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PURE-INJECTIVES RELATIVE TO A COTORSION PAIR: APPLICATIONS

Published online by Cambridge University Press:  02 August 2012

SERGIO ESTRADA
Affiliation:
Departamento de Matemática Aplicada, Universidad de Murcia, Campus del Espinardo, Espinardo (Murcia) 30100, Spain e-mail: sestrada@um.es
PEDRO A. GUIL ASENSIO
Affiliation:
Departamento de Matemáticas, Universidad de Murcia, Campus del Espinardo, Espinardo (Murcia) 30100, Spain e-mail: paguil@um.es
Corresponding
E-mail address:
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Abstract

Finitely accessible categories naturally arise in the context of the classical theory of purity. In this paper we generalise the notion of purity for a more general class and introduce techniques to study such classes in terms of indecomposable pure injectives related to a new notion of purity. We apply our results in the study of the class of flat quasi-coherent sheaves on an arbitrary scheme.

Type
Research Article
Copyright
Copyright © Glasgow Mathematical Journal Trust 2012

References

1.Adámek, J., Herrlich, H. and Strecker, G. E., Abstract and concrete categories. The joy of cats, Repr. Theory Appl. Categ. 17 (2006), 1507.Google Scholar
2.Adámek, J. and Rosický, J., Locally presentable and accessible categories, vol. 189 (Cambridge University Press, Cambridge, UK, 1994).CrossRefGoogle Scholar
3.Aldrich, S. T., Enochs, E., García-Rozas, J. R. and Oyonarte, L., Covers and envelopes in Grothendieck categories. Flat covers of complexes with applications, J. Algebra 243 (2001), 615630.CrossRefGoogle Scholar
4.Bican, L., El Bashir, R. and Enochs, E., All modules have flat covers, Bull. Lond. Math. Soc. 33 (4) (2001), 385390.CrossRefGoogle Scholar
5.Crawley-Boevey, W. W., Locally finitely presented additive categories, Comm. Algebra 22 (1994), 16411674.CrossRefGoogle Scholar
6.Dung, N. V. and García, J. L., Additive categories of locally finite representation type, J. Algebra 238 (2001), 200238.CrossRefGoogle Scholar
7.Eklof, P. C., Homological algebra and set theory, Trans. Amer. Math. Soc. 227 (1977), 207–225CrossRefGoogle Scholar
8.Enochs, E., Flat covers and flat cotorsion modules, Proc. Amer. Math. Soc. 97 (1984), 179184.CrossRefGoogle Scholar
9.Enochs, E. and Estrada, S., Relative homological algebra in the category of quasi-coherent sheaves, Adv. Math. 194 (2005), 284295.CrossRefGoogle Scholar
10.Enochs, E., Estrada, S. and García-Rozas, J. R., Gorenstein categories and Tate cohomology on projective schemes, Math. Nachr. 281 (2008), 525540.CrossRefGoogle Scholar
11.Enochs, E., Estrada, S., García Rozas, J. R. and Oyonarte, L., Flat and cotorsion quasi-coherent sheaves. Applications. Algebra Represent. Theory 7 (2004), 441456.CrossRefGoogle Scholar
12.Enochs, E., Estrada, S., García Rozas, J. R. and Oyonarte, L., Flat covers in the category of quasi-coherent sheaves over the projective line, Comm. Algebra 32 (2004), 14971508.CrossRefGoogle Scholar
13.Enochs, E. and Jenda, O. M. G., Relative homological algebra, GEM 30 (2000) (W. de Gruyter, Berlin, Germany).Google Scholar
14.Enochs, E. and López Ramos, J. A., Kaplansky classes, Rend. Sem. Mat. Univ. Padova 107 (2002), 6779.Google Scholar
15.Enochs, E. and Oyonarte, L., Flat covers and cotorsion envelopes of sheaves, Proc. Amer. Math. Soc. 130 (2001), 12851292.CrossRefGoogle Scholar
16.Estrada, S., Guil Asensio, P., Prest, M. and Trlifaj, J., Model category structures arising from Drinfeld vector bundles, arXiv: 0906.5213v1.Google Scholar
17.Harrison, D. K., Infinite abelian groups and homological methods, Ann. Math. 69 (1959), 366391.CrossRefGoogle Scholar
18.Murfet, D. and Salarian, S., Totally acyclic complexes over noetherian schemes, Adv. Math. 296 (2011), 10961133.CrossRefGoogle Scholar
19.Salce, L., Cotorsion theories for abelian groups, Symp. Math. 23 (1979), 1132.Google Scholar
20.Šťovíček, J., Deconstructibility and the Hill lemma in Grothendieck categories, Forum Math. (to appear).Google Scholar
21.Xu, J., Flat covers of modules (Lecture Notes in Mathematics, no. 1634 (Springer-Verlag, Berlin, Germany 1996).CrossRefGoogle Scholar
22.Ziegler, M., Model theory of modules, Ann. Pure Appl. Log. 26 (2) (1984), 149213.CrossRefGoogle Scholar

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