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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 20.48.5 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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