Purity in categories of sheaves


We consider categorical and geometric purity for sheaves of modules over a scheme satisfying some mild conditions, both for the category of all sheaves and for the category of quasicoherent sheaves. We investigate the relations between these four purities; for example, we give a characterisation of geometric pure-injectives in both the quasicoherent and non-quasicoherent case. We also compute a number of examples, in particular describing both the geometric and categorical Ziegler spectra for the category of quasicoherent sheaves over the projective line over a field.

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    Let us point out here that even though “there is no non-trivial covering of any open set”, the sheaf axiom in general has the extra consequence that sections over the empty set are the final object of the category. Therefore, e.g. sheaves of abelian groups over this two-point space form a proper subcategory of presheaves, which need not assign the zero group to the empty set (!). However, since we always assume \({\mathcal {O}}_X\) to be a sheaf of rings, its ring of sections over the empty set is the zero ring, over which any module is trivial.


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Correspondence to Alexander Slávik.

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Alexander Slávik’s research was supported from the Grant GA ČR 17-23112S of the Czech Science Foundation, from the Grant SVV-2017-260456 of the SVV project and from the grant UNCE/SCI/022 of the Charles University Research Centre.

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Prest, M., Slávik, A. Purity in categories of sheaves. Math. Z. 297, 429–451 (2021). https://doi.org/10.1007/s00209-020-02517-5

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  • Scheme
  • Sheaf
  • Pure-exact sequence
  • Ziegler spectrum

Mathematics Subject Classification

  • 14A15
  • 18E15
  • 18F20
  • 03C60
  • 16G20