## 76.47 Perfect morphisms

This section is the analogue of More on Morphisms, Section 37.61 for morphisms of schemes. The reader is encouraged to read up on perfect morphisms of schemes in that section first.

The property “perfect” of morphisms of schemes is étale local on the source-and-target. To see this use More on Morphisms, Lemmas 37.61.10 and 37.61.14 and Descent, Lemma 35.32.6. By Morphisms of Spaces, Lemma 67.22.1 we may define the notion of a perfect morphism of algebraic spaces as follows and it agrees with the already existing notion defined in More on Morphisms, Section 37.61 when the algebraic spaces in question are representable.

Definition 76.47.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$.

1. We say $f$ is perfect if the equivalent conditions of Morphisms of Spaces, Lemma 67.22.1 hold with $\mathcal{P} =$“perfect”.

2. Let $x \in |X|$. We say $f$ is perfect at $x$ if there exists an open neighbourhood $X' \subset X$ of $x$ such that $f|_{X'} : X' \to Y$ is perfect.

Note that a perfect morphism is pseudo-coherent, hence locally of finite presentation. Beware that a base change of a perfect morphism is not perfect in general.

Proof. Omitted. Hint: Use the schemes version of this lemma, see More on Morphisms, Lemma 37.61.3. $\square$

Proof. Omitted. Hint: Use the schemes version of this lemma, see More on Morphisms, Lemma 37.61.4. $\square$

Lemma 76.47.4. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent

1. $f$ is flat and perfect, and

2. $f$ is flat and locally of finite presentation.

Proof. Omitted. Hint: Use the schemes version of this lemma, see More on Morphisms, Lemma 37.61.5. $\square$

Lemma 76.47.5. Let $S$ be a scheme. Let $Y$ be a Noetherian algebraic space over $S$. Let $f : X \to Y$ be a perfect proper morphism of algebraic spaces. Let $E \in D(\mathcal{O}_ X)$ be perfect. Then $Rf_*E$ is a perfect object of $D(\mathcal{O}_ Y)$.

Proof. We claim that Derived Categories of Spaces, Lemma 75.22.1 applies. Conditions (1) and (2) are immediate. Condition (3) is local on $X$. Thus we may assume $X$ and $Y$ affine and $E$ represented by a strictly perfect complex of $\mathcal{O}_ X$-modules. Thus it suffices to show that $\mathcal{O}_ X$ has finite tor dimension as a sheaf of $f^{-1}\mathcal{O}_ Y$-modules on the étale site. By Derived Categories of Spaces, Lemma 75.13.4 it suffices to check this on the Zariski site. This is equivalent to being perfect for finite type morphisms of schemes by More on Morphisms, Lemma 37.61.11. $\square$

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