Adic finiteness: Bounding homology and applications

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 RΓ_𝔟(R) has finite flat dimension when viewed through some positive power of the Frobenius endomorphism.

KEYWORDS:

• Adic finiteness
• amplitude inequality
• co-support
• flat dimension
• Frobenius endomorphism
• injective dimension
• projective dimension
• support

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 13C12
• 13D05
• 13D07
• 13D09

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 supp_R(X) = V(𝔞)≠0, so we have X≄0, and thus sup(X),inf(X)∈ℤ.

³Note that complexes in [8] are indexed cohomologically, so one has to translate [8, Proposition 2.2] carefully.

⁴The following alternate proof of part (c) is worth noting. By Hom-evaluation [2, Lemma 4.4(I)], the assumptions on P provide an isomorphism $R{Hom}_R(P,Y)\simeq P^{\ast }{\otimes }_R^{L}Y$ where $P^{\ast }=R{Hom}_R(P,R)$ satisfies ${supp}_R\left(P^{\ast }\right)={supp}_R\left(P\right)$, as in the proof of Corollary 4.3 below. Now apply Fact 2.2.