Flat covers and cotorsion envelopes of sheaves

Enochs, Edgar; Oyonarte, Luis

doi:10.1090/S0002-9939-01-06190-1

Flat covers and cotorsion envelopes of sheaves
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by Edgar Enochs and Luis Oyonarte

Proc. Amer. Math. Soc. 130 (2002), 1285-1292

DOI: https://doi.org/10.1090/S0002-9939-01-06190-1

Published electronically: October 24, 2001

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Abstract:

In this paper we prove that any sheaf of modules over any topological space (in fact, any $\mathcal {O}$-module where $\mathcal {O}$ is a sheaf of rings on the topological space) has a flat cover and a cotorsion envelope. This result is very useful, as we shall explain later in the introduction, in order to compute cohomology, due to the fact that the category of sheaves ($\mathcal {O}$-modules) does not have in general enough projectives.

References

S.T. Aldrich, E. Enochs, J.R. García Rozas and L. Oyonarte, Covers and envelopes in Grothendieck categories. Flat covers of complexes with applications, Submitted.
L. Bican, R. El Bashir and E. Enochs, All modules have flat covers, To appear in Bull. London Math. Soc.
Paul C. Eklof, Homological algebra and set theory, Trans. Amer. Math. Soc. 227 (1977), 207–225. MR 453520, DOI 10.1090/S0002-9947-1977-0453520-X
P. Eklof and J. Trlifaj, How to make Ext vanish, Bull. London Math. Soc. 33 (2001), no. 1, 44–51.
Edgar E. Enochs, Injective and flat covers, envelopes and resolvents, Israel J. Math. 39 (1981), no. 3, 189–209. MR 636889, DOI 10.1007/BF02760849
Rüdiger Göbel and Saharon Shelah, Cotorsion theories and splitters, Trans. Amer. Math. Soc. 352 (2000), no. 11, 5357–5379. MR 1661246, DOI 10.1090/S0002-9947-00-02475-2
Mark Hovey, Model categories, Mathematical Surveys and Monographs, vol. 63, American Mathematical Society, Providence, RI, 1999. MR 1650134
Birger Iversen, Cohomology of sheaves, Universitext, Springer-Verlag, Berlin, 1986. MR 842190, DOI 10.1007/978-3-642-82783-9
Christian U. Jensen and Helmut Lenzing, Model-theoretic algebra with particular emphasis on fields, rings, modules, Algebra, Logic and Applications, vol. 2, Gordon and Breach Science Publishers, New York, 1989. MR 1057608
Daniel G. Quillen, Homotopical algebra, Lecture Notes in Mathematics, No. 43, Springer-Verlag, Berlin-New York, 1967. MR 0223432, DOI 10.1007/BFb0097438
Luigi Salce, Cotorsion theories for abelian groups, Symposia Mathematica, Vol. XXIII (Conf. Abelian Groups and their Relationship to the Theory of Modules, INDAM, Rome, 1977) Academic Press, London-New York, 1979, pp. 11–32. MR 565595
Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
A. Pultr and W. Tholen, Localic enrichments of categories, Cahiers Topologie Géom. Différentielle Catég. 41 (2000), no. 2, 121–142 (English, with French summary). MR 1769340
Jinzhong Xu, Flat covers of modules, Lecture Notes in Mathematics, vol. 1634, Springer-Verlag, Berlin, 1996. MR 1438789, DOI 10.1007/BFb0094173