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PreprintCC0arXiv

On the existence of Gorenstein injective covers

  • Yuhui Hua
  • Shun-Jie Li
  • Chencheng Zhang
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Abstract

For a locally finitely generated Grothendieck category, we characterise the Gorenstein injective covering property by several equivalent conditions, including closure under coproducts. To complete this characterisation, we construct a countable commutative Noetherian Cohen--Macaulay local ring of dimension one for which this property fails, answering negatively Iacob's question Question 1 whether character modules of Gorenstein injective modules over left Noetherian rings are Gorenstein flat. Using this characterisation, we prove that for Noetherian schemes the covering property is Zariski-local. Moreover, assuming that 𝒒ℐ(QCoh(X)) is a covering class, we establish compatible adjunctions and recollements for homotopy categories of injectives and totally acyclic injective complexes under open restriction. However, this property is not stalk-local: we construct a commutative Noetherian ring S such that 𝒒ℐ(Sp) forms a covering class for all prime ideals p, but 𝒒ℐ(S) do not form a covering class.

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Cite

BibTeXRIS
BibTeX
@misc{hua2026existence,
  title = {{On the existence of Gorenstein injective covers}},
  author = {Yuhui Hua and Shun-Jie Li and Chencheng Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.18620},
  eprint = {2610.18620},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.18620},
  note = {CC0},
}
RIS
TY  - UNPB
TI  - On the existence of Gorenstein injective covers
AU  - Yuhui Hua
AU  - Shun-Jie Li
AU  - Chencheng Zhang
PY  - 2026
DA  - 2026/10/13
PB  - arXiv
DO  - 10.48550/arxiv.2610.18620
UR  - https://arxiv.org/abs/2610.18620
AB  - For a locally finitely generated Grothendieck category, we characterise the Gorenstein injective covering property by several equivalent conditions, including closure under coproducts. To complete this characterisation, we construct a countable commutative Noetherian Cohen--Macaulay local ring of dimension one for which this property fails, answering negatively Iacob's question Question 1 whether character modules of Gorenstein injective modules over left Noetherian rings are Gorenstein flat. Using this characterisation, we prove that for Noetherian schemes the covering property is Zariski-local. Moreover, assuming that 𝒒ℐ(QCoh(X)) is a covering class, we establish compatible adjunctions and recollements for homotopy categories of injectives and totally acyclic injective complexes under open restriction. However, this property is not stalk-local: we construct a commutative Noetherian ring S such that 𝒒ℐ(S_p) forms a covering class for all prime ideals p, but 𝒒ℐ(S) do not form a covering class.
ER  -