The Stacks project

Lemma 15.66.15. Let $A \to B$ be a ring map. Let $K^\bullet $ be a complex of $B$-modules. Let $a, b \in \mathbf{Z}$. The following are equivalent

  1. $K^\bullet $ has tor amplitude in $[a, b]$ as a complex of $A$-modules,

  2. $K^\bullet _\mathfrak q$ has tor amplitude in $[a, b]$ as a complex of $A_\mathfrak p$-modules for every prime $\mathfrak q \subset B$ with $\mathfrak p = A \cap \mathfrak q$,

  3. $K^\bullet _\mathfrak m$ has tor amplitude in $[a, b]$ as a complex of $A_\mathfrak p$-modules for every maximal ideal $\mathfrak m \subset B$ with $\mathfrak p = A \cap \mathfrak m$.

Proof. Assume (3) and let $M$ be an $A$-module. Then $H^ i = H^ i(K^\bullet \otimes _ A^\mathbf {L} M)$ is a $B$-module and $(H^ i)_\mathfrak m = H^ i(K^\bullet _\mathfrak m \otimes _{A_\mathfrak p}^\mathbf {L} M_\mathfrak p)$. Hence $H^ i = 0$ for $i \not\in [a, b]$ by Algebra, Lemma 10.23.1. Thus (3) $\Rightarrow $ (1). We omit the proofs of (1) $\Rightarrow $ (2) and (2) $\Rightarrow $ (3). $\square$

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