The Stacks project

Lemma 30.20.4. Let $A$ be a Noetherian ring. Let $I \subset A$ be an ideal. Let $f : X \to \mathop{\mathrm{Spec}}(A)$ be a proper morphism. Let $\mathcal{F}$ be a coherent sheaf on $X$. Fix $p \geq 0$. There exists a $c \geq 0$ such that

  1. for all $n \geq c$ we have

    \[ \mathop{\mathrm{Ker}}(H^ p(X, \mathcal{F}) \to H^ p(X, \mathcal{F}/I^ n\mathcal{F})) \subset I^{n - c}H^ p(X, \mathcal{F}). \]
  2. the inverse system

    \[ \left(H^ p(X, \mathcal{F}/I^ n\mathcal{F})\right)_{n \in \mathbf{N}} \]

    satisfies the Mittag-Leffler condition (see Homology, Definition 12.31.2), and

  3. we have

    \[ \mathop{\mathrm{Im}}(H^ p(X, \mathcal{F}/I^ k\mathcal{F}) \to H^ p(X, \mathcal{F}/I^ n\mathcal{F})) = \mathop{\mathrm{Im}}(H^ p(X, \mathcal{F}) \to H^ p(X, \mathcal{F}/I^ n\mathcal{F})) \]

    for all $k \geq n + c$.

Proof. Let $c = \max \{ c_ p, c_{p + 1}\} $, where $c_ p, c_{p + 1}$ are the integers found in Lemma 30.20.3 for $H^ p$ and $H^{p + 1}$.

Let us prove part (1). Consider the short exact sequence

\[ 0 \to I^ n\mathcal{F} \to \mathcal{F} \to \mathcal{F}/I^ n\mathcal{F} \to 0 \]

From the long exact cohomology sequence we see that

\[ \mathop{\mathrm{Ker}}( H^ p(X, \mathcal{F}) \to H^ p(X, \mathcal{F}/I^ n\mathcal{F}) ) = \mathop{\mathrm{Im}}( H^ p(X, I^ n\mathcal{F}) \to H^ p(X, \mathcal{F}) ) \]

Hence by Lemma 30.20.3 part (2) we see that this is contained in $I^{n - c}H^ p(X, \mathcal{F})$ for $n \geq c$.

Note that part (3) implies part (2) by definition of the Mittag-Leffler systems.

Let us prove part (3). Fix an $n$. Consider the commutative diagram

\[ \xymatrix{ 0 \ar[r] & I^ n\mathcal{F} \ar[r] & \mathcal{F} \ar[r] & \mathcal{F}/I^ n\mathcal{F} \ar[r] & 0 \\ 0 \ar[r] & I^{n + m}\mathcal{F} \ar[r] \ar[u] & \mathcal{F} \ar[r] \ar[u] & \mathcal{F}/I^{n + m}\mathcal{F} \ar[r] \ar[u] & 0 } \]

This gives rise to the following commutative diagram

\[ \xymatrix{ H^ p(X, \mathcal{F}) \ar[r] & H^ p(X, \mathcal{F}/I^ n\mathcal{F}) \ar[r]_\delta & H^{p + 1}(X, I^ n\mathcal{F}) \ar[r] & H^{p + 1}(X, \mathcal{F}) \\ H^ p(X, \mathcal{F}) \ar[r] \ar[u]^1 & H^ p(X, \mathcal{F}/I^{n + m}\mathcal{F}) \ar[r] \ar[u]^\gamma & H^{p + 1}(X, I^{n + m}\mathcal{F}) \ar[u]^\alpha \ar[r]^-\beta & H^{p + 1}(X, \mathcal{F}) \ar[u]_1 } \]

with exact rows. By Lemma 30.20.3 part (4) the kernel of $\beta $ is equal to the kernel of $\alpha $ for $m \geq c$. By a diagram chase this shows that the image of $\gamma $ is contained in the kernel of $\delta $ which shows that part (3) is true (set $k = n + m$ to get it). $\square$

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