The Stacks project

Remark 76.42.7. Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$ and let $\mathcal{I}, \mathcal{K} \subset \mathcal{O}_ X$ be quasi-coherent sheaves of ideals. Let $\alpha : (\mathcal{F}_ n) \to (\mathcal{G}_ n)$ be a morphism of $\textit{Coh}(X, \mathcal{I})$. Given an affine scheme $U = \mathop{\mathrm{Spec}}(A)$ and a surjective étale morphism $U \to X$ denote $I, K \subset A$ the ideals corresponding to the restrictions $\mathcal{I}|_ U, \mathcal{K}|_ U$. Denote $\alpha _ U : M \to N$ of finite $A^\wedge $-modules which corresponds to $\alpha |_ U$ via Cohomology of Schemes, Lemma 30.23.1. We claim the following are equivalent

1. there exists an integer $t \geq 1$ such that $\mathop{\mathrm{Ker}}(\alpha _ n)$ and $\mathop{\mathrm{Coker}}(\alpha _ n)$ are annihilated by $\mathcal{K}^ t$ for all $n \geq 1$,

2. for any (or some) affine open $\mathop{\mathrm{Spec}}(A) = U \subset X$ as above the modules $\mathop{\mathrm{Ker}}(\alpha _ U)$ and $\mathop{\mathrm{Coker}}(\alpha _ U)$ are annihilated by $K^ t$ for some integer $t \geq 1$.

If these equivalent conditions hold we will say that $\alpha $ is a map whose kernel and cokernel are annihilated by a power of $\mathcal{K}$. To see the equivalence we refer to Cohomology of Schemes, Remark 30.25.1.