ABSTRACT
We prove the versions of amplitude inequalities of Iversen, Foxby and Iyengar, and Frankild and Sather-Wagstaff that replace finite generation conditions with adic finiteness conditions. As an application, we prove that a local ring R of prime characteristic is regular if and only if for some proper ideal 𝔟 the derived local cohomology complex RΓ𝔟(R) has finite flat dimension when viewed through some positive power of the Frobenius endomorphism.
Acknowledgements
We are grateful to Srikanth Iyengar, Liran Shaul, Amnon Yekutieli, and the anonymous referee for helpful comments about this work. Sean Sather–Wagstaff was supported in part by a grant from the NSA.
Notes
1In the literature, semi-flat complexes are sometimes called “K-flat” or “DG-flat.”
2The conditions Y∈𝒟−(R) and Y≄0 imply that sup(Y)∈ℤ. Note that we do not need to make a similar assumption for X, since the condition 𝔞≠R implies that suppR(X) = V(𝔞)≠0, so we have X≄0, and thus sup(X),inf(X)∈ℤ.
3Note that complexes in [Citation8] are indexed cohomologically, so one has to translate [Citation8, Proposition 2.2] carefully.
4The following alternate proof of part (c) is worth noting. By Hom-evaluation [Citation2, Lemma 4.4(I)], the assumptions on P provide an isomorphism where satisfies , as in the proof of Corollary 4.3 below. Now apply Fact 2.2.