The Stacks project

Lemma 36.10.5. Let $f : X \to S$ be a morphism of affine schemes corresponding to the ring map $R \to A$. Let $M^\bullet $ be a complex of $A$-modules and let $E$ be the corresponding object of $D(\mathcal{O}_ X)$. Then

  1. $E$ as an object of $D(f^{-1}\mathcal{O}_ S)$ has tor amplitude in $[a, b]$ if and only if $M^\bullet $ has tor amplitude in $[a, b]$ as an object of $D(R)$.

  2. $E$ locally has finite tor dimension as an object of $D(f^{-1}\mathcal{O}_ S)$ if and only if $M^\bullet $ has finite tor dimension as an object of $D(R)$.

Proof. Consider a prime $\mathfrak q \subset A$ lying over $\mathfrak p \subset R$. Let $x \in X$ and $s = f(x) \in S$ be the corresponding points. Then $(f^{-1}\mathcal{O}_ S)_ x = \mathcal{O}_{S, s} = R_\mathfrak p$ and $E_ x = M^\bullet _\mathfrak q$. Keeping this in mind we can see the equivalence as follows.

If $M^\bullet $ has tor amplitude in $[a, b]$ as a complex of $R$-modules, then the same is true for the localization of $M^\bullet $ at any prime of $A$. Then we conclude by Cohomology, Lemma 20.46.5 that $E$ has tor amplitude in $[a, b]$ as a complex of sheaves of $f^{-1}\mathcal{O}_ S$-modules. Conversely, assume that $E$ has tor amplitude in $[a, b]$ as an object of $D(f^{-1}\mathcal{O}_ S)$. We conclude (using the last cited lemma) that $M^\bullet _\mathfrak q$ has tor amplitude in $[a, b]$ as a complex of $R_\mathfrak p$-modules for every prime $\mathfrak q \subset A$ lying over $\mathfrak p \subset R$. By More on Algebra, Lemma 15.66.15 we find that $M^\bullet $ has tor amplitude in $[a, b]$ as a complex of $R$-modules. This finishes the proof of (1).

Since $X$ is quasi-compact, if $E$ locally has finite tor dimension as a complex of $f^{-1}\mathcal{O}_ S$-modules, then actually $E$ has tor amplitude in $[a, b]$ for some $a, b$ as a complex of $f^{-1}\mathcal{O}_ S$-modules. Thus (2) follows from (1). $\square$


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