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Module theory over Leavitt path algebras and K-theory

Profile image of Pere AraPere Ara

2010

https://doi.org/10.1016/J.JPAA.2009.10.001
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31 pages

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Abstract

Let k be a field and let E be a finite quiver. We study the structure of the finitely presented modules of finite length over the Leavitt path algebra L k (E) and show its close relationship with the finite-dimensional representations of the inverse quiver E of E, as well as with the class of finitely generated P k (E)-modules M such that Tor P k (E) q (k |E 0 | , M ) = 0 for all q, where P k (E) is the usual path algebra of E. By using these results we compute the higher K-theory of the von Neumann regular algebra Q k (E) = L k (E)Σ −1 , where Σ is the set of all square matrices over P k (E) which are sent to invertible matrices by the augmentation map ǫ : P k (E) → k |E 0 | .

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References (33)

  1. G. Abrams and G. Aranda Pino. The Leavitt path algebra of a graph. J. Algebra, 293(2):319-334, 2005.
  2. F. Albrecht. On projective modules over semi-hereditary rings. Proc. Amer. Math. Soc., 12:638- 639, 1961.
  3. P. Ara. Finitely presented modules over Leavitt algebras. J. Pure Appl. Algebra, 191(1-2):1-21, 2004.
  4. P. Ara and M. Brustenga. The regular algebra of a quiver. J. Algebra, 309(1):207-235, 2007.
  5. P. Ara and M. Brustenga. Mixed quiver algebras. 2008. Preprint.
  6. P. Ara, M. Brustenga, and G. Cortiñas. K-theory for Leavitt path algebras. 2009. Preprint.
  7. P. Ara and W. Dicks. Universal localizations embedded in power-series rings. Forum Math., 19(2):365-378, 2007.
  8. P. Ara, K. R. Goodearl, and E. Pardo. K 0 of purely infinite simple regular rings. K-Theory, 26(1):69-100, 2002.
  9. P. Ara, M. A. Moreno, and E. Pardo. Nonstable K-theory for graph algebras. Algebr. Represent. Theory, 10:157-178, 2007.
  10. M. Auslander, I. Reiten, and S. O. Smalø. Representation theory of Artin algebras, volume 36 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1997. Corrected reprint of the 1995 original.
  11. G. M. Bergman and W. Dicks. Universal derivations and universal ring constructions. Pacific J. Math., 79(2):293-337, 1978.
  12. A. J. Berrick and M. E. Keating. Categories and modules with K-theory in view, volume 67 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2000.
  13. P. M. Cohn. Free rings and their relations, volume 19 of London Mathematical Society Monographs. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London, second edition, 1985.
  14. J. Cuntz. Simple C * -algebras generated by isometries. Comm. Math. Phys., 57(2):173-185, 1977.
  15. J. Cuntz and W. Krieger. A class of C * -algebras and topological Markov chains. Invent. Math., 56(3):251-268, 1980.
  16. W. Dicks. Private communication. November 2002.
  17. M. Farber and P. Vogel. The Cohn localization of the free group ring. Math. Proc. Cambridge Philos. Soc., 111(3):433-443, 1992.
  18. S. M. Gersten. K-theory of free rings. Comm. Algebra, 1:39-64, 1974.
  19. T. Y. Lam. Lectures on modules and rings, volume 189 of Graduate Texts in Mathematics. Springer- Verlag, New York, 1999.
  20. W. G. Leavitt. Modules without invariant basis number. Proc. Amer. Math. Soc., 8:322-328, 1957.
  21. W. G. Leavitt. The module type of a ring. Trans. Amer. Math. Soc., 103:113-130, 1962.
  22. J. Lewin. Free modules over free algebras and free group algebras: The Schreier technique. Trans. Amer. Math. Soc., 145:455-465, 1969.
  23. A. Neeman. Nonconmutative localisation in algebraic K-theory. II. Adv. Math., 213(1):785-819, 2007.
  24. A. Neeman and A. Ranicki. Noncommutative localization and chain complexes I. Algebraic K- and L-theory. 2001. arXiv:math/0109118v1 [math.RA].
  25. A. Neeman and A. Ranicki. Noncommutative localisation in algebraic K-theory. I. Geom. Topol., 8:1385-1425 (electronic), 2004.
  26. A. Ranicki and D. Sheiham. Blanchfield and Seifert algebra in high-dimensional boundary link theory. I. Algebraic K-theory. Geom. Topol., 10:1761-1853 (electronic), 2006.
  27. J. Rosenberg. Algebraic K-theory and its applications, volume 147 of Graduate Texts in Mathe- matics. Springer-Verlag, New York, 1994.
  28. A. H. Schofield. Representation of rings over skew fields, volume 92 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 1985.
  29. D. Sheiham. Invariants of boundary link cobordism. II. The Blanchfield-Duval form. In Non- commutative localization in algebra and topology, volume 330 of London Math. Soc. Lecture Note Ser., pages 143-219. Cambridge Univ. Press, Cambridge, 2006.
  30. M. Siles Molina. Algebras of quotients of path algebras. J. Algebra, 319(12):5265-5278, 2008.
  31. B. Stenström. Rings of quotients. Springer-Verlag, New York, 1975. Die Grundlehren der Mathe- matischen Wissenschaften, Band 217, An introduction to methods of ring theory.
  32. R. G. Swan. Algebraic K-theory. Lecture Notes in Mathematics, No. 76. Springer-Verlag, Berlin, 1968.
  33. C.A. Weibel. An introduction to algebraic K-theory. A forthcoming graduate textbook, see http://www.math.rutgers.edu/~weibel/Kbook.html.

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