The Stacks project

Lemma 36.38.5. Let $X$ be a quasi-compact and quasi-separated scheme with the resolution property. Then the map $K_0(\textit{Vect}(X)) \to K_0(X)$ is an isomorphism.

Proof. This lemma will follow in a straightforward manner from Lemmas 36.37.2, 36.37.3, and 36.37.4 whose results we will use without further mention. Let us construct an inverse map

\[ c : K_0(X) = K_0(D_{perf}(\mathcal{O}_ X)) \longrightarrow K_0(\textit{Vect}(X)) \]

Namely, any object of $D_{perf}(\mathcal{O}_ X)$ can be represented by a bounded complex $\mathcal{E}^\bullet $ of finite locally free $\mathcal{O}_ X$-modules. Then we set

\[ c([\mathcal{E}^\bullet ]) = \sum (-1)^ i[\mathcal{E}^ i] \]

Of course we have to show that this is well defined. For the moment we view $c$ as a map defined on bounded complexes of finite locally free $\mathcal{O}_ X$-modules.

Suppose that $\mathcal{E}^\bullet \to \mathcal{F}^\bullet $ is a surjective map of bounded complexes of finite locally free $\mathcal{O}_ X$-modules. Let $\mathcal{K}^\bullet $ be the kernel. Then we obtain short exact sequences of $\mathcal{O}_ X$-modules

\[ 0 \to \mathcal{K}^ n \to \mathcal{E}^ n \to \mathcal{F}^ n \to 0 \]

which are locally split because $\mathcal{F}^ n$ is finite locally free. Hence $\mathcal{K}^\bullet $ is also a bounded complex of finite locally free $\mathcal{O}_ X$-modules and we have $c(\mathcal{E}^\bullet ) = c(\mathcal{K}^\bullet ) + c(\mathcal{F}^\bullet )$ in $K_0(\textit{Vect}(X))$.

Suppose given a bounded complex $\mathcal{E}^\bullet $ of finite locally free $\mathcal{O}_ X$-modules which is acyclic. Say $\mathcal{E}^ n = 0$ for $n \not\in [a, b]$. Then we can break $\mathcal{E}^\bullet $ into short exact sequences

\[ \begin{matrix} 0 \to \mathcal{E}^ a \to \mathcal{E}^{a + 1} \to \mathcal{F}^{a + 1} \to 0, \\ 0 \to \mathcal{F}^{a + 1} \to \mathcal{E}^{a + 2} \to \mathcal{F}^{a + 3} \to 0, \\ \ldots \\ 0 \to \mathcal{F}^{b - 3} \to \mathcal{E}^{b - 2} \to \mathcal{F}^{b - 2} \to 0, \\ 0 \to \mathcal{F}^{b - 2} \to \mathcal{E}^{b - 1} \to \mathcal{E}^ b \to 0 \end{matrix} \]

Arguing by descending induction we see that $\mathcal{F}^{b - 2}, \ldots , \mathcal{F}^{a + 1}$ are finite locally free $\mathcal{O}_ X$-modules, and

\[ c(\mathcal{E}^\bullet ) = \sum (-1)[\mathcal{E}^ n] = \sum (-1)^ n([\mathcal{F}^{n - 1}] + [\mathcal{F}^ n]) = 0 \]

Thus our construction gives zero on acyclic complexes.

It follows from the results of the preceding two paragraphs that $c$ is well defined. Namely, suppose the bounded complexes $\mathcal{E}^\bullet $ and $\mathcal{F}^\bullet $ of finite locally free $\mathcal{O}_ X$-modules represent the same object of $D(\mathcal{O}_ X)$. Then we can find quasi-isomorphisms $a : \mathcal{G}^\bullet \to \mathcal{E}^\bullet $ and $b : \mathcal{G}^\bullet \to \mathcal{F}^\bullet $ with $\mathcal{G}^\bullet $ bounded complex of finite locally free $\mathcal{O}_ X$-modules. We obtain a short exact sequence of complexes

\[ 0 \to \mathcal{E}^\bullet \to C(a)^\bullet \to \mathcal{G}^\bullet [1] \to 0 \]

see Derived Categories, Definition 13.9.1. Since $a$ is a quasi-isomorphism, the cone $C(a)^\bullet $ is acyclic (this follows for example from the discussion in Derived Categories, Section 13.12). Hence

\[ 0 = c(C(f)^\bullet ) = c(\mathcal{E}^\bullet ) + c(\mathcal{G}^\bullet [1]) = c(\mathcal{E}^\bullet ) - c(\mathcal{G}^\bullet ) \]

as desired. The same argument using $b$ shows that $0 = c(\mathcal{F}^\bullet ) - c(\mathcal{G}^\bullet )$. Hence we find that $c(\mathcal{E}^\bullet ) = c(\mathcal{F}^\bullet )$ and $c$ is well defined.

A similar argument using the cone on a map $\mathcal{E}^\bullet \to \mathcal{F}^\bullet $ of bounded complexes of finite locally free $\mathcal{O}_ X$-modules shows that $c(Y) = c(X) + c(Z)$ if $X \to Y \to Z$ is a distinguished triangle in $D_{perf}(\mathcal{O}_ X)$. Details omitted. Thus we get the desired homomorphism of abelian groups $c : K_0(X) \to K_0(\textit{Vect}(X))$.

It is clear that the composition $K_0(\textit{Vect}(X)) \to K_0(X) \to K_0(\textit{Vect}(X))$ is the identity. On the other hand, let $\mathcal{E}^\bullet $ be a bounded complex of finite locally free $\mathcal{O}_ X$-modules. Then the the existence of the distinguished triangles of “stupid truncations” (see Homology, Section 12.15)

\[ \sigma _{\geq n}\mathcal{E}^\bullet \to \sigma _{\geq n - 1}\mathcal{E}^\bullet \to \mathcal{E}^{n - 1}[-n + 1] \to (\sigma _{\geq n}\mathcal{E}^\bullet )[1] \]

and induction show that

\[ [\mathcal{E}^\bullet ] = \sum (-1)^ i[\mathcal{E}^ i[0]] \]

in $K_0(X) = K_0(D_{perf}(\mathcal{O}_ X))$ with apologies for the notation. Hence the map $K_0(\textit{Vect}(X)) \to K_0(D_{perf}(\mathcal{O}_ X)) = K_0(X)$ is surjective which finishes the proof. $\square$

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