The contraherent version of the theorem of Slavik and Stovicek

Abstract

This is a paper about contraherent cosheaves on non-semi-separated schemes. We prove two theorems, a negative one and a positive one. On the negative side, let X be a quasi-compact, quasi-separated scheme that is not semi-separated. We present a locally cotorsion contraherent cosheaf on X that does not have an admissible monomorphism into any locally injective locally contraherent cosheaf. The construction and proof follow the arguments of Slavik and Stovicek in arXiv:1902.05740. On the positive side, let X be a Noetherian scheme of finite Krull dimension. We prove that every W-locally contraherent cosheaf on X has an admissible monomorphism into a locally cotorsion W-locally contraherent cosheaf. Moreover, the cokernel is a flat contraherent cosheaf. The proof is based on the theorem of Raynaud-Gruson about the projective dimensions of flat modules and Enochs' classification of flat cotorsion modules.

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BibTeX

@misc{positselski2026contraherent,
  title = {{The contraherent version of the theorem of Slavik and Stovicek}},
  author = {Leonid Positselski},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.08491},
  eprint = {2609.08491},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.08491},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - The contraherent version of the theorem of Slavik and Stovicek
AU  - Leonid Positselski
PY  - 2026
DA  - 2026/09/16
PB  - arXiv
DO  - 10.48550/arxiv.2609.08491
UR  - https://arxiv.org/abs/2609.08491
AB  - This is a paper about contraherent cosheaves on non-semi-separated schemes. We prove two theorems, a negative one and a positive one. On the negative side, let X be a quasi-compact, quasi-separated scheme that is not semi-separated. We present a locally cotorsion contraherent cosheaf on X that does not have an admissible monomorphism into any locally injective locally contraherent cosheaf. The construction and proof follow the arguments of Slavik and Stovicek in arXiv:1902.05740. On the positive side, let X be a Noetherian scheme of finite Krull dimension. We prove that every W-locally contraherent cosheaf on X has an admissible monomorphism into a locally cotorsion W-locally contraherent cosheaf. Moreover, the cokernel is a flat contraherent cosheaf. The proof is based on the theorem of Raynaud-Gruson about the projective dimensions of flat modules and Enochs' classification of flat cotorsion modules.
ER  -

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