A new family of algebras whose representation schemes are smooth  [ Une nouvelle famille d’algèbres dont les schémas de représentations sont lisses ]
Annales de l'Institut Fourier, Tome 66 (2016) no. 3, pp. 1261-1277.

Dans cet article, nous fournissons une condition nécéssaire et suffisante pour la lissité du schéma qui paramétrise les représentations n-dimensionelles d’une algèbre associative, engendrée par un nombre fini d’éléments sur un corps algébriquement clos. En particulier, notre résultat implique que les points MRep A n (k) satisfaisant Ext A 2 (M,M)=0 sont réguliers. Ceci généralise aux algèbres engendrées par un nombre fini d’éléments des résultats connus sur les algèbres de dimension finie.

We give a necessary and sufficient smoothness condition for the scheme parameterizing the n-dimensional representations of a finitely generated associative algebra over an algebraically closed field. In particular, our result implies that the points MRep A n (k) satisfying Ext A 2 (M,M)=0 are regular. This generalizes well-known results on finite-dimensional algebras to finitely generated algebras.

Reçu le : 2014-08-01
Révisé le : 2015-04-09
Accepté le : 2015-09-10
Publié le : 2016-12-14
DOI : https://doi.org/10.5802/aif.3037
Classification : 14B05,  16E65,  16S38
Mots clés: Géométrie non-commutative, cohomologie de Hochschild, théorie des Représentations
@article{AIF_2016__66_3_1261_0,
     author = {Ardizzoni, Alessandro and Galluzzi, Federica and Vaccarino, Francesco},
     title = {A new family of algebras whose representation schemes are smooth},
     journal = {Annales de l'Institut Fourier},
     pages = {1261--1277},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {66},
     number = {3},
     year = {2016},
     doi = {10.5802/aif.3037},
     language = {en},
     url = {aif.centre-mersenne.org/item/AIF_2016__66_3_1261_0/}
}
Ardizzoni, Alessandro; Galluzzi, Federica; Vaccarino, Francesco. A new family of algebras whose representation schemes are smooth. Annales de l'Institut Fourier, Tome 66 (2016) no. 3, pp. 1261-1277. doi : 10.5802/aif.3037. https://aif.centre-mersenne.org/item/AIF_2016__66_3_1261_0/

[1] van den Bergh, Michel A relation between Hochschild homology and cohomology for Gorenstein rings, Proc. Amer. Math. Soc., Tome 126 (1998) no. 5, pp. 1345-1348 | Article

[2] Bockland, Raf Noncommutative Tangent Cones and Calabi Yau Algebras (http://arxiv.org/abs/0711.0179)

[3] Bongartz, Klaus A geometric version of the Morita equivalence, J. Algebra, Tome 139 (1991) no. 1, pp. 159-171 | Article

[4] Cartan, Henri; Eilenberg, Samuel Homological algebra, Princeton University Press, Princeton, N. J., 1956, xv+390 pages

[5] Crawley-Boevey, William Preprojective algebras, differential operators and a Conze embedding for deformations of Kleinian singularities, Comment. Math. Helv., Tome 74 (1999) no. 4, pp. 548-574 | Article

[6] Crawley-Boevey, William; Schröer, Jan Irreducible components of varieties of modules, J. Reine Angew. Math., Tome 553 (2002), pp. 201-220 | Article

[7] Cuntz, Joachim; Quillen, Daniel Algebra extensions and nonsingularity, J. Amer. Math. Soc., Tome 8 (1995) no. 2, pp. 251-289 | Article

[8] Gabriel, Peter Finite representation type is open, Proceedings of the International Conference on Representations of Algebras (Carleton Univ., Ottawa, Ont., 1974), Paper No. 10 (1974), 23 pp. Carleton Math. Lecture Notes, No. 9 pages

[9] Geiss, C Deformation Theory of finite-dimensional Modules and Algebras (2006) (Lectures given at ICTP)

[10] Geiss, Christoff; de la Peña, José Antonio On the deformation theory of finite-dimensional algebras, Manuscripta Math., Tome 88 (1995) no. 2, pp. 191-208 | Article

[11] Gerstenhaber, Murray On the deformation of rings and algebras, Ann. of Math. (2), Tome 79 (1964), pp. 59-103

[12] Ginzburg, Victor Calabi-Yau algebras (http://arxiv.org/abs/math/0612139)

[13] Ginzburg, Victor Lectures on Noncommutative Geometry (http://arxiv.org/abs/math/0506603)

[14] Grothendieck, A. Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. I, Inst. Hautes Études Sci. Publ. Math. (1964) no. 20, 259 pages

[15] Harrison, D. K. Commutative algebras and cohomology, Trans. Amer. Math. Soc., Tome 104 (1962), pp. 191-204

[16] Humphreys, James E. Introduction to Lie algebras and representation theory, Springer-Verlag, New York-Berlin, 1972, xii+169 pages (Graduate Texts in Mathematics, Vol. 9)

[17] Le Bruyn, Lieven Non-smooth algebra with smooth representation variety (asked in MathOverflow) (http://mathoverflow.net/questions/9738)

[18] Le Bruyn, Lieven Noncommutative geometry and Cayley-smooth orders, Pure and Applied Mathematics (Boca Raton), Tome 290, Chapman & Hall/CRC, Boca Raton, FL, 2008, lxiv+524 pages

[19] Mumford, David Lectures on curves on an algebraic surface, With a section by G. M. Bergman. Annals of Mathematics Studies, No. 59, Princeton University Press, Princeton, N.J., 1966, xi+200 pages

[20] Procesi, Claudio Rings with polynomial identities, Marcel Dekker, Inc., New York, 1973, viii+190 pages (Pure and Applied Mathematics, 17)

[21] Sernesi, Edoardo Deformations of algebraic schemes, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Tome 334, Springer-Verlag, Berlin, 2006, xii+339 pages

[22] Vaccarino, Francesco Linear representations, symmetric products and the commuting scheme, J. Algebra, Tome 317 (2007) no. 2, pp. 634-641 | Article

[23] Weibel, Charles A. An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, Tome 38, Cambridge University Press, Cambridge, 1994, xiv+450 pages | Article

[24] Zusmanovich, Pasha A converse to the second Whitehead lemma, J. Lie Theory, Tome 18 (2008) no. 2, pp. 295-299