奇点模型范畴的Quillen等价
|
摘要
令R是左Gorenstein环.我们构造了奇点反导出模型范畴和奇点余导出模型范畴(见文[Models for singularity categories,Adv Math.,2014,254:187-232])之间的Quillen等价.作为应用,给出了投射,内射模的正合复形的同伦范畴之间的一个具体的等价■.
Let R be a left-Gorenstein ring. We construct a Quillen equivalence between singular contraderived model category and singular coderived model category introduced in(see [Models for singularity categories, Adv. Math., 2014, 254: 187-232]). As an application, we explicitly give an equivalence ■ for the homotopy categories of exact complexes of projective and injective modules.
Let R be a left-Gorenstein ring. We construct a Quillen equivalence between singular contraderived model category and singular coderived model category introduced in(see [Models for singularity categories, Adv. Math., 2014, 254: 187-232]). As an application, we explicitly give an equivalence ■ for the homotopy categories of exact complexes of projective and injective modules.
引文
[1] Becker H., Models for singularity categories, Adv. Math.,2014, 254:187-232.
[2] Beilinson A. A., Bernstein J., Deligne P., Faisceaux Pervers, Asterisque, Vol.100, 1982.
[3] Beligiannis A., The homological theory of contravariantly finite subcategories:Auslander-Buchweitz contexts, Gorenstein categories and(co)stabilization, Comm. Algebra, 2000, 28:4547-4596.
[4] Beligiannis A., Reiten I., Homological and homotopical aspects of torsion theories, Mem. Amer. Math. Soc.,2007, 188(883).
[5] Bergh P. A., Jorgensen D. A., Oppermann S., The Gorenstein defect category, Quart. J. Math., 2015, 66:459-471.
[6] Chen X. W., Homotopy equivalences induced by balanced pairs, J. Algebra, 2010, 324:2718-2731.
[7] Christensen L. W., Frankild A., Holm H., On Gorenstein projective, injective and flat dimensions-A functorial description with applications, J. Algebra, 2006, 302:231-279.
[8] Dwyer W. G., Spalinski J., Homotopy Theories and Model Categories, Handbook of algebraic topology(Amsterdam), North-Holland, Amsterdam, pp. 73-126, 1995.
[9] Enochs E. E., Jenda O. M. G., Relative Homological Algebra, De Gruyter Expositions in Mathematics No.30,Walter De Gruyter, New York, 2000.
[10] Jorgensen P., The homotopy category of complexes of projective modules, Adv. Math.,2005, 193:223-232.
[11] Gillespie J.,The flat model structure on Ch(R), Trans. Amer. Math.Soc., 2004, 356:3369-3390.
[12] Gillespie J., Cotorsion pairs and degreewise homological model structures, Homol. Homotopy Appl.,2008,10:283-304.
[13] Gillespie J., Gorenstein complexes and recollements from cotorsion pairs, Adv. Math., 2016, 291:859-911.
[14] Hovey M., Model Categories, Mathematical Surveys and Monographs Vol.63, Amer. Math. Soc., 1999.
[15] Hovey M., Cotorsion pairs, model category structures, and representation theory. Math. Z.,2002,241:553-592.
[16] Iyengar S., Krause H., Acyclicity versus total acyclicity for complexes over Noetherian rings, Doc. Math.,2006, 11:207-240.
[17] Krause H., The stable derived category of a Noetherian scheme, Compos. Math., 2005, 141(5):1128-1162.
[18] Quillen D. G., Homotopical Algebra, Lecture Notes in Mathematics No. 43, Springer-Verlag, BerlinHeidelberg, 1967.
[19] Ren W., Gorenstein projective modules and Frobenius extensions, Sci. China Math.,2018, 61(7):1175-1186.
[20] Ren W., Applications of cotorsion triples, Comm. Algebra, in press. DOI:10.1080/00927872.2018.1539170
[2] Beilinson A. A., Bernstein J., Deligne P., Faisceaux Pervers, Asterisque, Vol.100, 1982.
[3] Beligiannis A., The homological theory of contravariantly finite subcategories:Auslander-Buchweitz contexts, Gorenstein categories and(co)stabilization, Comm. Algebra, 2000, 28:4547-4596.
[4] Beligiannis A., Reiten I., Homological and homotopical aspects of torsion theories, Mem. Amer. Math. Soc.,2007, 188(883).
[5] Bergh P. A., Jorgensen D. A., Oppermann S., The Gorenstein defect category, Quart. J. Math., 2015, 66:459-471.
[6] Chen X. W., Homotopy equivalences induced by balanced pairs, J. Algebra, 2010, 324:2718-2731.
[7] Christensen L. W., Frankild A., Holm H., On Gorenstein projective, injective and flat dimensions-A functorial description with applications, J. Algebra, 2006, 302:231-279.
[8] Dwyer W. G., Spalinski J., Homotopy Theories and Model Categories, Handbook of algebraic topology(Amsterdam), North-Holland, Amsterdam, pp. 73-126, 1995.
[9] Enochs E. E., Jenda O. M. G., Relative Homological Algebra, De Gruyter Expositions in Mathematics No.30,Walter De Gruyter, New York, 2000.
[10] Jorgensen P., The homotopy category of complexes of projective modules, Adv. Math.,2005, 193:223-232.
[11] Gillespie J.,The flat model structure on Ch(R), Trans. Amer. Math.Soc., 2004, 356:3369-3390.
[12] Gillespie J., Cotorsion pairs and degreewise homological model structures, Homol. Homotopy Appl.,2008,10:283-304.
[13] Gillespie J., Gorenstein complexes and recollements from cotorsion pairs, Adv. Math., 2016, 291:859-911.
[14] Hovey M., Model Categories, Mathematical Surveys and Monographs Vol.63, Amer. Math. Soc., 1999.
[15] Hovey M., Cotorsion pairs, model category structures, and representation theory. Math. Z.,2002,241:553-592.
[16] Iyengar S., Krause H., Acyclicity versus total acyclicity for complexes over Noetherian rings, Doc. Math.,2006, 11:207-240.
[17] Krause H., The stable derived category of a Noetherian scheme, Compos. Math., 2005, 141(5):1128-1162.
[18] Quillen D. G., Homotopical Algebra, Lecture Notes in Mathematics No. 43, Springer-Verlag, BerlinHeidelberg, 1967.
[19] Ren W., Gorenstein projective modules and Frobenius extensions, Sci. China Math.,2018, 61(7):1175-1186.
[20] Ren W., Applications of cotorsion triples, Comm. Algebra, in press. DOI:10.1080/00927872.2018.1539170