K-injective complexes in commutative algebra

Abstract

It is an unfortunate fact that even over commutative noetherian rings, localization of K-injective complexes need not be K-injective. However, not all is lost. Given a complex of injective modules J over a commutative noetherian ring A, we show that K-injectivity of J can be detected locally: if (A,m) is a noetherian local ring, then J is K-injective over A if and only if Hom_A(Â,J) is K-injective over Â, the m-adic completion of A; and for an arbitrary commutative noetherian ring A, J is K-injective over A if and only if Hom_A(A_m,J) is K-injective over A_m for every maximal ideal m in the singular locus of A. The local result is deduced from a more general theorem on faithfully flat descent of K-injectivity along maps to (possibly noncommutative, possibly non-noetherian) rings. The global result is established via the local-to-global principle in tensor-triangulated geometry. As an application, we generalize a theorem of Grothendieck on faithfully flat descent of regularity, extending it to faithfully flat extensions by noncommutative left coherent left regular rings.

Finitistic injective dimension exceeding finitistic projective dimension for a commutative ring
Liang Chen
arXiv16 Sep 2026PreprintarXiv licenceFull textAlgebra & number theoryarXiv
A simple C^*-algebra which is not K₁-injective
Andrew S. Toms
arXiv8 Sep 2026PreprintarXiv licenceFull textAnalysis & PDEsarXiv
New companions to the Andrews–Gordon identities motivated by commutative algebra
Pooneh Afsharijoo, Jehanne Dousse, Frédéric Jouhet +1
Advances in Mathematics9 Oct 2026Peer-reviewedLicence not stated8 citedAlgebra & number theoryHAL
Complexes of pattern-avoiding injective words
Sergi Elizalde, Philip Hanlon, Patricia Hersh
arXiv5 Oct 2026PreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv
Structure and Representations of a Diagrammatic Non-Commutative Tangloid Algebra
Hadeel B. Albeladi, Sofia Lambropoulou
arXiv29 Sep 2026PreprintCC BY 4.0Full textAlgebra & number theory Geometry & topologyarXiv

Cite

BibTeX RIS

BibTeX

@misc{shaul2026kinjective,
  title = {{K-injective complexes in commutative algebra}},
  author = {Liran Shaul},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.05884},
  eprint = {2610.05884},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.05884},
}

RIS

TY  - UNPB
TI  - K-injective complexes in commutative algebra
AU  - Liran Shaul
PY  - 2026
DA  - 2026/10/05
PB  - arXiv
DO  - 10.48550/arxiv.2610.05884
UR  - https://arxiv.org/abs/2610.05884
AB  - It is an unfortunate fact that even over commutative noetherian rings, localization of K-injective complexes need not be K-injective. However, not all is lost. Given a complex of injective modules J over a commutative noetherian ring A, we show that K-injectivity of J can be detected locally: if (A,m) is a noetherian local ring, then J is K-injective over A if and only if Hom_A(Â,J) is K-injective over Â, the m-adic completion of A; and for an arbitrary commutative noetherian ring A, J is K-injective over A if and only if Hom_A(A_m,J) is K-injective over A_m for every maximal ideal m in the singular locus of A. The local result is deduced from a more general theorem on faithfully flat descent of K-injectivity along maps to (possibly noncommutative, possibly non-noetherian) rings. The global result is established via the local-to-global principle in tensor-triangulated geometry. As an application, we generalize a theorem of Grothendieck on faithfully flat descent of regularity, extending it to faithfully flat extensions by noncommutative left coherent left regular rings.
ER  -

Abstract

Related

Cite