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The Stacks project

Lemma 15.83.4. Let R \to A be a flat ring map of finite presentation. Let K \in D(A). The following are equivalent

  1. K is R-perfect, and

  2. K is isomorphic to a finite complex of R-flat, finitely presented A-modules.

Proof. To prove (2) implies (1) it suffices by Lemma 15.83.2 to show that an R-flat, finitely presented A-module M defines an R-perfect object of D(A). Since M has finite tor dimension over R, it suffices to show that M is pseudo-coherent. By Algebra, Lemma 10.168.1 there exists a finite type \mathbf{Z}-algebra R_0 \subset R and a flat finite type ring map R_0 \to A_0 and a finite A_0-module M_0 flat over R_0 such that A = A_0 \otimes _{R_0} R and M = M_0 \otimes _{R_0} R. By Lemma 15.64.17 we see that M_0 is pseudo-coherent A_0-module. Choose a resolution P_0^\bullet \to M_0 by finite free A_0-modules P_0^ n. Since A_0 is flat over R_0, this is a flat resolution. Since M_0 is flat over R_0 we find that P^\bullet = P_0^\bullet \otimes _{R_0} R still resolves M = M_0 \otimes _{R_0} R. (You can use Lemma 15.61.2 to see this.) Hence P^\bullet is a finite free resolution of M over A and we conclude that M is pseudo-coherent.

Assume (1). We can represent K by a bounded above complex P^\bullet of finite free A-modules. Assume that K viewed as an object of D(R) has tor amplitude in [a, b]. By Lemma 15.66.2 we see that \tau _{\geq a}P^\bullet is a complex of R-flat, finitely presented A-modules representing K. \square


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