Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-26T06:29:58.540Z Has data issue: false hasContentIssue false
  • English
  • Français

Pure Injective and Absolutely Pure Sheaves

Part of: Algebraic geometry: Foundations Abelian categories Categories with structure Categories and theories Modules, bimodules and ideals Categories and geometry

Published online by Cambridge University Press:  20 November 2015

Edgar Enochs ,
Sergio Estrada  and
Sinem Odabaşi
Edgar Enochs
Affiliation:
Department of Mathematics, University of Kentucky, Lexington, KY 40506, USA (enochs@ms.uky.edu)
Sergio Estrada
Affiliation:
Departamento de Matemáticas, Universidad de Murcia, 30100 Murcia, Spain (sestrada@um.es; sinem.odabasi1@um.es)
Sinem Odabaşi
Affiliation:
Departamento de Matemáticas, Universidad de Murcia, 30100 Murcia, Spain (sestrada@um.es; sinem.odabasi1@um.es)
Get access Rights & Permissions [Opens in a new window]

Abstract

We study two notions of purity in categories of sheaves: the categorical and the geometric. It is shown that pure injective envelopes exist in both cases under very general assumptions on the scheme. Finally, we introduce the class of locally absolutely pure (quasi-coherent) sheaves with respect to the geometrical purity, and characterize locally Noetherian closed subschemes of a projective scheme in terms of the new class.

Keywords

local puritycategorical purityconcentrated schemelocally finitely presented categorypure injective sheafabsolutely pure sheaf

MSC classification

Secondary: 18E15: Grothendieck categories 18C35: Accessible and locally presentable categories 18D10: Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories 18F20: Presheaves and sheaves 14A15: Schemes and morphisms 16D90: Module categories ; module theory in a category-theoretic context; Morita equivalence and duality
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 59 , Issue 3 , August 2016 , pp. 623 - 640
Copyright
Copyright © Edinburgh Mathematical Society 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Cohn, P. M., On the free product of associative rings, Math. Z. 71 (1959), 380–398.Google Scholar
2. Conrad, B., Grothendieck duality and base change, Lecture Notes in Mathematics, Volume 1750 (Springer, 2000).Google Scholar
3. Crawley-Boevey, W., Locally finitely presented additive categories, Commun. Alg. 22 (1994), 1641–1674.Google Scholar
4. Crivei, S., Prest, M. and Torrecillas, B., Covers in finitely accessible categories, Proc. Am. Math. Soc. 138 (2010), 1213–1221.Google Scholar
5. El Bashir, R., Covers and directed colimits, Alg. Representat. Theory 9 (2006), 423–430.Google Scholar
6. Enochs, E. E. and Estrada, S., Relative homological algebra in the category of quasi-coherent sheaves, Adv. Math. 194 (2005), 284–295.Google Scholar
7. Enochs, E. and Jenda, O. M. G., Relative homological algebra I, 2nd edn, De Gruyter Expositions in Mathematics, Volume 30 (Walter De Gruyter, Berlin, 2011).Google Scholar
8. Estrada, S. and Saorín, M., Locally finitely presented categories with no flat objects, Forum Math. 27 (2015), 269–301.Google Scholar
9. Fieldhouse, D. J., Pure theories, Math. Annalen 184 (1969), 1–18.Google Scholar
10. García, J. L. and Martínez Hernández, J., Purity through Gabriel's functor rings, Bull. Soc. Math. Belg. A45(1–2) (1993), 137–152.Google Scholar
11. Garkusha, G., Classifying finite localizations of quasi-coherent sheaves, St Petersburg Math. J. 21(3) (2010), 433–458.Google Scholar
12. Grothendieck, A. and Dieudonné, J. A., Eléments de géométrie algébrique I, Grundlehren der Mathematischen Wissenschaften, Volume 166 (Springer, 1971).Google Scholar
13. Harris, M. E., Some results on coherent rings, Proc. Am. Math. Soc. 17 (1966), 474–479.Google Scholar
14. Hartshorne, R., Residues and duality, Lecture Notes in Mathematics, Volume 20 (Springer, 1966).CrossRefGoogle Scholar
15. Herzog, I., Pure-injective envelopes, J. Alg. Appl. 2(4) (2003), 397–402.Google Scholar
16. Lazard, D., Autour de la platitude, Bull. Soc. Math. France 97 (1969), 81–128.Google Scholar
17. Maddox, B., Absolutely pure modules, Proc. Am. Math. Soc. 18 (1967), 155–158.CrossRefGoogle Scholar
18. Megibben, C., Absolutely pure modules, Proc. Am. Math. Soc. 26 (1970), 561–566.CrossRefGoogle Scholar
19. Murfet, D., Modules over a scheme (available at http://therisingsea.org/notes/ModulesOverAScheme.pdf; 2006).Google Scholar
20. Pinzon, K. R., Absolutely pure modules, PhD Thesis, University of Kentucky (available at http://uknowledge.uky.edu/gradschool_diss/379/; 2005).Google Scholar
21. Prest, M., Purity, spectra and localisation, Encyclopedia of Mathematics and Its Applications, Volume 121 (Cambridge University Press, 2009).Google Scholar
22. Prest, M. and Ralph, A., Locally finitely presented categories of sheaves of modules (available at http://www.maths.manchester.ac.uk/~mprest/publications.html).Google Scholar
23. Prüfer, H., Untersuchungen über die Zerlegbarkeit der abzählbaren primären abelschen Gruppen, Math. Z. 17 (1923), 35–61.Google Scholar
24. Stenström, B., Pure submodules, Ark. Mat. 7 (1967), 159–171.Google Scholar
25. Stenström, B., Purity in functor categories, J. Alg. 8 (1968), 352–361.Google Scholar
26. Stenström, B., Coherent rings and FP-injective modules, J. Lond. Math. Soc. 2 (1970), 323–329.Google Scholar
27. Stenström, B., Rings of quotients, Die Grundlehren der mathematischen Wissenschaften, Volume 217 (Springer, 1975).Google Scholar
28. Swan, R. S., Cup products in sheaf cohomology, pure injectives, and a substitute for projective resolutions, J. Pure Appl. Alg. 144(2) (1999), 169–211.Google Scholar
29. Xu, J., Flat covers of modules, Lecture Notes in Mathematics, Volume 1634 (Springer, 1996).Google Scholar