The Stacks project

Proof. Let $\{ W_ j\} _{j \in J}$ be the collection of irreducible components of $\text{Supp}(\mathcal{F})$ which have $\delta$-dimension $k + 1$. Note that $\{ W_ j\}$ is a locally finite collection of closed subsets of $X$ by Lemma 42.10.1. For every $j$, let $\xi _ j \in W_ j$ be the generic point. Set

See Definition 42.68.13 for notation. We claim that

If we prove this then the lemma follows.

Let $Z \subset X$ be an integral closed subscheme of $\delta$-dimension $k$. To prove the equality above it suffices to show that the coefficient $n$ of $[Z]$ in $[H^0(\mathcal{F}, \varphi , \psi )]_ k - [H^1(\mathcal{F}, \varphi , \psi )]_ k$ is the same as the coefficient $m$ of $[Z]$ in $\sum (W_ j \to X)_*\text{div}(f_ j)$. Let $\xi \in Z$ be the generic point. Consider the local ring $A = \mathcal{O}_{X, \xi }$. Let $M = \mathcal{F}_\xi$ as an $A$-module. Denote $\varphi , \psi : M \to M$ the action of $\varphi , \psi$ on the stalk. By our choice of $\xi \in Z$ we have $\delta (\xi ) = k$ and hence $\dim (\text{Supp}(M)) = 1$. Finally, the integral closed subschemes $W_ j$ passing through $\xi$ correspond to the minimal primes $\mathfrak q_ i$ of $\text{Supp}(M)$. In each case the element $f_ j \in R(W_ j)^*$ corresponds to the element $\det _{\kappa (\mathfrak q_ i)}(M_{\mathfrak q_ i}, \varphi , \psi )$ in $\kappa (\mathfrak q_ i)^*$. Hence we see that

and

Thus the result follows from Proposition 42.68.43. $\square$