Lemma 21.22.2. Let $I$ be an ideal of a ring $A$. Let $\mathcal{C}$ be a site. Let

\[ \ldots \to \mathcal{F}_3 \to \mathcal{F}_2 \to \mathcal{F}_1 \]

be an inverse system of $A$-modules on $\mathcal{C}$ such that $\mathcal{F}_ n = \mathcal{F}_{n + 1}/I^ n\mathcal{F}_{n + 1}$. Let $p \geq 0$. Given $n$ define

\[ N_ n = \bigcap \nolimits _{m \geq n} \mathop{\mathrm{Im}}\left( H^{p + 1}(\mathcal{C}, I^ n\mathcal{F}_{m + 1}) \to H^{p + 1}(\mathcal{C}, I^ n\mathcal{F}_{n + 1}) \right) \]

If $\bigoplus N_ n$ satisfies the ascending chain condition as a graded $\bigoplus _{n \geq 0} I^ n/I^{n + 1}$-module, then the inverse system $M_ n = H^ p(\mathcal{C}, \mathcal{F}_ n)$ satisfies the Mittag-Leffler condition^{1}.

**Proof.**
The proof is exactly the same as the proof of Lemma 21.22.1. In fact, the result will follow from the arguments given there as soon as we show that $\bigoplus N_ n$ is a graded $\bigoplus _{n \geq 0} I^ n/I^{n + 1}$-submodule of $\bigoplus H^{p + 1}(\mathcal{C}, I^ n\mathcal{F}_{n + 1})$ and that the boundary maps $\delta _ n : M_ n \to H^{p + 1}(\mathcal{C}, I^ n\mathcal{F}_{n + 1})$ have image contained in $N_ n$.

Suppose that $\xi \in N_ n$ and $f \in I^ k$. Choose $m \gg n + k$. Choose $\xi ' \in H^{p + 1}(\mathcal{C}, I^ n\mathcal{F}_{m + 1})$ lifting $\xi $. We consider the diagram

\[ \xymatrix{ 0 \ar[r] & I^ n\mathcal{F}_{m + 1} \ar[d]_ f \ar[r] & \mathcal{F}_{m + 1} \ar[d]_ f \ar[r] & \mathcal{F}_ n \ar[d]_ f \ar[r] & 0 \\ 0 \ar[r] & I^{n + k}\mathcal{F}_{m + 1} \ar[r] & \mathcal{F}_{m + 1} \ar[r] & \mathcal{F}_{n + k} \ar[r] & 0 } \]

constructed as in the proof of Lemma 21.22.1. We get an induced map on cohomology and we see that $f \xi ' \in H^{p + 1}(\mathcal{C}, I^{n + k}\mathcal{F}_{m + 1})$ maps to $f \xi $. Since this is true for all $m \gg n + k$ we see that $f\xi $ is in $N_{n + k}$ as desired.

To see the boundary maps $\delta _ n$ have image contained in $N_ n$ we consider the diagrams

\[ \xymatrix{ 0 \ar[r] & I^ n\mathcal{F}_{m + 1} \ar[d] \ar[r] & \mathcal{F}_{m + 1} \ar[d] \ar[r] & \mathcal{F}_ n \ar[d] \ar[r] & 0 \\ 0 \ar[r] & I^ n\mathcal{F}_{n + 1} \ar[r] & \mathcal{F}_{n + 1} \ar[r] & \mathcal{F}_ n \ar[r] & 0 } \]

for $m \geq n$. Looking at the induced maps on cohomology we conclude.
$\square$

## Comments (0)