Perfect closure detects injective dimension
Abstract
Let R be a complete noetherian local ring of prime characteristic p, and let R∞ denote its perfect closure. We prove that a finitely generated R-module N has finite injective dimension if and only if ExtRi(R∞, N) = 0 for all i > 0. As a Gorenstein counterpart, we show that finiteness of the Gorenstein injective (or projective) dimension of R∞ forces R to be Gorenstein, and we relate this to a non noetherian Cohen factorization of R → R∞. Applications include preservation of Gorensteinness under weakly etale extensions, structural results on F-coherent and weakly F-nilpotent rings, along with the ascent of Frobenius closure of a parameter ideal to its powers. Assuming ExtiR(R/m,R∞)=0 for some i>dim(R), we show R is regular. This has some applications. We study the rationality problem of Hilbert--Kunz multiplicity by linking it to R∞. Recall that the global dimension can be viewed as a uniform bound on the injective dimension of modules. Finally, we determine gldim(R∞) by presenting a new bound.
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BibTeX
@misc{asgharzadeh2026perfect,
title = {{Perfect closure detects injective dimension}},
author = {Mohsen Asgharzadeh},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.30416},
eprint = {2606.30416},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.30416},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Perfect closure detects injective dimension
AU - Mohsen Asgharzadeh
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2606.30416
UR - https://arxiv.org/abs/2606.30416
AB - Let R be a complete noetherian local ring of prime characteristic p, and let R^∞ denote its perfect closure. We prove that a finitely generated R-module N has finite injective dimension if and only if Ext_R^i(R^∞, N) = 0 for all i > 0. As a Gorenstein counterpart, we show that finiteness of the Gorenstein injective (or projective) dimension of R^∞ forces R to be Gorenstein, and we relate this to a non noetherian Cohen factorization of R → R^∞. Applications include preservation of Gorensteinness under weakly etale extensions, structural results on F-coherent and weakly F-nilpotent rings, along with the ascent of Frobenius closure of a parameter ideal to its powers. Assuming Ext^i_R(R/m,R^∞)=0 for some i>dim(R), we show R is regular. This has some applications. We study the rationality problem of Hilbert--Kunz multiplicity by linking it to R^∞. Recall that the global dimension can be viewed as a uniform bound on the injective dimension of modules. Finally, we determine gldim(R^∞) by presenting a new bound.
ER -