Abstract
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.
Similar content being viewed by others
Notes
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.
References
Beĭlinson, A.A.: Coherent sheaves on \(\mathbb{P}^n\) and problems in linear algebra. Funktsional. Anal. i Prilozhen. 12(3), 68–69 (1978)
Chen, X., Krause, H.: Introduction to coherent sheaves on weighted projective lines. arXiv:0911.4473
Cohn, P.M.: On the free product of associative rings. Math. Zeit. 71(1), 380–398 (1959)
Čoupek, P., Šťovíček, J.: Cotilting sheaves on Noetherian schemes. Math. Zeit. (2019). https://doi.org/10.1007/s00209-019-02404-8
Enochs, E., Estrada, S., Odabaşı, S.: Pure injective and absolutely pure sheaves. Proc. Edinburgh Math. Soc. 59(3), 623–640 (2016). https://doi.org/10.1017/S0013091515000462
Enochs, E., Gillespie, J., Odabasi, S.: Pure exact structures and the pure derived category of a scheme. Math. Proc. Camb. Philos. Soc. 163(2), 251–264 (2017). https://doi.org/10.1017/S0305004116000980
Garkusha, G.: Classifying finite localizations of quasi-coherent sheaves. St Petersburg Math. J. 21(3), 433–458 (2010)
Hartshorne, R.: Algebraic Geometry. Graduate Texts in Mathematics, vol. 52. Springer, New York (1997)
Iversen, B.: Cohomology of Sheaves. Springer, New York (1986)
Jensen, C.U., Lenzing, H.: Model Theoretic Algebra; with Particular Emphasis on Fields. Gordon and Breach, Rings and Modules, Philadelphia (1989)
Murfet, D.: Modules over a scheme. http://therisingsea.org/notes/ModulesOverAScheme.pdf (2006)
Murfet, D.: Concentrated schemes. http://therisingsea.org/notes/ConcentratedSchemes.pdf (2006)
Prest, M.: Model Theory and Modules. London Math. Soc. Lect. Note Ser., vol. 130. Cambridge University Press, Cambridge (1988)
Prest, M.: Purity, Spectra and Localisation. Encyclopedia of Mathematics and its Applications, vol. 121. Cambridge University Press, Cambridge (2009)
Prest, M.: Definable additive categories: purity and model theory. Mem. Am. Math. Soc. 210, 987 (2011)
Prest, M.: Abelian categories and definable additive categories. arXiv:1202.0426v1 (2012)
Prest, M.: Multisorted modules and their model theory. pp. 115–151 in Model Theory of Modules, Algebras and Categories, Contemporary Mathematics, vol. 730 Amer. Math.Soc. https://doi.org/10.1090/conm/730 (2019)
Prest, M., Ralph, A.: Locally Finitely Presented Categories of Sheaves of Modules. Preprint. https://personalpages.manchester.ac.uk/staff/mike.prest/publications.html (2010). Accessed 26 Mar 2020
Prüfer, H.: Untersuchungen über die Zerlegbarkeit der abzählbaren primären abelschen Gruppen. Math. Zeit. 17(1), 35–61 (1923)
Reynders, G.: Ziegler Spectra over Serial Rings and Coalgebras. Ph.D. thesis. https://personalpages.manchester.ac.uk/staff/mike.prest/publications.html (1998). Accessed 26 Mar 2020
Thomason, T.W., Trobaugh, T.: Higher algebraic K-theory of schemes and of derived categories. The Grothendieck Festschrift, Vol. III. Progr. Math. 88, 247–435 (1990)
The Stacks Project Authors. Stacks project. http://stacks.math.columbia.edu. Accessed 26 Mar 2020
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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.
Rights and permissions
About this article
Cite this article
Prest, M., Slávik, A. Purity in categories of sheaves. Math. Z. 297, 429–451 (2021). https://doi.org/10.1007/s00209-020-02517-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-020-02517-5