Equivariant motivic Hall algebras
Poguntke, Thomas
Advisor : Prof. Dr. Tobias Dyckerhoff
Introduction
Slope filtrations and Hall algebras
Let L|K be a finite, totally ramified extension of complete discretely valued fields of characteristic (0, p) with
perfect residue field k. A smooth projective variety X over L comes with the following linear algebraic data.
It is explained in [1] how to express the aforementioned notions of semistability in our categorical setting.
• The crystalline cohomology groups of the special fibre Xk become K-vector spaces after inverting p.
• These carry a natural Frobenius action, compatibly with a fixed lift σ of the Frobenius of k to K.
Definition 2: Let E be a quasi-abelian K-linear category, equipped with a pair of maps as follows.
• The rank function rk : π0 (E) → N, additive on exact sequences, such that rk(M ) = 0 ⇔ M = 0.
• A degree function deg : K0 (E) → Z, with deg(M ) ≤ deg(N ) for all M → N with (co-)kernel = 0.
• By comparison with the de Rham cohomology of X, they inherit a (Hodge) filtration over L.
Then M is semistable, if µ(N ) ≤ µ(M ) for all 0 6= N ≤ M , where µ(M ) =
Definition 1: Let isocK denote the category of F -isocrystals over K, that is, the category of pairs (V, ϕ),
∼→ V a σ-semilinear
where V ∈ vectK is a finite dimensional vector space over K, and ϕ : V ⊗K,σ K −−
automorphism of V . The category of filtered F -isocrystals over L|K is defined to be the fibre product
Z-filL|K ×vectK isocK = Z-filL ×vectL isocK ,
(1)
where Z-filL|K is the category of finite dimensional K-vector spaces together with a Z-filtration over L.
In fact, the p-adic analogue of a complex period domain, parametrizing Hodge structures, is a moduli space
for semistable filtered F -isocrystals over K, cf. Definition 2. They form an abelian subcategory of (1), which
Colmez and Fontaine describe in terms of certain representations of the absolute Galois group,
cris
∼
(Z-filL ×vectL isocK )ss
0 −−→ repK (GL ).
Example 2: (a) For the category E of vector bundles on a (connected) smooth projective curve X over
the field K, we have the usual notions of rank and degree. Note that OX OX (1) has (co-)kernel = 0.
P
(b) Let Q be a (connected, acyclic) quiver. Then rk(M ) = i∈Q0 dimK (Mi ) for M ∈ E = repK (Q), and
∼→ ZQ0 3 θ defines a degree function deg (M ) = P
any map Hom(K0 (E), Z) −−
θ
i∈Q0 θi · dimK (Mi ).
(c) Let E = Z-filL|K with rank function rk(V, F • ) = dimK (V ). The degree is weighted by the jumps of F • ,
deg• (V, F • ) =
X
λ · dimL (F λ V /F λ−1 V ) ∈ Z.
λ∈Z
On isocK , define degσ (V, ϕ) := − valp (det ϕ). The fibre product in (2) is endowed with deg := deg• + degσ .
Several further examples appear in the survey article [1], where the following is proved in this generality.
• Originally, by Harder-Narasimhan [4], in the context of moduli spaces of vector bundles on curves.
• Reineke [6] counts Fq -points of quiver moduli spaces in order to infer their Betti numbers (over C).
• Joyce [5] refines this point count to motivic measures, for more general moduli spaces, over any K.
Our goal is to generalize this approach to accommodate for the equivariant setting of [2]. Let K be a field.
Suppose B is a semisimple abelian K-linear category, and E, D are quasi-abelian categories over K. Let
ω
∈ Q q {∞} is the slope.
The full subcategory Eλss of semistable objects of E of slope ∈ {λ, ∞} is inherently abelian.
(2)
The cohomology of p-adic period domains is studied in [2]. A similar strategy is pursued in different settings.
deg M
rk M
ν
E −−→ B ←−− D
Proposition 1: There is a unique filtration F • : Qop × E → E, such that 0 ( F λ1 M ( . . . ( F λn M = M ,
for M ∈ E, is uniquely determined by F λi M/F λi−1 M being semistable of decreasing slopes λ1 > . . . > λn .
If K = Fq , and E is finitary, we can express Proposition 1 as an equation in the Hall algebra of E. This is
the convolution algebra H(E) = Q[π0 (E)] of finitely supported functions on π0 (E) with values in Q, that is,
X
(ϕ ∗ ψ)(E) =
ϕ(E/N )ψ(N ), for ϕ, ψ ∈ Q[π0 (E)].
N ≤E
be two K-linear, exact isofibrations. Then for any field extension L|K, we replace (1) by the fibre product
(EL ×BL B) ×B D = EL ×BL D, where EL := E ⊗K L.
(3)
More precisely, if we complete H(E) with respect to its K0 (E)-grading, where 1M lies in degree [M ], then
X
b
1π0 (E) =
1π0 (Eλss ) ∗ . . . ∗ 1π0 (Eλssn ) ∈ H(E).
(4)
1
λ1 >...>λn
L
Example 1: (a) The fibre functor ω : repB (Q) → B, M 7→ i∈Q0 Mi , for a quiver Q is an isofibration.
(b) Similarly, this applies to the forgetful functor on the category of representations in B of a group G. By
arguing pointwise, this extends to (affine) group schemes over K, for B = vectK , and therefore, the fibre
functor of any neutral (quasi-)Tannakian category over K is an (exact, linear) isofibration. In particular,
ω : Z-filK −→ vectK ←− isocK : ν.
If E is hereditary, the Euler form χ(M, N ) = dim Hom(M, N )−dim Ext1 (M, N ) on K0 (E)×2 defines a twisted
group ring Qhχi [K0 (E)], with [M ][N ] = q −χ(M,N ) [M ⊕ N ]. By [6], Lemma 6.1, the integral is a ring morphism
Z
1
b
[M ].
(5)
: H(E)
−→ Qhχi [[K0 (E)]], 1M 7−→ # Aut(M
)
E
For now, assume ν = idB . Then (3) at least specializes to [2] over finite fields, as well as to [5], when B = 0.
Then (4) yields a formula (6) for the number of points of the moduli stack of objects Mα of class α ∈ K0 (E).
Equivariant motivic Hall algebras
Further directions
Over an arbitrary field K, the basic idea is to replace the number of points q = #A1 (Fq ) by the affine line
itself. To this end, we understand it as an element L = [A1K ] ∈ K0 (Var/K) of the Grothendieck ring of
varieties, a naı̈ve version of the Grothendieck ring of (effective) Chow motives over K.
There are two different immediate ways to broaden the types of situations in which our formalism applies.
Definition 3: Let Z be a stack in groupoids on the big étale site of affine schemes over K. Let St/Z be
the category over Z formed by Artin stacks, locally of finite type, with affine stabilizer groups over K.
Then the Grothendieck ring of stacks K0 (St/Z) is the free Z-module on π0 (St/Z) modulo the relations
0
0
[X q X ] = [X] + [X ],
[X] = [X ] for all geometric equivalences X → X0 ,
[X1 ] = [X2 ] for all Zariski fibrations Xi → X0 with equivalent fibres.
0
∼→ K (St/K). As above, there is a HN-recursion
This ensures that K0 (Var/K)[L−1 , (Ln − 1)−1 | n ∈ N] −−
0
X
P
ss
[Mα ] =
L− i<j χ(αi ,αj ) [Mss
(6)
α1 ] · · · [Mαn ] ∈ K0 (St/K).
(1) The structure groups appearing in (9). Namely, if E carries an exact duality, there is a module over
the Hall algebra of E on isometry classes of selfdual objects [7], essentially replacing GLn by On or Spn .
This has a natural interpretation in the context of [3], which allows us to define an equivariant motivic
Hall module. In effect, there should be an analogue of Theorem 1, generalizing the results of [7].
(2) The categories (3), for ν 6= idB . Constructing an appropriate Hall algebra is not a problem. In fact, the
Waldhausen S-construction associated with (3) is a 2-Segal stack [3], hence an algebra under K0D (St/−).
However, establishing (4) raises technical issues. For instance, we need to replace Definition 3 with a
ring of (germs of) analytic stacks (on the étale site of affinoid spaces) over a non-Archimedean field.
Definition 4: Let G be an affine group scheme over K, and H ⊆ G a closed subgroup scheme. Then (8)
defines an algebra structure on H(G, H) := K0 (St/[H\G/H]), with respect to the correspondence
(∂0 ,∂2 )
(∂0 ,∂2 )
∂
1
M ×K M ←−−−− S2 (E) −−−
→ M,
(7)
where S2 (E) is the moduli stack of short exact sequences in E, which are mapped in (7) to their outer terms
and their middle term, respectively. That is, multiplication in H(E) is defined as the composition
(∂0 ,∂2 )∗
(∂1 )∗
K0 (St/M) ⊗ K0 (St/M) −−−K−→ K0 (St/M ×K M) −−−−→ K0 (St/S2 (E)) −−−→ K0 (St/M).
(10)
We call H(G, H) the motivic Hecke algebra, as taking K-points in (10) yields the classical Hecke algebra.
The motivic Hall algebra H(E) := K0 (St/M) of E is the convolution algebra along the correspondence
−× −
∂
1
[G\(G/H)2 ] ×K [G\(G/H)2 ] ←−−−− [G\(G/H)3 ] −−−
→ [G\(G/H)2 ].
α1 >...>αn
α1 +...+αn =α
There are many related constructions (e.g. in geometric Langlands) which it would be interesting to compare.
References
[1] Y. André. Slope filtrations. (2009).
(8)
[2] J.-F. Dat, S. Orlik, M. Rapoport. Period domains over finite and p-adic fields. (2010).
K0B (St/−),
Let N be the moduli stack of B. By replacing K0 (St/−) with its B-equivariant variant
we obtain
M
Aut(V )
Hω (E) := K0B (St/S1ω (E)) ∼
(9)
K0
(St/S1ω (E)), where S1ω (E) = M ×N N (K),
=
V ∈π0 (B)
the equivariant motivic Hall algebra of E, with product between different summands via parabolic induction.
H
Theorem 1: There is a natural morphism of simplicial stacks • : S•ω (E) → K0 (S• (E)), whose pushforward
Z
[3] T. Dyckerhoff, M. Kapranov. Higher Segal Spaces: Part I. (2012).
[4] G. Harder, M. S. Narasimhan. On the Cohomology of Moduli Spaces of Vector Bundles on Curves. (1975).
[5] D. Joyce. Configurations in abelian categories. II. Ringel-Hall algebras. (2007).
[6] M. Reineke. The Harder-Narasimhan system in quantum groups and cohomology of quiver moduli. (2003).
[7] M. B. Young. The Hall module of an exact category with duality. (2016).
ω
b ω (E) −→ K0B (St/K)hχi [[K0 (E)]]
:H
E
is an algebra morphism if E is hereditary. For B = 0, this recovers the motivic version of (5) from [5].
2016 Mathematical Institute
thomas.poguntke@hcm.uni-bonn.de