Remark 30.25.1. Let $X$ be a Noetherian scheme 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 open $\mathop{\mathrm{Spec}}(A) = U \subset X$ with $\mathcal{I}|_ U, \mathcal{K}|_ U$ corresponding to ideals $I, K \subset A$ denote $\alpha _ U : M \to N$ of finite $A^\wedge$-modules which corresponds to $\alpha |_ U$ via 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 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 $t \geq 1$, and

3. there exists a finite affine open covering $X = \bigcup U_ i$ such that the conclusion of (2) holds for $\alpha _{U_ i}$.

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 use the following commutative algebra fact: suppose given an exact sequence

$0 \to T \to M \to N \to Q \to 0$

of $A$-modules with $T$ and $Q$ annihilated by $K^ t$ for some ideal $K \subset A$. Then for every $f, g \in K^ t$ there exists a canonical map $"fg": N \to M$ such that $M \to N \to M$ is equal to multiplication by $fg$. Namely, for $y \in N$ we can pick $x \in M$ mapping to $fy$ in $N$ and then we can set $"fg"(y) = gx$. Thus it is clear that $\mathop{\mathrm{Ker}}(M/JM \to N/JN)$ and $\mathop{\mathrm{Coker}}(M/JM \to N/JN)$ are annihilated by $K^{2t}$ for any ideal $J \subset A$.

Applying the commutative algebra fact to $\alpha _{U_ i}$ and $J = I^ n$ we see that (3) implies (1). Conversely, suppose (1) holds and $M \to N$ is equal to $\alpha _ U$. Then there is a $t \geq 1$ such that $\mathop{\mathrm{Ker}}(M/I^ nM \to N/I^ nN)$ and $\mathop{\mathrm{Coker}}(M/I^ nM \to N/I^ nN)$ are annihilated by $K^ t$ for all $n$. We obtain maps $"fg" : N/I^ nN \to M/I^ nM$ which in the limit induce a map $N \to M$ as $N$ and $M$ are $I$-adically complete. Since the composition with $N \to M \to N$ is multiplication by $fg$ we conclude that $fg$ annihilates $T$ and $Q$. In other words $T$ and $Q$ are annihilated by $K^{2t}$ as desired.

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