三角矩阵环上的余挠对与稳定范畴

doi:10.12386/A20210007

鏁板瀛︽姤鈥衡�� 2022, Vol. 65 鈥衡�� Issue (5): 849-858.DOI: 10.12386/A20210007

鈥� 璁烘枃鈥� 涓婁竴绡� 涓嬩竴绡�

涓夎鐭╅樀鐜笂鐨勪綑鎸犲涓庣ǔ瀹氳寖鐣�

浠讳紵, 寮犳槬闇�

閲嶅簡甯堣寖澶у鏁板绉戝瀛﹂櫌閲嶅簡 401331

鏀剁鏃ユ湡:2021-01-13 淇洖鏃ユ湡:2021-06-24 鍑虹増鏃ユ湡:2022-09-15 鍙戝竷鏃ユ湡:2021-05-15
閫氳浣滆��: 寮犳槬闇�,E-mail:cxzhang@cqnu.edu.cn
浣滆�呯畝浠�:浠讳紵,E-mail:wren@cqnu.edu.cn
鍩洪噾璧勫姪:
鍥藉鑷劧绉戝鍩洪噾璧勫姪椤圭洰锛�11871125锛夛紱閲嶅簡甯傝嚜鐒剁瀛﹀熀閲戯紙cstc2018jcyjAX0541锛夊強閲嶅簡甯傛暀濮旂瀛︽妧鏈爺绌堕」鐩紙KJQN201800509锛�

Cotorsion Pairs and Stable Categories over Triangular Matrix Rings

Wei REN Chun, Xia ZHANG

School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, P. R. China

Received:2021-01-13 Revised:2021-06-24 Online:2022-09-15 Published:2021-05-15

鎽樿/Abstract

鎽樿锛� 閫氬父鐢ㄢ�滃皬鐜��$A$锛�$B$涓婃ā鐨勬�ц川鍒荤敾涓夎鐭╅樀鐜�$T=\left (\begin{smallmatrix}A&U\\0&B\end{smallmatrix}\right)$涓婄殑妯�.鎴戜滑鍙嶈繃鏉ョ敤鈥滃ぇ鐜��$T$涓婄殑妯¤寖鐣寸殑妯″瀷缁撴瀯锛屽埢鐢讳簡鈥滃皬鐜��$A$锛�$B$涓婄殑Gorenstein鎶曞皠妯＄殑绋冲畾鑼冪暣$\underline{\mathcal{GP}}(A)$锛�$\underline{\mathcal{GP}}(B)$.涓烘锛岄鍏堣寮曞叆Gorenstein鎶曞皠$T$-妯¤寖鐣寸殑涓や釜瀛愯寖鐣达紝骞舵瀯閫犱簡涓庝箣瀵瑰簲鐨勪袱涓畬澶囦綑鎸犲.姝ゅ锛屽皢妯＄殑浣欐尃瀵规帹骞垮埌澶嶅舰鑼冪暣锛屽苟鐮旂┒浜嗗褰㈢殑鍚屼鸡鑼冪暣涓殑绛変环涓庣矘鍚�.

鍏抽敭璇�: 浣欐尃瀵�, Gorenstein鎶曞皠妯�, 绋冲畾鑼冪暣, 鍚屼鸡鑼冪暣

Abstract: In general, the properties of modules over a triangular matrix ring $T=\left(\begin{smallmatrix}A & U \\0 & B\end{smallmatrix}\right)$ are studied via modules over diagonal "small rings" $A$ and $B$. However, we use model structures on the category of $T$-modules to characterize the stable categories $\underline{\mathcal{GP}}(A)$, $\underline{\mathcal{GP}}(B)$ of Gorenstein projective modules over $A$ and $B$. To this end, we introduce two subcategories of Gorenstein $T$-modules, and obtain two corresponding complete cotorsion pairs. Moreover, cotorsion pairs of modules are lifted to $T$-complexes, and the equivalences and recollements of homotopy categories of complexes are studied.

Key words: cotorsion pair, Gorenstein projective module, stable category, homotopy category

涓浘鍒嗙被鍙�:

O154.2

浠讳紵, 寮犳槬闇�. 涓夎鐭╅樀鐜笂鐨勪綑鎸犲涓庣ǔ瀹氳寖鐣碵J]. 鏁板瀛︽姤, 2022, 65(5): 849-858.

Wei REN Chun, Xia ZHANG. Cotorsion Pairs and Stable Categories over Triangular Matrix Rings[J]. Acta Mathematica Sinica, Chinese Series, 2022, 65(5): 849-858.

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閾炬帴鏈枃: https://actamath.cjoe.ac.cn/Jwk_sxxb_cn/CN/10.12386/A20210007

https://actamath.cjoe.ac.cn/Jwk_sxxb_cn/CN/Y2022/V65/I5/849

鍙傝�冩枃鐚�

[1] Aldrich S. T., Enochs E. E., Envelopes and covers by modules of finite injective and projective dimensions, J. Algebra, 2001, 242:447-459.
[2] Auslander M., Reiten I., Smalø S. O., Representation Theory of Artin Algebras, Cambridge Studies in Advanced Mathematics, Vol. 36, Cambridge University Press, Cambridge, 1995.
[3] Becker H., Models for singularity categories, Adv. Math., 2014, 254:187-232.
[4] Beligiannis A., Reiten I., Homological and Homotopical Aspects of Torsion Theories, Mem. Amer. Math. Soc., 188(883), 2007.
[5] Bravo D., Gillespie J., Hovey M., The stable category of a general ring, arXiv:1405.5768.
[6] Chen X. W., Singularity categories, Schur functors and triangular matrix rings, Algebra Represent. Theory, 2009, 12:181-191.
[7] Eklof P., Trlifaj J., How to make Ext vanish, Bull. London Math. Soc., 2001, 23:41-51.
[8] Enochs E. E., Cortés-Izurdiaga M., Torrecillas B., Gorenstein conditions over trianglar matrix rings, J. Pure Appl. Algebra, 2014, 218:1544-1554.
[9] Enochs E. E., Jenda O. M. G., Relative homological algebra, De Gruyter Expositions in Mathematics, No. 30, New York:Walter De Gruyter, 2000.
[10] Enochs E. E., López-Ramos J. A., Kaplansky classes, Rend. Sem. Mat. Univ. Padova, 2002, 107:67-79.
[11] Eshraghi H., Hafezi R., Salarian S., et al., Gorenstein projective modules over triangular matrix rings, Algebra Colloq., 2016, 23(1):97-104.
[12] Gillespie J., Kaplansky classes and derived categories, Math. Z., 2007, 257:811-843.
[13] Gillespie J., Gorenstein complexes and recollements from cotorsion pairs, Adv. Math., 2016, 291:859-911.
[14] Haghany A., Varadarajan K., Study of modules over formal triangular matrix rings, J. Pure Appl. Algebra, 2000, 147(1):41-58.
[15] Hovey M., Model Categories, Mathematical Surveys and Monographs, Vol. 63, Amer. Math. Soc., 1999.
[16] Hovey M., Cotorsion pairs, model category structures, and representation theory, Math. Z., 2002, 241:553-592.
[17] Kaplansky I., Projective modules, Ann. Math., 1958, 68:372-377.
[18] Li H. H., Zheng Y. F., Hu J. S., et al., Gorenstein projective modules and recollement over triangular matrix rings, Comm. Algebra, 2020, 48(11):4932-4947.
[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, 2019, 47(7):2727-2741.
[21] Zhang P., Gorenstein-projective modules and symmetric recollements, J. Algebra, 2013, 388:65-80.
[22] Zhang P., Triangulated Categories and Derived Categories (in Chinese), Science Press, Beijing, 2015.
[23] Zhu R. M., Peng Y. Y., Ding N. Q., Recollements associated to cotorsion pairs over upper triangular matrix rings, Publ. Math. Debrecen, 2021, 98:83-113.

涓夎鐭╅樀鐜笂鐨勪綑鎸犲涓庣ǔ瀹氳寖鐣�

Cotorsion Pairs and Stable Categories over Triangular Matrix Rings

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[5]	缃楃鑺�, 灞呰吘闇�, 鍚寸編浜�. rep(A₃, I, A)涓殑Gorenstein鎶曞皠妯�[J]. 鏁板瀛︽姤, 2015, 58(1): 115-124.
[6]	闄堝悏璞�. 娑夊強杩瑰悓浼﹁寖鐣寸殑鍑犲浼撮殢鍑藉瓙[J]. 鏁板瀛︽姤, 1998, 41(3): -.