环扩张下的Gorenstein平坦模型结构

任伟, 张春霞

数学学报 ›› 2017, Vol. 60 ›› Issue (5) : 859-864.

PDF(431 KB)
PDF(431 KB)
数学学报 ›› 2017, Vol. 60 ›› Issue (5) : 859-864. DOI: 10.12386/A2017sxxb0073
论文

环扩张下的Gorenstein平坦模型结构

    任伟, 张春霞
作者信息 +

Gorenstein Flat Model Structures Under Extensions of Rings

    Wei REN, Chun Xia ZHANG
Author information +
文章历史 +

摘要

研究了环扩张下的Gorenstein平坦模型结构及其同伦范畴.设RS是满足一些条件的平坦扩张.我们证明了若fMNS-模范畴的Gorenstein平坦模型结构中是上纤维化(纤维化,弱等价),则fMNR-模范畴中亦如此;若RS是优越扩张,反过来也成立,即在优越扩张下Gorenstein平坦模型结构是不变的.进而,相关的稳定范畴是等价的,当且仅当对任意Gorenstein平坦S-模M,Coker(ηM)是平坦的,其中η表示S-模范畴和R-模范畴间的Quillen伴随函子的单位.

Abstract

The Gorenstein flat model structures and resulting homotopy categories under extensions of rings are studied. Along the flat extension RS satisfying a few conditions, we show that if f:MN is a cofibration (resp. fibration, weak equivalence) in the Gorenstein flat model structure of S-Mod, then f is so in R-Mod; furthermore, the converse holds if RS is an excellent extension. That is, Gorenstein flat model structure are invariant under excellent extensions. Moreover, the associated stable categories are equivalent if and only if Coker(ηM) is flat for any Gorenstein flat S-module M, where η is unit of the Quillen adjunction between S-Mod and R-Mod.

关键词

Gorenstein 平坦模型结构 / 平坦扩张 / 优越扩张

Key words

Gorenstein flat model structure / flat extension / excellent extension

引用本文

导出引用
任伟, 张春霞. 环扩张下的Gorenstein平坦模型结构. 数学学报, 2017, 60(5): 859-864 https://doi.org/10.12386/A2017sxxb0073
Wei REN, Chun Xia ZHANG. Gorenstein Flat Model Structures Under Extensions of Rings. Acta Mathematica Sinica, Chinese Series, 2017, 60(5): 859-864 https://doi.org/10.12386/A2017sxxb0073

参考文献

[1] Bennis D., A note on Gorenstein flat dimension, Algebra Colloq., 2011, 18: 155-161.
[2] Bonami L., On the Structure of Skew Group Rings, Algebra Berichte 48, Verlag Reinhard Ficher, Munchen, 1984.
[3] Dwyer W. G., Hirschhorn P. S., Kan D. M., et al., Homotopy Limit Functors on Model Categories and Homotopical Categories, Mathematical Surveys and Monographs, vol. 113, Amer. Math. Soc., 2004.
[4] Enochs E. E., Jenda O. M. G., Relative Homological Algebra, De Gruyter Expositions in Mathematics no. 30, Walter De Gruyter, New York, 2000.
[5] Enochs E. E., Jenda O. M. G., Lopez-Ramos J. A., The existence of Gorenstein flat covers, Math. Scand., 2004, 94: 46-62.
[6] Enochs E. E., Jenda O. M. G., Torrecillas B., Gorenstein flat modules, Nanjing Daxue Xuebao Shuxue Bannian Kan, 1993, 10: 1-9.
[7] Fang H. J., Normalizing extensions and modules, J. Math. Res. Exp., 1992, 12: 401-406.
[8] Gillespie J., The flat stable module category of a coherent ring, J. Pure Appl. Algebra, 2016. doi: 10.1016/j.jpaa.2016.10.012.
[9] Hovey M., Model Categories, Mathematical Surveys and Monographs vol. 63, Amer. Math. Soc., 1999.
[10] Hovey M., Cotorsion pairs, model category structures, and representation theory, Math. Z., 2002, 241: 553-592.
[11] Huang Z. Y., Sun J. X., Invariant properties of representations under excellent extensions, J. Algebra, 2012, 358: 87-101.
[12] Liu Z. K., Excellent extensions and homological dimensions, Comm. Algebra, 1994, 22: 1741-1745.
[13] Mao L. X., Ding N. Q., The cotorsion dimension of modules and rings, Lecture Notes Pure Appl. Math., Abelian groups, rings, modules and homological algebra, 2005, 249: 217-233.
[14] Passman D. S., The Algebraic Structure of Group Rings, Wiley-Interscience, New York, 1977.
[15] Passman D. S., It's essentially Maschke's theorem, Rocky Mountain J. Math., 1983, 13: 37-54.

基金

国家自然科学基金资助项目(11401475,11401476);中国博士后科学基金资助项目(2016M591592)

PDF(431 KB)

Accesses

Citation

Detail

段落导航
相关文章

/