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On a Deformation Theory of Finite Dimensional Modules Over Repetitive Algebras
- Source :
- Algebras and Representation Theory. 26:1-22
- Publication Year :
- 2021
- Publisher :
- Springer Science and Business Media LLC, 2021.
-
Abstract
- Let Λ be a basic finite dimensional algebra over an algebraically closed field $\Bbbk $ , and let $\widehat {\Lambda }$ be the repetitive algebra of Λ. In this article, we prove that if $\widehat {V}$ is a left $\widehat {\Lambda }$ -module with finite dimension over $\Bbbk $ , then $\widehat {V}$ has a well-defined versal deformation ring $R(\widehat {\Lambda },\widehat {V})$ , which is a local complete Noetherian commutative $\Bbbk $ -algebra whose residue field is also isomorphic to $\Bbbk $ . We also prove that $R(\widehat {\Lambda }, \widehat {V})$ is universal provided that $\underline {\text {End}}_{\widehat {\Lambda }}(\widehat {V})=\Bbbk $ and that in this situation, $R(\widehat {\Lambda }, \widehat {V})$ is stable after taking syzygies. We apply the obtained results to finite dimensional modules over the repetitive algebra of the 2-Kronecker algebra, which provides an alternative approach to the deformation theory of objects in the bounded derived category of coherent sheaves over $\mathbb {P}^{1}_{\Bbbk }$ .
- Subjects :
- Noetherian
Derived category
Deformation ring
Mathematics::Commutative Algebra
General Mathematics
Dimension (graph theory)
Lambda
Coherent sheaf
Combinatorics
High Energy Physics::Theory
Residue field
Mathematics::Quantum Algebra
FOS: Mathematics
Representation Theory (math.RT)
Algebraically closed field
Mathematics - Representation Theory
Mathematics
Subjects
Details
- ISSN :
- 15729079 and 1386923X
- Volume :
- 26
- Database :
- OpenAIRE
- Journal :
- Algebras and Representation Theory
- Accession number :
- edsair.doi.dedup.....9242e712f36486a02be31cff02ea5d1d