The Stacks project

42.23 Chow groups and K-groups

In this section we are going to compare $K_0$ of the category of coherent sheaves to the chow groups.

Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be a scheme locally of finite type over $S$. We denote $\textit{Coh}(X) = \textit{Coh}(\mathcal{O}_ X)$ the category of coherent sheaves on $X$. It is an abelian category, see Cohomology of Schemes, Lemma 30.9.2. For any $k \in \mathbf{Z}$ we let $\textit{Coh}_{\leq k}(X)$ be the full subcategory of $\textit{Coh}(X)$ consisting of those coherent sheaves $\mathcal{F}$ having $\dim _\delta (\text{Supp}(\mathcal{F})) \leq k$.

Lemma 42.23.1. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be a scheme locally of finite type over $S$. The categories $\textit{Coh}_{\leq k}(X)$ are Serre subcategories of the abelian category $\textit{Coh}(X)$.

Proof. The definition of a Serre subcategory is Homology, Definition 12.10.1. The proof of the lemma is straightforward and omitted. $\square$

Lemma 42.23.2. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be a scheme locally of finite type over $S$. The maps

\[ Z_ k(X) \longrightarrow K_0(\textit{Coh}_{\leq k}(X)/\textit{Coh}_{\leq k - 1}(X)), \quad \sum n_ Z[Z] \mapsto \left[\bigoplus \nolimits _{n_ Z > 0} \mathcal{O}_ Z^{\oplus n_ Z}\right] - \left[\bigoplus \nolimits _{n_ Z < 0} \mathcal{O}_ Z^{\oplus -n_ Z}\right] \]

and

\[ K_0(\textit{Coh}_{\leq k}(X)/\textit{Coh}_{\leq k - 1}(X)) \longrightarrow Z_ k(X),\quad \mathcal{F} \longmapsto [\mathcal{F}]_ k \]

are mutually inverse isomorphisms.

Proof. Note that if $\sum n_ Z[Z]$ is in $Z_ k(X)$, then the direct sums $\bigoplus \nolimits _{n_ Z > 0} \mathcal{O}_ Z^{\oplus n_ Z}$ and $\bigoplus \nolimits _{n_ Z < 0} \mathcal{O}_ Z^{\oplus -n_ Z}$ are coherent sheaves on $X$ since the family $\{ Z \mid n_ Z > 0\} $ is locally finite on $X$. The map $\mathcal{F} \to [\mathcal{F}]_ k$ is additive on $\textit{Coh}_{\leq k}(X)$, see Lemma 42.10.4. And $[\mathcal{F}]_ k = 0$ if $\mathcal{F} \in \textit{Coh}_{\leq k - 1}(X)$. By part (1) of Homology, Lemma 12.11.3 this implies that the second map is well defined too. It is clear that the composition of the first map with the second map is the identity.

Conversely, say we start with a coherent sheaf $\mathcal{F}$ on $X$. Write $[\mathcal{F}]_ k = \sum _{i \in I} n_ i[Z_ i]$ with $n_ i > 0$ and $Z_ i \subset X$, $i \in I$ pairwise distinct integral closed subschemes of $\delta $-dimension $k$. We have to show that

\[ [\mathcal{F}] = [\bigoplus \nolimits _{i \in I} \mathcal{O}_{Z_ i}^{\oplus n_ i}] \]

in $K_0(\textit{Coh}_{\leq k}(X)/\textit{Coh}_{\leq k - 1}(X))$. Denote $\xi _ i \in Z_ i$ the generic point. If we set

\[ \mathcal{F}' = \mathop{\mathrm{Ker}}(\mathcal{F} \to \bigoplus \xi _{i, *}\mathcal{F}_{\xi _ i}) \]

then $\mathcal{F}'$ is the maximal coherent submodule of $\mathcal{F}$ whose support has dimension $\leq k - 1$. In particular $\mathcal{F}$ and $\mathcal{F}/\mathcal{F}'$ have the same class in $K_0(\textit{Coh}_{\leq k}(X)/\textit{Coh}_{\leq k - 1}(X))$. Thus after replacing $\mathcal{F}$ by $\mathcal{F}/\mathcal{F}'$ we may and do assume that the kernel $\mathcal{F}'$ displayed above is zero.

For each $i \in I$ we choose a filtration

\[ \mathcal{F}_{\xi _ i} = \mathcal{F}_ i^0 \supset \mathcal{F}_ i^1 \supset \ldots \supset \mathcal{F}_ i^{n_ i} = 0 \]

such that the successive quotients are of dimension $1$ over the residue field at $\xi _ i$. This is possible as the length of $\mathcal{F}_{\xi _ i}$ over $\mathcal{O}_{X, \xi _ i}$ is $n_ i$. For $p > n_ i$ set $\mathcal{F}_ i^ p = 0$. For $p \geq 0$ we denote

\[ \mathcal{F}^ p = \mathop{\mathrm{Ker}}\left(\mathcal{F} \longrightarrow \bigoplus \xi _{i, *}(\mathcal{F}_{\xi _ i}/\mathcal{F}_ i^ p)\right) \]

Then $\mathcal{F}^ p$ is coherent, $\mathcal{F}^0 = \mathcal{F}$, and $\mathcal{F}^ p/\mathcal{F}^{p + 1}$ is isomorphic to a free $\mathcal{O}_{Z_ i}$-module of rank $1$ (if $n_ i > p$) or $0$ (if $n_ i \leq p$) in an open neighbourhood of $\xi _ i$. Moreover, $\mathcal{F}' = \bigcap \mathcal{F}^ p = 0$. Since every quasi-compact open $U \subset X$ contains only a finite number of $\xi _ i$ we conclude that $\mathcal{F}^ p|_ U$ is zero for $p \gg 0$. Hence $\bigoplus _{p \geq 0} \mathcal{F}^ p$ is a coherent $\mathcal{O}_ X$-module. Consider the short exact sequences

\[ 0 \to \bigoplus \nolimits _{p > 0} \mathcal{F}^ p \to \bigoplus \nolimits _{p \geq 0} \mathcal{F}^ p \to \bigoplus \nolimits _{p > 0} \mathcal{F}^ p/\mathcal{F}^{p + 1} \to 0 \]

and

\[ 0 \to \bigoplus \nolimits _{p > 0} \mathcal{F}^ p \to \bigoplus \nolimits _{p \geq 0} \mathcal{F}^ p \to \mathcal{F} \to 0 \]

of coherent $\mathcal{O}_ X$-modules. This already shows that

\[ [\mathcal{F}] = [\bigoplus \mathcal{F}^ p/\mathcal{F}^{p + 1}] \]

in $K_0(\textit{Coh}_{\leq k}(X)/\textit{Coh}_{\leq k - 1}(X))$. Next, for every $p \geq 0$ and $i \in I$ such that $n_ i > p$ we choose a nonzero ideal sheaf $\mathcal{I}_{i, p} \subset \mathcal{O}_{Z_ i}$ and a map $\mathcal{I}_{i, p} \to \mathcal{F}^ p/\mathcal{F}^{p + 1}$ on $X$ which is an isomorphism over the open neighbourhood of $\xi _ i$ mentioned above. This is possible by Cohomology of Schemes, Lemma 30.10.6. Then we consider the short exact sequence

\[ 0 \to \bigoplus \nolimits _{p \geq 0, i \in I, n_ i > p} \mathcal{I}_{i, p} \to \bigoplus \mathcal{F}^ p/\mathcal{F}^{p + 1} \to \mathcal{Q} \to 0 \]

and the short exact sequence

\[ 0 \to \bigoplus \nolimits _{p \geq 0, i \in I, n_ i > p} \mathcal{I}_{i, p} \to \bigoplus \nolimits _{p \geq 0, i \in I, n_ i > p} \mathcal{O}_{Z_ i} \to \mathcal{Q}' \to 0 \]

Observe that both $\mathcal{Q}$ and $\mathcal{Q}'$ are zero in a neighbourhood of the points $\xi _ i$ and that they are supported on $\bigcup Z_ i$. Hence $\mathcal{Q}$ and $\mathcal{Q}'$ are in $\textit{Coh}_{\leq k - 1}(X)$. Since

\[ \bigoplus \nolimits _{i \in I} \mathcal{O}_{Z_ i}^{\oplus n_ i} \cong \bigoplus \nolimits _{p \geq 0, i \in I, n_ i > p} \mathcal{O}_{Z_ i} \]

this concludes the proof. $\square$

Lemma 42.23.3. Let $\pi : X \to Y$ be a finite morphism of schemes locally of finite type over $(S, \delta )$ as in Situation 42.7.1. Then $\pi _* : \textit{Coh}(X) \to \textit{Coh}(Y)$ is an exact functor which sends $\textit{Coh}_{\leq k}(X)$ into $\textit{Coh}_{\leq k}(Y)$ and induces homomorphisms on $K_0$ of these categories and their quotients. The maps of Lemma 42.23.2 fit into a commutative diagram

\[ \xymatrix{ Z_ k(X) \ar[d]^{\pi _*} \ar[r] & K_0(\textit{Coh}_{\leq k}(X)/\textit{Coh}_{\leq k - 1}(X)) \ar[d]^{\pi _*} \ar[r] & Z_ k(X) \ar[d]^{\pi _*} \\ Z_ k(Y) \ar[r] & K_0(\textit{Coh}_{\leq k}(Y)/\textit{Coh}_{\leq k - 1}(Y)) \ar[r] & Z_ k(Y) } \]

Proof. A finite morphism is affine, hence pushforward of quasi-coherent modules along $\pi $ is an exact functor by Cohomology of Schemes, Lemma 30.2.3. A finite morphism is proper, hence $\pi _*$ sends coherent sheaves to coherent sheaves, see Cohomology of Schemes, Proposition 30.19.1. The statement on dimensions of supports is clear. Commutativity on the right follows immediately from Lemma 42.12.4. Since the horizontal arrows are bijections, we find that we have commutativity on the left as well. $\square$

Lemma 42.23.4. Let $X$ be a scheme locally of finite type over $(S, \delta )$ as in Situation 42.7.1. There is a canonical map

\[ \mathop{\mathrm{CH}}\nolimits _ k(X) \longrightarrow K_0(\textit{Coh}_{\leq k + 1}(X)/\textit{Coh}_{\leq k - 1}(X)) \]

induced by the map $Z_ k(X) \to K_0(\textit{Coh}_{\leq k}(X)/\textit{Coh}_{\leq k - 1}(X))$ from Lemma 42.23.2.

Proof. We have to show that an element $\alpha $ of $Z_ k(X)$ which is rationally equivalent to zero, is mapped to zero in $K_0(\textit{Coh}_{\leq k + 1}(X)/\textit{Coh}_{\leq k - 1}(X))$. Write $\alpha = \sum (i_ j)_*\text{div}(f_ j)$ as in Definition 42.19.1. Observe that

\[ \pi = \coprod i_ j : W = \coprod W_ j \longrightarrow X \]

is a finite morphism as each $i_ j : W_ j \to X$ is a closed immersion and the family of $W_ j$ is locally finite in $X$. Hence we may use Lemma 42.23.3 to reduce to the case of $W$. Since $W$ is a disjoint union of intregral scheme, we reduce to the case discussed in the next paragraph.

Assume $X$ is integral of $\delta $-dimension $k + 1$. Let $f$ be a nonzero rational function on $X$. Let $\alpha = \text{div}(f)$. We have to show that $\alpha $ is mapped to zero in $K_0(\textit{Coh}_{\leq k + 1}(X)/\textit{Coh}_{\leq k - 1}(X))$. Let $\mathcal{I} \subset \mathcal{O}_ X$ be the ideal of denominators of $f$, see Divisors, Definition 31.23.10. Then we have short exact sequences

\[ 0 \to \mathcal{I} \to \mathcal{O}_ X \to \mathcal{O}_ X/\mathcal{I} \to 0 \]

and

\[ 0 \to \mathcal{I} \xrightarrow {f} \mathcal{O}_ X \to \mathcal{O}_ X/f\mathcal{I} \to 0 \]

See Divisors, Lemma 31.23.9. We claim that

\[ [\mathcal{O}_ X/\mathcal{I}]_ k - [\mathcal{O}_ X/f\mathcal{I}]_ k = \text{div}(f) \]

The claim implies the element $\alpha = \text{div}(f)$ is represented by $[\mathcal{O}_ X/\mathcal{I}] - [\mathcal{O}_ X/f\mathcal{I}]$ in $K_0(\textit{Coh}_{\leq k}(X)/\textit{Coh}_{\leq k - 1}(X))$. Then the short exact sequences show that this element maps to zero in $K_0(\textit{Coh}_{\leq k + 1}(X)/\textit{Coh}_{\leq k - 1}(X))$.

To prove the claim, let $Z \subset X$ be an integral closed subscheme of $\delta $-dimension $k$ and let $\xi \in Z$ be its generic point. Then $I = \mathcal{I}_\xi \subset A = \mathcal{O}_{X, \xi }$ is an ideal such that $fI \subset A$. Now the coefficient of $[Z]$ in $\text{div}(f)$ is $\text{ord}_ A(f)$. (Of course as usual we identify the function field of $X$ with the fraction field of $A$.) On the other hand, the coefficient of $[Z]$ in $[\mathcal{O}_ X/\mathcal{I}] - [\mathcal{O}_ X/f\mathcal{I}]$ is

\[ \text{length}_ A(A/I) - \text{length}_ A(A/fI) \]

Using the distance fuction of Algebra, Definition 10.121.5 we can rewrite this as

\[ d(A, I) - d(A, fI) = d(I, fI) = \text{ord}_ A(f) \]

The equalities hold by Algebra, Lemmas 10.121.6 and 10.121.7. (Using these lemmas isn't necessary, but convenient.) $\square$

Remark 42.23.5. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be a scheme locally of finite type over $S$. We will see later (in Lemma 42.69.3) that the map

\[ \mathop{\mathrm{CH}}\nolimits _ k(X) \longrightarrow K_0(\textit{Coh}_{k + 1}(X)/\textit{Coh}_{\leq k - 1}(X)) \]

of Lemma 42.23.4 is injective. Composing with the canonical map

\[ K_0(\textit{Coh}_{k + 1}(X)/\textit{Coh}_{\leq k - 1}(X)) \longrightarrow K_0(\textit{Coh}(X)/\textit{Coh}_{\leq k - 1}(X)) \]

we obtain a canonical map

\[ \mathop{\mathrm{CH}}\nolimits _ k(X) \longrightarrow K_0(\textit{Coh}(X)/\textit{Coh}_{\leq k - 1}(X)). \]

We have not been able to find a statement or conjecture in the literature as to whether this map is should be injective or not. It seems reasonable to expect the kernel of this map to be torsion. We will return to this question (insert future reference).

Lemma 42.23.6. Let $X$ be a locally Noetherian scheme. Let $Z \subset X$ be a closed subscheme. Denote $\textit{Coh}_ Z(X) \subset \textit{Coh}(X)$ the Serre subcategory of coherent $\mathcal{O}_ X$-modules whose set theoretic support is contained in $Z$. Then the exact inclusion functor $\textit{Coh}(Z) \to \textit{Coh}_ Z(X)$ induces an isomorphism

\[ K'_0(Z) = K_0(\textit{Coh}(Z)) \longrightarrow K_0(\textit{Coh}_ Z(X)) \]

Proof. Let $\mathcal{F}$ be an object of $\textit{Coh}_ Z(X)$. Let $\mathcal{I} \subset \mathcal{O}_ X$ be the quasi-coherent ideal sheaf of $Z$. Consider the descending filtration

\[ \ldots \subset \mathcal{F}^ p = \mathcal{I}^ p \mathcal{F} \subset \mathcal{F}^{p - 1} \subset \ldots \subset \mathcal{F}^0 = \mathcal{F} \]

Exactly as in the proof of Lemma 42.23.4 this filtration is locally finite and hence $\bigoplus _{p \geq 0} \mathcal{F}^ p$, $\bigoplus _{p \geq 1} \mathcal{F}^ p$, and $\bigoplus _{p \geq 0} \mathcal{F}^ p/\mathcal{F}^{p + 1}$ are coherent $\mathcal{O}_ X$-modules supported on $Z$. Hence we get

\[ [\mathcal{F}] = [\bigoplus \nolimits _{p \geq 0} \mathcal{F}^ p/\mathcal{F}^{p + 1}] \]

in $K_0(\textit{Coh}_ Z(X))$ exactly as in the proof of Lemma 42.23.4. Since the coherent module $\bigoplus _{p \geq 0} \mathcal{F}^ p/\mathcal{F}^{p + 1}$ is annihilated by $\mathcal{I}$ we conclude that $[\mathcal{F}]$ is in the image. Actually, we claim that the map

\[ \mathcal{F} \longmapsto c(\mathcal{F}) = [\bigoplus \nolimits _{p \geq 0} \mathcal{F}^ p/\mathcal{F}^{p + 1}] \]

factors through $K_0(\textit{Coh}_ Z(X))$ and is an inverse to the map in the statement of the lemma. To see this all we have to show is that if

\[ 0 \to \mathcal{F} \to \mathcal{G} \to \mathcal{H} \to 0 \]

is a short exact sequence in $\textit{Coh}_ Z(X)$, then we get $c(\mathcal{G}) = c(\mathcal{F}) + c(\mathcal{H})$. Observe that for all $q \geq 0$ we have a short exact sequence

\[ 0 \to (\mathcal{F} \cap \mathcal{I}^ q\mathcal{G})/ (\mathcal{F} \cap \mathcal{I}^{q + 1}\mathcal{G}) \to \mathcal{G}^ q/\mathcal{G}^{q + 1} \to \mathcal{H}^ q/\mathcal{H}^{q + 1} \to 0 \]

For $p, q \geq 0$ consider the coherent submodule

\[ \mathcal{F}^{p, q} = \mathcal{I}^ p\mathcal{F} \cap \mathcal{I}^ q\mathcal{G} \]

Arguing exactly as above and using that the filtrations $\mathcal{F}^ p = \mathcal{I}^ p\mathcal{F}$ and $\mathcal{F} \cap \mathcal{I}^ q\mathcal{G}$ are locally finite, we find that

\[ [\bigoplus \nolimits _{p \geq 0} \mathcal{F}^ p/\mathcal{F}^{p + 1}] = [\bigoplus \nolimits _{p, q \geq 0} \mathcal{F}^{p, q}/(\mathcal{F}^{p + 1, q} + \mathcal{F}^{p, q + 1})] = [\bigoplus \nolimits _{q \geq 0} (\mathcal{F} \cap \mathcal{I}^ q\mathcal{G})/ (\mathcal{F} \cap \mathcal{I}^{q + 1}\mathcal{G})] \]

in $K_0(\textit{Coh}(Z))$. Combined with the exact sequences above we obtain the desired result. Some details omitted. $\square$


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