The Stacks project

Lemma 36.35.3. Let $f : X \to S$ be a morphism of schemes which is flat and locally of finite presentation. Let $E$ be an object of $D_\mathit{QCoh}(\mathcal{O}_ X)$. The following are equivalent

  1. $E$ is $S$-perfect,

  2. for any affine open $U \subset X$ mapping into an affine open $V \subset S$ the complex $R\Gamma (U, E)$ is $\mathcal{O}_ S(V)$-perfect.

  3. there exists an affine open covering $S = \bigcup V_ i$ and for each $i$ an affine open covering $f^{-1}(V_ i) = \bigcup U_{ij}$ such that the complex $R\Gamma (U_{ij}, E)$ is $\mathcal{O}_ S(V_ i)$-perfect.

Proof. Being pseudo-coherent is a local property and “locally having finite tor dimension” is a local property. Hence this lemma immediately reduces to the statement: if $X$ and $S$ are affine, then $E$ is $S$-perfect if and only if $K = R\Gamma (X, E)$ is $\mathcal{O}_ S(S)$-perfect. Say $X = \mathop{\mathrm{Spec}}(A)$, $S = \mathop{\mathrm{Spec}}(R)$ and $E$ corresponds to $K \in D(A)$, i.e., $K = R\Gamma (X, E)$, see Lemma 36.3.5.

Observe that $K$ is $R$-perfect if and only if $K$ is pseudo-coherent and has finite tor dimension as a complex of $R$-modules (More on Algebra, Definition 15.83.1). By Lemma 36.10.2 we see that $E$ is pseudo-coherent if and only if $K$ is pseudo-coherent. By Lemma 36.10.5 we see that $E$ has finite tor dimension over $f^{-1}\mathcal{O}_ S$ if and only if $K$ has finite tor dimension as a complex of $R$-modules. $\square$

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