Geodesic
Invariants and rings of quotients of $H$-semiprime $H$-module algebra satisfying a~polynomial identity
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 5 (2016), pp. 22-40.

Voir la notice de l'article provenant de la source Math-Net.Ru

We consider an action of a finite-dimensional Hopf algebra $H$ on a PI-algebra. We prove that an $H$-semiprime $H$-module algebra $A$ has a Frobenius artinian classical ring of quotients $Q$ if $A$ has a finite set of $H$-prime ideals with zero intersection. The ring of quotients $Q$ is an $H$-semisimple $H$-module algebra and finitely generated module over the subalgebra of central invariants. Moreover, if the algebra $A$ is projective module of constant rank over its center then $A$ is integral over the subalgebra of central invariants.
Keywords: Hopf algebras, invariant theory, PI-algebras, rings of quotients. 
@article{IVM_2016_5_a1,
     author = {M. S. Eryashkin},
     title = {Invariants and rings of quotients of $H$-semiprime $H$-module algebra satisfying a~polynomial identity},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {22--40},
     publisher = {mathdoc},
     number = {5},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2016_5_a1/}
}
TY  - JOUR
AU  - M. S. Eryashkin
TI  - Invariants and rings of quotients of $H$-semiprime $H$-module algebra satisfying a~polynomial identity
JO  - Izvestiâ vysših učebnyh zavedenij. Matematika
PY  - 2016
SP  - 22
EP  - 40
IS  - 5
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IVM_2016_5_a1/
LA  - ru
ID  - IVM_2016_5_a1
ER  - 
%0 Journal Article
%A M. S. Eryashkin
%T Invariants and rings of quotients of $H$-semiprime $H$-module algebra satisfying a~polynomial identity
%J Izvestiâ vysših učebnyh zavedenij. Matematika
%D 2016
%P 22-40
%N 5
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IVM_2016_5_a1/
%G ru
%F IVM_2016_5_a1
M. S. Eryashkin. Invariants and rings of quotients of $H$-semiprime $H$-module algebra satisfying a~polynomial identity. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 5 (2016), pp. 22-40. http://geodesic.mathdoc.fr/item/IVM_2016_5_a1/

[1] Sweedler M. E., Hopf algebras, Benjamin, New York, 1969 | MR | Zbl

[2] Burbaki N., Kommutativnaya algebra, Mir, M., 1971 | MR

[3] Montgomery S., Hopf algebras and their actions on rings, CBMS Reg. Conf. Ser. Math., 85, Amer. Math. Soc., Providence, RI, 1993 | DOI | MR

[4] Skryabin S. M., “Invariants of finite Hopf algebras”, Advances in Math., 183:2 (2004), 209–239 | DOI | MR | Zbl

[5] Artamonov V. A., “Invarianty algebr Khopfa”, Vestn. MGU. Ser. 1, matem. mekhan., 1996, no. 4, 45–49 ; “Исправление”, Вестн. МГУ. Сер. 1, матем. механ., 1997, No 2, 64 | MR | Zbl | MR

[6] Zhu Shenglin, “Integrality of module algebras over its invariants”, J. Algebra, 180:1 (1996), 187–205 | DOI | MR | Zbl

[7] Totok A. A., “Ob invariantakh konechnomernykh tochechnykh algebr Khopfa”, Vestn. MGU. Ser. 1, matem. mekhan., 1997, no. 3, 31–34 | MR

[8] Etingof P., “Galois bimodules and integrality of $\mathrm{PI}$ comodule algebras over invariants”, J. of noncommutative geometry, 9:2 (2015), 567–602 | DOI | MR | Zbl

[9] Eryashkin M. S., “Invarianty deistviya poluprostoi konechnomernoi algebry Khopfa na algebrakh spetsialnogo vida”, Izv. vuzov. Matem., 2011, no. 8, 14–22 | MR | Zbl

[10] Eryashkin M. S., “Koltsa Martindeila i $H$-modulnye algebry, obladayuschie invariantnymi kharakteristicheskimi mnogochlenami”, Sib. matem. zhurn., 53:4 (2012), 822–838 | MR

[11] Cohen M., “Smash products, inner actions and quotient rings”, Pacific J. Math., 125:1 (1986), 45–66 | DOI | MR

[12] Montgomery S., Schneider H. J., “Prime ideals in Hopf Galois extensions”, Israel J. Math., 112:1 (1999), 187–235 | DOI | MR | Zbl

[13] Passman D., Infinite crossed products, Academic Press, London, 1989 | MR | Zbl

[14] Skryabin S. M., “Structure of $H$-semiprime artinian algebras”, Algebr. Represent. Theor., 14:5 (2011), 803–822 | DOI | MR | Zbl

[15] Lomp C., “A central closure construction for certain algebra extensions. Applications to Hopf actions”, J. Pure Appl. Algebra, 198:1–3 (2005), 297–316 | DOI | MR | Zbl

[16] Robson J. C., Small L. W., “Liberal extensions”, Proc. London Math. Soc., 42:1 (1981), 87–103 | DOI | MR | Zbl

[17] McConnell J. C., Robson J. C., Noncommutative Noetherian rings, Amer. Math. Soc., Providence, RI, 2001 | MR | Zbl

[18] Atya M., Makdonald I., Vvedenie v kommutativnuyu algebru, Mir, M., 1972 | MR

[19] Larson R. G., Sweedler M. E., “An associative orthogonal bilinear form for Hopf algebras”, Amer. J. Math., 91:1 (1969), 75–94 | DOI | MR | Zbl