The Stacks project

Lemma 42.69.2. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be a scheme locally of finite type over $S$. Let $\mathcal{F}$ be a coherent sheaf on $X$. Let

\[ \xymatrix{ \ldots \ar[r] & \mathcal{F} \ar[r]^\varphi & \mathcal{F} \ar[r]^\psi & \mathcal{F} \ar[r]^\varphi & \mathcal{F} \ar[r] & \ldots } \]

be a complex as in Homology, Equation ( Assume that

  1. $\dim _\delta (\text{Supp}(\mathcal{F})) \leq k + 1$.

  2. $\dim _\delta (\text{Supp}(H^ i(\mathcal{F}, \varphi , \psi ))) \leq k$ for $i = 0, 1$.

Then we have

\[ [H^0(\mathcal{F}, \varphi , \psi )]_ k \sim _{rat} [H^1(\mathcal{F}, \varphi , \psi )]_ k \]

as $k$-cycles on $X$.

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

\[ f_ j = \det \nolimits _{\kappa (\xi _ j)} (\mathcal{F}_{\xi _ j}, \varphi _{\xi _ j}, \psi _{\xi _ j}) \in R(W_ j)^*. \]

See Definition 42.68.13 for notation. We claim that

\[ - [H^0(\mathcal{F}, \varphi , \psi )]_ k + [H^1(\mathcal{F}, \varphi , \psi )]_ k = \sum (W_ j \to X)_*\text{div}(f_ j) \]

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

\[ n = - e_ A(M, \varphi , \psi ) \]


\[ m = \sum \text{ord}_{A/\mathfrak q_ i} (\det \nolimits _{\kappa (\mathfrak q_ i)}(M_{\mathfrak q_ i}, \varphi , \psi )) \]

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

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