Skip to main content
SpringerLink
Log in

The Injective and Projective Leavitt Complexes

  • Conference paper
  • First Online:
Leavitt Path Algebras and Classical K-Theory

Part of the book series: Indian Statistical Institute Series ((INSIS))

  • 477 Accesses

Abstract

For a certain finite graph E, we consider the corresponding finite dimensional algebra A with radical square zero. An explicit compact generator for the homotopy category of acyclic complexes of injective (resp. projective) modules over A, called the injective (resp. projective) Leavitt complex of E, was constructed in [18] (resp. [19]). We overview the connection between the injective (resp. projective) Leavitt complex and the Leavitt path algebra of E. A differential graded bimodule structure, which is right quasi-balanced, is endowed to the injective (resp. projective) Leavitt complex in [18] (resp. [19]). We prove that the injective (resp. projective) Leavitt complex is not left quasi-balanced.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. A. Alahmadi, H. Alsulami, S.K. Jain, E. Zelmanov, Leavitt path algebras of finite Gelfand–Kirillov dimension. J. Algebra Appl. 11(6), 6 (2012)

    Article  MathSciNet  Google Scholar 

  2. G. Abrams, G. Aranda Pino, The Leavitt path algebra of a graph. J. Algebra 293(2), 319–334 (2005)

    MathSciNet  MATH  Google Scholar 

  3. P. Ara, M.A. Moreno, E. Pardo, Nonstable \(\mathbf{K}\)-theory for graph algebras. Algebr. Represent. Theory 10(2), 157–178 (2007)

    Article  MathSciNet  Google Scholar 

  4. R.O. Buchweitz, Maximal Cohen-Macaulay Modules and Tate-Cohomology Over Gorenstein Rings, (unpublished manuscript) (1987). http://hdl.handle.net/1807/16682

  5. A.I. Bondal, M.M. Kapranov, Enhanced triangulated categories. Mat. Sb. 181(5), 669–683 (1990); (English translation Math. USSR-Sb. 70(1), 93–107 (1990))

    Google Scholar 

  6. M. Bökstedt, A. Neeman, Homotopy limits in triangulated categories. Compos. Math. 86, 209–234 (1993)

    MathSciNet  MATH  Google Scholar 

  7. X.W. Chen, D. Yang, Homotopy categories, Leavitt path algebras, and Gorenstein projective modules. Int. Math. Res. Not. 10, 2597–2633 (2015)

    Article  MathSciNet  Google Scholar 

  8. J. Cuntz, W. Krieger, A class of \(C^*\)-algebras and topological Markov chains. Invent. Math. 63, 25–40 (1981)

    Article  MathSciNet  Google Scholar 

  9. J.A. Drozd, Tame and wild matrix problems, in Representation theory, II (Proceedings of the Second International Conference on Carleton University, Ottawa, 1979). Lecture Notes in Mathematics, vol. 832. (Springer, Berlin, 242–258, 1980)

    Chapter  Google Scholar 

  10. A. Grothendieck, The cohomology theory of abstract algebraic varieties, in Proceedings of the International Congress on Mathematics (Edinburgh, 1958) (Cambridge University Press, New York, 1960), pp. 103–118

    Google Scholar 

  11. D. Happel, On the derived category of a finite-dimensional algebra. Comment. Math. Helv. 62, 339–389 (1987)

    Article  MathSciNet  Google Scholar 

  12. D. Happel, Triangulated Categories in the Representation Theory of Finite-Dimensional Algebras. London Mathematical Society Lecture Note Series, vol. 119 (Cambridge University Press, Cambridge, 1988)

    Google Scholar 

  13. R. Hazrat, R. Preusser, Applications of normal forms for weighted Leavitt path algebras: simple rings and domains. Algebr. Represent. Theory, https://doi.org/10.1007/s10468-017-9674-3

    Article  MathSciNet  Google Scholar 

  14. B. Keller, Deriving DG categories. Ann. Sci. Éc. Norm. Supér. (4) 27(1), 63–102 (1994)

    Google Scholar 

  15. G.M. Kelly, Chain maps inducing zero homology maps. Proc. Camb. Philos. Soc. 61, 847–854 (1965)

    Article  MathSciNet  Google Scholar 

  16. H. Krause, The stable derived category of a Noetherian scheme. Compos. Math. 141, 1128–1162 (2005)

    Article  MathSciNet  Google Scholar 

  17. A. Kumjian, D. Pask, I. Raeburn, J. Renault, Graphs, groupoids, and Cuntz-Krieger algebras. J. Funct. Anal. 144, 505–541 (1997)

    Article  MathSciNet  Google Scholar 

  18. H. Li, The injective Leavitt complex. Algebr. Represent Theor. 21, 833–858 (2018)

    Article  MathSciNet  Google Scholar 

  19. H. Li, The projective Leavitt complex. Proc. Edinb. Math. Soc. 61, 1155–1177 (2018)

    Article  MathSciNet  Google Scholar 

  20. A. Neeman, The Grothendieck duality theorem via Bousfield’s techniques and Brown representability. J. Am. Math. Soc. 9, 205–236 (1996)

    Article  MathSciNet  Google Scholar 

  21. A. Neeman, Triangulated Categories, Annals of Mathematics Studies, vol. 148 (Princeton University Press, Princeton, NJ, 2001)

    Google Scholar 

  22. D.O. Orlov, Triangulated categories of sigularities and D-branes in Landau–Ginzburg models. Proc. Steklov Inst. Math. 246(3), 227–248 (2004)

    Google Scholar 

  23. I. Raeburn, Graph Algebras. CBMS Regional Conference Series in Mathematics, vol. 103 (The American Mathematical Society, Providence, RI, 2005)

    Google Scholar 

  24. J. Rickard, Derived categories and stable equivalence. J. Pure Appl. Algebra 61, 303–317 (1989)

    Article  MathSciNet  Google Scholar 

  25. J. Rickard, Morita theory for derived categories. J. Lond. Math. Soc. 39(2), 436–456 (1989)

    Article  MathSciNet  Google Scholar 

  26. J. Rickard, Derived equivalences as derived functors. J. Lond. Math. Soc. 43(2), 37–48 (1991)

    Article  MathSciNet  Google Scholar 

  27. A.V. Roiter, Matrix problems, in Proceedings of the International Congress of Mathematicians (Helsinki, 1978), vol. 1 (Academia Scientiarum Fennica, Helsinki 1980), pp. 319–322

    Google Scholar 

  28. S.P. Smith, Category equivalences involving graded modules over path algebras of quivers. Adv. Math. 230, 1780–1810 (2012)

    Article  MathSciNet  Google Scholar 

  29. J.-L. Verdier, Catégories dérivées, in SGA \(4\frac{1}{2}\). Lecture Notes in Mathematics, vol. 569 (Springer, Berlin, 1977), pp. 262–311

    Google Scholar 

Download references

Acknowledgements

The author thanks Xiao-Wu Chen for many helpful discussions and encouragement. This project was supported by the National Natural Science Foundation of China (No.s 11522113 and 11571329). The author thanks Dr. Ambily Ambattu Asokan for hospitality when the author was in Cochin University of Science and Technology. The author thanks the support from CIMPA and the support from Centre for Research in Mathematics and Data Science. The author thanks Roozbeh Hazrat for many helpful discussions and encouragement. The author also would like to acknowledge the support of the Australian Research Council grant DP160101481. The author thanks Pere Ara, Jie Du, and Steffen Koenig for their support and help. The author thanks the anonymous referees for their very helpful suggestions to improve this paper.

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Nanjing Normal University, Nanjing, China

    Huanhuan Li

  2. Centre for Research in Mathematics, Western Sydney University, Sydney, NSW, Australia

    Huanhuan Li

Authors
  1. Huanhuan Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Huanhuan Li .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Cochin University of Science and Technology, Cochin, Kerala, India

    A. A. Ambily

  2. Centre for Research in Mathematics, Western Sydney University, Sydney, NSW, Australia

    Roozbeh Hazrat

  3. Statistics and Mathematics Unit, Indian Statistical Institute, Bengaluru, Karnataka, India

    B. Sury

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Singapore Pte Ltd.

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Li, H. (2020). The Injective and Projective Leavitt Complexes. In: Ambily, A., Hazrat, R., Sury, B. (eds) Leavitt Path Algebras and Classical K-Theory. Indian Statistical Institute Series. Springer, Singapore. https://doi.org/10.1007/978-981-15-1611-5_4

Download citation

Publish with us

Policies and ethics

Search

Navigation