On deformations of associative algebras

• 成果类型: Article
• 署名作者: Bezrukavnikov, Roman; Ginzburg, Victor
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2007.166.533
• 发表日期: 2007
• 页码: 533-548
• 关键词: dg categories; COHOMOLOGY

摘要

In a classic paper, Gerstenhaber showed that first order deformations of an associative k-algebra a are controlled by the second Hochschild cohomology group of a. More generally, any n-parameter first order deformation of a gives, due to commutativity of the cup-product on Hochschild cohomology, a graded algebra morphism Sym(center dot)(k(n)) -> Ext(a-bimod)(2 center dot)(a, a). We prove that any extension of the n-parameter first order deformation of a to an infinite order formal deformation provides a canonical 'lift' of the graded algebra morphism above to a dg-algebra morphism Sym(center dot)(k(n)) -> RHom(center dot)(a, a), where the symmetric algebra Sym(center dot)(k(n)) is viewed as a dg-algebra (generated by the vector space k(n) placed in degree 2) equipped with zero differential.