Covers in finitely accessible categories

Crivei, Septimiu; Prest, Mike; Torrecillas, Blas

doi:10.1090/S0002-9939-09-10178-8

Covers in finitely accessible categories
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by Septimiu Crivei, Mike Prest and Blas Torrecillas PDF

Proc. Amer. Math. Soc. 138 (2010), 1213-1221 Request permission

Abstract:

We show that in a finitely accessible additive category every class of objects closed under direct limits and pure epimorphic images is covering. In particular, the classes of flat objects in a locally finitely presented additive category and of absolutely pure objects in a locally coherent category are covering.

References

Jiří Adámek and Jiří Rosický, Locally presentable and accessible categories, London Mathematical Society Lecture Note Series, vol. 189, Cambridge University Press, Cambridge, 1994. MR 1294136, DOI 10.1017/CBO9780511600579
S. Tempest Aldrich, Edgar E. Enochs, J. R. García Rozas, and Luis Oyonarte, Covers and envelopes in Grothendieck categories: flat covers of complexes with applications, J. Algebra 243 (2001), no. 2, 615–630. MR 1850650, DOI 10.1006/jabr.2001.8821
M. Auslander and Sverre O. Smalø, Preprojective modules over Artin algebras, J. Algebra 66 (1980), no. 1, 61–122. MR 591246, DOI 10.1016/0021-8693(80)90113-1
Silvana Bazzoni, When are definable classes tilting and cotilting classes?, J. Algebra 320 (2008), no. 12, 4281–4299. MR 2464106, DOI 10.1016/j.jalgebra.2008.08.028
L. Bican, R. El Bashir, and E. Enochs, All modules have flat covers, Bull. London Math. Soc. 33 (2001), no. 4, 385–390. MR 1832549, DOI 10.1017/S0024609301008104
William Crawley-Boevey, Locally finitely presented additive categories, Comm. Algebra 22 (1994), no. 5, 1641–1674. MR 1264733, DOI 10.1080/00927879408824927
William Crawley-Boevey, Infinite-dimensional modules in the representation theory of finite-dimensional algebras, Algebras and modules, I (Trondheim, 1996) CMS Conf. Proc., vol. 23, Amer. Math. Soc., Providence, RI, 1998, pp. 29–54. MR 1648602
S. Crivei, A note on flat covers of comodules, Mathematica (Cluj) 50 (73) (2008), 187–190.
Septimiu Crivei and José Luis García, Gruson-Jensen duality for idempotent rings, Comm. Algebra 33 (2005), no. 11, 3949–3966. MR 2183972, DOI 10.1080/00927870500261082
Juan Cuadra and Daniel Simson, Flat comodules and perfect coalgebras, Comm. Algebra 35 (2007), no. 10, 3164–3194. MR 2356149, DOI 10.1080/00914030701409908
Sorin Dăscălescu, Constantin Năstăsescu, and Şerban Raianu, Hopf algebras, Monographs and Textbooks in Pure and Applied Mathematics, vol. 235, Marcel Dekker, Inc., New York, 2001. An introduction. MR 1786197
Nguyen Viet Dung and José Luis García, Additive categories of locally finite representation type, J. Algebra 238 (2001), no. 1, 200–238. MR 1822190, DOI 10.1006/jabr.2000.8620
Paul C. Eklof and Jan Trlifaj, How to make Ext vanish, Bull. London Math. Soc. 33 (2001), no. 1, 41–51. MR 1798574, DOI 10.1112/blms/33.1.41
Robert El Bashir, Covers and directed colimits, Algebr. Represent. Theory 9 (2006), no. 5, 423–430. MR 2252654, DOI 10.1007/s10468-006-9030-5
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
Edgar Enochs, S. Estrada, J. R. García Rozas, and L. Oyonarte, Flat covers in the category of quasi-coherent sheaves over the projective line, Comm. Algebra 32 (2004), no. 4, 1497–1508. MR 2100370, DOI 10.1081/AGB-120028794
Edgar Enochs, S. Estrada, J. R. García-Rozas, and L. Oyonarte, Flat cotorsion quasi-coherent sheaves. Applications, Algebr. Represent. Theory 7 (2004), no. 4, 441–456. MR 2096670, DOI 10.1023/B:ALGE.0000042145.72104.cc
Edgar E. Enochs and Overtoun M. G. Jenda, Relative homological algebra, De Gruyter Expositions in Mathematics, vol. 30, Walter de Gruyter & Co., Berlin, 2000. MR 1753146, DOI 10.1515/9783110803662
Edgar Enochs and Luis Oyonarte, Flat covers and cotorsion envelopes of sheaves, Proc. Amer. Math. Soc. 130 (2002), no. 5, 1285–1292. MR 1879949, DOI 10.1090/S0002-9939-01-06190-1
J. L. García, P. L. Gómez Sánchez, and J. Martínez Hernández, Locally finitely presented categories and functor rings, Osaka J. Math. 42 (2005), no. 1, 173–187. MR 2132009
J. L. Garcia and D. Simson, On rings whose flat modules form a Grothendieck category, Colloq. Math. 73 (1997), no. 1, 115–141. MR 1436954, DOI 10.4064/cm-73-1-115-141
J. R. García Rozas and B. Torrecillas, Preserving and reflecting covers by functors. Applications to graded modules, J. Pure Appl. Algebra 112 (1996), no. 1, 91–107. MR 1402398, DOI 10.1016/0022-4049(95)00132-8
James Gillespie, The flat model structure on $\textrm {Ch}(R)$, Trans. Amer. Math. Soc. 356 (2004), no. 8, 3369–3390. MR 2052954, DOI 10.1090/S0002-9947-04-03416-6
James Gillespie, The flat model structure on complexes of sheaves, Trans. Amer. Math. Soc. 358 (2006), no. 7, 2855–2874. MR 2216249, DOI 10.1090/S0002-9947-06-04157-2
Ivo Herzog, Pure-injective envelopes, J. Algebra Appl. 2 (2003), no. 4, 397–402. MR 2020947, DOI 10.1142/S0219498803000581
Mark Hovey, Cotorsion pairs, model category structures, and representation theory, Math. Z. 241 (2002), no. 3, 553–592. MR 1938704, DOI 10.1007/s00209-002-0431-9
P. Jørgensen and H. Holm, Covers, precovers, and purity, Illinois J. Math. 52 (2008), 691–703.
Henning Krause, Exactly definable categories, J. Algebra 201 (1998), no. 2, 456–492. MR 1612398, DOI 10.1006/jabr.1997.7252
C. Năstăsescu, M. Van den Bergh, and F. Van Oystaeyen, Separable functors applied to graded rings, J. Algebra 123 (1989), no. 2, 397–413. MR 1000494, DOI 10.1016/0021-8693(89)90053-7
Katherine Pinzon, Absolutely pure covers, Comm. Algebra 36 (2008), no. 6, 2186–2194. MR 2418384, DOI 10.1080/00927870801952694
M. Prest, Purity, spectra and localisation, Cambridge University Press, 2009.
M. Prest, Definable additive categories: Purity and model theory, Mem. Amer. Math. Soc., to appear. MIMS EPrint: 2006.218, July 2006.
Mike Prest, Vera Puninskaya, and Alexandra Ralph, Some model theory of sheaves of modules, J. Symbolic Logic 69 (2004), no. 4, 1187–1199. MR 2135662, DOI 10.2178/jsl/1102022218
Juan Rada and Manuel Saorin, Rings characterized by (pre)envelopes and (pre)covers of their modules, Comm. Algebra 26 (1998), no. 3, 899–912. MR 1606190, DOI 10.1080/00927879808826172
Bo Stenström, Purity in functor categories, J. Algebra 8 (1968), 352–361. MR 229697, DOI 10.1016/0021-8693(68)90064-1
Robert Wisbauer, Foundations of module and ring theory, Revised and translated from the 1988 German edition, Algebra, Logic and Applications, vol. 3, Gordon and Breach Science Publishers, Philadelphia, PA, 1991. A handbook for study and research. MR 1144522
Jinzhong Xu, Flat covers of modules, Lecture Notes in Mathematics, vol. 1634, Springer-Verlag, Berlin, 1996. MR 1438789, DOI 10.1007/BFb0094173