Gorenstein injective, projective and flat (pre)covers
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Abstract
We prove that if the ring R is left noetherian andif the class of Gorenstein injective modules, GI, is closed under ltrations, then GI is precovering. We extend this result to the category of complexes. We also prove that when R is commutative noetherian and such that the character modules of Gorenstein injective modules are Gorenstein at, the class of Gorenstein injective complexes is both covering and enveloping. This is the case when the ring is commutative noetherian with a dualizing complex. The second part of the paper deals with Gorenstein projective and at complexes. We prove the existence of special Gorenstein projective precovers over commutative noetherian rings of nite Krull dimension.
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Iacob, A., Enochs, E., & Estrada, S.
(2014).
Gorenstein injective, projective and flat (pre)covers.
Acta Mathematica Universitatis Comenianae, 83(2), 217-230.
Retrieved from http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/amuc/article/view/90/72
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References
[1] D. Bennis and N. Mahdou. Strongly Gorenstein projective, injective and at modules. J. Pure Appl. Algebra, (210):1709-1718, 2007.
[2] D. Bravo and E.E. Enochs and A. Iacob and O.M.G. Jenda and J. Rada. Cotorsion pairs in C(R-Mod). Rocky Mountain J. Math., 42(6): 1787-1802, 2012.
[3] L.W. Christensen, A. Frankild, and H. Holm On Gorenstein projective, injective and at dimensions - A functorial description with applications. J. Algebra,
302(1): 231-279, 2006.
[4] L.W. Christensen, H.B. Foxby, and H. Holm Beyond totally reflexive modules and back. A survey on Gorenstein dimensions. book chapter in "Commutative Algebra: Noetherian and non-Noetherian perspectives", 101-143. Springer-Verlag, 2011.
[5] P. Eklof. Homological algebra and set theory. Trans. American Math. Soc., (227):207-225, 1977.
[6] P. Eklof, J. Trlifaj, How to make Ext vanish. Bull. Lond. Math. Soc. (33):41-51, 2001.
[7] E.E. Enochs. Injective and at covers and resolvents. Israel J. Math. 39:189-209, 1981.
[8] E.E. Enochs. Shortening ltrations. Science China Math., (55):687-693, 2012.
[9] E.E. Enochs and S. Estrada and A. Iacob. Cotorsion pairs, model structures and homotopy categories. Houston Journal of Mathematics, to appear.
[10] E.E. Enochs and S. Estrada and A. Iacob. Gorenstein projective and at complexes over noetherian rings. Mathematische Nachrichten, 7:834-851, 2012.
[11] E.E. Enochs and O.M.G. Jenda. Gorenstein injective and projective modules. Mathematische Zeitschrift, (220):611-633, 1995.
[12] E.E. Enochs and O.M.G. Jenda. Relative Homological Algebra. Walter de Gruyter, 2000. De Gruyter Exposition in Math.
[13] E.E. Enochs, and O.M.G. Jenda, and J.A. Lopez-Ramos The existence of Gorenstein at covers. Math. Scand., 94:46-62, 2004.
[14] E.E. Enochs and O.M.G. Jenda and B. Torrecillas. Gorenstein at modules. Journal Nanjing University 10:1-9, 1993.
[15] E.E. Enochs and A. Iacob. Gorenstein injective covers and envelopes over noetherian rings. Proc. Amer. Math. Soc., to appear.
[16] E.E. Enochs, and J.A. Lopez-Ramos Kaplansky classes. Rend. Sem. Univ. Padova, 107:67-79, 2002.
[17] R. Gobel and J. Trlifaj. Approximations and endomorphism algebras of modules. Walter de Gruyter, 2006. De Gruyter Exposition in Math. 41
[18] J.R. Garca Rozas. Covers and evelopes in the category of complexes of modules. CRC Press LLC, 1999.
[19] J. Gillespie. Cotorsion pairs and degreewise homological model structures. Homotopy, homology and applications, 10(1):283-304, 2008.
[20] J. Gillespie. Gorenstein complexes and recollements from cotorsion pairs. preprint, arXiv. 1210.0196.
[21] J. Gillespie. Kaplansky classes and derived categories. Math. Z., 257(4):811-843, 2007.
[22] H. Holm. Gorenstein homological dimensions. Journal of pure and applied algebra, 189:167-193, 2004.
[23] H. Holm and P. Jrgensen. Cotorsion pairs induced by duality pairs. Journal of Commutative Algebra, (1):621-633, 2009.
[24] P. Jrgensen. Existence of Gorenstein projective resolutions and Tate cohomology. J. Eur. Math. Soc. 9:59-76, 2007.
[25] Z. Liu and C. Zhang. Gorenstein injective complexes of modules over noetherian rings. J. Algebra 321: 1546-1554, 2009.
[26] D. Murfet and S. Salarian. Totally acyclic complexes over noetherian schemes. Adv. Math., 226(2):1096-1133, 2011.
[27] J. Šťovíček. Deconstructibility and the Hill lemma in Grothendieck categories. Forum Math., 25:193-219, 2013.
[2] D. Bravo and E.E. Enochs and A. Iacob and O.M.G. Jenda and J. Rada. Cotorsion pairs in C(R-Mod). Rocky Mountain J. Math., 42(6): 1787-1802, 2012.
[3] L.W. Christensen, A. Frankild, and H. Holm On Gorenstein projective, injective and at dimensions - A functorial description with applications. J. Algebra,
302(1): 231-279, 2006.
[4] L.W. Christensen, H.B. Foxby, and H. Holm Beyond totally reflexive modules and back. A survey on Gorenstein dimensions. book chapter in "Commutative Algebra: Noetherian and non-Noetherian perspectives", 101-143. Springer-Verlag, 2011.
[5] P. Eklof. Homological algebra and set theory. Trans. American Math. Soc., (227):207-225, 1977.
[6] P. Eklof, J. Trlifaj, How to make Ext vanish. Bull. Lond. Math. Soc. (33):41-51, 2001.
[7] E.E. Enochs. Injective and at covers and resolvents. Israel J. Math. 39:189-209, 1981.
[8] E.E. Enochs. Shortening ltrations. Science China Math., (55):687-693, 2012.
[9] E.E. Enochs and S. Estrada and A. Iacob. Cotorsion pairs, model structures and homotopy categories. Houston Journal of Mathematics, to appear.
[10] E.E. Enochs and S. Estrada and A. Iacob. Gorenstein projective and at complexes over noetherian rings. Mathematische Nachrichten, 7:834-851, 2012.
[11] E.E. Enochs and O.M.G. Jenda. Gorenstein injective and projective modules. Mathematische Zeitschrift, (220):611-633, 1995.
[12] E.E. Enochs and O.M.G. Jenda. Relative Homological Algebra. Walter de Gruyter, 2000. De Gruyter Exposition in Math.
[13] E.E. Enochs, and O.M.G. Jenda, and J.A. Lopez-Ramos The existence of Gorenstein at covers. Math. Scand., 94:46-62, 2004.
[14] E.E. Enochs and O.M.G. Jenda and B. Torrecillas. Gorenstein at modules. Journal Nanjing University 10:1-9, 1993.
[15] E.E. Enochs and A. Iacob. Gorenstein injective covers and envelopes over noetherian rings. Proc. Amer. Math. Soc., to appear.
[16] E.E. Enochs, and J.A. Lopez-Ramos Kaplansky classes. Rend. Sem. Univ. Padova, 107:67-79, 2002.
[17] R. Gobel and J. Trlifaj. Approximations and endomorphism algebras of modules. Walter de Gruyter, 2006. De Gruyter Exposition in Math. 41
[18] J.R. Garca Rozas. Covers and evelopes in the category of complexes of modules. CRC Press LLC, 1999.
[19] J. Gillespie. Cotorsion pairs and degreewise homological model structures. Homotopy, homology and applications, 10(1):283-304, 2008.
[20] J. Gillespie. Gorenstein complexes and recollements from cotorsion pairs. preprint, arXiv. 1210.0196.
[21] J. Gillespie. Kaplansky classes and derived categories. Math. Z., 257(4):811-843, 2007.
[22] H. Holm. Gorenstein homological dimensions. Journal of pure and applied algebra, 189:167-193, 2004.
[23] H. Holm and P. Jrgensen. Cotorsion pairs induced by duality pairs. Journal of Commutative Algebra, (1):621-633, 2009.
[24] P. Jrgensen. Existence of Gorenstein projective resolutions and Tate cohomology. J. Eur. Math. Soc. 9:59-76, 2007.
[25] Z. Liu and C. Zhang. Gorenstein injective complexes of modules over noetherian rings. J. Algebra 321: 1546-1554, 2009.
[26] D. Murfet and S. Salarian. Totally acyclic complexes over noetherian schemes. Adv. Math., 226(2):1096-1133, 2011.
[27] J. Šťovíček. Deconstructibility and the Hill lemma in Grothendieck categories. Forum Math., 25:193-219, 2013.