Lemma 37.71.2. Consider a cartesian diagram
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.
Lemma 37.71.2. Consider a cartesian diagram
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 37.71.1 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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