Frontiers of Mathematics in China >
On the homotopy category of AC-injective complexes
Received date: 08 Nov 2015
Accepted date: 07 Mar 2016
Published date: 01 Feb 2017
Copyright
Let R be any ring. We motivate the study of a class of chain complexes of injective R-modules that we call AC-injective complexes, showing that K(AC-Inj), the chain homotopy category of all AC-injective complexes, is always a compactly generated triangulated category. In general, all DGinjective complexes are AC-injective and in fact there is a recollement linking K(AC-Inj) to the usual derived category D(R). This is based on the author’s recent work inspired by work of Krause and Stovicek. Our focus here is on giving straightforward proofs that our categories are compactly generated.
Key words: AC-injective; recollement; compactly generated; triangulated category
James GILLESPIE . On the homotopy category of AC-injective complexes[J]. Frontiers of Mathematics in China, 2017 , 12(1) : 97 -115 . DOI: 10.1007/s11464-016-0551-x
1 |
Bravo D, Enochs E E, Iacob A, Jenda O, Rada J. Cotorsion pairs in C(R-Mod). Rocky Mountain J Math, 2012, 42(6): 1787–1802
|
2 |
Bravo D, Gillespie J. Absolutely clean, level, and Gorenstein AC-injective complexes. Comm Algebra, 2016, 44(5): 2213–2233
|
3 |
Bravo D, Gillespie J, Hovey M. The stable module category of a general ring (submitted)
|
4 |
Bühler T. Exact categories. Expo Math, 2010, 28(1): 1–69
|
5 |
Enochs E E, and Jenda O M G. Relative Homological Algebra. de Gruyter Exp Math, Vol 30. Berlin: Walter de Gruyter & Co, 2000
|
6 |
Gillespie J. The flat model structure on Ch(R). Trans Amer Math Soc, 2004, 356(8): 3369–3390
|
7 |
Gillespie J. Cotorsion pairs and degreewise homological model structures. Homology, Homotopy Appl, 2008, 10(1): 283–304
|
8 |
Gillespie J. Model structures on exact categories. J Pure Appl Algebra, 2011, 215: 2892–2902
|
9 |
Gillespie J. How to construct a Hovey triple from two cotorsion pairs. Fund Math, 2015, 230(3): 281–289
|
10 |
Gillespie J. Models for homotopy categories of injective and Gorenstein injectives. 2015, arXiv: 1502.05530
|
11 |
Gillespie J. Gorenstein complexes and recollements from cotorsion pairs. Adv Math, 2016, 291: 859–911
|
12 |
Gillespie J. Models for mock homotopy categories of projectives. Homology, Homotopy Appl, 2016, 18(1): 247–263
|
13 |
Hovey M. Cotorsion pairs, model category structures, and representation theory. Math Z, 2002, 241: 553–592
|
14 |
Keller B. Derived categories and their uses. In: Handbook of Algebra, Vol 1. Amsterdam: North-Holland, 1996, 671–701
|
15 |
Krause H. The stable derived category of a Noetherian scheme. Compos Math, 2005, 141(5): 1128–1162
|
16 |
Neeman A. The derived category of an exact category. J Algebra, 1990, 135: 388–394
|
17 |
Neeman A. Triangulated Categories. Ann of Math Studies, Vol 148. Princeton: Princeton University Press, 2001
|
18 |
Stovicek J. Exact model categories, approximation theory, and cohomology of quasicoherent sheaves. In: Advances in Representation Theory of Algebras (ICRA Bielefeld, Germany, 8-17 August, 2012). EMS Series of Congress Reports. Zürich: Eur Math Soc Publishing House, 2014, 297–367
|
19 |
Stovicek J. On purity and applications to coderived and singularity categories. arXiv: 1412.1615
|
20 |
Weibel C A. An Introduction to Homological Algebra. Cambridge Stud Adv Math, Vol 38. Cambridge: Cambridge University Press, 1994
|
21 |
Yang G, Liu Z K. Cotorsion pairs and model structures on Ch(R). Proc Edinb Math Soc (2), 2011, 54(3): 783–797
|
22 |
Yang X Y, Ding N Q. On a question of Gillespie. Forum Math, 2015, 27(6): 3205–3231
|
/
〈 | 〉 |