A survey on Han's conjecture
DOI:
https://doi.org/10.14244/lajm.v2i02.18Keywords:
Hochschild homology, Han's conjecture, global dimension, homology of associative algebras, finite-dimensional algebrasAbstract
In 1989, D. Happel pointed out for a possible connection between the global dimension of a finite-dimensional algebra and its Hochschild cohomology: is it true that the vanishing of Hochschild cohomology higher groups is sufficient to deduce that the global dimension is finite? After the discovery of a counterexample, Y. Han proposed, in 2006, to reformulate this question to homology. In this survey, after introducing the concepts and results involved, I present the efforts made until now towards the comprehension of Han's conjecture; which includes: examples of algebras that have been proven to satisfy it and extensions to preserve it.
References
Hochschild G. On the Cohomology Groups of an Associative Algebra. Annals of Mathematics. 1945;46(1):58-67. Available from: https://doi.org/10.2307/1969145. DOI: https://doi.org/10.2307/1969145
Cartan H, Eilenberg S. Homological Algebra. Princeton University Press; 1956. DOI: https://doi.org/10.1515/9781400883844
Happel D. Hochschild cohomology of finite—dimensional algebras. In: Séminaire d’Algèbre Paul Dubreil et Marie-Paul Malliavin. Springer Berlin Heidelberg; 1989. p. 108-26. Available from: https://doi.org/10.1007/BFb0084073. DOI: https://doi.org/10.1007/BFb0084073
Buchweitz RO, Green EL, Madsen D, Solberg Ø. Finite Hochschild cohomology without finite global dimension. Mathematical Research Letters. 2005;12(6):805-16. Available from: https://doi.org/10.4310/MRL.2005.v12.n6.a2. DOI: https://doi.org/10.4310/MRL.2005.v12.n6.a2
Skoldberg E. The Hochschild Homology of Truncated and Quadratic Monomial Algebras. Journal of the London Mathematical Society. 1999 02;59(1):76-86. Available from: https://doi.org/10.1112/S0024610799007036. DOI: https://doi.org/10.1112/S0024610799007036
Han Y. Hochschild (Co)Homology Dimension. Journal of the London Mathematical Society. 2006 06;73(3):657-68. Available from: https://doi.org/10.1112/S002461070602299X. DOI: https://doi.org/10.1112/S002461070602299X
Weibel C A. An Introduction to Homological Algebra. vol. 38 of Cambridge Studiesin Advanced Mathematics. Cambridge University Press; 1994. DOI: https://doi.org/10.1017/CBO9781139644136
Auslander M. On the Dimension of Modules and Algebras (III): Global Dimension. Nagoya Mathematical Journal. 1955;9:67-77. Available from: https://doi.org/10.1017/S0027763000023291. DOI: https://doi.org/10.1017/S0027763000023291
Lam TY. A First Course in Noncommutative Rings. vol. 131 of Graduate Texts in Mathematics. 2nd ed. Springer New York, NY; 2001. Available from: https://doi.org/10.1007/978-1-4419-8616-0. DOI: https://doi.org/10.1007/978-1-4419-8616-0
Eilenberg S, Nagao H, Nakayama T. On the Dimension of Modules and Algebras, IV: Dimension of Residue Rings of Hereditary Rings. Nagoya Mathematical Journal. 1956;10:87-95. Available from: https://doi.org/10.1017/ S002776300000009X. DOI: https://doi.org/10.1017/S002776300000009X
Assem I, Skowronski A, Simson D. Elements of the Representation Theory of Associative Algebras 1: Techniques of Representation Theory. vol. 65 of London Mathematical Society Student Texts. Cambridge University Press; 2006. DOI: https://doi.org/10.1017/CBO9780511614309
Lam TY. Lectures on Modules and Rings. vol. 198 of Graduate Texts in Mathematics. Springer New York, NY; 1999. Available from: https://doi. org/10.1007/978-1-4612-0525-8. DOI: https://doi.org/10.1007/978-1-4612-0525-8
Kassel C. Homology and cohomology of associative algebras. A concise introduction to cyclic homology. 2004. Advanced School on Non-commutative Geometry ICTP, Trieste, August 2004. Available from: https://cel. archives-ouvertes.fr/cel-00119891/.
Loday JL. Cyclic Homology. vol. 301 of Grundlehren der mathematischen Wissenschaften. 2nd ed. Springer Berlin, Heidelberg; 1998. Available from: https://doi.org/10.1007/978-3-662-11389-9. DOI: https://doi.org/10.1007/978-3-662-11389-9
Benson DJ. Representations and Cohomology II. vol. 31 of Cambridge Studies in Advanced Mathematics. Cambridge University Press; 1991.
Benson DJ. Representations and Cohomology I. vol. 30 of Cambridge Studies in Advanced Mathematics. Cambridge University Press; 1991.
Rickard J. Derived Equivalences As Derived Functors. Journal of the London Mathematical Society. 1991;s2-43(1):37-48. Available from: https://doi.org/10.1112/jlms/s2-43.1.37. DOI: https://doi.org/10.1112/jlms/s2-43.1.37
Keller B. Invariance of cyclic homology under derived equivalence. In: Representation Theory of Algebras: Seventh International Conference on Representations of Algebras, August 22-26, 1994, Cocoyoc, Mexico. vol. 18. American Mathematical Soc.; 1996. p. 353-61.
Berg CF. Structure Theorems for Basic Algebras. arXiv; 2011. Available from: https://arxiv.org/abs/1102.1100.
Keller B. Invariance and localization for cyclic homology of DG algebras. Journal of Pure and Applied Algebra. 1998;123(1):223-73. Available from: https:// doi.org/10.1016/S0022-4049(96)00085-0. DOI: https://doi.org/10.1016/S0022-4049(96)00085-0
Swan RG. The nontriviality of the restriction map in the cohomology of groups. Proc Amer Math Soc. 1960;11:885-7. Available from: https://doi.org/10. 1090/S0002-9939-1960-0124050-2. DOI: https://doi.org/10.1090/S0002-9939-1960-0124050-2
Burghelea D. The cyclic homology of the group rings. Commentarii mathematici Helvetici. 1985;60:354-65. Available from: http://eudml.org/doc/ 140019. DOI: https://doi.org/10.1007/BF02567420
Cibils C. Hochschild homology of an algebra whose quiver has no oriented cycles. In: Representation Theory I: Finite Dimensional Algebras. vol. 1177 of Lecture Notes in Mathematics. Springer-Verlag Berlin Heidelberg; 1986. p. 55-9. Available from: https://link.springer.com/content/pdf/10. 1007/BFb0075258.pdf. DOI: https://doi.org/10.1007/BFb0075258
Avramov LL, Vigue ́-Poirrier M. Hochschild homology criteria for smoothness. International Mathematics Research Notices. 1992 01;1992(1):17-25. Available from: https://doi.org/10.1155/S1073792892000035. DOI: https://doi.org/10.1155/S1073792892000035
BACH (The Buenos Aires Cyclic Homology Group). A Hochschild homology criterium for the smoothness of an algebra. Commentarii Mathematici Helvetici. 1994;69:163-8. Available from: https://doi.org/10.1007/ BF02564480. DOI: https://doi.org/10.1007/BF02564480
Han Y, Xu Y. Hochschild (Co)homology of Exterior Algebras. Communications in Algebra. 2006;35(1):115-31. Available from: https://doi.org/10.1080/00927870601041375. DOI: https://doi.org/10.1080/00927870601041375
Bergh P, Erdmann K. Homology and cohomology of quantum complete intersections. Algebra & Number Theory. 2008;2(5):501-22. Available from: https://doi.org/10.2140/ant.2008.2.501. DOI: https://doi.org/10.2140/ant.2008.2.501
Bergh PA, Madsen D. Hochschild homology and global dimension. Bulletin of the London Mathematical Society. 2009;41(3):473-82. Available from: https://doi.org/10.1112/blms/bdp018. DOI: https://doi.org/10.1112/blms/bdp018
Solotar A, Vigué-Poirrier M. Two classes of algebras with infinite Hochschild homology. Proceedings of the American Mathematical Society. 2010;138(3):861-9. Available from: https://doi.org/10.1090/ S0002-9939-09-10168-5. DOI: https://doi.org/10.1090/S0002-9939-09-10168-5
Solotar A, Suárez-Alvarez M, Vivas Q. Hochschild homology and cohomology of Generalized Weyl algebras: the quantum case. Annales de l’Institut Fourier. 2013;63(3):923-56. Available from: https://doi.org/10.5802/aif. 2780. DOI: https://doi.org/10.5802/aif.2780
Bergh PA, Madsen DO. Hochschild homology and trivial extension algebras. Proc Amer Math Soc. 2017;145(4):1475-80. ArXiv:1509.09039. Available from: https://doi.org/10.1090/proc/13363. DOI: https://doi.org/10.1090/proc/13363
Avramov LL, Iyengar S. Gaps in Hochschild cohomology imply smoothness for commutative algebras. Mathematical Research Letters. 2005;12(6):789-804. Available from: https://doi.org/10.4310/MRL.2005.v12.n6.a1. DOI: https://doi.org/10.4310/MRL.2005.v12.n6.a1
Xu Y, Han Y, Jiang W. Hochschild cohomology of truncated quiver algebras. Science in China Series A: Mathematics. 2007;50:727-36. Available from: https: //doi.org/10.1007/s11425-007-2085-x. DOI: https://doi.org/10.1007/s11425-007-2085-x
Drinfel’d VG. Quantum groups; 1987. Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798-820.
Beilinson A, Ginzburg V, Soergel W. Koszul Duality Patterns in Representation Theory. J Amer Math Soc. 1996;9:473-527. Available from: https://doi. org/10.1090/S0894-0347-96-00192-0. DOI: https://doi.org/10.1090/S0894-0347-96-00192-0
Igusa K. Notes on the no loops conjecture. Journal of Pure and Applied Algebra. 1990;69(2):161-76. Available from: https://doi.org/10.1016/0022-4049(90)90040-O. DOI: https://doi.org/10.1016/0022-4049(90)90040-O
Goodearl KR, Hodges TJ, Lenagan TH. Krull and global dimensions of Weyl algebras over division rings. Journal of Algebra. 1984;91(2):334-59. Available from: https://doi.org/10.1016/0021-8693(84)90107-8. DOI: https://doi.org/10.1016/0021-8693(84)90107-8
Farinati M A, Solotar A L, Suárez-Álvarez M. Hochschild homology and cohomology of generalized Weyl algebras. Annales de l’Institut Fourier. 2003;53(2):465-88. Available from: https://doi.org/10.5802/aif.1950. DOI: https://doi.org/10.5802/aif.1950
Cibils C, Redondo MJ, Solotar A. Han’s conjecture and Hochschild homology for null-square projective algebras. Indiana Univ Math J. 2021;70(2):639-68. Available from: https://doi.org/10.1512/iumj.2021.70.8402. DOI: https://doi.org/10.1512/iumj.2021.70.8402
Cibils C, Lanzilotta M, Marcos EN, Solotar A. Han’s conjecture for bounded extensions. Journal of Algebra. 2022;598:48-67. Available from: https://doi.org/10.1016/j.jalgebra.2022.01.022. DOI: https://doi.org/10.1016/j.jalgebra.2022.01.022
Iusenko K, MacQuarrie J. Homological properties of extensions of abstract and pseudocompact algebras; 2021. Preprint from arXiv:2108.12923.
Cibils C, Lanzilotta M, Marcos EN, Solotar A. Split bounded extension algebras and Han’s conjecture. Pacific Journal of Mathematics. 2020;307:63-77. Available from: https://doi.org/10.2140/pjm.2020.307.63. DOI: https://doi.org/10.2140/pjm.2020.307.63
Cibils C, Lanzilotta M, Marcos EN, Solotar A. Jacobi-Zariski long nearly exact sequences for associative algebras. Bulletin of the London Mathematical Society. 2021;53(6):1636-50. Available from: https://doi.org/10.1112/blms.12516. DOI: https://doi.org/10.1112/blms.12516
Cibils C, Lanzilotta M, Marcos EN, Solotar A. Deleting or adding arrows of a bound quiver algebra and Hochschild (co)homology. Proc Amer Math Soc. 2020;148:2421-32. Available from: https://doi.org/10.1090/proc/14936. DOI: https://doi.org/10.1090/proc/14936
Hochschild G. Relative Homological Algebra. Transactions of the American Mathematical Society. 1956;82(1):246-69. Available from: https://doi.org/ 10.2307/1992988. DOI: https://doi.org/10.1090/S0002-9947-1956-0080654-0
Cruz GC. Homologia de Álgebras Pseudocompactas: as fronteiras da conjectura de Han. Universidade de São Paulo; 2023. Available from: https://doi.org/ 10.11606/D.45.2023.tde-20062023-140944.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2023 Guilherme da Costa Cruz
This work is licensed under a Creative Commons Attribution 4.0 International License.
Authors may enter into separate contractual agreements for the non-exclusive distribution of the journal's published version of the work (for example, posting it in an institutional repository or publishing it in a book), with an acknowledgment of its initial publication in this journal.
