Geodesic
Hochschild (co)homology of Yoneda algebras of reconstruction algebras of type ${\mathbf {A}}_{1}$
Czechoslovak Mathematical Journal, Tome 65 (2015) no. 4, pp. 1085-1099
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

The reconstruction algebra is a generalization of the preprojective algebra, and plays important roles in algebraic geometry and commutative algebra. We consider the homological property of this class of algebras by calculating the Hochschild homology and Hochschild cohomology. Let $\Lambda _{t}$ be the Yoneda algebra of a reconstruction algebra of type ${\mathbf {A}}_{1}$ over a field $\k $. In this paper, a minimal projective bimodule resolution of $\Lambda _{t}$ is constructed, and the $\k $-dimensions of all Hochschild homology and cohomology groups of $\Lambda _{t}$ are calculated explicitly.
The reconstruction algebra is a generalization of the preprojective algebra, and plays important roles in algebraic geometry and commutative algebra. We consider the homological property of this class of algebras by calculating the Hochschild homology and Hochschild cohomology. Let $\Lambda _{t}$ be the Yoneda algebra of a reconstruction algebra of type ${\mathbf {A}}_{1}$ over a field $\k $. In this paper, a minimal projective bimodule resolution of $\Lambda _{t}$ is constructed, and the $\k $-dimensions of all Hochschild homology and cohomology groups of $\Lambda _{t}$ are calculated explicitly.
DOI : 10.1007/s10587-015-0229-7
Classification : 16E40, 16G10
Keywords: Hochschild cohomology; reconstruction algebra; Yoneda algebra 
@article{10_1007_s10587_015_0229_7,
     author = {Hou, Bo and Guo, Yanhong},
     title = {Hochschild (co)homology of {Yoneda} algebras of reconstruction algebras of type ${\mathbf {A}}_{1}$},
     journal = {Czechoslovak Mathematical Journal},
     pages = {1085--1099},
     year = {2015},
     volume = {65},
     number = {4},
     doi = {10.1007/s10587-015-0229-7},
     mrnumber = {3441337},
     zbl = {06537712},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1007/s10587-015-0229-7/}
}
TY  - JOUR
AU  - Hou, Bo
AU  - Guo, Yanhong
TI  - Hochschild (co)homology of Yoneda algebras of reconstruction algebras of type ${\mathbf {A}}_{1}$
JO  - Czechoslovak Mathematical Journal
PY  - 2015
SP  - 1085
EP  - 1099
VL  - 65
IS  - 4
UR  - http://geodesic.mathdoc.fr/articles/10.1007/s10587-015-0229-7/
DO  - 10.1007/s10587-015-0229-7
LA  - en
ID  - 10_1007_s10587_015_0229_7
ER  - 
%0 Journal Article
%A Hou, Bo
%A Guo, Yanhong
%T Hochschild (co)homology of Yoneda algebras of reconstruction algebras of type ${\mathbf {A}}_{1}$
%J Czechoslovak Mathematical Journal
%D 2015
%P 1085-1099
%V 65
%N 4
%U http://geodesic.mathdoc.fr/articles/10.1007/s10587-015-0229-7/
%R 10.1007/s10587-015-0229-7
%G en
%F 10_1007_s10587_015_0229_7
Hou, Bo; Guo, Yanhong. Hochschild (co)homology of Yoneda algebras of reconstruction algebras of type ${\mathbf {A}}_{1}$. Czechoslovak Mathematical Journal, Tome 65 (2015) no. 4, pp. 1085-1099. doi: 10.1007/s10587-015-0229-7

[1] Avramov, L. L., Vigué-Poirrier, M.: Hochschild homology criteria for smoothness. Int. Math. Res. Not. 1992 (1992), 17-25. | DOI | MR | Zbl

[2] Beilinson, A., Ginzburg, V., Soergel, W.: Koszul duality patterns in representation theory. J. Am. Math. Soc. 9 (1996), 473-527. | DOI | MR | Zbl

[3] Bergh, P. A., Madsen, D.: Hochschild homology and global dimension. Bull. Lond. Math. Soc. 41 (2009), 473-482. | DOI | MR | Zbl

[4] Brieskorn, E.: Rationale Singularitäten komplexer Flächen. Invent. Math. 4 German (1968), 336-358. | DOI | MR | Zbl

[5] Buchweitz, R.-O., Green, E. L., Madsen, D., Solberg, {Ø.: Finite Hochschild cohomology without finite global dimension. Math. Res. Lett. 12 (2005), 805-816. | DOI | MR | Zbl

[6] Butler, M. C. R., King, A. D.: Minimal resolutions of algebras. J. Algebra 212 (1999), 323-362. | DOI | MR | Zbl

[7] Cartan, H., Eilenberg, S.: Homological Algebra. Princeton Mathematical Series, Vol. 19 Princeton University Press 15, Princeton (1956). | MR | Zbl

[8] Cibils, C.: Rigidity of truncated quiver algebras. Adv. Math. 79 (1990), 18-42. | DOI | MR | Zbl

[9] Gerstenhaber, M.: On the deformation of rings and algebras. Ann. Math. (2) 79 (1964), 59-103. | DOI | MR | Zbl

[10] Green, E. L.: Noncommutative Gröbner bases, and projective resolutions. Computational Methods for Representations of Groups and Algebras. Proc. of the Euroconf., Essen, Germany, 1997 Progr. Math. 173 Birkhäuser, Basel (1999), 29-60 P. Dr{ä}xler et al. | MR | Zbl

[11] Green, E. L., Hartman, G., Marcos, E. N., Solberg, {Ø.: Resolutions over Koszul algebras. Arch. Math. 85 (2005), 118-127. | DOI | MR | Zbl

[12] Green, E., Huang, R. Q.: Projective resolutions of straightening closed algebras generated by minors. Adv. Math. 110 (1995), 314-333. | DOI | MR | Zbl

[13] Han, Y.: Hochschild (co)homology dimension. J. Lond. Math. Soc., (2) 73 (2006), 657-668. | DOI | MR | Zbl

[14] Happel, D.: Hochschild cohomology of finite-dimensional algebras. Séminaire D'Algèbre Paul Dubreil et Marie-Paul Malliavin, Proc. of the Seminar, Paris, 1987-1988 Lecture Notes in Math. 1404 Springer, Berlin (1989), 108-126 M. -P. Malliavin. | MR | Zbl

[15] Hou, B., Xu, Y.: Hochschild (co)homology of ${\mathbb{Z}}_{n}$-Galois coverings of exterior algebras in two variables. Acta Math. Sin., Chin. Ser. 51 (2008), 241-252. | MR

[16] Igusa, K.: Notes on the no loops conjecture. J. Pure Appl. Algebra 69 (1990), 161-176. | DOI | MR | Zbl

[17] Iyama, O., Wemyss, M.: The classification of special Cohen-Macaulay modules. Math. Z. 265 (2010), 41-83. | DOI | MR | Zbl

[18] Loday, J. L.: Cyclic Homology. Grundlehren der Mathematischen Wissenschaften 301, Springer Berlin (1992). | MR | Zbl

[19] Skowroński, A.: Simply connected algebras and Hochschild cohomology. Representations of Algebras. Proc. of the 6. Int. Conf., Carleton University, Ottawa, Canada, 1992, CMS Conf. Proc. 14 AMS, Providence V. Dlab et al. (1993), 431-447. | MR

[20] Snashall, N., Taillefer, R.: The Hochschild cohomology ring of a class of special biserial algebras. J. Algebra Appl. 9 (2010), 73-122. | DOI | MR | Zbl

[21] Wemyss, M.: Reconstruction algebras of type {$D$} (II). Hokkaido Math. J. 42 (2013), 293-329. | DOI | MR

[22] Wemyss, M.: Reconstruction algebras of type {$D$} (I). J. Algebra 356 (2012), 158-194. | DOI | MR | Zbl

[23] Wemyss, M.: Reconstruction algebras of type {$A$}. Trans. Am. Math. Soc. 363 (2011), 3101-3132. | DOI | MR | Zbl

[24] Wemyss, M.: The {$ GL(2,\Bbb C)$} McKay correspondence. Math. Ann. 350 (2011), 631-659. | MR

[25] Wunram, J.: Reflexive modules on cyclic quotient surface singularities. Singularities, Representation of Algebras, and Vector Bundles, Proc. of the Symp., Lambrecht, Germany, 1985 Lecture Notes in Math. 1273 Springer, Berlin (1987), 221-231 G. -M. Greuel et al. | MR | Zbl

Cité par Sources :