On deformations of associative algebras


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


Roman Bezrukavnikov

Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, United States

Victor Ginzburg

Department of Mathematics, University of Chicago, Chicago, IL 60637, United States