Hochschild Cohomology of skew group rings and invariants

E. Marcos; R. Martínez-Villa; Ma. Martins

Open Mathematics (2004)

  • Volume: 2, Issue: 2, page 177-190
  • ISSN: 2391-5455

Abstract

top
Let A be a k-algebra and G be a group acting on A. We show that G also acts on the Hochschild cohomology algebra HH ⊙ (A) and that there is a monomorphism of rings HH ⊙ (A) G→HH ⊙ (A[G]). That allows us to show the existence of a monomorphism from HH ⊙ (Ã) G into HH ⊙ (A), where à is a Galois covering with group G.

How to cite

top

E. Marcos, R. Martínez-Villa, and Ma. Martins. "Hochschild Cohomology of skew group rings and invariants." Open Mathematics 2.2 (2004): 177-190. <http://eudml.org/doc/268925>.

@article{E2004,
abstract = {Let A be a k-algebra and G be a group acting on A. We show that G also acts on the Hochschild cohomology algebra HH ⊙ (A) and that there is a monomorphism of rings HH ⊙ (A) G→HH ⊙ (A[G]). That allows us to show the existence of a monomorphism from HH ⊙ (Ã) G into HH ⊙ (A), where à is a Galois covering with group G.},
author = {E. Marcos, R. Martínez-Villa, Ma. Martins},
journal = {Open Mathematics},
keywords = {16E40; 16D20; 16S37},
language = {eng},
number = {2},
pages = {177-190},
title = {Hochschild Cohomology of skew group rings and invariants},
url = {http://eudml.org/doc/268925},
volume = {2},
year = {2004},
}

TY - JOUR
AU - E. Marcos
AU - R. Martínez-Villa
AU - Ma. Martins
TI - Hochschild Cohomology of skew group rings and invariants
JO - Open Mathematics
PY - 2004
VL - 2
IS - 2
SP - 177
EP - 190
AB - Let A be a k-algebra and G be a group acting on A. We show that G also acts on the Hochschild cohomology algebra HH ⊙ (A) and that there is a monomorphism of rings HH ⊙ (A) G→HH ⊙ (A[G]). That allows us to show the existence of a monomorphism from HH ⊙ (Ã) G into HH ⊙ (A), where à is a Galois covering with group G.
LA - eng
KW - 16E40; 16D20; 16S37
UR - http://eudml.org/doc/268925
ER -

References

top
  1. [1] I. Assem: Algèbres et modules. Enseignement des Mathématiques, Les Presses de l'Université, d'Ottawa, Masson, 1997. 
  2. [2] R. Berger: Koszulity for nonquadratic algebras, J. Algebra, Vol. 239, (2001), pp. 705–734. http://dx.doi.org/10.1006/jabr.2000.8703 Zbl1035.16023
  3. [3] K. Bongartz and P. Gabriel: “Covering spaces in Representation-Theory”, Invent. Math, Vol. 65, (1982), pp. 331–378. http://dx.doi.org/10.1007/BF01396624 Zbl0482.16026
  4. [4] C. Cibils and M.J. Redondo: Cartan-Leray spectral sequence for Galois coverings of categories, preprint. 
  5. [5] M. Cohen and S. Montgomery: “Group-graded rings, smash products, and group actions”, Trans. Amer. Math. Soc., Vol. 279, No. 1, (1984), pp. 237–258. http://dx.doi.org/10.2307/1999586 Zbl0533.16001
  6. [6] W. Crawley-Boevey and M.P. Holland: “Noncommutative deformations of Kleinian singularities”, Duke Math. J., Vol. 92, (1998), pp. 605–635. http://dx.doi.org/10.1215/S0012-7094-98-09218-3 Zbl0974.16007
  7. [7] K. Erdmann and N. Snashall: “On Hochschild cohomology of preprojective algebras II”, J. Algebra, Vol. 205, (1998), pp. 413–434. http://dx.doi.org/10.1006/jabr.1997.7329 
  8. [8] K. Erdmann and N. Snashall: “Preprojective algebras of Dynkin type, periodicity and the second Hochschild cohomology”, In: CMS Conf. Proc.: Algebras and Modules, II. Geiranger, 1998, Vol. 24, AMS, 1998, pp. 183–193. Zbl1034.16501
  9. [9] M. Gerstenhaber: “On the deformation of rings of algebras”, Ann. Math., Vol. 79, (1964), pp. 59–103. http://dx.doi.org/10.2307/1970484 Zbl0123.03101
  10. [10] E. Green: “Graphs with relations, coverings and group-graded algebras”, Trans. Amer. Math. Soc., Vol. 279, No. 1, (1983), pp. 297–310. http://dx.doi.org/10.2307/1999386 Zbl0536.16001
  11. [11] E. Green, R. Martinez-Villa, E.N. Marcos and P. Zhang: “D-Koszul algebras”, to appear in J. Pure and Appl. Algebra. Zbl1075.16013
  12. [12] E. Green, E.N. Marcos and O. Solberg: “Representation and almost split sequences of Hopf algebras”, In: CMS Conf. Proceedings: Representation theory of algebras, Cocoyoc, Mexico, August 22–26, 1994, Vol. 18, AMS, Providence, 1996. Zbl0855.16034
  13. [13] D. Happel: “Hochschild cohomology of finite-dimensional algebras”, In: Séminaire d'Algèbre P. Dubreil et M.-P. Malliavin, 39ème Année, Lecture Notes in Math, Vol. 1404, Springer-Verlag, 1989, pp. 108–126. 
  14. [14] H. Lenzing: “Curve singularities arizing from the representation theory of tame hereditary algebras”, In: Rep. Theory I. Otawa, 1984, Lecture Notes in Math, Vol. 1177, Springer-Verlag, 1986, pp. 199–231. 
  15. [15] E.N. Marcos: Singularidades, Módulos sobre álgebras de Artin e álgebras de Hopf, Tese de Livre-Docencia, IME-USP, São Paulo, Brasil, 1996. 
  16. [16] R. Martinez-Villa: “Skew group algebras and their Yoneda algebras”, Mathematical Journal of Okayama Univ., Vol. 43, (2001), pp. 1–16. Zbl1023.16013
  17. [17] R. Martínez-Villa and J.A., de la Peña “The universal cover of a quiver with relations”, J. Pure Appl. Algebra, Vol. 30, (1983), pp. 277–292. http://dx.doi.org/10.1016/0022-4049(83)90062-2 Zbl0522.16028
  18. [18] I. Reiten and M. Van Den Bergh: “Two-dimensional tame and maximal orders of finite representation type”, Mem. Amer. Math. Soc., Vol. 80, (1989). Zbl0677.16002
  19. [19] J. Rickard: “Morita theory for derived category 1”, J. London Math. Soc., Vol. 39, (1989), pp. 436–456. Zbl0642.16034
  20. [20] C.A. Weibel: An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, Vol. 38, Cambridge University Press, Cambridge, 1994. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.