The Stacks project

Lemma 37.68.2. Consider a cartesian diagram

\[ \xymatrix{ X \ar[r]_ i \ar[d]_ f & X' \ar[d]^{f'} \\ Y \ar[r]^ j & Y' } \]

of schemes. Assume $X' \to Y'$ is flat and locally of finite presentation and $Y \to Y'$ is a finite order thickening. Let $E' \in D(\mathcal{O}_{X'})$. If $E = Li^*(E')$ is $Y$-perfect, then $E'$ is $Y'$-perfect.

Proof. Recall that being $Y$-perfect for $E$ means $E$ is pseudo-coherent and locally has finite tor dimension as a complex of $f^{-1}\mathcal{O}_ Y$-modules (Derived Categories of Schemes, Definition 36.35.1). By Lemma 37.68.1 we find that $E'$ is pseudo-coherent. In particular, $E'$ is in $D_\mathit{QCoh}(\mathcal{O}_{X'})$, see Derived Categories of Schemes, Lemma 36.10.1. To prove that $E'$ locally has finite tor dimension we may work locally on $X'$. Hence we may assume $X'$, $S'$, $X$, $S$ are affine, say given by rings $A'$, $R'$, $A$, $R$. Then we reduce to the commutative algebra version by Derived Categories of Schemes, Lemma 36.35.3. The commutative algebra version in More on Algebra, Lemma 15.83.8. $\square$

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