lemma 2.2.3. Let λ beanobjectof Rind L- tor , f : Y → X beasmoothmorphismofschemesin
_
p
fp
p : |X |→ Z [{ + ∞} beafunction. Then f ! carries D ≥ 0 ( X,λ ) to 2 D ≥ 0 ( Y,λ ) . Moreover,if p isaweak
_
_
≤ q ≤ f 2p
perversityfunctionon X and q isaweakperversityfunctionon
Y satisfying f 0p
+ 2 dim f ,
!
then f : D ( X,λ ) → D ( Y,λ ) is t -exactwithrespecttothe t -structuresassociatedto p and q.
Proof. The first assertion follows from the above lemma.
Poincaré duality f ! ' f _ hdim f i .
The second assertion follows from A) and
_
Let X be an L-coprime higher Artin (resp. a higher Deligne–Mumford) stack equipped with a perversity
evaluation p, and λ be an object of Rind L- tor (resp. Rind). For an atlas (resp. étale atlas) u : X 0 → X
p
p
qc . sep
with X 0 a scheme in Sch
, we denote by D ≤u 0 ( X,λ ) _ D ( X,λ ) (resp. D ≥u 0 ( X,λ ) _ D ( X,λ )) the
pu
p
_
≤0
full subcategory spanned by complexesK such that u K is in D ( X,λ
) (resp. u D ≥ 0 ( X,λ
)).
0
0
p
p
Lemma 2.2.4. The pair of subcategories ( D ≤u 0 ( X,λ ) , D ≥u 0 ( X,λ )) donotdependonthechoiceof
u.
p
p
p
p
In what follows, we will write ( D ≤ 0 ( X,λ ) , D ≥ 0 ( X,λ )) for ( D ≤u 0 ( X,λ ) , D ≥u 0 ( X,λ )) .
Proposition 2.2.5. Let X bean L-coprimehigherArtin(resp.ahigherDeligne–Mumford)stackequipped
withaperversitysmooth(resp.étale)evaluation
p,and λ beanobjectof Rind L- tor (resp. Rind ). Then
p ≤0
p ≥0
A) The pair of subcategories ( D ( X,λ ) , D ( X,λ )) determine a right complete accessible t structureon D ( X,λ ) ,whichisweaklyleftcompleteif
p takesvaluesin Z .
B) If f : Y → X is a smooth (resp. étale) morphism, then f _ : D ( X,λ ) → D ( Y,λ ) is t -exact with
respecttothe t -structuresassociatedto p and f _ p.
ProofofLemma2.2.4andProposition2.2.5.
There exists k ≥ 2 such that X and Y are in Chpk - Ar (resp.
k -DM
Chp
). We proceed by induction on k . The case k = k 2 follows from Gabber’s theorem and Lemma
2.2.3. The induction step follows from the proofs of [18, 4.3.7, 4.3.8].
_
ENHANCED ADIC FORMALISM FOR ARTIN STACKS
23
Remark 2.2.6. We call the t -structure in Proposition 2.2.5 the perverset -structure with respect to p and
denote by p τ ≤ 0 and p τ ≥ 0 the corresponding truncation functors respectively. The for every (étale) atlas
qc . sep
, u_ ◦ p τ ≤ 0 ' pu τ ≤ 0 ◦ u and u_ ◦ p τ ≥ 0 ' pu τ ≥ 0 ◦ u.
u : X 0 → X with X 0 a scheme inSch
If p = 0, we recover the usual t -structure. If X is a higher Deligne-Mumford stack and p is a
perversity smooth evaluation, then thet -structure associated top coincides with the t -structure associated
to p | Vo´et( X ) . If X is in Schqc . sep , then the t -structure associated to p coincides with the t -structure
defined by Gabber associated to the function pid X .
By definition, the perverse t -structure can be described as follows.
Proposition 2.2.7. Let X bean L-coprimehigherArtinstack(resp.ahigherDeligne–Mumfordstack)
equippedwithaperversitysmooth(resp.étale)evaluation p,and λ beanobjectof Rind L- tor (resp. Rind ).
Let K beacomplexin D ( X,λ ) .
p
A) K belongsto D ≤ n ( X,λ ) ifandonlyifforeverypointedsmooth(resp.étale)schematicneigh_
_
≤ p( X,u,x
0
0
0 )+ n
borhood( X,u,x
( x,λ
).
0
0
0 ) of X , i x 0 u 0 K 2 D
0
p ≥n
B) K belongs to D ( X,λ ) if and only if K 2 D (+) ( X,λ ) and for every pointed smooth (resp.
_
≥ p( X,u,x
!
0
0
0 )+ n
étale)schematicneighborhood ( X,u,x
( x,λ
).
0
0
0 ) of X , i x 0 u0 K 2 D
0
At the end of the section, we study the restriction of perverset -structures constructed above to various
subcategory of constructible complexes. We fix an L-coprime base schemeS that is a disjoint union of
schemes which are excellent, quasi-compact, and admit a global dimension function for which we fix one.
Ar
Proposition 2.2.8. Let λ = (Ξ , Λ) be an object of Rind L- dual . For an object f : X → S of Chplft / S
p ≤0
equippedwithan admissible perversityevaluation p thatislocallybounded,thetruncationfunctors
τ ,
p ≥0
?
k
D
preserve
(
)
for
?
=
(+)
(
)
(b)
orempty.
τ
,
,
cons X,λ
Proof. In [9].
_
2.3. Adic perverse t -structures. For the adic formalism, we define
p
D ≤ n ( X,λ ) = p D ≤ n ( X,λ ) ∩ D ( X,λ ) ,
≤
p
D ≥ n ( X,λ ) = p D ≤ n • 1 ( X,λ ) ? _ D ( X,λ ) .
≥
p
0
p
0
Then the pair ( D ( X,λ ) , D ( X,λ )) defines a t -structure, called the adicperverse t -structure with
respect to p, on D ( X,λ ) . Denote p τ ≤ 0 and p τ ≥ 0 the corresponding truncation functors respectively. We
first have the following.
Lemma 2.3.1. Let X be an L-coprime higher Artin stack (resp. a higher Deligne–Mumford stack) ep,and λ beanobjectof Rind L- tor (resp. Rind ).
quippedwithaperversitysmooth(resp.étale)evaluation
Let K be a complex in D ( X,λ ) . Let u : X 0 → X be an atlas (resp. étale atlas) with X 0 a scheme in
≤
≥
≤
Schqc . sep . Then K belongsto p D n ( X,λ ) (resp. p D n ( X,λ ) ) ifandonlyif u_ K belongsto pu D n ( X,λ
)
0
≥n
pu
(resp. D ( X,λ
).
)
0
≤
p
n
Proof. We only need to show that u_ is t -exact. By definition, we obviously have u_ D ( X,λ ) _
≤n
pu
p >n
D ( X,λ
). For the other direction, assume K 2 D ( X,λ ), that is, Hom( L , K ) = 0 for all
0
L 2 D ( X,λ ) ∩ p D ≤ n ( X,λ ) . By Poincaré duality, we only need to show that for
L 0 2 D ( X,λ
)∩
0
pu
≤ n • 2 dim u
0 !
0
D
( X,λ
), Hom(L ,u K ) = 0, or equivalently, Hom( u! L , K ) = 0. This follows from the
0
p
fact that u! preserves adic objects andu! L 0 2 D ≤ n ( X,λ ) .
_
We have the following description in terms of the cohomology on stalks, that is similar to the Proposition 2.2.7.
Proposition 2.3.2. Let X bean L-coprimehigherArtinstack(resp.ahigherDeligne–Mumfordstack)
equippedwithaperversitysmooth(resp.étale)evaluation p,and λ beanobjectof Rind L- tor (resp. Rind ).
Let K beacomplexin D ( X,λ ) .
≤
p
n
A) K belongsto D ( X,λ ) ifandonlyifforeverypointedsmooth(resp.étale)schematicneigh≤ p( X,u,x
_
_
0
0
0 )+ n
borhood( X,u,x
( x,λ
).
0
0
0 ) of X , i x 0 u0 K 2 D
0
24
YIFENG LIU AND WEIZHE ZHENG
≥
p
n
K 2 D (+) ( X,λ )
B) Assume p is locally bounded. Then K belongsto D ( X,λ ) ifand onlyif
!
_
andforeverypointedsmooth(resp.étale)schematicneighborhood ( X,u,x
0
0
0 ) of X , i x 0 u0 K 2
D
≥ p( X,u,x
0
0
0 )+
n
( x,λ
).
0
≤
p
n
p
Proof. For A), by definition, K belongs to D ( X,λ ) if and only if K 2 D ≤ n ( X,λ ) , viewed as
object of D ( X,λ ) . By Proposition 2.2.7 A), it is equivalent to say that for (
Y,u,y ) of X , iu_y _ K 2
D ≤ p( Y,u,y )+ n ( y,λ ). We only need to notice that iu_y _ K is adic.
qc . sep
For B), by Lemma 2.3.1, we may assume X 2 Sch
is quasi-compact and p = p is a
p ≥n
L 2
bounded weak perversity function.
Then K 2 D ( X,λ ) is equivalent to that for every
p <n
D ( X,λ ) , Hom ( L , K ) 2 D > 0 ( X,λ ), which is then equivalent to that Hom ( L , K ) 2 D + ( X,λ )
and i !x Hom ( L , K ) ' Hom ( i _x L ,i !x K ) ' Hom ( i _x L ,i x ! K ) 2 D > 0 ( x,λ ) for every geometric point x
p <n
<α + n
p ≥n
of X . Assume α<p<β
. Then D ( X,λ ) _ D
( X,λ ) . Therefore, K 2 D ( X,λ ) implies
≥ α+ n
≥ p ( x )+ n
!
K 2 D
( X,λ ) and i x K 2 D
( x,λ ) for every geometric point x of X . Conversely, assume
≥γ
≥
+
!
K 2 D ( X,λ ) , say in D ( X,λ ), and i x K 2 D p( x )+ n ( x,λ ) for every geometric point x of X . We
only need to show that Hom ( L , K ) 2 D + ( X,λ ). In fact, we have Hom ( L , K ) 2 D ≥ γ • β • n ( X,λ ). _
Remark 2.3.3. Let p, q be two perversity smooth (resp. étale) evaluations on an L-coprime higher Artin
stack (resp. a higher Deligne–Mumford stack) X . Let λ be an object of Rind L- tor (resp. Rind).
(+)
A) Since p τ ≥ 0 (resp. p τ ≥ 0 ) preserves D (+) ( X,λ ) (resp. D ( X,λ )), so does p τ ≤ 0 (resp. p τ ≤ 0 ).
p ≤0
p ≥0
p ≤0
p ≥0
Therefore, the intersection of ( D ( X,λ ) , D ( X,λ )) (resp. ( D ( X,λ ) , D ( X,λ )) ) with
(+)
(+)
D ( X,λ ) (resp. D ( X,λ ) )induces a t -structure on the latter ∞ -category.
B) If p ≤ q, then
q
q ≤0
p
q
(a) p τ ≤ 0 (resp. p τ ≤ 0 ) preserves D ≤ 0 ( X,λ ) (resp. D ( X,λ ) ) since D ≤ 0 ( X,λ ) _ D ≤ 0 ( X,λ )
≤
≤
p
0
q
0
(resp. D ( X,λ ) _ D ( X,λ ));
p
p ≥0
q
p
q ≥0
q ≥0
(b) τ
(resp. τ ) preserves D ≥ 0 ( X,λ ) (resp. D ( X,λ ) ) since D ≥ 0 ( X,λ ) _ D ≥ 0 ( X,λ )
q ≥0
p ≥0
(resp. D ( X,λ ) _ D ( X,λ ));
q
(c) p τ ≥ 0 (resp. p τ ≥ 0 ) is equivalent to the identity function when restricted to D ≥ 0 ( X,λ ) (resp.
q ≥0
q ≥0
p ≥0
q ≥0
p ≥0
D ( X,λ ) ) since D ( X,λ ) _ D ( X,λ ) (resp. D ( X,λ ) _ D ( X,λ ));
p
(d) q τ ≤ 0 (resp. q τ ≤ 0 ) is equivalent to the identity function when restricted to D ≤ 0 ( X,λ ) (resp.
p ≤0
p ≤0
q ≤0
p ≤0
q ≤0
D ( X,λ ) ) since D ( X,λ ) _ D ( X,λ ) (resp. D ( X,λ ) _ D ( X,λ ));
q ≥0
D ( X,λ ) (resp.
(e) p τ < 0 (resp. p τ < 0 ) is equivalent to the null function when restricted to
≥
q
D 0 ( X,λ ) );
p ≤0
D ( X,λ ) (resp.
(f) q τ > 0 (resp. q τ > 0 ) is equivalent to the null function when restricted to
p ≤0
D ( X,λ ) );
p ≤0
D ( X,λ ) , p D ≥ 0 ( X,λ )) (resp.
p is locally bounded, then the intersection of (
C) By 2 (a), if
p ≤0
p ≥0
(•)
(b)
(•)
(b)
( D ( X,λ ) , D ( X,λ )) ) with D ( X,λ ) or D ( X,λ ) (resp. D ( X,λ ) or D ( X,λ ))induces
a t -structure on the latter ∞ -category.
D) By 2 (e) and (f), if X is quasi-compact and p is bounded, there exist constant integers α<β
p
p
p 0
p 0
such that H0 = H0 ◦ τ [α,β ] (resp. H = H ◦ τ [ α,β ] ).
3. The m-adic formalism and constructibility
In this chapter, we introduce a special case of the adic formalism, namely, the m-adic formalism on
which there is a good notion of constructibility. Such formalism is enough for most applications. The
basic notion of the m-adic formalism is given in §3.1. In §3.2, we introduce some finiteness conditions
under which we may refine the construction of the usual t -structure. Then we define the category of
constructible adic complexes in this setting in §3.3, and on which the constructible adic perverse
tstructure in §3.4. The last section §3.5 is dedicated to proving the compatibility between our theory and
Laszlo–Olsson [14,15] under their restrictions.
3.1. The m-adic formalism.
ENHANCED ADIC FORMALISM FOR ARTIN STACKS
25
Definition 3.1.1. Define a category PRing as follows. The objects are pairs (Λ , m), where _ is a small
ring and m _ Λ is a principal ideal, such that
• m is generated by an element that is not a zero divisor;
n +1
• the natural homomorphism _ → lim
( n 2 N).
k n Λn is an isomorphism, where _n = Λ / m
A morphism from (_ 0, m0) to (_ , m) is a ring homomorphism φ : Λ 0 → Λ satisfying φ• 1 ( m) = m0. We
denote by PRing L- tor _ PRing the full subcategory spanned by (Λ, m) such that ( N, Λ •) 2 Rind L-tor .
π
We have a natural functorPRing → Fun([1], Rind) sending (Λ, m) to ( N, Λ•) k→( _, Λ). In what follows,
we simply write _ • for the ringed diagram ( N, Λ •).
Let A be an Abelian category. An object M • in Fun( Nop , A ) is called essentiallynull if for each n 2 N,
there is an r 2 N such that M r + n → M n is the zero morphism. If A admits sequential limits, we have a
op
A → A . Given a topos T , we have a pair of adjoint functors
left exact functor lim:
k Fun( N , )
_
π : Mod( T, Λ) → Mod( T N, Λ •);
π_ : Mod( T N, Λ•) → Mod( T, Λ)
N
◦
induced by the morphism π: ( N, Λ •) → ( _, Λ) . Then π_ = lim
k ν , where ν : Mod( T , Λ•)
op
Fun(N , Mod( T, Λ)) is the obvious forgetful functor, which is exact.
Lemma 3.1.2. Let F• beamodulein Mod( T N, Λ•) suchthat νF • isessentiallynull. Then
forall n ≥ 0.
Rn πF
_
•
→
=0
Proof. Note that R n πF
( 7→Hn( U N,F • )) , where U runs over
_ • is the sheaf associated to the presheaf U
objects of T . Let a : ( U N, Λ •) → ( _N , Λ •) be the morphism of ringed topoi. Since R q aF
_ • is essentially
aF
_
null for all q, we have R•(U N,F • ) ' R limR
_ • = 0.
k
Let X be a higher Artin stack. We have a pair of adjoint functors
_
_
L π = L πX : D ( X, Λ) → D ( X, Λ •);
R π_ = R πX _ : D ( X, Λ•) → D ( X, Λ) .
Let D 0 ( X, Λ •) be the full subcategory of D ( X, Λ •) spanned by complexes whose cohomology sheaves
(+)
are all essentially null. Let D 0 ( X, Λ •) = D (+) ( X, Λ •) ∩ D 0 ( X, Λ •). Both are stable subcategories.
Lemma 3.1.3. For
(+)
K 2 D 0 ( X, Λ•) ,wehaveR π_ K = 0 .
Proof. We may assume that X is a scheme and K
Lemma 3.1.2.
2 D +0 ( X, Λ •).
Then the statement follows from
_
Lemma 3.1.4. The cohomological amplitude of π_ iscontainedin [k 1, 0].
× λ n +1
Proof. By assumption, m = ( λ ), where λ 2 Λ is not a zero divisor. Since Λ kkkk→ Λ is a resolution of
Λ n, the tor-dimension of the Λ-module Λ n is ≤ 1.
_
Definition 3.1.5 (Normalized complex). A complex K 2 D ( X, Λ•) is called normalized (resp. essentially
(+)
normalized) if the cofiber3[20, 1.1.1.6] of the adjunction mapL π_ R π_ K → K is 0 (resp. inD 0 ( X, Λ•)).
Let D n ( X, Λ •) (resp. D en ( X, Λ •)) be the full subcategory of D ( X, Λ •) spanned by normalized (resp.
essentially normalized) complexes, which is a stable subcategory.
Note that D ( X, Λ) = D ( X, Λ) , so that the image of L π_ = π_ is contained in D ( X, Λ •). In particular,
we have D n ( X, Λ •) _ D ( X, Λ •). For the other direction, we have the following lemma.
Lemma 3.1.6. We have D ( X, Λ •) ∩ D (+) ( X, Λ •) _ D n ( X, Λ •) .
Proof. The proof is similar to [27, 4.13].
_
Lemma 3.1.7. The image of the functor L π_ ◦ R π_ | D en ( X, Λ •) iscontainedin D n ( X, Λ•) . Moreover,
D n ( X, Λ •) _ D en ( X, Λ •) .
theinducedfunctor D en ( X, Λ•) → D n ( X, Λ•) isrightadjointtotheinclusion
3Theunderlyingobjectintheordinarytriangulatedcategoryisacone[20,1.1.2.10].
26
YIFENG LIU AND WEIZHE ZHENG
Proof. For the fist assertion, we need to show that L π_ R π_ L π_ R π_ K → L π_ R π_ K is an equivalence
(+)
for K 2 D en ( X, Λ •). By de_nition, the co_ber of L π_ R π_ K → K is contained in D 0 ( X, Λ•). The
assertion then follows from Lemma 3.1.3.
For the second assertion, we need to show that the natural transformation L π_ ◦ R π_ → id induces a
homotopy equivalence (i.e. an equivalence inH )
_
Map D n ( X, Λ •) ( K , L π R π_ L ) → Map D en ( X, Λ •) ( K , L ) ,
for every object K (resp. L ) of D n ( X, Λ •) (resp. D en ( X, Λ •)). By definition, the cofiber L 0 of
(+)
L π_ R π_ L → L is in D 0 ( X, Λ•), and K is equivalent to L π_ R π_ K . Therefore, the assertion follows
from the fact that
_
0
0
Map D en ( X, Λ •) ( L π R π_ K , L ) ' MapD ( X, Λ) ( R π_ K , R π_ L ) '{_}
Here in the second equivalence we have used the fact that
3.1.3.
Lemma 3.1.8. Let K 2 D ( X, Λ •) ∩D
and Hom(K , L ) = 0 .
(•)
( X, Λ •) , L 2
(+)
D0
0
R π_ L
.
= 0, which follows from Lemma
_
( X, Λ•) . ThenHom ( K , L ) 2
(+)
D0
( X, Λ •)
_
Proof. The proof is similar to [27, 4.19].
≥n
Lemma 3.1.9. For every K 2 D n ( X, Λ •) and n 2 Z , τ K isin D en ( X, Λ •) . Moreover,thefiberof
≥n
K → τ ≥ n K isconcentratedindegree n k 1.
theadjunctionmapL π_ R πτ
_
Proof. This is essentially proved in [27, 4.14].
Let us recall the arguments.
The fiber of the map
a : L πτ_ ≥ n R π_ K → τ ≥ n L π_ R π_ K ' τ ≥ n K is concentrated in degreen k 1 and belongs to D 0 ( X, Λ •).
Consider the diagram
L π_ R π_ L πτ_
≥n
R π_ K
b
_
L π R πa
_
/L πτ_
/
≥n
R π_ K
a
_
_
c
≥n
/τ ≥ n _K .
K
L π_ R πτ
__
/
By Lemma 3.1.2, L π_ R πa
_ is an equivalence. By Lemma 3.1.6,b is an equivalence. Therefore, the fiber
of c is equivalent to the fiber of a.
_
We denote Moden ( X, Λ•) the full subcategory of D en ( X, Λ•) spanned by complexes that are concentrated at degree 0, and Mod (0 X, Λ•) the full subcategory of Mod en ( X, Λ•) spanned by essentially null
modules. Then Mod 0( X, Λ•) is closed under sub-objects, quotients and extensions.
Proposition 3.1.10. For
n 2 Z , let D ≤enn ( X, Λ•) = D ≤ n ( X, Λ •) ∩ D en ( X, Λ •) and D ≥enn ( X, Λ •) =
≥n
D ( X, Λ •) ∩ D en ( X, Λ •) . Then ( D ≤en0 ( X, Λ •) , D ≥en0 ( X, Λ •)) defines a t -structure on D en ( X, Λ •) with
heart Mod en ( X, Λ •) ,and Mod en ( X, Λ•) is(equivalenttothenerveof)afullsubcategoryof
Mod( X, Λ •) ,
closedunderkernels,cokernelsandextensions.
τ ≤ 0 and τ ≥ 0 preserve the full subcategory D en ( X, Λ •). Since
Proof. We only need to show that
D en ( X, Λ •) is a stable full subcategory, we only need to prove this for τ ≥ 0 , that is, the cofiber of the
(+)
≥0
K → τ ≥ 0 K is in D 0 ( X, Λ •) for every object K of D en ( X, Λ •). Consider
adjunction map L π_ R πτ
_
the diagram
L π_ R πτ
_
≥0
L π_ R π_ K
_
L π R πa
_
_
L π_ R πτ
__
b
/τ ≥ 0 L π_ R π_ K
/
a
_
c
≥0
/τ ≥ 0_K .
K
/
(+)
(+)
By definition, the cofiber of L π_ R π_ K → K is in D 0 ( X, Λ •), so that the co_ber of a is in D 0 ( X, Λ •).
_
It follows that L π_ R πa
_ is an equivalence, by Lemma 3.1.3. By Lemma 3.1.7,L π R π_ K 2 D n ( X, Λ •).
(+)
Thus, by the first part of Lemma 3.1.9, the cofiber of b is in D 0 ( X, Λ •). Therefore, by the octahedral
(+)
_
axiom, the cofiber of c is in D 0 ( X, Λ •) as well.
ENHANCED ADIC FORMALISM FOR ARTIN STACKS
27
Corollary 3.1.11. The essential image of L π_ ◦ R π_ | D ≥enn ( X, Λ •) is right perpendicular to the full
subcategoryD n ( X, Λ•) ∩ D <n ( X, Λ •) of D n ( X, Λ •) .
DM
Let f : Y → X be a smooth (resp. étale) morphism in ChpAr
), (Λ , m) be an object of
L (resp. Chp
_
PRingL- tor (resp. PRing). By Poincaré duality, f : D ( X, Λ •) → D ( Y, Λ•) preserves normalized complexes.
Moreover, if f is surjective, then K 2 D ( X, Λ •) is normalized if and only if f _ K is normalized.
(Λ , m) beanobjectof PRing, K , L 2 D n ( X, Λ •) .
Proposition 3.1.12. Let X beahigherArtinstack,
ThenHom ( K , L ) isadic.
Proof. The proof is similar to [27, 4.18].
_
Proposition 3.1.13. Let f : Y → X beamorphismofhigherArtinstacks,
Then f _ : D ( Y, Λ) → D ( X, Λ) preservesnormalizedcomplexes.
Proof. Let K 2 D n ( Y, Λ) . Then L π_ R πf_
_
L π R πf_
_K
_K
_
' L πf
→ f _ K is equivalent to the composite
_ R π_ K
_
→ f _ L π_ R π_ K k→f _ K .
Therefore, it su_ces to show that for all L 2 D ( Y, Λ) , L πf_
family ( e_n ) n 2 N is conservative, it su_ces to show that _
Therefore, it su_ces to show that the diagram
D ( X, Λ)
O
_
f O
D ( Y, Λ)
is left adjointable.
(Λ , m) beanobjectof PRing.
Λn • Λ
_L
L
n
Λ
→ f _ L π_ L is an equivalence. Since the
f _ L → f _ (Λ n
L
Λ
L ) is an equivalence.
/D ( X, Λ n)
O
/
Of _
Λn • Λ
/D ( Y, Λ n)
/
For this, we may assume that f is a morphism of schemes. Using the resolution
L
× λ n +1
Λ kkkk→ Λ of Λ ,n we obtain an equivalence Λ
trivial.
n
k' Λ
RHom Λ (Λ n , k )[ 1]. The assertion is then
_
Similarly, we have the following.
ChpAr
Proposition 3.1.14. Let f : Y → X beamorphismlocallyoffinitetypeof
L (resp.locallyoffinite
DM
DM
type of Chp , resp. locally quasi-finite of Chp ), (Λ , m) be an object of PRingL- tor (resp. PRingtor ,
resp. PRing). Then f ! : D ( X, Λ) → D ( Y, Λ) preservesnormalizedcomplexes.
3.2. Finiteness conditions and the usual t -structure. Let X be a higher Artin stack and (_ , m) be
an object of PRing. Recall that D n ( X, Λ •) _ D ( X, Λ•).
Definition 3.2.1. The pair ( X, (Λ , m)) is said to be admissible if D ( X, Λ •) _ D n ( X, Λ •) (so that
D ( X, Λ •) = D n ( X, Λ •)), that is, for every K 2 D ( X, Λ •), the adjunction map L π_ R π_ K → K is
an equivalence.
Proposition 3.2.2. Let ( X, (Λ , m)) be an admissible pair. Then RX | D en ( X, Λ•) '
≥
D n ( X, Λ •) istheessentialimageofL π_ ◦ R π_ | D ≥enn ( X, Λ •) .
L π_ ◦ R π_ , and
Proof. The first assertion follows from Lemma 3.1.7. We let D 0 denote the essential image of L π_ ◦
≥n
R π_ | D ≥enn ( X, Λ •) in the second assertion. Then D 0 _ D ( X, Λ •) by Corollary ??. Moreover, since
_
L π : D ( X, Λ) → D ( X, Λ •) is left t -exact, R π_ | D ( X, Λ •): D ( X, Λ •) → D ( X, Λ) is right t -exact. Thus,
≥n
for K 2 D ( X, Λ •), the _ber of K → τ ≥ n K (where τ ≥ n is the truncation functor in D ( X, Λ•)),
equivalent to the fiber of L π_ R π_ K ' L πτ_ ≥ n R π_ K → τ ≥ n L π_ R π_ K , is concentrated in degree
≥n
K , which belongs to D 0 by
n k 1 and belongs to D 0 ( X, Λ •). Therefore, K ' L π_ R π_ K ' L π_ R πτ
_
_
Lemma 3.1.9.
0
Remark 3.2.3. Let Mod en ( X, Λ•) be the full subcategory of Mod en ( X, Λ •) spanned by complexes (modules) K such that H i L π_ R π_ K = 0 for i> 0, which is an exact category. Then the projection functor
♥
0
D ( X, Λ•) → Mod en ( X, Λ •) / Mod 0( X, Λ •) from the heart of the usual t -structure to the full subcategory
0
of Mod en ( X, Λ •) / Mod 0( X, Λ •) spanned by the image of Moden ( X, Λ •) is an equivalence of categories. In
0
_
fact, L π ◦ R π_ | Mod en ( X, Λ •) induces a quasi-inverse.
28
YIFENG LIU AND WEIZHE ZHENG
Proposition 3.2.4. Let f : Y → X beamorphismofhigherArtinstacks.
_
A) Let (Λ , m) be an object of PRing such that ( X, (Λ , m)) is admissible. Then f : D ( X, Λ•) →
D ( Y, Λ•) is t -exactwithrespecttotheusual t -structures.
B) Assume f islocallyoffinitetype(resp.locallyoffinitetype,resp.alocallyquasi-finitemorphism
ofDeligne–Mumfordstacks). Let (Λ , m) beanobjectof PRingL- tor (resp. PRingtor ,resp. PRing)
suchthat ( Y, (Λ , m)) isadmissible. Then f ! : D ( Y, Λ •) → D ( X, Λ •) isright t -exactwithrespect
totheusual t -structures.
≥
≤
•
_
n
n 1
Proof. A) We only need to show the right t -exactness off . Let K 2 D ( X, Λ •), L 2 D
( Y, Λ •).
Consider the fiber sequence τ ≤ n • 1 K → K → τ ≥ n K in D ( X, Λ•). It induces a fiber sequence
_
f τ_ ≤ n • 1 K → f K → f τ_ ≥ n K in D ( Y, Λ•). By assumption and Proposition 3.2.2, τ ≤ n • 1 K 2
(+)
(+)
D 0 ( X, Λ•). Thus f τ_ ≤ n • 1 K 2 D 0 ( Y, Λ •), and Hom( L ,f τ_ ≤ n • 1 K ) = 0 by 3.1.8. It follows that
_
_
≥n
Hom(L ,f K ) = 0. Therefore, f K 2 D ( Y, Λ•).
B) Similar to A).
_
Ar
DM
Let f : Y → X be a smooth surjective morphism in ChpL (resp. Chp ), (Λ , m) be an object of
PRingL- tor (resp. PRing). By Poincaré duality, if ( Y, (Λ , m)) is admissible, then ( X, (Λ , m)) is locally
admissible. This applies in particular to the case whereY is an algebraic space. In this case, admissibility
is related to the following finiteness condition on cohomological dimension.
Definition 3.2.5. Let X be a higher Artin (resp. Deligne–Mumford) stack, R be a ring. We say X is
locally R -bounded, if there exists an atlas (resp. étale atlas) ` i 2 I X i → X with X i algebraic spaces such
that for every i 2 I , and every schemeU étale and of finite presentation over X i ,
cdR( U ) := max { n | H n( U,F ) 6
= 0 for some F 2 Mod( U,R ) } < ∞ .
Proposition 3.2.6. Let X beanalgebraicspace, (Λ , m) beanobjectof PRing. Considerthefollowing
conditions:
A) Thepair ( X, (Λ , m)) isadmissible.
B) Forevery K 2 D ( X, Λ •) ,R π_←•
( F Λ • K ) = 0 ,where F • 2 Mod( X, Λ •) ' D ♥ ( X, Λ•) is
0
0
0
∙∙∙k→ Λ/ m k→∙∙∙k→ Λ / m.
C) R π_ K = 0 forevery K 2 D 0 ( X, Λ •) .
D) There exists an étale cover ` i 2 I X i → X by algebraic spaces such that, for every i 2 I , the
cohomologicaldimensionof π_ : Mod( X i,N´et, Λ•) → Mod( X i, ´et, Λ) isfinite.
E) Thealgebraicspace X islocally (Λ / m) -bounded.
Wehave E) ) D) ) C) ) B) , A) .
Proof. E) ) D) : By étale base change, we can assume that for every scheme
U étale and of finite type
over X , cd Λ / m ( U ) = N< ∞ . Since for n 2 N, every _ n = Λ / mn +1 -module is a successive extension of
N , Λ•), R i πF
Λ / m-modules, we have cdΛ n ( U ) = N . For a sheaf F • 2 Mod( X ´et
_ • is the sheaf associated to
i
N
the presheaf U 7→H ( U´et,F • ). Thus, from the exact sequence
N ,F • )
/R 1lim Hi • 1 ( U´et,F n )
/H i ( U´et
/lim Hi ( U´et,F n )
/0,
kn
kn
/
/
/
/
we know that R i πF
+ 1.
_ • = 0 for i>N
D) ) C) : We can assume X to be quasi-compact. Then this follows from Lemma 3.1.3 and the
following standard observation. Let f : B → A be a left exact additive functor of Grothendieck Abelian
categories, such that R i f = 0 for i>d
where d is a non-negative integer. Then R f sends D≤ n ( B )
≤ n+ d
to D
( A ) . In fact, let X be an element of D ≤ n ( B ). By [12, 14.3.4], we can compute R fX by
fY , where Y is any resolution of X with f -acyclic components. We can take Y to be τ ≤ n + d+1 of
a homotopically injective resolution with injective components (fibrant replacement) of
X . Then Y
belongs to C≤ n + d +1 ( B ) . This shows that R f sends D≤ n ( B ) to D ≤ n + d+1 ( A ) . It follows that R f sends
D ≤ n ( B ) to D ≤ n + d ( A ) by truncation.
C) ) B) : In fact, for every K 2 D ( X, Λ •), F• Λ • K belongs to D 0 ( X, Λ •).
0
ENHANCED ADIC FORMALISM FOR ARTIN STACKS
29
B) , A) : Let K 2 D ( X, Λ•). We need to show that R π_←•
( F Λ • K ) = 0 if and only if the adjunction
map L π_ R π_ K → K is an equivalence. Since Q n 2 N e_n is conservative, the latter is equivalent to the
L
condition that the morphism _: Λ n Λ R π_ K → K n := e_n K is an isomorphism in the derived category
D( X, Λ n) for all n 2 N. The morphism _ can be decomposed as
Λn
L
Λ
L
α
_
R π_ K k→R π_ (L π Λn
Λ•
L
β
K ) k→R π_ ( π_ Λn
Λ•
γ
K )
k→R( π≥ n ) _ ( π≥_ n Λn
L
Λ• ,≥
δ
n
K ≥ n ) k→R( π≥ n ) _ ( en _ K n ) ' K n ,
where π≥ n : ( N≥ n , Λ • , ≥ n ) → ( _, Λ) , en : ( { n } , Λ) → ( N≥ n , Λ• , ≥ n ) . Here N≥ n _ N is the full subcategory
spanned by integers≥ n .
By assumption, m is generated by an element λ that is not a zero divisor.
Thus we have a finite
× λ n +1
free resolution [_ kkkk→ Λ] of Λ n as an _-module. Therefore, L π_ Λ n is represented by the complex of
× λ n +1
Λ •-modules [Λ • kkkk→ Λ•] (in degrees k 1 and 0). This implies that L π_ Λ n
× λ n +1
mapping cone ofK kkkk→ K , which is a fibrant object. Then _ n
both represented by π_ K
by the identity.
Consider the diagram
0 ×λ
n +1
0
0
kkkk→ π_ K , where K
Λ R π_ K
Λ•
K is represented by the
and Rπ_ (L π_ Λn
L
Λ•
K ) are
is a fibrant replacement of K , and α is represented
j
( N≥ n , Λ• , ≥ n )
_
( N≥ n ,λ
_ ≥ n)
j
/ ( N, Λ • )
/
0
_
/( N,λ
_)
/
π
0
π≥
L
L
0
n
&_
Λ,
&_
where λ is the constant ring with value _. By the cofinality of N≥ n in N, the natural transformation π_0 →
_
_
( π≥0 n ) _ ◦ j 0 is an isomorphism. Sincej _0 admits an exact left adjoint, it follows that R π_0 → R( π≥0 n ) _ ◦ j 0
_
is an isomorphism. Thus the natural transformation R π_ → R π≥ n ◦ j is an isomorphism. Therefore, γ
is an isomorphism.
L
L
The morphism δ is induced by the morphism π≥_ n Λ n Λ • , ≥ n K ≥ n → ene_ _n ( π≥_ n Λ n Λ • , ≥ n K ≥ n ) '
en _ K n , which is an isomorphism since K is adic. Therefore, _ is an isomorphism if and only if β is an
isomorphism.
By the above resolution of _, the cone of L π_ Λ n → Λn is Gn• [k 2], where Gnm = Λ / mmin( m,n )+1 and
the transition maps are multiplication by λ , so that G0• = F• . Thus, if β is an isomorphism for n = 0,
n•1
then R π_←•
( F Λ • K ) = 0. For n ≥ 1, Gn• is an extension of F • by G• +1 . Thus, if R π_←•
( F Λ • K ) = 0,
then, by the above, β is an isomorphism for all n 2 N.
_
3.3. Constructible adic complexes. Let (_ , m) be an object of PRing such that Λ/ mn +1 is Noetherian
for all n . For a higher Artin stack X , we define
D cons ( X, Λ •) = D ( X, Λ •) ∩ D cons ( X, Λ •) ,
D (+)
cons ( X,
Λ•) = D ( X, Λ•) ∩ D (+)
cons ( X, Λ •) ,
•)
•)
D (cons
( X, Λ •) = D ( X, Λ •) ∩ D (cons
( X, Λ •) .
The following is an immediate consequence of the definitions and [18, 6.2.3].
Lemma 3.3.1. Let f : Y → X beamorphismofhigherArtinstacks. Thentheoperationsin§1.4restrict
tothefollowing:
_
1L 0: f : D cons ( X, Λ •) → D cons ( Y, Λ •) ;
(•)
(• )
(•)
3L 0: kk X : D cons ( X, Λ •) × D cons ( X, Λ •) → D cons ( X, Λ•) .
•
( )
Inparticular, D cons ( X,λ )
[20, 2.2.1] isasymmetricmonoidalcategory.
30
YIFENG LIU AND WEIZHE ZHENG
S be
As in [18, 6.2], to state the results for the other operations, we work in a relative setting. Let
an L-coprime higher Artin stack. Assume that there exists an atlas S → S, where S is either a quasiexcellent scheme or a regular scheme of dimension≤ 1. Combining [18, 6.2.4] and Propositions 3.1.12,
3.1.13, 3.1.14, we have the following.
Proposition 3.3.2. Let f : Y → X beamorphismof ChpAr
lft / S. Thentheoperationsin§1.4restrictto
thefollowing:
(+)
(+)
1R 0: f _ : D cons ( Y, Λ •) → D cons ( X, Λ •) ,if f isquasi-compactandquasi-separated,and ( X, (Λ , m))
and ( Y, (Λ , m)) areadmissible;
(• )
(•)
2L 0: f ! : D cons ( Y, Λ •) → D cons ( X, Λ •) , if f is quasi-compact and quasi-separated,
and
λ 2 PRingL- tor ;
!
2R 0: f : D cons ( X, Λ •) → D cons ( Y, Λ •) , if ( X, (Λ , m)) and ( Y, (Λ , m)) are admissible, and λ 2
PRingL- tor .
(•)
(+)
(+)
3R 0: Hom X ( k , k ) : D cons ( X, Λ•) op × D cons ( X, Λ•) → D cons ( X, Λ•) ,if ( X, (Λ , m)) isadmissible.
Let X be a scheme in Schqc . sep . Recall that a complex K 2 D ( X, Λ •) is a λ -complex [14, 3.0.6] if
H K is constructible and almost adic. In particular, K 2 D cons ( X, Λ•).
n
qc . sep
Lemma 3.3.3. Let X beaschemein Sch
suchthatConditionC)inProposition3.2.6holdsfor
0
thepair ( X, (Λ , m)) . Let D cons ( X, Λ•) bethefullsubcategoryof D en ( X, Λ •) spannedby λ -complexes. We
have
τ ≥ n and τ ≤ n .
A) D 0cons ( X, Λ•) isclosedunderthetruncationfunctors
_
0
B) The essential image of L π R π_ D cons ( X, Λ •) coincides with D cons ( X, Λ •) = D ( X, Λ•) ∩
D cons ( X, Λ •) .
Proof. Essentially in [27].
_
≤n
D cons
( X, Λ•) =
≤ n•1
of D cons ( X, Λ•) ?
Proposition 3.3.4. Let the assumptions be as in the above lemma.
Put
≤
≥n
D n ( X, Λ •) ∩D cons ( X, Λ •) . Thentherightperpendicularfullsubcategory D cons
( X, Λ•)
in D cons ( X, Λ •) istheessentialimageofL π_ R π_ ( D 0cons ( X, Λ•) ∩ D ≥enn ( X, Λ•)) . Moreover,thetruncation
functors τ ≤ n ' L π_ ◦ R π_ ◦ τ ≤ n and τ ≥ n ' L π_ ◦ R π_ ◦ τ ≥ n .
Proof.
_
Corollary 3.3.5. Let X beahigherArtinstackthatislocally
(Λ / m) -bounded. Thenthefullsubcategory
D cons ( X, Λ•) ispreservedunderthetruncationfunctors
τ ≤ n andhence τ ≥ n on D ( X, Λ •) .
Proof.
_
3.4. Constructible adic perverse t -structures. We fix an L-coprime base schemeS that is a disjoint
union of schemes which are excellent, quasi-compact, finite dimensional, and admit a global dimension
function for which we fix one. We fix also an object (_
, m) of PRingL-tor such that _/ mn +1 is an (L-torsion)
Gorenstein ring of dimension 0 for every n 2 N and S is locally (_ / m) -bounded.
Proposition 3.4.1. For an object f : X → S of ChpAr
lft / S equippedwithan admissible perversityevaluation
?
p,thetruncationfunctors p τ ≤ 0 , p τ ≥ 0 preserveD cons ( X, Λ •) for ? = (+) , ( k ) , (b) orempty.
Proof. By Lemma 2.3.1, we may assume that X is a quasi-compact, separated (and excellent, finite
dimensional) scheme which is (_/ m) -bounded, and p = p is an admissible perversity function on X . In
particular, p is bounded. We prove by Noetherian induction. We may further assume X is irreducible.
b
For a complex K 2 D cons ( X, Λ •), we may assume K 2 D cons ( X, Λ•) _ D b ( X, Λ •) by Remark 2.3.3 C).
Choose a dense open subset
U of X such that
• U is essentially smooth;
• p( x ) ≤ p( η) + codimX ( x ) for x 2 |U |, where η is the unique generic point of X ;
• p( x ) ≥ p( η) for x 2 |U |;
• The complex K U := K | U , viewed as an element of D b ( U, Λ•), has smooth almost adic cohomology sheaves.
ENHANCED ADIC FORMALISM FOR ARTIN STACKS
31
Then the perverse truncation forK U is simply the usual truncation (up to a shift byp( η)), which preserves
constructibility by Corollary 3.3.5.
_
3.5. Compatibility with Laszlo–Olsson. We prove the compatibility between our adic formalism and
Laszlo–Olsson’s [14], under their assumptions.
Let L = { ` } . Let S be an L-coprime scheme satisfying that
A) It is a_ne excellent and finite-dimensional;
B) For every S-scheme X of finite type, there exists an étale cover X 0 → X such that cd`( Y ) < ∞ 4
for every schemeY étale and of finite type over X 0;
C) It admits a global dimension function and we fix such a function (see [18, 6.3.1]).
Fix a complete discrete valuation ring _ with the maximal ideal m and residue characteristic ` such
that _ = limk n Λ n, where Λ n = Λ / mn +1 , as in [14]. In particular, (_ , m) is an object of PRing. For every
stack X in ChpLMB
lft / S , the pair X is locally (_ / m)-b ounded.
From the definition of D cons ( X , Λ•), which is the full subcategory of D ( X , Λ •) spanned by constructible
adic complexes, [14, 3.0.10, 3.0.14, 3.0.18], and [18, 5.3.6], we have a canonical equivalence between
categories
hD cons ( X , Λ •) ' D c ( X , Λ) ,
C.1)
where the latter one is defined in [14, 3.0.6].
Proposition 3.5.1. For a morphism f : Y → X offinitetypein
betweenfunctors:
hf
_
' L f _ : D c ( X , Λ) → D c ( Y, Λ) ,
hf
_
•
•
hf ! ' R f ! : D (c ) ( Y, Λ) → D (c ) ( X , Λ) ,
h( kk
) ' (k )
L
ChpLMB
lft / S ,therearenaturalisomorphisms
(+)
' R f _ : D (+)
c ( Y, Λ) → D c ( X , Λ);
!
hf ' R f ! : D c ( X , Λ) → D c ( Y, Λ);
•
•
•
( k ) : D (c ) ( X , Λ) × D (c ) ( X , Λ) → D (c ) ( X , Λ) ,
•
(+)
hHom ( k , k ) ' R hom Λ ( k , k ) : D (c ) ( X , Λ) opp × D (+)
c ( X , Λ) → D c ( X , Λ) .
thatarecompatiblewith
(3.1).
By Lemma 3.3.1 and Proposition 3.3.2, the six operations on the left side in the above proposition do
have the correct range.
Proof. The isomorphisms for tensor product, internal Hom andf _ simply follow from the same definitions
here and in [14, §ü, 6]. The isomorphism for f _ follows from the adjunction and that for f _ ([18, 6.3.2]).
The isomorphism for f ! will follows from the adjunction and that for f ! which will be proved below.
By the compatibility of dualizing complexes and the isomorphisms for internal Hom, we have natural
isomorphisms DX ' D X and D Y ' D Y. Moreover, D ( X , Λ •) dual contains D cons ( X , Λ •). Therefore, by
[14, 9.1], to prove the isomorphism for f ! , we only need to show that our functors satisfy
!
hf ' D Y ◦ hf
_
◦ DX .
In fact, by the biduality isomorphism, we have
!
!
hf ' hf ◦ D X ◦ D X( k )
!
= h f ◦ hHom X (h Hom X ( k , ΩX) , ΩX)
' hR Y ◦ hf ! ◦ hHom X (h Hom X ( k , ΩX) , ΩX)
' hR Y ◦ hHom Y (h f _ ◦ hHom X ( k , ΩX) ,f ! ΩX)
' hHom Y (h f
_
◦ hHom X ( k , ΩX) , ΩY)
_
= D Y ◦ hf ◦ D X .
_
4Accordingtoournotation,cd isnothingbutcd
`
F`
.
32
YIFENG LIU AND WEIZHE ZHENG
Remark 3.5.2. In view of the above compatibility, we prove all the expected properties of the six operations, in particular the Base Change Theorem, in the adic case of Laszlo–Olsson [14].
Our definition of the constructible adic perverse t -structure coincides with Laszlo–Olsson [15] under
their restrictions, where in particular X is a locally Noetherian (1-)Artin stack over a field k (that is, S =
Speck ) with cd `( k ) < ∞ , and p is the middle perversity smooth evaluation, that is, the unique perverse
qc . sep
smooth evaluation such that for every atlas u : X 0 → X with X 0 a scheme in Sch
, pu = ( f ◦ u) _1 p0 ,
where f : X → S is the structure morphism and p0 is the zero perverse function on S = Spec k .
4. Descent properties
In §4.1, we prove that our construction of derived ∞ -categories, as well as those of adic complexes, of
(higher) Artin stacks satisfies not only the smooth descent, but also the smooth hyperdescent
.
4.1. Hyperdescent. The étale ∞ -topos of an a_ne scheme is not hypercomplete in general. On the
contrary, the stable ∞ -categories we constructed satisfy (smooth) hyperdescent. We begin with a general
definition.
Definition 4.1.1 ( F -descent). Let C, D be ∞ -categories, F : Cop → D be a functor, X •+ : N( Δ + ) op → C
be an augmented simplicial object of C.
A) We say X •+ is an augmentationof F -descent if F ◦ ( X •+ ) op is a limit diagram in D .
B) Assume C admits pullbacks. We say X •+ is a hypercovering for universal F -descent if X q+ →
(coskq• 1 ( X •+/X •+ 1 )) q is a morphism of universal F -descent for all q ≥ 0.
In particular, a morphism f : X 0+ → X •+ 1 in C is of F -descent [18, 3.1.1] if and only if the Čech nerve of
f is an augmentation of F -descent.
By definition, a morphism of C is of F -descent if and only if its Čech nerve is an augmentation of
F -descent. We will give several criteria for B) ) A) .
Proposition 4.1.2. Let C bean ∞ -categoryadmittingpullbacks, D bean n -categoryadmittingfinitelimits, F : Cop → D beafunctor. Theneveryhypercovering X •+ foruniversal F -descentisanaugmentation
of F -descent.
Lemma 4.1.3. Let C, D be ∞ -categories such that C admits finite limits, F : Cop → D be a functor,
C suchthat V• → e isan
e beafinal objectof C, f • : U• → V• beamorphism ofsimplicial objectsof
augmentationof F -descentand f q isamorphismof F -descentforall q. Assumethereexists n ≥ 0 such
that U• is n -coskeletal, V• is ( n k 1)-coskeletal,and f q isanequivalencefor q<n . Then U• → e isan
augmentationof F -descent.
Proof. We may assumeF ( e) is an initial object of D . Let W+ : N( Δ + × Δ ) op → be a Čech nerve off • ,
W = W+ | N( Δ × Δ ) op . For every q ≥ 0, W+ | N( Δ + ×{ [q]} ) op is a Čech nerve off q , which is a morphism
op
of F -descent by assumption. It follows that F ◦ W+ | N( Δ + ×{ [q]} ) is a limit diagram. We may thus
op
op
identify the limit of F ◦ W with the limit F ◦ W+ | N( { [k 1]}× Δ s ). Since W+ | N( { [k 1]}× Δ ) op
op
can be identified with V• , the limit of F ◦ W can be identified with F ( e). Let D • = W ◦ δ, where
op
δ : N( Δ ) op → N( Δ × Δ ) op is the diagonal map. Since N(Δ ) op is sifted [19, 5.5.8.4], the limit of F ◦ D •
op
op
can be identified with F ( e). The proof of [19, 6.5.3.9] exhibits U• | N( Δ s ) as a retract of D • | N( Δ s ) .
op
It follows that the limit of F ◦ U• is a retract of F ( e) , and hence is F ( e) .
_
Lemma 4.1.4. Let C, D be ∞ -categoriessuchthat C admitspullbacks, F : Cop → D beafunctor, n ≥ 0,
X •+ bean n -coskeletalhypercoveringforuniversal F -descent. Then X •+ isanaugmentationof F -descent.
Proof. Since morphisms of universal F -descent are stable under pullback and composition,
coskm ( X •+/X •+ 1 ) → coskm • 1 ( X •+/X •+ 1 ) satisfies the assumptions of Lemma 4.1.3.
It follows by
+
+
induction that cosk n( X •/X • 1 ) is an augmentation of F -descent.
_
Lemma 4.1.5. Let D bean n -categoryadmittingfinitecolimits, f • : Y• → X • beamorphismofsemisimD suchthat Yq → X q isanequivalencefor q ≤ n . Thentheinduced
plicial(resp.simplicial)objectsof
morphismbetweengeometricrealizations |f • | : |Y• |→| X • | isanequivalencein D .
ENHANCED ADIC FORMALISM FOR ARTIN STACKS
33
The existence of the geometric realizations is guaranteed by [20, 1.3.3.10].
Proof. The semisimplicial case follows from the simplicial case by taking left Kan extensions. The simplicial case follows from the proof of [20, 1.3.3.10].
_
ProofofProposition??. It su_ces to apply the dual version of Lemma 4.1.5 to h : X •+
coskn ( X •+/X •+ 1 ) and Lemma 4.1.4.
→
_
The following can be used to simplify the proof of [23, 2.2.4].
Proposition 4.1.6. Let C bean ∞ -categoryadmittingpullbacks, D beanstable ∞ -categoryendowedwith
aweaklyrightcomplete t -structurethateitheradmitscountablelimitsorisrightcomplete,
F : Cop → D
+
op
beafunctor, X • : N( Δ + ) → C beahypercoveringforuniversal F -descentsuchthat F ◦ ( X •+ ) op factors
through D ≥ 0 . Then X •+ isanaugmentationof F -descent.
Proof. Let n ≥ 0. By Lemma 4.1.4, Y•+ = cosk n( X •+/X •+ 1 ) is an augmentation of F -descent, so that
it su_ces to show that the morphism h• : X •+ → Y•+ induces an isomorphism c : K = limk p2 Δ F ( X p ) →
lim
k p2 Δ F ( Yp ) = L is an isomorphism. By [20, 1.2.4.4, 1.2.4.5], we have a morphism of converging spectral
sequences
pq
+H p+ q K
E1 = H q F ( X p )
3
pq
p+ q
c1
H
c
_
_
pq
+Hp+_q L,
E01 = H_q F ( Yp )
3
pq
concentrated in the first quadrant. For p ≤ n , since hp is an equivalence,c1 is an isomorphism for all q.
≤
•
n 1
It follows that cpq
c is an equivalence. Sincen is arbitrary
r is an isomorphism for p + q ≤ n k 1, and τ
D
and is weakly right complete, c is an equivalence.
_
We let Pr Lst ,t (resp. Pr R
st ,t ) denote the ∞ -category defined as follows.
• Objects of Pr Lst ,t (resp. Pr R
st ,t ) are presentable stable∞ -categoriesC equipped with a t -structure.
• Morphisms of Pr Lst ,t (resp. Pr R
st ,t ) are t -exact functors admitting right (resp. left) adjoints.
The ∞ -categoriesPr Lst ,t (resp. Pr R
st ,t ) admit small limits, and those limits are preserved by the forgetful
L
L
R
L
R
C ≤ 0 (resp.
functor Pr st ,t → Pr st (resp. Pr st ,t → Pr R
st ). For a diagram K → Pr st ,t or K → Pr st ,t , (lim
k k)
≤0
≥0
≥0
C
C
(lim
by objects whose image inCk is in Ck (resp. Ck ).
k k ) ) is the full subcategory of lim
k k spanned
C 2 I → lim C2k I .
For an interval I , we have an equivalence (lim
k k)
k
L
R
We denote by Pr st ,t, wrc (resp. Pr st ,t, rc , wlc ) the full subcategory of Pr Lst ,t (resp. Pr R
st ,t ) spanned by those
C that are weakly right complete (resp. right complete and weakly left complete). These full subcategories
are stable under small limits in Pr R
st ,t .
Proposition 4.1.7. Consider a diagram
( D 0) op
j
op
F
/Pr R
st ,t, rc , wlc
/
P
_
_
G
/Cat_∞
D op
_
/
of ∞ -categories, where D admits pullbacks, j is an inclusion satisfying the right lifting property with
D 0 arestableunder
respectto ∂ Δ n _ Δ n for n ≥ 2, P istheforgetfulfunctor. Assumethatthearrowsin
0
+
op
pullbacksin D byarrowsin D . Let X • : N( Δ + ) → D beahypercoveringforuniversal G-descentsuch
that X •+ | N( Δ s+ ) op factorsthrough j . Then X •+ isanaugmentationof G-descent.
Proof. By the right completeness of F ( X p+ ) for p ≥k 1, it su_ces to show ( F ◦ ( X •+ ) op | N( Δ s + )) ≤ 0 is a
+ op
◦
| N( Δ s ) = C.
limit diagram. Let f ! be a left adjoint of the t -exact functor f _ : F ( X •+ 1 ) → lim
k F (≤X0 • )
+ ≤0
≤0
_≤ 0
The restrictions of these provide adjoint functors (f ! ) and (f ) : F ( X • 1 ) → C . Let us first show
that a : ff! K_ → K is an equivalence for all K 2 F ( X •+ 1 ) ≤ 0 , namely that ( f _≤) 0 is fully faithful. This
34
YIFENG LIU AND WEIZHE ZHENG
is similar to Proposition 4.1.6. Let n ≥ 0. The morphism h• : X •+ → coskn ( X •+/X
diagram
c
ff! K_
a
+
• 1)
= Y•+ induces a
_
/ggK
!
/
b
" |
"K |
_
+ op
◦
| N( Δ s ) . By Lemma 4.1.4,
where g! is a left adjoint of the t -exact functor g : F ( X •+ 1 ) → lim
k F ( Y• )
_
+
Y• is an augmentation of G-descent, so that b is an equivalence. Moreover,c = lim(
f pf !Kp_ → gpgK
! p ),
k
→
+
+
where f p! is a left adjoint of f p_ : F ( X • 1 ) → F ( X p+ ) , gp ! is a left adjoint of gp_ : F ( Y• 1 ) → F ( Yp+ ) ,
_
_
f pf ! p → gpg! p is induced by hp . By [20, 1.2.4.6, 1.2.4.7], we have a morphism of converging spectral
sequences
pq
+H p+ q ff! K_
E1 = H q( f • p! f •_ p K )
3
pq
c1
Hp+ q c
__
+Hp + q ggK,
_!
3
pq
concentrated in the third quadrant. For p ≥k n , since hp is an equivalence , c1 is an isomorphism for
≥ 1• n
pq
all q. It follows that cr is an isomorphism for p + q ≥ 1 k n , and τ
c is an equivalence. Therefore,
τ ≥ 1• n a is an equivalence. Sincen is arbitrary and F ( X •+ 1 ) is weakly left complete, a is an equivalence.
_
It remains to show that d : L → f fL
is an equivalence for every L 2 C≤ 0 . Since C is weakly left
!
≥ 1• n
d is an equivalence for every n ≥ 1. For this, we may assume
complete, it su_ces to show that τ
L 2 C[1 • n, 0] . We will show that L is in the essential image of ( f _≤) 0 . Since (f _≤) 0 is fully faithful,
[1 • n, 0]
this proves that d is an equivalence. Let H : Pr R
. It
st ,t, rc , wlc → Cat n be the functor sending F to F
su_ces to show that H ◦ F ◦ ( X •+ ) op | N( Δ s + ) is a limit diagram. Since Cat n is an ( n + 1)-category, we
may assume that X •+/X •+ 1 is (n + 1)-coskeletal by Lemma 4.1.5 applied toX •+ → coskn +1 ( X •+/X •+ 1 ) . In
_
this case, F ◦ ( X •+ ) op | N( Δ s+ ) is a limit diagram by Lemma 4.1.4.
_
pq
g• p ! g•_ p K )
E01 = H q( _
The following variant of Proposition 4.1.7 will be used to establish proper hyperdescent. To state it
conveniently, we introduce a bit of terminology. Let C be an ∞ -category admitting pullbacks, F : Cop →
Cat ∞ be a functor. We say a morphism f of C is F -conservative if F ( f ) is conservative. We say f is
universally F -conservative if every pullback of f in C is F -conservative. We say an augmented simplicial
object X •+ of C is a hypercovering for universal F -conservativenessif X n+ → (coskn • 1 ( X •+/X •+ 1 )) n is
universally F -conservative for all n ≥ 0.
C be an ∞ -category admitting pullbacks, F : Cop → Pr Lst ,t, wrc be a functor,
Proposition 4.1.8. Let
L
a be an integer, G : Pr st ,t, wrc → Cat ∞ be the functor sending C to C≥ a (resp. C+ = S n C≥ n ), X •+
be a hypercovering for universal ( G ◦ F ) -descent (resp. and universal ( P ◦ F ) -conservativeness, where
P : Pr Lst ,t, wrc → Cat ∞ istheforgetfulfunctor). Then
X •+ isanaugmentationof ( G ◦ F ) -descent.
Proof. The proof of the case of C≥ a is similar to the proof of Proposition 4.1.7. In the case of C+ , the
+ op
◦
→ lim G ◦ F ◦ ( X •+ ) op is an equivalence. The rest of the
conservativeness implies thatG(lim
k F (X • ) )
k
_
proof is similar.
Consider the functors [18, þ.4]
Chp Ar
_
Ar
L
EO : N( Chp ) op → Fun(N( Rind) op , Pr st , cl ) ,
C hp Ar
L
op
L
EO! : N( ChpAr
L ) → Fun(N( Rind L -tor ) , Pr st ) ,
and their adic version in §1.4
Chp Ar
_
EO : N( ChpAr ) op → Fun(N( Rind) op , Pr Lst , cl ) ,
C hp Ar
L
op
L
EO! : N( ChpAr
L ) → Fun(N( Rind L -tor ) , Pr st ) .
ENHANCED ADIC FORMALISM FOR ARTIN STACKS
35
∞ -DM
Ar
A hypercovering X •+ in Chp (or Chp
introduced below) is a smooth hypercoveringif X q+ →
+
(coskq• 1 ( X •+/X • 1 )) q is smoothandsurjective for all q ≥ 0.
Proposition 4.1.9. Every smooth hypercovering in ChpAr (resp. ChpAr
L ) is an augmentation of
_
op
_
op
Ar EO -descent(resp.
Ar EO
Ar EO -descent(resp.
-descent)and
EO! -descent).
!
Chp
Chp L
C hp
C hp Ar
L
Together with [20, 1.2.4.5] and its dual version, it implies Theorem 0.1.2.
Ar
Ar
Proof. Let X •+ be an augmented simplicial object ofChp (resp. ChpL ). It su_ces to apply Proposition
Ar
Ar
4.1.7 to the full subcategory Chpsm /X h 1 _ Chp/X h 1 spanned by higher Artin stacks smooth over X • 1 .
_
The adic version follows by restricting to the subcategories of adic complexes.
∞ - DM
Definition 4.1.10. The ∞ -category of ∞ -DM stacks Chp
is the ∞ -category Sch(G´et( Z )) of G´et( Z ) schemes in the sense of [21, 2.3.9, 2.6.11].
Using Proposition 4.1.7, we can adapt the DESCENT program [18, ü] to define enhanced operation
maps for ∞ -DM stacks:
Chp ∞
Chp
-DM
∞ -DM
∞ - DM cart
) F 0,A
_
EO: δ2, { 2} Fun(Δ 1 , Chp
∞ - DM op
_
EO : ( Chp
)
→
PrL
→ Fun(N( Rind tor ) op , Mon P f st ( Cat ∞ )) ,
→ Fun(N( Rind) op , Pr Lst , cl ) ,
compatible with the enhanced operation maps constructed for higher DM stacks [18, þ.5].
Definition 1.4.1, we obtain similarly the enhanced adic operation maps for ∞ -DM stacks:
Chp ∞
Chp ∞
-DM
-DM
∞ - DM cart
) F 0,A
_
EO: δ2, { 2} Fun(Δ 1 , Chp
∞ - DM op
_
EO : ( Chp
)
→
Applying
PrL
→ Fun(N( Rind tor ) op , Mon P f st ( Cat ∞ )) ,
→ Fun(N( Rind) op , Pr Lst , cl ) ,
compatible with the enhanced adic operation maps constructed for higher DM stacks in §1.4. Recall by
restriction, we obtain functors:
C hp ∞
-DM
EO! , Chp ∞
-DM
∞ - DM
EO! : ( Chp
) F → Fun(N( Rind tor ) op , Pr Lst ) .
∞
_
Proposition 4.1.11. Every smooth hypercovering in Chp - DM is an augmentation of Chp ∞ - DM EO op
_
op
descentand(resp. Chp ∞ - DM EO! -descent)and Chp ∞ - DM EO -descent(resp. Chp ∞ - DM EO! -descent).
4.2. Proper descent. The following criterion is useful for checking condition A) of [20, 6.2.4.3].
Proposition 4.2.1. Let C, D bestable ∞ -categoriesequippedwith t -structuressuchthat D isweaklyleft
C isrightcomplete), F : C → D beaconservative t -exact
complete(resp.andweaklyrightcompleteand
functor. Assumeoneofthefollowing
A) C isleftcomplete;
B) C admitscountablecolimits.
Then C≤ 0 (resp. C) admits F -splitgeometricrealizations,andthosegeometricrealizationsarepreserved
by F .
Proof. Let X • be an F -split simplicial object of C, Y• : N( Δ ∙
the unnormalized cochain complex
) op → D be an extension ofF ◦ X • . Then
∙∙∙→ H q Y2 → Hq Y1 → H q Y0 → Hq Y• 1 → 0
is acyclic. It follows that the unnormalized cochain complex
θq
∙∙∙→ Hq X 2 → Hq X 1 k→ Hq X 0
is an acyclic resolution of the object A q = coker( θq ) in the heart of C and FA q ' Hq Y• 1 . The lemma
then follows from [20, 1.2.4.9 (resp. 1.2.4.10)] in case A) and the following variant in case B).
_
36
YIFENG LIU AND WEIZHE ZHENG
Lemma 4.2.2. Let C beastable ∞ -categoryadmittingcountablecolimitsandequippedwitha t -structure
(resp. right complete t -structure), X • be a semisimplicial object of C≤ 0 (resp. C) such that for every
integer q,theunnormalizedcochaincomplex
θq
∙∙∙→ Hq X 2 → Hq X 1 k→ Hq X 0
is an acyclic resolution of the object A q = coker( θq ) in the heart of C. Then there exists a geometric
realization X = |X • | in C, and for every integer q, the map Hq X 0 → H q X induces an isomorphism
A q ' Hq X .
Proof. One can repeat the proof of [20, 1.2.4.11 (resp. 1.2.4.12)] verbatim.
_
Consider the functors
≥0
_
EO :
C hp Ar
L
Chp ∞
≥0
-DM
Ar
N( ChpL ) op → Fun(N( Rind L-tor ) op , Cat ∞ ) ,
_
∞
EO : N( Chp
-DM op
)
→ Fun(N( Rind tor ) op , Cat ∞ )
sending X to λ 7→D ≥ 0 ( X,λ ) . Note that we also put restrictions on the categories of ringed diagrams
here.
Proposition 4.2.3. Let S bean L-coprime locally Noetherian higherArtinstack(thatis, thereexists
anatlas S → S where S isalocallyNoetherianscheme).
A) Foreveryobject λ of Rind L- tor andeveryCartesiansquare
W
g
q
/Z
/ p
_ f
_
/X_
Y_
/
Ar
in ChpAr
L (resp. Chplft / S) with p properoffinitediagonal(proper,1-Artin,ofseparateddiagonal),
theinducedsquare
D ≥ 0 ( Z,λ ) o
o
_
p
_
D ≥ 0 ( X,λ )
g
f
_
q
D ≥ 0 ( W,λ
_ )o
o
_
_
_
D ≥ 0 (_Y,λ )
isrightadjointable.
Ar
B) Every proper finite-diagonal hypercovering in ChpL (resp. proper, 1-Artin, separated-diagonal
≥0
_
Ar
hypercoveringin Chplft / S) isanaugmentationof Chp Ar EO -descent.
L
A similar result holds for
Chp ∞
≥0
-DM
_
EO .
Proof. Sketch: We apply [18, 3.3.6, 4.3.6] to prove A) and the descent part of B). Assumption A) of
[18, 3.3.6] follows from the dual of Proposition 4.2.1 A) and 2.2.1.
Assumption B) follows from the
existence of a finite cover [26, Theorem B] (resp. proper cover [22, 1.1]). The assertion on hypercovering
_
follows from Proposition 4.1.8 applied to the functor sending X to D ( X,λ ) for a fixed λ .
Similarly, we have the following.
Proposition 4.2.4. Let S be an L-coprime higher Artin stack such that there exists an atlas S → S
where S isalocallyNoetherianschemeandsuchthateverya_neschemeadmittingafinitelypresented
mapto S is R -cohomologicallyfiniteforevery L-torsionring R .
A) Foreveryobject λ of Rind L- tor andeveryCartesiansquare
W
g
q
_
Y_
f
/Z
/ p
_
/X_
/
ENHANCED ADIC FORMALISM FOR ARTIN STACKS
37
Ar
in Chplft / S with p proper,1-Artin,ofseparateddiagonal,theinducedsquare
D ( Z,λ ) o
o
_
p
_
D ( X,λ )
g
_
_
q
D ( W,λ
_ )o
o
f
_
_
D ( Y,λ
_ )
isrightadjointable.
Ar
B) Every proper, 1-Artin, separated-diagonal and surjective morphism in Chplft / S is of
descent.
A similar result holds for
Chp
∞ -DM
Chp Ar
L
_
EO -
_
EO .
4.3. Flat descent.
Lemma 4.3.1. Let f : ` i 2 I X i → X beasurjectivemorphismofDMstackssuchthatforevery
≥0
_
X i → X isastrictlocalizationof
X . Then f isamorphismofuniversal
-descent.
Chp DM EO
i 2 I,
This follows from [20, 6.2.4.3].
Proposition 4.3.2. Any _at and locally _nitely presented hypercovering of higher Artin stacks (resp.
≥
≥
∞ -DM stacks)isanaugmentationof Chp Ar0 EO_ -descent(resp. Chp ∞ - DM0 EO_ -descent).
This is an analogue of flat cohomological descent [5, 7.4].
Proof. By Proposition 4.1.8, we are reduced to show that any surjective flat and locally finitely presented
morphism f : Y → X of higher Artin stacks (resp. ∞ -DM stacks) is of universal descent. By [18, 4.1.2],
we are reduced to the case of schemes. Let X 0 be a disjoint union of strict localizations of
X , such
that the morphism g : X 0 → X is surjective. By [1, IV 17.16.2, 18.5.11], there exists a finite surjective
morphism g0: Z → X 0, such that the composite map Z → X factorizes through f . Since g and g0
≥0
_
are both of universal Sch EO -descent by [20, 6.2.4.3], it follows from [18, 4.1.2] that f is of universal
≥0
_
_
Sch EO -descent.
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Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, US
E-mailaddress : liuyf@math.mit.edu
Morningside Center of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of
Sciences, Beijing 100190, China
E-mailaddress : wzheng@math.ac.cn