Processing math: 100%

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


Comments (0)


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.