The Stacks project

Lemma 51.2.2. Let $A$ be a ring and let $I \subset A$ be a finitely generated 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 quasi-coherent $\mathcal{O}_ U$-module. Then

  1. there exists an $A$-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. we may take $M = H^0(U, \mathcal{F})$ in which case we have $H^0_ Z(M) = H^1_ Z(M) = 0$.

Proof. The existence of $M$ follows from Properties, Lemma 28.22.1 and the fact that quasi-coherent sheaves on $X$ correspond to $A$-modules (Schemes, Lemma 26.7.5). Then we look at the distinguished triangle

\[ R\Gamma _ Z(X, \widetilde{M}) \to R\Gamma (X, \widetilde{M}) \to R\Gamma (U, \widetilde{M}|_ U) \to R\Gamma _ Z(X, \widetilde{M})[1] \]

of Cohomology, Section 20.21. Since $X$ is affine we have $R\Gamma (X, \widetilde{M}) = M$ by Cohomology of Schemes, Lemma 30.2.2. By our choice of $M$ we have $\mathcal{F} = \widetilde{M}|_ U$ and hence this produces an exact sequence

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

and isomorphisms $H^ p(U, \mathcal{F}) = H^{p + 1}_ Z(X, \widetilde{M})$ for $p \geq 1$. By Lemma 51.2.1 we have $H^ i_ Z(M) = H^ i_ Z(X, \widetilde{M})$ for all $i$. Thus (1) and (2) do hold. Finally, setting $M' = H^0(U, \mathcal{F})$ we see that the kernel and cokernel of $M \to M'$ are $I$-power torsion. Therefore $\widetilde{M}|_ U \to \widetilde{M'}|_ U$ is an isomorphism and we can indeed use $M'$ as predicted in (3). It goes without saying that we obtain zero for both $H^0_ Z(M')$ and $H^0_ Z(M')$. $\square$


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