The Stacks project

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.

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0888. Beware of the difference between the letter 'O' and the digit '0'.