Relative global dimensions and stable homotopy categories. (English) Zbl 1456.18009
Let \(R\) be a ring. An \(F\) a left \(R\)-module is type \(FP_\infty\) if \(F\) has projective resolution with finitely generated terms. A left (resp. right) \(R\)-modules \(A\) is absolutely clean (resp. level) if it satisfies \(\mathrm{Ext}_R^1(F,A)=0\) (resp. \(\mathrm{Tor}_1^R(A,F)=0\)) for all left \(R\)-modules of type \(FP_\infty\), \(F\). With this notion D. Bravo et al. [“The stable module category of a general ring”, Preprint, arXiv:1405.5768] define Gorenstein AC-projective (resp. AC- injective and AC-flat) modules extending the classical notions. The authors study the dimensions of modules using resolutions by modules of the preceding classes, thus they introduce Gorenstein AC-projective (resp. injective and flat) dimension of left \(R\)-modules and the global dimension of the ring associated. Several relations among these dimension are obtained. It is shown that left (resp. right) coherent rings have finite global Gorenstein AC-injective (resp. AC-projective) dimension. Next, the compactly generated properties of the singularity categories and stable categories with respect to Gorenstein AC-homological modules are considered. For a ring with finite global Gorentein AC-injective dimension, the stable category of Gorenstein AC-injective left modules is compactly generated. As a consequence, it is obtained that for a left coherent ring with finite Gorenstein global dimension, the stable category of Gorenstein injective is compactly generated. This results was proved by A. Beligiannis for right coherent and left perfect or left Morita ring with finite Gorenstein global dimension [Math. Scand. 89, No. 1, 5–45 (2001; Zbl 1023.55009)] and for Iwanaga-Gorenstein rings by M. Hovey [Math. Z. 241, No. 3, 553–592 (2002; Zbl 1016.55010)] and X.-W. Chen [Math. Nachr. 284, No. 2–3, 199–212 (2011; Zbl 1244.18014)]. Several properties of Gorenstein AC-flat modules are studied in the Appendix.
Reviewer: Blas Torrecillas (Almería)
MSC:
| 18G25 | Relative homological algebra, projective classes (category-theoretic aspects) |
| 18G20 | Homological dimension (category-theoretic aspects) |
Keywords:
Gorenstein AC-projective modules; Gorenstein AC-injective modules; Gorenstein AC-flat modules; Gorenstein dimensions; singularity derived category; compactly generatedReferences:
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