This note discusses a framework for the investigation of the prime spectrum of an associative alg... more This note discusses a framework for the investigation of the prime spectrum of an associative algebra A that is equipped with an action of a Hopf algebra H . In particular, we study a notion of H -rationality for ideals of A and comment on a possible Dixmier-Moeglin equivalence for H -prime ideals of A .
Recently there has been some interest in so-called" Additivity Principles"(21 w... more Recently there has been some interest in so-called" Additivity Principles"(21 which, for a ring extension S c R and a prime ideal P of R, relate the Goldie rank of R/P to the Goldie ranks of S/Q, for all primes Q of S which are minimal over P n S. In this note, we prove such a ...
Let k be an algebraically closed field of characteristic 0 and let D be a division algebra whose ... more Let k be an algebraically closed field of characteristic 0 and let D be a division algebra whose center F contains k. We shall say that D can be reduced to r parameters if D = D_0 tensor_{F_0} F, where D_0 is a division algebra, the center F_0 of D_0 contains k and trdeg(F_0/k) = r. We show that every division algebra of odd degree n >= 5 can be reduced to at most (n-1)(n-2)/2 parameters. Moreover, every crossed product division algebra of degree n >= 4 can be reduced to at most (log_2(n) - 1)n + 1 parameters. Our proofs of these results rely on lattice-theoretic techniques.
Http Dx Doi Org 10 1080 00927879208824353, Jun 27, 2007
ABSTRACT Let be an algebra that is graded by a group G. Then the Hochschild and cyclic homologies... more ABSTRACT Let be an algebra that is graded by a group G. Then the Hochschild and cyclic homologies of S have canonical decompositions with components labeled by the set T(G) of conjugacy classes of G : for Hochschild homology and similarly for cyclic homology. In this article, we describe the components of Hochschild homology in the case where S is strongly G graded.The description is given in terms of a spectral sequence where Hq (R,Sg ) is the Hochschild homology of the identity component R = Se of S with coefficients in the bimodule Sg and Hp (CG (g),.) is the group homology of the centralizer CG (g) of g in G. If R is a separable algebra then the spectral sequence degenerates and yields an isomorphism
This note presents some results on projective modules and the Grothendieck groups K_0 and G_0 for... more This note presents some results on projective modules and the Grothendieck groups K_0 and G_0 for Frobenius algebras and for certain Hopf Galois extensions. Our principal technical tools are the Higman trace for Frobenius algebras and a product formula for Hattori-Stallings ranks of projectives over Hopf Galois extensions. Comment: 15 pages
This note discusses a framework for the investigation of the prime spectrum of an associative alg... more This note discusses a framework for the investigation of the prime spectrum of an associative algebra A that is equipped with an action of a Hopf algebra H . In particular, we study a notion of H -rationality for ideals of A and comment on a possible Dixmier-Moeglin equivalence for H -prime ideals of A .
Recently there has been some interest in so-called" Additivity Principles"(21 w... more Recently there has been some interest in so-called" Additivity Principles"(21 which, for a ring extension S c R and a prime ideal P of R, relate the Goldie rank of R/P to the Goldie ranks of S/Q, for all primes Q of S which are minimal over P n S. In this note, we prove such a ...
Let k be an algebraically closed field of characteristic 0 and let D be a division algebra whose ... more Let k be an algebraically closed field of characteristic 0 and let D be a division algebra whose center F contains k. We shall say that D can be reduced to r parameters if D = D_0 tensor_{F_0} F, where D_0 is a division algebra, the center F_0 of D_0 contains k and trdeg(F_0/k) = r. We show that every division algebra of odd degree n >= 5 can be reduced to at most (n-1)(n-2)/2 parameters. Moreover, every crossed product division algebra of degree n >= 4 can be reduced to at most (log_2(n) - 1)n + 1 parameters. Our proofs of these results rely on lattice-theoretic techniques.
Http Dx Doi Org 10 1080 00927879208824353, Jun 27, 2007
ABSTRACT Let be an algebra that is graded by a group G. Then the Hochschild and cyclic homologies... more ABSTRACT Let be an algebra that is graded by a group G. Then the Hochschild and cyclic homologies of S have canonical decompositions with components labeled by the set T(G) of conjugacy classes of G : for Hochschild homology and similarly for cyclic homology. In this article, we describe the components of Hochschild homology in the case where S is strongly G graded.The description is given in terms of a spectral sequence where Hq (R,Sg ) is the Hochschild homology of the identity component R = Se of S with coefficients in the bimodule Sg and Hp (CG (g),.) is the group homology of the centralizer CG (g) of g in G. If R is a separable algebra then the spectral sequence degenerates and yields an isomorphism
This note presents some results on projective modules and the Grothendieck groups K_0 and G_0 for... more This note presents some results on projective modules and the Grothendieck groups K_0 and G_0 for Frobenius algebras and for certain Hopf Galois extensions. Our principal technical tools are the Higman trace for Frobenius algebras and a product formula for Hattori-Stallings ranks of projectives over Hopf Galois extensions. Comment: 15 pages
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