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New approaches to finite generation of cohomology rings.
- Source :
-
Journal of Algebra . Dec2021, Vol. 587, p390-428. 39p. - Publication Year :
- 2021
-
Abstract
- In support variety theory, representations of a finite dimensional (Hopf) algebra A can be studied geometrically by associating any representation of A to an algebraic variety using the cohomology ring of A. An essential assumption in this theory is the finite generation condition for the cohomology ring of A and that for the corresponding modules. In this paper, we introduce various approaches to study the finite generation condition. First, for any finite dimensional Hopf algebra A , we show that the finite generation condition on A -modules can be replaced by a condition on any affine commutative A -module algebra R under the assumption that R is integral over its invariant subring R A. Next, we use a spectral sequence argument to show that a finite generation condition holds for certain filtered, smash and crossed product algebras in positive characteristic if the related spectral sequences collapse. Finally, if A is defined over a number field over the rationals, we construct another finite dimensional Hopf algebra A ′ over a finite field, where A can be viewed as a deformation of A ′ , and prove that if the finite generation condition holds for A ′ , then the same condition holds for A. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 00218693
- Volume :
- 587
- Database :
- Academic Search Index
- Journal :
- Journal of Algebra
- Publication Type :
- Academic Journal
- Accession number :
- 152446432
- Full Text :
- https://doi.org/10.1016/j.jalgebra.2021.07.027