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Recollement of colimit categories and its applications. (English) Zbl 07285986
Summary: 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.
##### MSC:
 18A30 Limits and colimits (products, sums, directed limits, pushouts, fiber products, equalizers, kernels, ends and coends, etc.) 19D50 Computations of higher $$K$$-theory of rings
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##### References:
 [1] Abrams, G.; Pino, G. Aranda, The Leavitt path algebra of a graph, J. Algebra 293 (2005), 319-334 [2] Hügel, L. Angeleri; Koenig, S.; Liu, Q., Recollements and tilting objects, J. Pure Appl. Algebra 215 (2011), 420-438 [3] Ara, P.; Moreno, M. A.; Pardo, E., Nonstable $$K$$-theory for graph algebras, Algebr. Represent. Theory 10 (2007), 157-178 [4] Asadollahi, J.; Hafezi, R.; Vahed, R., On the recollements of functor categories, Appl. Categ. Struct. 24 (2016), 331-371 [5] Barot, M.; Lenzing, H., One-point extensions and derived equivalence, J. Algebra 264 (2003), 1-5 [6] Beilinson, A. A.; Bernstein, J.; Deligne, P., Faisceaux pervers, Analysis and Topology on Singular Spaces I Astérisque 100. Société Mathématique de France, Paris (1982), 5-171 French [7] Bergman, G. M., Direct limits and fixed point sets, J. Algebra 292 (2005), 592-614 [8] Chen, Q.; Lin, Y., Recollements of extension algebras, Sci. China, Ser. A 46 (2003), 530-537 [9] Cline, E.; Parshall, B.; Scott, L., Algebraic stratification in representation categories, J. Algebra 117 (1988), 504-521 [10] Franjou, V.; Pirashvili, T., Comparison of abelian categories recollements, Doc. Math. 9 (2004), 41-56 [11] Fuchs, L.; Göbel, R.; Salce, L., On inverse-direct systems of modules, J. Pura Appl. Algebra 214 (2010), 322-331 [12] Grothendieck, A., 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 French [13] Guo, X. J.; Li, L. B., $$K_1$$ group of finite dimensional path algebra, Acta Math. Sin., Engl. Ser. 17 (2001), 273-276 [14] Happel, D., Reduction techniques for homological conjectures, Tsukuba J. Math. 17 (1993), 115-130 [15] Li, L. P., Derived equivalences between triangular matrix algebras, Commun. Algebra 46 (2018), 615-628 [16] Mahmood, S. J., Limimts and colimits in categories of d. g. near-rings, Proc. Edinb. Math. Soc., II. Ser. 23 (1980), 1-7 [17] Miyachi, J., Localization of triangulated categories and derived categories, J. Algebras 141 (1991), 463-483 [18] Parshall, B. J.; Scott, L. L., 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 [19] Quillen, D., Higher algebraic $$K$$-theory. I, Higher $$K$$-Theories Lecture Notes in Mathematics 341. Springer, Berlin (1973) [20] Xue, R.; Yan, Y.; Chen, Q., On colimit-categories, J. Math., Wuhan Univ. 32 (2012), 439-446
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