ABOUT ALGEBRAIC K-THEORY AND ALGEBRAIC
CYCLES
MARKUS SEVERITT
Contents
Introduction
1. Filtration by (Co)Dimension of Support
1.1. The d-Cycle of a Coherent Sheaf
1.2. Calculations modulo K0 (X)1
2. Naturality with Respect to Proper Push-Forwards
3. The Surjectivity
4. Rational Equivalences
5. Naturality with Respect to Flat Pull-Backs
Appendix A. The K-Group
A.1. The K-Group of an Abelian Category
A.2. The K-Group of Modules of Finite Lengths
A.3. The K-Group of a Scheme
A.4. Support and Closed Subschemes
Appendix B. Algebraic Cycles
B.1. Cycle groups
B.2. Chow groups
References
1
3
3
4
5
7
9
11
12
12
12
13
14
17
17
19
20
Introduction
The aim of this paper is to understand the following natural graded surjective morphism
φ
Z∗ (X) −
→ Gr∗ K0 (X)
which induces a natural graded surjective morphism
φ
CH∗ (X) −
→ Gr∗ K0 (X)
where Z∗ is the cycle group, CH∗ the Chow group and Gr∗ K0 the graded
group associated to the filtration by dimension of support. Natural means
that it commutes with proper push-forwards and flat pull-backs of relative
dimension n. In the latter case the grading shifts by n. In degree d the
inducing morphism reads as
M
φd
Z −→
K0 (X)d /K0 (X)d−1
x∈Xd
1
2
MARKUS SEVERITT
given by the assignement
1x 7→ [(ix )∗ (OVx )]
where ix : Vx ,→ X is the d-dimensional subvariety generated by x. Here
K0 (X)d = K0 (X)dim(X)−d denotes the subgroup of K0 (X) generated by the
coherent sheaves of dimSupp ≤ d (codimSupp ≥ dim(X) − d respectively).
For a given coherent sheaf M on X with dimSupp(M ) ≤ d a preimage of
[M ] under φd is given by the d-cycle of M
X
Zd (M ) =
lOX,x (Mx ) ∈ Zd (X)
x∈Xd
In order to understand this we need that in the top groups (i.e. for d =
dim(X)) of φ we have in fact an isomorphism which comes out of the split
exact sequence
ψ
0 → K0 (X)1 → K0 (X) −
→ Zd (X) → 0
where ψ([M ]) = Zd (M ) is given by the cycle above and ψ is split by φ. A
few words of caution: There are problems with the well-definedness of the
assignment [M ] 7→ Zd (M ) on K0 (X)d for d < dim(X) since there a relations
involving coherent sheaves whose support may have higher dimension. The
problems remain even after passing to CHd (X) but it is okay after passing
to CHd (X) ⊗ Q. This problem is captured by the Brown-Gersten-Quillen
spectral sequence
M
E1p,q =
K−p−q (k(x)) ⇒ G−p−q (X)
x∈X p
where Gn (X) = Kn (Coh(X)) are Quillen’s higher K-Groups where Coh(X)
denotes the abelian category of coherent sheaves on X.
Overmore, also the exact sequence above is encoded in the BGQ spectral
sequence: By denoting Coh(X)d the full subcategory of Coh(X) of sheaves
of dimSupp ≤ d, we have a functor
a
Coh(X)d →
A(OX,x )
x∈Xd
by taking stalks where A(R) is the abelian category of R-modules of finite
length. This functor clearly induces a functor
a
Coh(X)d /Coh(X)d−1 →
A(OX,x )
x∈Xd
This functor is in fact an equivalence of abelian categories which is stated
but not proven in the proof of [Sri96, Theorem 5.20]. The functor induces
a right exact sequence
M
K0 (Coh(X)d−1 ) → K0 (Coh(X)d ) →
K0 (A(OX,x ) → 0
x∈Xd
which extends to the left by Quillen’s higher K-groups (this is in fact the sequence which leads to the BGQ spectral sequence). Note that K0 (A(OX,x )) ∼
=
Z by taking lengths. Thus we recover a version of ψ. Further note that
K0 (X)d = Im(K0 (Coh(X)d ) → K0 (Coh(X)))
ABOUT ALGEBRAIC K-THEORY AND ALGEBRAIC CYCLES
3
Thus for d = dim(X) we get our demanded exact sequence
ψ
0 → K0 (X)1 → K0 (X) −
→ Zd (X) → 0
We will show the split exactness of the latter exact sequence elementary.
The idea of the proof can easily be adapted to give an elementary proof of the
general right exact sequence mentioned above without using the equivalence
of abelian categories (this is left to the reader :-)). An additional chapter
should be added which proves that the functor
a
Coh(X)d /Coh(X)d−1 →
A(OX,x )
x∈Xd
is indeed an equivalence of categories. This is work in progress.
All stuff which is going to be presented are elaborated ideas and statements from [Ful98] and [Sri96].
1. Filtration by (Co)Dimension of Support
Troughout this section let k be fixed field and by a k-scheme X we mean
X of finite type over k. In particular, X is noetherian.
1.1. The d-Cycle of a Coherent Sheaf. The first aim is to understand
why the d-cycle of M
X
Zd (M ) :=
lOX,x (Mx ) · 1x
x∈Xd
makes sense where M is a module with dimSupp(M ) ≤ d. That is we need
to know that the appearing lengths are finite.
Lemma 1.1. Suppose that Z is an irreducible component of X, η the generic
point of Z and M a coherent module on X. Then Mη has finite length over
OX,η .
Proof. Locally X = Spec(A) with A noetherian, M is a finitely generated
A-module and η is minimal prime ideal P . Thus Mη = MP is a finitely generated (OX,η = AP )-module. But AP is a noetherian local ring with Krull
dimension 0 (localization at a minimal prime ideal). Hence it is Artinian by
[AM69, Theorem 8.5] and the claim follows.
Lemma 1.2. Let M be a coherent sheaf on X with dimSupp(M ) ≤ d, then
Mx has finite length over OX,x for all x ∈ Xd .
Proof. Let i : Z = Supp(M ) ,→ X be the support of M . Then by Corollary
A.13 there is a coherent sheaf N on Z such that i∗ N = M . Now let x ∈ Xd .
If x ∈
/ Z, then Mx = 0. If x ∈ Z, it is a generic point of Z and we know that
Nx has finite lenght over OZ,x by the previous lemma. But OX,z OZ,x is
a quotient map of local rings and thus Mx = (i∗ N )x ∼
= Nx has finite length
over OX,x .
Lemma 1.3. Let i : Z ,→ X be a closed embedding. Then for a coherent
sheaf M on Z with dimSupp(M ) ≤ d we have
i∗ (Zd (M )) = Zd (i∗ (M )) ∈ Zd (X)
4
MARKUS SEVERITT
Proof. This follows immediately from the definitions since for a point z ∈ Z
and its image i(z) ∈ X the residue fields are isomorphic and thus the degree
showing up in the push-forwards of cycles is 1. Furthermore the stalks of
i∗ (M ) vanish outside of Z.
Remark 1.4. We will see in Corollary 3.3 that this rule holds in fact for
arbitrary finite morphisms f : X → Y , that is,
f∗ (Zd (M )) = Zd (f∗ (M )) ∈ Zd (Y )
for a coherent sheaf M on X with dimSupp(M ) ≤ d.
1.2. Calculations modulo K0 (X)1 . The next aim is to understand a sufficient condition for two coherent sheaves being congruent in K0 (X) modulo
K0 (X)1 . It turns out that we only need to know that they are isomorphic
at all generic points of dimension dim(X). One important tool for this is
the folllowing lemma.
Lemma 1.5. Let η be a general point of X and M , N two coherent sheaves
on X with Mη ∼
= Nη . Then M|U ∼
= N|U for an open neighbourhood U of η.
Proof. Without loss of generality let X = Spec(A) with A noetherian and
η = P is a prime ideal of A, such that there are finite presentations
φ
Am −
→ An → M → 0
and
ψ
Al −
→ Ak → N → 0
These induce finite presentations for the stalks MP and NP . Now the isomorphism f : MP → NP can be extended on the free AP -modules of the
finite presentation, such that we obtain a commutative diagram
Am
P
g
AlP
φP
/ An
P
h
ψP
/ Ak
P
/ MP
/0
f ∼
=
/ NP
/0
Since AP is the localization at the prime ideal P and all the morphisms
between the free AP -modules are determined by matrixes, we can choose
the coefficients of the matrixes to live in a localization Aa away from an
element a ∈ A − P . That is, we get an induced map F : Ma → Na such that
F ⊗Aa AP = f and thus
Ker(F ) ⊗Aa AP = Ker(f ) = 0
and
Coker(F ) ⊗Aa AP = Coker(f ) = 0
Since A and thus Aa is noetherian, Ker(F ) and Coker(F ) are finitely generated Aa -modules. Hence using standard arguments in localization theory
of modules there is an element b ∈ A − P such that
Ker(F ) ⊗Aa Aab = 0
and
Coker(F ) ⊗Aa Aab = 0
ABOUT ALGEBRAIC K-THEORY AND ALGEBRAIC CYCLES
5
whence F ⊗Aa Aab : Mab → Nab is an isomorphism. Since Spec(Aab ) = W
is open in Spec(A), the claim follows.
Proposition 1.6. Let X be of dimension ≤ d and X1 , . . . , Xn the irreducible
components of dimension d (possibly n = 0) with generic points ηi . Let M ,
N be coherent sheaves on X such that Mηi ∼
= Nηi for all i. Then
[M ] ≡ [N ]
mod K0 (X)d−1
in K0 (X) = K0 (X)d .
Proof. By the previous lemma, we obtain open neighbourhoods Vi of ηi , such
that M|Vi ∼
= N|Vi for all i. By setting Ui := Vi − (X1 ∪ . . . ∪ X̂i ∪ . . . ∪ Xn ), we
obtain pairwise disjoint open subschemes where M and N are isomorphic.
Since M and N are sheaves,
these isomorphisms glue to an isomorphism
S
M|U ∼
= N|U where U = Ui which contains all ηi . Denote by j : U ,→ X the
inclusion and by V the coherent sheaf M|U on U . Thus we have morphisms
M → j∗ V and N → j∗ V which induce M ⊕ N → j∗ V . Denote by I the
image of the latter morphism which is a coherent sheaf on X. The kernels
and cokernels of the morphisms M → I and N → I have dimSupp < d
since they are isomorphisms at all stalks of the generic points of dimension
d. Thus [N ] − [I], [M ] − [I] ∈ K0 (X)d−1 whence [M ] − [N ] ∈ K0 (X)d−1 . An immediate consequence is the following corollary which tells us how to
reconstruct a coherent sheaf out of the stalks at the generic points modulo
K0 (X)1 .
Corollary 1.7. Let X be reduced with dim(X) ≤ d and X1 , . . . , Xn the
irreducible components of dimension d with generic points ηi and embeddings
ji : Xi ,→ X where the Xi are considered with the reduced closed subscheme
structure. Then for every coherent sheaf M on X we have
n
X
lOX,ηi (Mηi )[(ji )∗ OXi ] ≡ [M ]
mod K0 (X)d−1
i=1
in K0 (X) = K0 (X)d .
Proof. Denote li := lOX,ηi (Mηi ). Then note that the left hand side can be
written as the class
" n
#
M
((ji )∗ OXi )li
i=1
Ln
li
li
and the stalk of
i=1 ((ji )∗ OXi ) at ηi is isomorphic to OXi ,ηi . Further
note that OX,ηi → OXi ,ηi is an isomorphism of fields since X is reduced.
Hence Mηi is a OXi ,ηi -vector space of dimension li and thus also isomorphic
li
to OX
. Thus the claim follows by the last proposition.
i ,ηi
2. Naturality with Respect to Proper Push-Forwards
The aim of this section and section 4 is to understand why the map
M
1x 7→[(ix )∗ (OVx )]
φd : Zd (X) =
Z −−−−−−−−−−
→ K0 (X)d /K0 (X)d−1
x∈Xd
6
MARKUS SEVERITT
induces a map
φd : CHd (X) → K0 (X)d /K0 (X)d−1
such that both maps are natural with respect to proper push-forwards. For
this, we use the approach given by [Ful98, Example 1.6.4, Example 15.1.5].
In this section, we will start by giving a conceptional approach to the map
φ and its naturality.
Notation 2.1. Let Sch/k be the category of finite type k-schemes with
proper morphisms. Furthermore denote
Grd K0 (X) := K0 (X)d /K0 (X)d−1
and
Z∗ =
M
Zn , CH∗ :=
n≥0
M
CHn , Gr∗ K0 :=
n≥0
M
Grn K0
n≥0
which are functors
Sch/k → Ab
(cf. Appendix A.3 and Appendix B).
Remark 2.2. For the functors Grd K0 note that for a proper morphism
f : X → Y we have f∗ (K0 (X)d−1 ) ⊂ K0 (Y )d−1 by Lemma A.14 and hence
we get an induced map
f∗ : Grd K0 (X) → Grd K0 (Y )
Let us consider functors H : Sch/k → Ab with the property that for all
varieties V in Sch/k there is given a class clH (V ) ∈ H(V ) such that for a
surjective morphism f : V → W between varieties in Sch/k we have
(1)
H(f )(clH (V )) = deg(V /W )clH (W )
where
(
0
, dim(W ) < dim(V )
deg(V /W ) =
[k(V ) : k(W )] , dim(W ) = dim(V )
Proposition 2.3. The functor
Z∗ : Sch/k → Ab
together with
clZ (V ) = [V ] = 1ηV ∈ Zdim(V ) (V ) ⊂ Z∗ (V )
where ηV is the generic point of V , is the universal functor among those
functors H : Sch/k → Ab together with clH satisfying (1). That is, there is
a unique natural transformation t : Z∗ ⇒ H satisfying t(clZ (V )) = clH (V ).
Proof. By the very definition of the push-forwards of Z∗ , this functor satisfies
(1). Now assume that such a natural transformation t exists. We will show
the uniqueness. By definition, the group Z∗ (X) for a k-scheme X is freely
generated by the fundamental cycles of the closed subvarieties of X. Let [V ]
be such a generator and i : V ,→ X the inclusion. Then [V ] = i∗ (clZ (V )).
Since t is a natural transformation, we obtain
t([V ]) = H(i)(t(clZ (V )) = H(i)(clH (V ))
ABOUT ALGEBRAIC K-THEORY AND ALGEBRAIC CYCLES
7
whence t is unique with the property t(clZ (V )) = clH (V ). Furthermore one
easily checks that this equation above defines a natural transformation t if
H satisfies (1).
The following lemma tells us that Gr∗ K0 together with clGr (V ) = [OV ]
fits into the picture and that we therefore obtain a natural transformation
Z∗ ⇒ Gr∗ K0
with the property that for a point x ∈ Xd
t(1x ) =
=
=
=
t((ix )∗ (clZ (Vx )))
(ix )∗ t(clZ (Vx ))
(ix )∗ (clGr (Vx ))
[(ix )∗ OVx ]
Thus, we recover our map φ.
Lemma 2.4. The functor
Gr∗ K0 : Sch/k → Ab
together with
clGr (V ) = [OV ] ∈ K0 (V )/K0 (V )1 ⊂ Gr∗ K0 (V )
satisfies (1).
Proof. We consider a surjective morphism f : V → W between k-varieties.
Let d = dim(V ). We have to show that
f∗ (clGr (V )) = deg(V /W )clGr (W ) ∈ K0 (W )d /K0 (W )d−1
In the case that dim(W ) < dim(V ), we have K0 (W )d = K0 (W )d−1 and
hence both sides are 0. If dim(W ) = dim(V ), we have that f is finite.
Hence the left hand side is just [f∗ OV ] and the right hand side reads as
deg(V /W )
[OW
]. Let η be the generic point of W . Then we have (f∗ OV )η ∼
=
deg(V /W )
deg(V
/W
)
∼
k(V ) and (OW
)η = k(W )
= k(V ). Now the claim follows by
Proposition 1.6.
3. The Surjectivity
The aim of this section is to see why the d-cycle of M
X
Zd (M ) :=
lOX,x (Mx ) · 1x
x∈Xd
provides a preimage of [M ] in K0 (X)d /K0 (X)d−1 under φd . So let us take a
coherent sheaf M on X with dimSupp(M ) ≤ d. By Corollary A.13 there is a
coherent sheaf N on Z := Supp(M ) such that i∗ (N ) = M where i : Z ,→ X
is the inclusion. As a closed embedding, i is finite. Thus i∗ ([N ]) = [M ] ∈
8
MARKUS SEVERITT
K0 (X)d . Furthermore we know the naturality of φ with respect to pushforwards by the last section and hence we get the commutative diagram
Zd (Z)
i∗
Zd (X)
φd
φd
/ K0 (Z)d /K0 (Z)d−1
i∗
/ K0 (X)d /K0 (X)d−1
Recall that we have
i∗ (Zd (N )) = Zd (i∗ (N )) = Zd (M )
by Lemma 1.3. Thus we have the equality
φ(Zd (M )) = φ(i∗ (Zd (N ))) = i∗ (φ(Zd (N )))
If dim(Z) < d, we know that K0 (Z)d = K0 (Z)d−1 and thus the desired
equality
φ(Zd (M )) = 0 = [i∗ (N )] = [M ]
holds. Therefore we are reduced to the case d = dim(Z).
For this we will show the following proposition.
Proposition 3.1. Let dim(X) = d. Then the sequence
ψ
0 → K0 (X)1 → K0 (X) −
→ Zd (X) → 0
is exact and ψ is split by φ, that is, we obtain an isomorphism
ψ : K0 (X)/K0 (X)1 Zd (X) : φ
Proof. First of all, rewrite ψ as the composition
P
M
[M ]7→ [Mx ] M
f M
K0 (X) −−−−−−−−→
K0 (A(OX,x )) ←
−
K0 (κ(x)) ∼
Z
=
x∈Xd
x∈Xd
x∈Xd
where f is the morphism induced by the surjections OX,x κ(x). Confer
Appendix A.2 for the fact that f is an isomorphism and the composition is
indeed ψ.
If dimSupp(M ) < d and x ∈ Xd , we clearly have Mx = 0 since otherwise
My 6= 0 for all y ∈ x = Vx and hence Vx ⊂ Supp(M ) which contradicts
the assumption. That is, we have K0 (X)1 ⊆ Ker(ψ) and thus an induced
morphism
K0 (X)/K0 (X)1 → Zd (X)
which we also call ψ by abuse of notation. Now the aim is to check that
ψ ◦ φ = id and that φ ◦ ψ ≡ id mod K0 (X)1 .
The equation ψ ◦ φ = id follows directly from the fact that for x ∈ Xd
and ix : Vx = xred ,→ X the stalks ((ix )∗ OVx )y for y ∈ Xd are either 0
if x 6= y (since then y ∈
/ Vx ) or κ(x) for x = y which is the free generator of K0 (A(OX,x ) identified with 1 ∈ Z under the chosen isomorphism
K0 (A(OX,x ) ∼
= Z (cf. Appendix A.2). Hence φ splits ψ and ψ is surjective.
The equation φ ◦ ψ ≡ id mod K0 (X)1 for a coherent sheaf M on X reads
as
X
φd (Zd (M )) =
lOX,x (Mx ) · [(ix )∗ (OVx )] ≡ [M ] mod K0 (X)1
x∈Xd
ABOUT ALGEBRAIC K-THEORY AND ALGEBRAIC CYCLES
9
By Corollary A.17 and the already known facts that both ψ and φ commute
with finite push-forwards, we can assume that X is reduced and then this
equation is precisely the one in Corollary 1.7.
As described above we get the following corollary.
Corollary 3.2. Let M be a coherent sheaf on X of dimSupp(M ) ≤ d. Then
φd (Zd (M )) = [M ] ∈ K0 (X)d /K0 (X)d−1
Hence φd is surjective.
Corollary 3.3. Let f : X → Y be a finite morphism and M a coherent
sheaf on X with dimSupp(M ) ≤ d. Then
f∗ (Zd (M )) = Zd (f∗ (M )) ∈ Zd (Y )
Proof. By Lemma 1.3 and the fact that M comes from its support, we
can assume that dim(X) = d. Furthermore by replacing Y by its schemetheoretic image, we can also assume dim(Y ) = dim(X) = d since f is finite.
By the compatibility with push-forwards we have a commutative diagram
Zd (X)
f∗
Zd (Y )
φX
φY
/ K0 (X)d /K0 (X)d−1
f∗
/ K0 (Y )d /K0 (Y )d−1
where both φX and φY are bijections by the proposition. Hence our claim
is equivalent to
f∗ (φX (Zd (M ))) = φY (Zd (f∗ (M )))
But we know by the preceeding corollary that Zd (M ) is the preimage of [M ]
under φX and Zd (f∗ (M )) is the preimage of [f∗ (M )] = f∗ ([M ]) under φY
which gives the desired equality.
4. Rational Equivalences
We are now going to continue the conceptional approach of section 2.
That is we want to understand why our natural map
φ : Z∗ (X) → Gr∗ K0 (X)
factors as
φ : CH∗ (X) → Gr∗ K0 (X)
For this, we consider functors H : Sch/k → Ab with classes clH for
varieties satisfying (1) (thus we have t : Z∗ ⇒ H by Proposition 2.3 with
t(clZ (V )) = clH (V ) for a variety V ) and the additional property, that for a
normal variety V in Sch/k and a dominant morphism f : V → P1 we have
(2)
t([f −1 (0)]) = t([f −1 (∞)])
Proposition 4.1. The functor
CH∗ : Sch/k → Ab
together with the classes clCH induced by clZ is universal among those functors H : Sch/k → Ab satisfying (1) and (2), that is, there is a unique natural
transformation t : CH∗ ⇒ H satisfying t(clCH (V )) = clH (V ).
10
MARKUS SEVERITT
Proof. By Proposition 2.3, we know that we have a unique transformation
t : Z∗ ⇒ H
satisfying t(clZ (V )) = clH (V ) if and only if H satisfies property (1). We also
know that the classes clCH are given by the classes clZ . Thus the only thing
left is to check, that t factors through CH∗ if and only if H additionally
satisfies property (2). But a factorization trough CH∗ exists if and only if t
respects rational equivalence.
So let us assume that t respects rational equivalence and consider the
situation of property (2). That is, we consider a dominant morphism f :
V → P1 where V is a normal variety. Now we consider the closure of the
graph
Γf : V → V × P 1
Hence [f −1 (0)] − [f −1 (∞)] is rationally equivalent to zero in Z∗ (V ) by the
definition of rational equivalence (cf. Appendix B.2). By assumption, t
respects rational equivalence and hence t([f −1 (0)]) = t([f −1 (∞)]). That is,
H satisfies property (2).
Now assume that H satisfies property (2). By Remark B.7, the subgroup
of cycles in Z∗ (X) rationally equivalent to zero is generated by the following
elements: For a normal variety V with proper morphism h : V → X and
a dominant morphism f : V → P1 we take [f −1 (0)] − [f −1 (∞)] ∈ Z∗ (V )
and then the push-forward of this cycle along h. Furthermore t is a natural
transformation and we thus have a commutative diagram
Z∗ (V )
h∗
Z∗ (X)
/ H(V )
t
t
H(h)
/ H(X)
By assumption t([f −1 (0)] − [f −1 (∞)]) = 0 ∈ H(V ) and thus also
t(h∗ ([f −1 (0)] − [f −1 (∞)])) = 0 ∈ H(X)
That is, t respects rational equivalence.
By Lemma 2.4 we already know that the functor Gr∗ K0 satisfies property
1 and that the induced natural transformation t : Z∗ ⇒ Gr∗ K0 is just our
natural map φ. The next lemma tells us that it also satisfies property 2
which gives us the desired factorization of φ trough CH∗ by the preceding
proposition.
Lemma 4.2. The functor
Gr∗ K0 : Sch/k → Ab
together with
clGr (V ) = [OV ] ∈ K0 (V )/K0 (V )1 ⊂ Gr∗ K0 (V )
satisfies (2).
Proof. We consider the induced natural transformation
t : Z∗ ⇒ Gr∗ K0
ABOUT ALGEBRAIC K-THEORY AND ALGEBRAIC CYCLES
11
determined by t([V ]) = [OV ]. Furthermore consider a normal variety V
and a dominant morphism f : V → P1 . Let Di = f −1 (i) with inclusions
ji : Di ,→ V for i = 0, ∞. We have to show that t([D0 ]) = t([D∞ ]). Consider
{0}, {∞} ⊂ P1 as effective Cartier divisors whose associated line bundles are
both O(1) since Pic(P1 ) ∼
= Z is generated by any hyperplane section which
is just a closed point like 0 and ∞. Then Di = f ∗ (i) as effective Cartier
divisors and the Di are defined by the ideal sheaves
L(−f ∗ (i)) ∼
= f ∗ L(−{i}) ∼
= f ∗ O(−1)
Thus we have exact sequences
0 → f ∗ O(−1) → OV → (ji )∗ ODi → 0
whence [(j0 )∗ OD0 ] = [(j∞ )∗ OD∞ ] in K0 (X). Furthermore
[Di ] = Zdim(X)−1 ((ji )∗ ODi ) ∈ Z∗ (X)
(cf. Remark B.4). By the preceding section, we know that the d-cycle
Zd (M ) provides a preimage of [M ] ∈ K0 (X)d /K0 (X)d−1 under φ = t for
dimSupp(M ) ≤ d. Therefore we get
t([D0 ]) = [(j0 )∗ OD0 ] = [(j∞ )∗ OD∞ ] = t([D∞ ])
as claimed.
5. Naturality with Respect to Flat Pull-Backs
Now we are going to see another nice property of our map
φ : Z∗ (X) → Gr∗ K0 (X)
Namely it is natural with respect to pull-backs for flat morphism f : X → Y
of relative dimension n, that is we obtain a commutative diagram
Z∗ (Y )
f∗
Z∗+n (X)
φ
φ
/ Gr∗ K0 (Y )
f∗
/ Gr∗+n K0 (X)
Note that this diagram makes sense because of Lemma A.15. This naturality
follows from the already seen fact that for M ∈ Coh(X) with dimSupp(M ) ≤
d we have
φ(Zd (M )) = [M ] ∈ K0 (X)d /K0 (X)d−1
The argument is the following: Let y ∈ Yd be a generator of Zd (Y ). By the
definition of pull-backs of cycles and the fact that f is of relative dimension
n, we have
f ∗ (1y ) = [f −1 (Vy )] = Zd+n (j∗ Of −1 (Vy ) )
where j : f −1 (Vy ) ,→ X is the inclusion. Now we use the fact that φ
commutes with push-forwards and the fact mentioned above to obtain
φ(f ∗ (1y )) = [j∗ Of −1 (Vy ) ] = [f ∗ ((iy )∗ OVy )] = f ∗ (φ(1y ))
12
MARKUS SEVERITT
Appendix A. The K-Group
A.1. The K-Group of an Abelian Category.
Definition A.1. Let A be a small abelian category. Then K0 ((A) is the
group generated by the objects of A modulo the relation: For a short exact
sequence
0→A→B→C→0
we set [B] = [A] + [C].
Remark A.2. Note that we have [A ⊕ B] = [A] + [B].
The K0 -construction is functorial in additive exact functors. That is, if
F : A → B is an additive exact functor between small abelian categories,
we obtain a group homomorphism
F∗ : K0 (A) → K0 (B)
by setting F∗ [A] = [F (B)].
A.2. The K-Group of Modules of Finite Lengths.
Lemma A.3. Let R be a commutative ring and A(R) the category of modules of finite length. Then we obtain an isomorphism
M
f M
K0 (A(R)) ←
−
K0 (R/m) ∼
Z
=
m
m
where the sum is indexed over all maximal ideals m of R, induced by the
surjections πm : R R/m with inverse map g given by
X
g([M ]) =
rm (M )
where rm (M ) is the multiplicity of R/m in a composition series of M . In
other words, the abelian group K0 (A(R)) is freely generated by the classes
[R/m] where m is a maximal ideal of R.
Proof. Let M ∈ A(R) and
0 = M0 ⊂ . . . ⊂ Mn = M
a composition series. It is well known that Mi+1 /Mi ∼
= R/m for a maximal
ideal m of R. Thus we get
X
[M ] =
rm (M )[R/m]
m
which means that f ◦ g = id. Furthermore (g ◦ f )(1m ) = g([R/m]) = 1m
and thus g ◦ f = id.
Remark A.4. If (R, m) is a local ring, we obtain an isomorphism
K0 (A(R)) ← K0 (R/m) ∼
=Z
whith inverse map
[M ] 7→ lR (M )
ABOUT ALGEBRAIC K-THEORY AND ALGEBRAIC CYCLES
13
A.3. The K-Group of a Scheme. Troughout this section let k be fixed
field and by a k-scheme X we mean X of finite type over k. In particular,
X is noetherian.
Notation A.5. For a k-scheme X let Coh(X) denote the abelian category
of coherent sheaves on X.
Definition A.6. For a k-scheme X let
K0 (X) := K0 (Coh(X))
If for a map f : X → Y every sheaf f∗ M is coherent for all M ∈ Coh(X)
and the functor f∗ : Coh(X) → Coh(Y ) is exact, we clearly obtain a group
homomorphism
f∗ : K0 (X) → K0 (Y )
by assigning [M ] 7→ [f∗ M ]. E.g. this is the case for f finite by the following
lemma.
Lemma A.7. Let f : X → Y be a finite morphism. Then f∗ is exact and
maps coherent sheaves to coherent sheaves.
Proof. As an affine morphism, f∗ is exact since locally we have a ringhomomorphism f : B → A and for an A−-module M the B-module f∗ M is
just the same abelian group as M . Furthermore, the coherence is also a
local question and we are in the same situation where f : B → A is a finite
B-algebra. Thus for a finitley generated A-module M , the B-module f∗ M
is also finitely generated which gives the claim.
In the case that for f : X → Y the functor f∗ might not be exact, it is
always left exact and we have the higher direct image sheaves Rn f∗ (the right
derived functors of f∗ ) with computation Rn f∗ (M ) is the sheaf associated
to the presheaf
V 7→ H n (f −1 (V ), M|f −1 (V ) )
Example A.8. For the structure map πX : X → Spec(k), we obtain
Rn (πX )∗ (M ) = H n (X, M )
the usual sheaf cohomology.
In this situation where f∗ might not be exact, it is possible to define a
push-forward f∗ on K-groups by using the higher direct image sheaves. But
for this, we need that Rn f∗ of a coherent sheaf on X is a coherent sheaf on
Y for all n ≥ 0. This is in fact true for proper morphisms by [EGA III,
Theorem 3.2.1]. Thus we can look at the assignment
X
[M ] 7→
(−1)n [Rn f∗ (M )]
n≥0
for a coherent sheaf M on X. By the long exact sequence for the higher
direct image sheaves, this gives a well defined map
f∗ : K0 (X) → K0 (Y )
To see that (f ◦ g)∗ = f∗ ◦ g∗ on K-groups the Leray-spectral sequence
E2p,q = (Rp f∗ ◦ Rq g∗ )(M ) ⇒ Rp+q (f ◦ g)∗ (M )
is used. Thus K0 is functorial in proper morphisms.
14
MARKUS SEVERITT
Example A.9. If we again look at the structure map πX : X → Spec(k)
with X proper over k and use the isomorphism K0 (k) ∼
= Z by taking dimension, we obtain
(πX )∗ ([M ]) = χ(M ) ∈ Z
the Euler-characteristic of M according to the computation Rn (πX )∗ (M ) =
H n (X, M ).
Now we want to look at possible pullback maps. If f : X → Y is flat,
then clearly
f ∗ : Coh(Y ) → Coh(X)
is exact and functorially induces a group homomorphism
f ∗ : K0 (Y ) → K0 (X)
The push-forward and pull-back maps commute for pullback diagrams. That
is, for a pullback diagram
X0
f0
g0
g
/X
f
/Y
Y0
with f proper and g flat (hence f 0 is proper and g 0 is flat), we have
f∗0 ◦ g 0∗ = g ∗ ◦ f∗
on K0 -groups which follows from the well known fact that the diagram
Coh(X 0 ) o
f∗0
g 0∗
Coh(X)
Coh(Y 0 ) o
g∗
f∗
Coh(Y )
commutes up to functor isomorphism and hence also
(Rn f 0 ) ◦ g 0∗ ∼
= g ∗ ◦ (Rn f∗ )
∗
for all n ∈ N by the exactness of g ∗ and g 0∗ : One checks
(Rn f 0 ) ◦ g 0∗ ∼
= Rn (f 0 ◦ g 0∗ )
∗
∗
and
g ∗ ◦ (Rn f∗ ) ∼
= Rn (g ∗ ◦ f∗ )
by the usual argument: They agree for n = 0, the right hand side is a universal δ-functor and the left hand side is exact and effaceable. The effaceabilty
follows from the fact that the pull-back respects flasque sheaves.
A.4. Support and Closed Subschemes. Troughout this section let k
be fixed field and by a k-scheme X we mean X of finite type over k. In
particular, X is noetherian.
Recall that for a coherent module M on X, the support of M
Supp(M ) = {x ∈ X|Mx 6= 0}
is the closed subscheme defined by the annulator ideal Ann(M ).
ABOUT ALGEBRAIC K-THEORY AND ALGEBRAIC CYCLES
15
Definition A.10. For d ∈ N define K0 (X)d to be the subgroup of K0 (X)
generated by those coherent sheaves whose support has dimension ≤ d.
Remark A.11. Note that
K0 (i)
K0 (X)d = Im(K0 (Coh(X)d ) −−−→ K0 (Coh(X)))
where i : Coh(X)d ⊂ Coh(X) denotes the full subcategory consisting of the
coherent sheaves with dimSupp ≤ d. Be aware of the fact that K0 (i) is not
an inclusion in general since there are relations in K0 (X) between coherent
sheaves with dimSupp ≤ d where sheaves of higher dimensions are involved.
One of our aims is to show that proper push-forwards and flat pull-backs
respect this filtration of K0 (X).
Lemma A.12. Let i : Z ,→ X be a closed subscheme defined by an ideal
sheaf I. Then
i∗ : Coh(Z) → Coh(X)
is a fully faithful exact functor whose image contains precisely the modules
M on X such that I · M = 0.
Proof. As an imbedding of a closed subscheme i is finite and thus induces
the exact functor i∗ with image contained in Coh(X) according to Lemma
A.7. The fully faithfulness is clear since i∗ (G)(U ) = G(Z ∩ U ). For the
image consider the local situation
i : Spec(A/I) ,→ Spec(A)
Then for an A/I-Module M , the A-module i∗ M is just the same abelian
group as M together with the scalar mutiplication induced by π : A A/I.
It is clear that i∗ M is killed by I. Futhermore for a given A module M ,
the scalar multiplication descents to A/I via π if and only if M is killed by
I.
Corollary A.13. Let M be a coherent module on X. Let i : Z = Supp(M ) ,→
X be the inclusion. Then M lies in the image of the functor
i∗ : Coh(Z) → Coh(X)
and thus in the image of the map
i∗ : K0 (Z) → K0 (X)
Proof. Since Z is defined by the ideal sheaf I = Ann(M ) and thus I · M = 0,
the claim follows by the previous lemma.
We are now ready to show that proper push-forwards and flat pull-backs
respect the filtration K0 (X)d ⊂ K0 (X).
Lemma A.14. Let f : X → Y be a proper map. Then
f∗ (K0 (X)d ) ⊂ K0 (Y )d
Proof. Let us first consider the case that f is an inclusion of a closed subscheme. Then for a generating sheaf M ∈ Coh(X)d the stalks of f∗ M at
points outside of X vanish and the stalks at points of X are the same as
the stalks of M . Thus f∗ M ∈ Coh(Y )d and the assertion follows since
f∗ [M ] = [f∗ M ] by the finiteness of f (cf. Lemma A.7).
16
MARKUS SEVERITT
Now let f be proper and M ∈ Coh(X)d a generating sheaf of K0 (X)d .
By the previous corollary there is N ∈ Coh(Z) with Z = Supp(M ) such
that M = i∗ N and i : Z ,→ X denotes the inclusion. Further denote by
j : f (Z) ,→ Y the scheme-theoretic image of Z under f and f 0 : Z → f (Z)
the restriction of f to Z. That is, we have a commutative diagram
f
XO
/Y
O
j
i
f0
Z
/ f (Z)
which induces a commutative diagram
f∗
K0 (X)
/ K0 (Y )
O
O
j∗
i∗
f∗0
K0 (Z)
/ K0 (f (Z))
Note that we have dim(Z) ≤ d and also dim(f (Z)) ≤ dim(Z) ≤ d. Thus
K0 (Z)d = K0 (X) and K0 (f (Z))d = K0 (f (Z)). That is we obtain our claim
f∗ [M ] ∈ K0 (Y )d since we already know that i and j as closed inclusions
respect the filtration and i∗ [N ] = [M ].
Lemma A.15. Let f : X → Y be a flat map of relative dimension n. Then
f ∗ (K0 (Y )d ) ⊂ K0 (X)d+n
Proof. Let M ∈ Coh(Y )d be a generating sheaf of K0 (Y )d . Again denote
by i : Z ,→ Y its support such that there is N Coh(Z) with i∗ [N ] = [M ].
Then by denoting f −1 (Z) = Z ×Y X the scheme-theoretic preimage and
j : f −1 (Z) ,→ X its inclusion we obtain a cartesian diagram
f −1 (Z)
f0
j
/X
f
Z
i
/Y
where f 0 is also flat of relative dimension n. That is, we know that dim(f −1 (Z)) ≤
dim(Z) + n ≤ d + n. This diagram induces a commutative diagram
K0 (f −1 (Z))
i∗
O
f∗
f 0∗
K0 (Z)
/ K0 (X)
O
j∗
/ K0 (Y )
We know that K0 (Z)d = K0 (Z) and K0 (f −1 (Z))d+n = K0 (f −1 (Z)). Further we know that i∗ and j∗ respect the filtration by the previous lemma
which gives us that f ∗ [M ] ∈ K0 (X)d+n .
Proposition A.16. Let i : Z ,→ X be a closed subscheme defined by a
nilpotent ideal sheaf I. Then the functor i∗ induces an isomorphism
i∗ : K0 (Z) → K0 (X)
ABOUT ALGEBRAIC K-THEORY AND ALGEBRAIC CYCLES
17
Proof. For a coherent sheaf M on X the nilpotent ideal sheaf I gives a finite
filtration
n
M
I i M/I i+1 M
i=0
where every factor is killed by I where I n+1 = 0. Thus there are coherent sheaves Ni ∈ Coh(Z) with i∗ Ni = I i M/I i+1 M . This defines an exact
functor
F : Coh(X) → Coh(Z)
L
by assigning M 7→
Ni and thus a map
F∗ : K0 (X) → K0 (Z)
Since
"
[M ] =
n
M
#
I i M/I i+1 M
i=0
and the I-filtration for an i∗ N is just i∗ N , we get that F∗ and i∗ are inverse
to each other.
Corollary A.17. The inclusion i : Xred ,→ X defined by the nilradical N
induces an isomorphism
i
∗
K0 (Xred ) −
→
K0 (X)
Proof. Since X is noetherian, it is quasi-compact. Thus it can be covered by
finitely many open affine subschemes Spec(Ai ) where Ai is noetherian. The
ideal sheaf defining i restricted to Spec(Ai ) is the nilradical N (Ai ) which is
nilpotent since Ai is noetherian. Thus the ideal sheaf N is nilpotent and
the claim follows by the previous proposition.
Appendix B. Algebraic Cycles
Troughout this section let k be a fixed field and by a k-scheme X we mean
X of finite type over k. In particular, X is noetherian.
B.1. Cycle groups.
Notation B.1. For a surjective morphism f : V → W between two kvarieties we denote by
(
0
, dim(W ) < dim(V )
deg(V /W ) =
[k(V ) : k(W )] , dim(W ) = dim(V )
Remark B.2. Since f is surjective, we obtain a map k(V ) ← k(W ) of kalgebras which is a field extension. In the case that dim(V ) = dim(W ), we
have that f and hence k(V )/k(W ) is finite.
Definition B.3. Let d ∈ N. Then we define the d-th cycle group
M
Zd (X) :=
Z
x∈Xd
18
MARKUS SEVERITT
where x ∈ Xd means a point of dimension d, that is, the generated subvariety
Vx = xred of X has dimension d. Further denote
M
Z∗ (X) :=
Zd (X)
d≥0
If X1 , . . . , Xn are the irreducible components of X considered as reduced
subschemes with generic points ηi ∈ Xdi we have the fundamental cycle
[X] =
n
X
lOX,ηi (OX,ηi ) · 1ηi ∈ Z∗ (X)
i=1
where these lengths are finite by Lemma 1.1. If Z ⊂ X is a closed subscheme,
we get Z∗ (Z) ⊂ Z∗ (X) with a shift of codim(Z, X) and thus we can consider
the fundamental cycle [Z] in Z∗ (X).
Remark B.4. For a subvariety V of X, the fundamental cycle is just
[V ] = 1η
where η is the generic point of V since OV,η = k(V ) is the function field of
V which has of course selflength 1.
Recall that for a coherent sheaf M on X with dimSupp(M ) ≤ d we can
consider the cycle
X
Zd (M ) =
lOX,x (Mx ) ∈ Zd (X)
x∈Xd
by Lemma 1.2.
Lemma B.5. If i : D ⊂ X is an effective Cartier divisor with X a k-variety
of dimension d, we have
[D] = Zd−1 (i∗ (OD ))
which is nothing else than
X
ordVx (D)
x∈Xd−1
the cycle canonically associated to an effective Cartier divisor (cf. [Ful98,
Chapter 2.1]).
Proof. The first claim is clear. For the second recall that the effective Cartier
divisor D ⊂ X is locally on U defined by a function f ∈ OU (U ). If x ∈
Xd−1 ∩ D and U ⊂ X open with Vx ∩ U 6= ∅ and D on U is defined by f
then
ordVx (D) = ordVx (f ) = lOV,x (OD,x )
by the definition of ordV (f ) (cf. [Ful98, Chapter 1.2]).
For a proper morphism f : X → Y , we want to define a group homomorphism
f∗ : Zd (X) → Zd (Y )
For this note that for a closed subvariety V = Vx of X, the image f (V ) =
Vf (x) ⊂ Y is a closed subvariety of Y . So we can make the assignment
1x 7→ deg(Vx /Vf (x) ) · 1f (x)
ABOUT ALGEBRAIC K-THEORY AND ALGEBRAIC CYCLES
19
which functorially induces a group homomorphism
f∗ : Zd (X) → Zd (Y )
for all d ∈ N. The reason why the degree shows up is that these pushforwards on the cycle groups should induce push-fowards on the Chow groups
which are going to be introduced in the next subsection. That is, these pushforwards should respect rational equivalence.
Furthermore for a flat morphism f : X → Y of relative dimension n there
are functorial pull-back maps
f ∗ : Zd (Y ) → Zd+n (X)
defined by assigning
1x = [Vx ] 7→ [f −1 (Vx )]
where f −1 (Vx ) = Vx ×Y X. The functorialty follows since f ∗ ([Z]) = [f −1 (Z)]
for any subscheme Z of X by [Ful98, Lemma 1.7.1]. Fortunately, also these
pull-backs respect rational equivalence and thus induce pull-backs on Chow
groups.
The push-forward and pull-back maps commute for pullback diagrams.
That is, for a pullback diagram
X0
f0
Y0
g0
g
/X
f
/Y
with f proper and g flat of relative dimension n (hence f 0 is proper and g 0
is flat of relative dimension n), we have
f∗0 ◦ g 0∗ = g ∗ ◦ f∗
by [Ful98, Proposition 1.7].
B.2. Chow groups. We are now going to define the Chow groups which
are just the quotient groups of the cycle groups by rational equivalence.
Thus we have to make the following
Definition B.6 (Rational Equivalence). A cycle α ∈ Zd is rational equivalent to 0 if and only if there are are finitely many d + 1-dimensional subvarieties V1 , . . . , Vn of X × P1 with dominant projections fi : Vi → P1 , such
that
n
X
α=
[Vi (0)] − [Vi (∞)]
i=1
−1
where Vi (p) = πX (fi (p)) for p = 0, ∞ is the
scheme-theoretic fibre fi−1 (p) ⊂ X ×{p} under the
isomorphic image of the
projection πX : X ×P1 X.
Remark B.7. By replacing the Vi by their normalizations Wi one can actually work with normal varieties and dominant maps gi : Wi → P1 together
with proper maps hi : Wi → X: For this note that we already have a proper
morphism πi : Vi → X. Denote by ni : Ṽi → Vi the normalization maps.
20
MARKUS SEVERITT
Then we get our claim by setting Wi := Ṽi , hi := πi ◦ ni and gi := fi ◦ ni :
Consider p = 0, ∞ as effective Cartier divisors. By Lemma B.5 we have
[gi−1 (p)] = [gi∗ (p)]
and also for fi where the left hand side is the fundamental cycle of a closed
subvariety and the right hand side the canonical cycle of an effective Cartier
divisor. Then by the following lemma we get
[V (p)] = (πi )∗ [fi∗ (p)] = (hi )∗ [gi∗ (p)]
for p = 0, ∞. Hence
α=
n
X
(hi )∗ ([gi−1 (0)] − [gi−1 (∞)])
i=1
Lemma B.8. Let V be a k-variety and n : Ṽ → V the normalization in the
function field k(V ). Then for an effective Cartier divisor D of V we have
n∗ [n∗ (D)] = [D]
where the push-forward is defined since n is finite.
Proof. Recall that
[D] =
X
ordVx (D)
x∈Vd−1
for dim(X) = d. Now fix a x ∈ Vd−1 . If f ∈ R is a local function defining D
in Vx , the image of f in n : R ,→ R̃ ⊂ k(V ) defines n∗ (D) in Vy for all y ∈ Ṽ
with n(y) = x. That is, the same functions appear in the cycle [n∗ (D)] as
in [D]. By the definition of the push-forward n∗ of cycles, the coefficient of
n∗ [n∗ (D)] at x is
X
ordVy (f ) · deg(Vy /Vx )
y∈Ṽd−1 ,n(y)=x
By [Ful98, Example 1.2.3 and Example A.3.1], this is the same as ordVx (f )
which is the coefficient of [D] at x. This gives the claim.
Note that there is another definition of rational equivalence which uses
rational maps f ∈ k(W )∗ for subvarieties W ⊂ X and their canonical divisors div(f ) which consists of orders of zeros and poles along codimension 1
subvarieties V of W (cf. [Ful98, Chapters 1.2, 1.3]). This definition is used
to proof the compatibility of the proper push-forwards of cycles and rational equivalence in [Ful98, Theorem 1.4]. Our definition is used to proof the
compatibility of the flat pull-back maps with rational equivalence in [Ful98,
Theorem 1.7]. That is, we also have functorial proper push-forwards and
flat pull-backs on Chow groups together with a commutativity for pullback
diagrams like for the cycle groups.
References
[AM69] M. F. Atiyah and I. G. Macdonald. Introduction to commutative algebra. AddisonWesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969.
[Ful98] William Fulton. Intersection Theory, volume 2 of Ergebnisse der Mathematik und
ihrer Grenzgebiete. 3. Folge. Springer Verlag, Berlin, second edition, 1998.
ABOUT ALGEBRAIC K-THEORY AND ALGEBRAIC CYCLES
21
[Sri96] V. Srinivas. Algebraic K-theory, volume 90 of Progress in Mathematics.
Birkhäuser Boston Inc., Boston, MA, second edition, 1996.