Lemma 15.77.6. Let A \to B be a local ring homomorphism. Let a, b \in \mathbf{Z}. Let d \geq 0. Let K^\bullet be a complex of B-modules. Assume
the ring map A \to B is flat,
the ring B/\mathfrak m_ AB is regular of dimension d,
K^\bullet is pseudo-coherent as a complex of B-modules, and
K^\bullet has tor amplitude in [a, b] as a complex of A-modules, in fact it suffices if H^ i(K^\bullet \otimes _ A^\mathbf {L} \kappa (\mathfrak m_ A)) is nonzero only for i \in [a, b].
Then K^\bullet is perfect as a complex of B-modules with tor amplitude in [a - d, b].
Proof.
By (3) we may assume that K^\bullet is a bounded above complex of finite free B-modules. We compute
\begin{align*} K^\bullet \otimes _ B^{\mathbf{L}} \kappa (\mathfrak m_ B) & = K^\bullet \otimes _ B \kappa (\mathfrak m_ B) \\ & = (K^\bullet \otimes _ A \kappa (\mathfrak m_ A)) \otimes _{B/\mathfrak m_ A B} \kappa (\mathfrak m_ B) \\ & = (K^\bullet \otimes _ A \kappa (\mathfrak m_ A)) \otimes ^{\mathbf{L}}_{B/\mathfrak m_ A B} \kappa (\mathfrak m_ B) \end{align*}
The first equality because K^\bullet is a bounded above complex of flat B-modules. The second equality follows from basic properties of the tensor product. The third equality holds because K^\bullet \otimes _ A \kappa (\mathfrak m_ A) = K^\bullet / \mathfrak m_ A K^\bullet is a bounded above complex of flat B/\mathfrak m_ A B-modules. Since K^\bullet is a bounded above complex of flat A-modules by (1), the cohomology modules H^ i of the complex K^\bullet \otimes _ A \kappa (\mathfrak m_ A) are nonzero only for i \in [a, b] by assumption (4). Thus the spectral sequence of Example 15.62.1 and the fact that B/\mathfrak m_ AB has finite global dimension d (by (2) and Algebra, Proposition 10.110.1) shows that H^ j(K^\bullet \otimes _ B^{\mathbf{L}} \kappa (\mathfrak m_ B)) is zero for j \not\in [a - d, b]. This finishes the proof by Lemma 15.77.2.
\square
Comments (0)