Czechoslovak Mathematical Journal, Vol. 70, No. 4, pp. 1147-1160, 2020


Recollement of colimit categories and its applications

Ju Huang, Qinghua Chen, Chunhuan Lai

Received May 29, 2019.   Published online September 18, 2020.

Abstract:  We give an explicit recollement for a cocomplete abelian category and its colimit category. We obtain some applications on Leavitt path algebras, derived equivalences and $K$-groups.
Keywords:  colimit category; recollement; Leavitt path algebra; $K_i$ group
Classification MSC:  18A30, 19D50
DOI:  10.21136/CMJ.2020.0240-19


References:
[1] G. Abrams, G. Aranda Pino: The Leavitt path algebra of a graph. J. Algebra 293 (2005), 319-334. DOI 10.1016/j.jalgebra.2005.07.028 | MR 2172342 | Zbl 1119.16011
[2] L. Angeleri Hügel, S. Koenig, Q. Liu: Recollements and tilting objects. J. Pure Appl. Algebra 215 (2011), 420-438. DOI 10.1016/j.jpaa.2010.04.027 | MR 2738361 | Zbl 1223.18008
[3] P. Ara, M. A. Moreno, E. Pardo: Nonstable $K$-theory for graph algebras. Algebr. Represent. Theory 10 (2007), 157-178. DOI 10.1007/s10468-006-9044-z | MR 2310414 | Zbl 1123.16006
[4] J. Asadollahi, R. Hafezi, R. Vahed: On the recollements of functor categories. Appl. Categ. Struct. 24 (2016), 331-371. DOI 10.1007/s10485-015-9399-6 | MR 3516076 | Zbl 1360.18020
[5] M. Barot, H. Lenzing: One-point extensions and derived equivalence. J. Algebra 264 (2003), 1-5. DOI 10.1016/S0021-8693(03)00124-8 | MR 1980681 | Zbl 1060.16011
[6] A. A. Beilinson, J. Bernstein, P. Deligne: Faisceaux pervers. Analysis and Topology on Singular Spaces I. Astérisque 100. Société Mathématique de France, Paris (1982), 5-171. (In French.) MR 0751966 | Zbl 1390.14055
[7] G. M. Bergman: Direct limits and fixed point sets. J. Algebra 292 (2005), 592-614. DOI 10.1016/j.jalgebra.2005.08.002 | MR 2172170 | Zbl 1088.18001
[8] Q. Chen, Y. Lin: Recollements of extension algebras. Sci. China, Ser. A 46 (2003), 530-537. DOI 10.1007/BF02884025 | MR 2014485 | Zbl 1215.18014
[9] E. Cline, B. Parshall, L. Scott: Algebraic stratification in representation categories. J. Algebra 117 (1988), 504-521. DOI 10.1016/0021-8693(88)90123-8 | MR 0957457 | Zbl 0659.18011
[10] V. Franjou, T. Pirashvili: Comparison of abelian categories recollements. Doc. Math. 9 (2004), 41-56. MR 2054979 | Zbl 1060.18008
[11] L. Fuchs, R. Göbel, L. Salce: On inverse-direct systems of modules. J. Pura Appl. Algebra 214 (2010), 322-331. DOI 10.1016/j.jpaa.2009.05.003 | MR 2558741 | Zbl 1181.13009
[12] A. Grothendieck: Groupes de classes des categories abeliennes et triangulees. Complexes parfaits. Séminaire de Géométrie Algébrique du Bois-Marie 1965-66 SGA 5. Lecture Notes in Mathematics 589. Springer, Berlin (1977), 351-371. (In French.) DOI 10.1007/BFb0096809 | MR 0491704 | Zbl 0345.00011
[13] X. J. Guo, L. B. Li: $K_1$ group of finite dimensional path algebra. Acta Math. Sin., Engl. Ser. 17 (2001), 273-276. DOI 10.1007/s101149900010 | MR 1830928 | Zbl 0986.19002
[14] D. Happel: Reduction techniques for homological conjectures. Tsukuba J. Math. 17 (1993), 115-130. DOI 10.21099/tkbjm/1496162134 | MR 1233117 | Zbl 0809.16021
[15] L. P. Li: Derived equivalences between triangular matrix algebras. Commun. Algebra 46 (2018), 615-628. DOI 10.1080/00927872.2017.1327051 | MR 3764883 | Zbl 06875436
[16] S. J. Mahmood: Limimts and colimits in categories of d. g. near-rings. Proc. Edinb. Math. Soc., II. Ser. 23 (1980), 1-7. DOI 10.1017/S0013091500003539 | MR 0582016 | Zbl 0414.16023
[17] J. Miyachi: Localization of triangulated categories and derived categories. J. Algebras 141 (1991), 463-483. DOI 10.1016/0021-8693(91)90243-2 | MR 1125707 | Zbl 0739.18006
[18] B. J. Parshall, L. L. Scott: Derived categories, quasi-hereditary algebras, and algebraic groups. Proceedings of the Ottawa-Moosonee Workshop in Algebra. Mathematical Lecture Note Series. Carlton University, Ottawa (1988), 1-104. Zbl 0711.18002
[19] D. Quillen: Higher algebraic $K$-theory. I. Higher $K$-Theories. Lecture Notes in Mathematics 341. Springer, Berlin (1973). DOI 10.1007/BFb0067053 | MR 0338129 | Zbl 0292.18004
[20] R. Xue, Y. Yan, Q. Chen: On colimit-categories. J. Math., Wuhan Univ. 32 (2012), 439-446. MR 2963903 | Zbl 1265.18002

Affiliations:   Ju Huang, School of Mathematics and Statistics, Minnan Normal University, Jiangbin Rd, Xiangcheng District, Zhangzhou, 363000, P. R. China, e-mail: Hj20140429@163.com; Qinghua Chen (corresponding author), College of Mathematics and Informatics, Qishan Campus, Fujian Normal University, Minhou Shang, Fuzhou 350117, P. R. China; Chunhuan Lai, College of Mathematics and Informatics, Qishan Campus, Fujian Normal University, Minhou Shang, Fuzhou 350117, and No. 5 Middle School, 49 Guitan Alley, Licheng District, Quanzhou, 362000, P. R. China, e-mail: 920665376@qq.com


 
PDF available at: