Abstract
For a finite quiver Q without sinks, we consider the corresponding finite dimensional algebra A with radical square zero. We construct an explicit compact generator for the homotopy category of acyclic complexes of injective A-modules. We call such a generator the injective Leavitt complex of Q. This terminology is justified by the following result: the differential graded endomorphism algebra of the injective Leavitt complex of Q is quasi-isomorphic to the Leavitt path algebra of Q. Here, the Leavitt path algebra is naturally \(\mathbb {Z}\)-graded and viewed as a differential graded algebra with trivial differential.
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Alahmadi, A., Alsulami, H., Jain, S.K., Zelmanov, E.: Leavitt path algebras of finite Gelfand-Kirillov dimension. J. Algebra Appl. 11(6), 6 (2012)
Abrams, G., Aranda Pino, G.: The Leavitt path algebra of a graph. J. Algebra 293(2), 319–334 (2005)
Ara, P., Moreno, M.A., Pardo, E.: Nonstable K-theory for graph algebras. Algebr. Represent. Theory 10(2), 157–178 (2007)
Assem, I., Simson, D., Skowroński, A.: Elements of the Representation Theory of Associative Algebras. Techniques of Representation Theory. London Math. Soc. Stud. Texts, vol. 1, p 65. Cambridge University Press, Cambridge (2006)
Buchweitz, R.O.: Maximal Cohen-Macaulay Modules and Tate-Cohomology over Gorenstein Rings, unpublished manuscrip, available at: http://hdl.handle.net/1807/16682 (1987)
Bökstedt, M., Neeman, A.: Homotopy limits in triangulated categories. Compos. Math. 86, 209–234 (1993)
Chen, X.W.: The sigularity category of an algebra with radical square zero. Doc. Math. 16, 921–936 (2011)
Chen, X.W.: Irreducible representations of Leavitt path algebras. Forum Math. 27(1), 549–574 (2015)
Chen, X.W., Yang, D.: Homotopy categories, Leavitt path algebras, and Gorenstein projective modules. Int. Math. Res. Not. 10, 2597–2633 (2015)
Cuntz, J., Krieger, W.: A class of C ∗-algebras and topological Markov chains. Invent. Math. 63, 25–40 (1981)
Keller, B.: Deriving DG categories. Ann. Sci. Éc. Norm. Supér. (4) 27(1), 63–102 (1994)
Krause, H.: The stable derived category of a Noetherian scheme. Compos. Math. 141, 1128–1162 (2005)
Kumjian, A., Pask, D., Raeburn, I., Renault, J.: Graphs, groupoids, and Cuntz–Krieger algebras. J. Funct. Anal. 144, 505–541 (1997)
Neeman, A.: The Grothendieck duality theorem via Bousfield’s techniques and Brown representability. J. Amer. Math. Soc. 9, 205–236 (1996)
Orlov, D.O.: Triangulated categories of sigularities and D-branes in Landau-Ginzburg models. Proc. Steklov Inst. Math. 246(3), 227–248 (2004)
Raeburn, I.: Graph Algebras CBMS Regional Conference Series in Mathematics, vol. 103. The American Mathematical Society, Providence (2005)
Smith, S.P.: Category equivalences involving graded modules over path algebras of quivers. Adv. Math. 230, 1780–1810 (2012)
Thompson, P.: Stable local cohomology. Comm. Algebra 45(1), 198–226 (2017)
Tomforde, M.: Uniqueness theorems and ideal structure for Leavitt path algebras. J. Algebra 318, 270–299 (2007)
Acknowledgements
The author thanks her supervisor Professor Xiao-Wu Chen for inspiring discussions and encouragement. This project was supported by the National Natural Science Foundation of China (No.s 11522113 and 11571329). The author also gratefully acknowledges the support of Australian Research Council grant DP160101481.
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Presented by Henning Krause.
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Li, H. The Injective Leavitt Complex. Algebr Represent Theor 21, 833–858 (2018). https://doi.org/10.1007/s10468-017-9741-9
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DOI: https://doi.org/10.1007/s10468-017-9741-9