The Stacks project

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

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

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

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

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

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

and

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

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

and the short exact sequence

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

this concludes the proof. $\square$

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. $\square$

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

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 integral 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

and

See Divisors, Lemma 31.23.9. We claim that

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

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

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

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

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

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

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

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

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

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

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