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Hochschild cohomology of special biserial algebras

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Abstract

Based on a four-term exact sequence, the formulae on the dimensions of the first and the second Hochschild cohomology groups of special biserial algebras with normed bases are obtained in terms of combinatorics.

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References

  1. Hochschild G. On the cohomology groups of an associative algebra. Ann Math, 46: 58–67 (1945)

    Article  MathSciNet  Google Scholar 

  2. Gerstenhaber M. On the deformation of rings and algebras. Ann Math, 79: 59–103 (1964)

    Article  MathSciNet  Google Scholar 

  3. Assem I, de la Peña J A. The foundamental groups of a triangular algebra. Comm Algebra, 24(1): 187–208 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  4. Happel D. Hochschild cohomology of finite dimensional algebras. Lecture Notes in Mathematics, Vol 1404. Berlin: Springer, 1989, 108–126

    Google Scholar 

  5. Skowroński A. Simply connected algebras and Hochschild cohomologies. Proc ICRA VI, Ottawa, 1992, Canad Math Proc, 14: 431–447 (1993)

    Google Scholar 

  6. Geiss Ch, de la Peña J A. On the deformation theory of algebras. Manuscripta Math, 88: 191–208 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  7. Cibils C. Cohomology of incidence algebras and simplicial complexes. J Pure Appl Algebra, 56: 221–232 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  8. Gerstenhaber M, Schack S P. Simplicial cohomology is Hochschild cohomology. J Pure Appl Algebra, 30: 143–156 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  9. Cibils C. Rigidity of truncated quiver algebras. Adv Math, 79: 18–42 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  10. Locateli A C. Hochschild cohomology of truncated quiver algebras. Comm Algebra, 27: 645–664 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  11. Zhang P. Hochschild cohomology of truncated algebras. Sci China Ser A-Math (in Chinese), 24: 1121–1125 (1994)

    Google Scholar 

  12. Zhang P. Hochschild cohomology of truncated basic cycle. Sci China Ser A-Math, 40: 1272–1278 (1997)

    Article  MATH  Google Scholar 

  13. Xu Y G, Han Y, Jiang W F. On Hochschild cohomology of truncated algebras. Sci China Ser A-Math, 50(5): 727–736 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  14. Xu Y G, Han Y. Hochschild (co)homology of exterior algebras. Comm Algebra, 35(1): 115–131 (2007)

    MATH  MathSciNet  Google Scholar 

  15. Bautista R, Gabriel P, Roiter A V, Salmerón L. Representation-finite algebras and multiplicative bases. Invent Math, 81: 217–285 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  16. Ringel C M. The indecomposable representations of the dihedral 2-group. Math Ann, 214: 19–34 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  17. Skowroński A, Waschbüsch J. Representation-finite biserial algebras. J Reine Angew Math, 345: 172–181 (1983)

    MATH  MathSciNet  Google Scholar 

  18. Wald W, J Waschbüsch. Tame biserial algebras. J Algebra, 95: 480–500 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  19. Erdmann K. Blocks of tame representation type and related algebras. Lecture Notes in Mathematics, Vol 1428. Berlin: Springer-Verlag, 1990

    Google Scholar 

  20. Gelfand I M, Ponomarev V A. Indecomposable representations of the Lorentz group. Usp Math Nask, 23: 3–60 (1968)

    MathSciNet  Google Scholar 

  21. Baues H J, Hennes M. The homotopy classfication of (n-1)-connected (n + 3)-dimensional polyhedra, n ⩾ 4. Topol ogy, 30: 373–408 (1992)

    Article  MathSciNet  Google Scholar 

  22. Drozd Yu A, Greuel G M. Vector bundles on singular projective curves. In: Applications of Algebraic Geometry to Coding Theory, Physics and Computation. Kluwer Academic Publishers, 2001, 1-15

  23. Huisgen-Zimmermann B. The phantom menace in representation theory. Contemp Math, 259: 247–278 (2000)

    MathSciNet  Google Scholar 

  24. Schröer J. On the Krull-Gabriel dimension of an algebra. Math Z, 233: 287–303 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  25. Brown P. The Ext-algebra of a representation-finite biserial algebra. J Algebra, 221: 611–629 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  26. Butler M C R, King A D. Minimal resolutions of algebras. J Algebra, 212: 323–362 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  27. Cibils C. Rigid monomial algebras. Math Ann, 289: 95–109 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  28. Gerstenhaber M, Schack S P. Relative Hochschild cohomology, rigid algebras, and the bockstein. J Pure Appl Algebra, 43: 53–74 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  29. Ringel C M. Combinatorial representation theory: history and future. Proc ICRA IX, Beijing, 2000. Beijing: Beijing Normal University Press, 2002, 122–144

    Google Scholar 

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Authors and Affiliations

  1. Faculty of Mathematics & Computer Science, Hubei University, Wuhan, 430062, China

    Yun-ge Xu

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  1. Yun-ge Xu
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Correspondence to Yun-ge Xu.

Additional information

This work was partially supported by the National Natural Science Foundation of China (Grant No. 10501010) and the Important Foundation of Hubei Provincial Department of Education (D200510005)

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Xu, Yg. Hochschild cohomology of special biserial algebras. SCI CHINA SER A 50, 1117–1128 (2007). https://doi.org/10.1007/s11425-007-0063-y

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