The Stacks project

Lemma 37.55.12. Let $i : Z \to X$ be a perfect closed immersion of schemes. Then $i_*\mathcal{O}_ X$ is a perfect $\mathcal{O}_ X$-module, i.e., it is a perfect object of $D(\mathcal{O}_ X)$.

Proof. This is more or less immediate from the definition. Namely, let $U = \mathop{\mathrm{Spec}}(A)$ be an affine open of $X$. Then $i^{-1}(U) = \mathop{\mathrm{Spec}}(A/I)$ for some ideal $I \subset A$ and $A/I$ has a finite resolution by finite projective $A$-modules by More on Algebra, Lemma 15.81.2. Hence $i_*\mathcal{O}_ Z|_ U$ can be represented by a finite length complex of finite locally free $\mathcal{O}_ U$-modules. This is what we had to show, see Cohomology, Section 20.46. $\square$


Comments (1)

Comment #6652 by Brian Shin on

Very small typo: should be


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