The Stacks project

Lemma 74.42.1. Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$ and let $\mathcal{I} \subset \mathcal{O}_ X$ be a quasi-coherent sheaf of ideals.

  1. The category $\textit{Coh}(X, \mathcal{I})$ is abelian.

  2. Exactness in $\textit{Coh}(X, \mathcal{I})$ can be checked étale locally.

  3. For any flat morphism $f : X' \to X$ of Noetherian algebraic spaces the functor $f^* : \textit{Coh}(X, \mathcal{I}) \to \textit{Coh}(X', f^{-1}\mathcal{I}\mathcal{O}_{X'})$ is exact.

Proof. Proof of (1). Choose an affine scheme $U_0$ and a surjective étale morphism $U_0 \to X$. Set $U_1 = U_0 \times _ X U_0$ and $U_2 = U_0 \times _ X U_0 \times _ X U_0$ as in our discussion of étale descent above. The categories $\textit{Coh}(U_ i, \mathcal{I}_ i)$ are abelian (Cohomology of Schemes, Lemma 30.23.2) and the pullback functors are exact functors $\textit{Coh}(U_0, \mathcal{I}_0) \to \textit{Coh}(U_1, \mathcal{I}_1)$ and $\textit{Coh}(U_1, \mathcal{I}_1) \to \textit{Coh}(U_2, \mathcal{I}_2)$ (Cohomology of Schemes, Lemma 30.23.9). The lemma then follows formally from the description of $\textit{Coh}(X, \mathcal{I})$ as a category of descent data. Some details omitted; compare with the proof of Groupoids, Lemma 39.14.6.

Part (2) follows immediately from the discussion in the previous paragraph. In the situation of (3) choose a commutative diagram

\[ \xymatrix{ U' \ar[d] \ar[r] & U \ar[d] \\ X' \ar[r] & X } \]

where $U'$ and $U$ are affine schemes and the vertical morphisms are surjective étale. Then $U' \to U$ is a flat morphism of Noetherian schemes (Morphisms of Spaces, Lemma 65.30.5) whence the pullback functor $\textit{Coh}(U, \mathcal{I}\mathcal{O}_ U) \to \textit{Coh}(U', \mathcal{I}\mathcal{O}_{U'})$ is exact by Cohomology of Schemes, Lemma 30.23.9. Since we can check exactness in $\textit{Coh}(X, \mathcal{O}_ X)$ on $U$ and similarly for $X', U'$ the assertion follows. $\square$

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