The Stacks project

Lemma 51.8.2. Let $A$ be a Noetherian ring and let $I \subset A$ be an ideal. Set $X = \mathop{\mathrm{Spec}}(A)$, $Z = V(I)$, $U = X \setminus Z$, and $j : U \to X$ the inclusion morphism. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ U$-module. Then

  1. there exists a finite $A$-module $M$ such that $\mathcal{F}$ is the restriction of $\widetilde{M}$ to $U$,

  2. given $M$ there is an exact sequence

    \[ 0 \to H^0_ Z(M) \to M \to H^0(U, \mathcal{F}) \to H^1_ Z(M) \to 0 \]

    and isomorphisms $H^ p(U, \mathcal{F}) = H^{p + 1}_ Z(M)$ for $p \geq 1$,

  3. given $M$ and $p \geq 0$ the following are equivalent

    1. $R^ pj_*\mathcal{F}$ is coherent,

    2. $H^ p(U, \mathcal{F})$ is a finite $A$-module,

    3. $H^{p + 1}_ Z(M)$ is a finite $A$-module,

  4. if the equivalent conditions in (3) hold for $p = 0$, we may take $M = \Gamma (U, \mathcal{F})$ in which case we have $H^0_ Z(M) = H^1_ Z(M) = 0$.

Proof. By Properties, Lemma 28.22.4 there exists a coherent $\mathcal{O}_ X$-module $\mathcal{F}'$ whose restriction to $U$ is isomorphic to $\mathcal{F}$. Say $\mathcal{F}'$ corresponds to the finite $A$-module $M$ as in (1). Note that $R^ pj_*\mathcal{F}$ is quasi-coherent (Cohomology of Schemes, Lemma 30.4.5) and corresponds to the $A$-module $H^ p(U, \mathcal{F})$. By Lemma 51.2.1 and the general facts in Cohomology, Section 20.21 we obtain an exact sequence

\[ 0 \to H^0_ Z(M) \to M \to H^0(U, \mathcal{F}) \to H^1_ Z(M) \to 0 \]

and isomorphisms $H^ p(U, \mathcal{F}) = H^{p + 1}_ Z(M)$ for $p \geq 1$. Here we use that $H^ j(X, \mathcal{F}') = 0$ for $j > 0$ as $X$ is affine and $\mathcal{F}'$ is quasi-coherent (Cohomology of Schemes, Lemma 30.2.2). This proves (2). Parts (3) and (4) are straightforward from (2); see also Lemma 51.2.2. $\square$


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