The Stacks project

Lemma 42.46.1. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $E \in D(\mathcal{O}_ X)$ be an object such that there exists a locally bounded complex $\mathcal{E}^\bullet $ of finite locally free $\mathcal{O}_ X$-modules representing $E$. Then a slight generalization of the above constructions

\[ c(\mathcal{E}^\bullet ) \in \prod \nolimits _{p \geq 0} A^ p(X),\quad ch(\mathcal{E}^\bullet ) \in \prod \nolimits _{p \geq 0} A^ p(X) \otimes \mathbf{Q},\quad P_ p(\mathcal{E}^\bullet ) \in A^ p(X) \]

are independent of the choice of the complex $\mathcal{E}^\bullet $.

Proof. We prove this for the total Chern class; the other two cases follow by the same arguments using Lemma 42.45.2 instead of Lemma 42.40.3.

As in Remark 42.38.10 in order to define the total chern class $c(\mathcal{E}^\bullet )$ we decompose $X$ into open and closed subschemes

\[ X = \coprod \nolimits _{i \in I} X_ i \]

such that the rank $\mathcal{E}^ n$ is constant on $X_ i$ for all $n$ and $i$. (Since these ranks are locally constant functions on $X$ we can do this.) Since $\mathcal{E}^\bullet $ is locally bounded, we see that only a finite number of the sheaves $\mathcal{E}^ n|_{X_ i}$ are nonzero for a fixed $i$. Hence we can define

\[ c(\mathcal{E}^\bullet |_{X_ i}) = \prod \nolimits _ n c(\mathcal{E}^ n|_{X_ i})^{(-1)^ n} \in \prod \nolimits _{p \geq 0} A^ p(X_ i) \]

as above. By Lemma 42.35.4 we have $A^ p(X) = \prod _ i A^ p(X_ i)$. Hence for each $p \in \mathbf{Z}$ we have a unique element $c_ p(\mathcal{E}^\bullet ) \in A^ p(X)$ restricting to $c_ p(\mathcal{E}^\bullet |_{X_ i})$ on $X_ i$ for all $i$.

Suppose we have a second locally bounded complex $\mathcal{F}^\bullet $ of finite locally free $\mathcal{O}_ X$-modules representing $E$. Let $g : Y \to X$ be a morphism locally of finite type with $Y$ integral. By Lemma 42.35.3 it suffices to show that with $c(g^*\mathcal{E}^\bullet ) \cap [Y]$ is the same as $c(g^*\mathcal{F}^\bullet ) \cap [Y]$ and it even suffices to prove this after replacing $Y$ by an integral scheme proper and birational over $Y$. Then first we conclude that $g^*\mathcal{E}^\bullet $ and $g^*\mathcal{F}^\bullet $ are bounded complexes of finite locally free $\mathcal{O}_ Y$-modules of constant rank. Next, by More on Flatness, Lemma 38.40.3 we may assume that $H^ i(Lg^*E)$ is perfect of tor dimension $\leq 1$ for all $i \in \mathbf{Z}$. This reduces us to the case discussed in the next paragraph.

Assume $X$ is integral, $\mathcal{E}^\bullet $ and $\mathcal{F}^\bullet $ are bounded complexes of finite locally free modules of constant rank, and $H^ i(E)$ is a perfect $\mathcal{O}_ X$-module of tor dimension $\leq 1$ for all $i \in \mathbf{Z}$. We have to show that $c(\mathcal{E}^\bullet ) \cap [X]$ is the same as $c(\mathcal{F}^\bullet ) \cap [X]$. Denote $d_\mathcal {E}^ i : \mathcal{E}^ i \to \mathcal{E}^{i + 1}$ and $d_\mathcal {F}^ i : \mathcal{F}^ i \to \mathcal{F}^{i + 1}$ the differentials of our complexes. By More on Flatness, Remark 38.40.4 we know that $\mathop{\mathrm{Im}}(d_\mathcal {E}^ i)$, $\mathop{\mathrm{Ker}}(d_\mathcal {E}^ i)$, $\mathop{\mathrm{Im}}(d_\mathcal {F}^ i)$, and $\mathop{\mathrm{Ker}}(d_\mathcal {F}^ i)$ are finite locally free $\mathcal{O}_ X$-modules for all $i$. By additivity (Lemma 42.40.3) we see that

\[ c(\mathcal{E}^\bullet ) = \prod \nolimits _ i c(\mathop{\mathrm{Ker}}(d_\mathcal {E}^ i))^{(-1)^ i} c(\mathop{\mathrm{Im}}(d_\mathcal {E}^ i))^{(-1)^ i} \]

and similarly for $\mathcal{F}^\bullet $. Since we have the short exact sequences

\[ 0 \to \mathop{\mathrm{Im}}(d_\mathcal {E}^ i) \to \mathop{\mathrm{Ker}}(d_\mathcal {E}^ i) \to H^ i(E) \to 0 \quad \text{and}\quad 0 \to \mathop{\mathrm{Im}}(d_\mathcal {F}^ i) \to \mathop{\mathrm{Ker}}(d_\mathcal {F}^ i) \to H^ i(E) \to 0 \]

we reduce to the problem stated and solved in the next paragraph.

Assume $X$ is integral and we have two short exact sequences

\[ 0 \to \mathcal{E}' \to \mathcal{E} \to \mathcal{Q} \to 0 \quad \text{and}\quad 0 \to \mathcal{F}' \to \mathcal{F} \to \mathcal{Q} \to 0 \]

with $\mathcal{E}$, $\mathcal{E}'$, $\mathcal{F}$, $\mathcal{F}'$ finite locally free. Problem: show that $c(\mathcal{E})c(\mathcal{E}')^{-1} \cap [X] = c(\mathcal{F})c(\mathcal{F}')^{-1} \cap [X]$. To do this, consider the short exact sequence

\[ 0 \to \mathcal{G} \to \mathcal{E} \oplus \mathcal{F} \to \mathcal{Q} \to 0 \]

defining $\mathcal{G}$. Since $\mathcal{Q}$ has tor dimension $\leq 1$ we see that $\mathcal{G}$ is finite locally free. A diagram chase shows that the kernel of the surjection $\mathcal{G} \to \mathcal{F}$ maps isomorphically to $\mathcal{E}'$ in $\mathcal{E}$ and the kernel of the surjection $\mathcal{G} \to \mathcal{E}$ maps isomorphically to $\mathcal{F}'$ in $\mathcal{F}$. (Working affine locally this follows from or is equivalent to Schanuel's lemma, see Algebra, Lemma 10.109.1.) We conclude that

\[ c(\mathcal{E})c(\mathcal{F}') = c(\mathcal{G}) = c(\mathcal{F})c(\mathcal{E}') \]

as desired. $\square$

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