On deformations of associative algebras. (Q2477076)

Let \(k\) be a field of characteristic zero, let \(T\) be a finite dimensional vector space over \(k\), and let \(\mathcal O=k\oplus T^*\) be the commutative local \(k\)-algebra with unit \(1\in k\) and with maximal ideal \(T^*\subset\mathcal O\) such that \((T^*)^2=0\). Given a unital associative \(k\)-algebra \(a\), an \(\mathcal O\)-deformation of \(a\) is a free \(\mathcal O\)-algebra \(A\) together with a \(k\)-algebra isomorphism \(\psi\colon A/T^*\cdot A\cong a\). Two \(\mathcal O\)-deformations of \(a\) are said to be equivalent if there is an \(\mathcal O\)-algebra isomorphism between them respecting the induced isomorphisms with \(a\). For each \(\mathcal O\)-deformation \((A,\psi)\) of \(a\), there is an exact sequence of \(a\)-bimodules \[ 0\to a\otimes_kT^*\to k\otimes_{\mathcal O}\Omega_A\otimes_{\mathcal O}k\to a\otimes_ka\to a\to 0 \] with corresponding extension class \[ \text{deform}(A,\psi)\in\text{Ext}^2_{a\text{-bimod}}(a,a\otimes_{k}T^*) =\Hom(T,\text{Ext}^2_{a\text{-bimod}}(a,a)). \] In this paper, the authors first give an invariant and multiparameter generalization of a classical result by \textit{M. Gerstenhaber} [Ann. Math. (2) 79, No. 1, 59-103 (1964; Zbl 0123.03101)], which says that the map assigning the class \(\text{deform}(A,\psi)\) to an \(\mathcal O\)-deformation \((A,\psi)\) gives a canonical bijection between the set of equivalence classes of \(\mathcal O\)-deformations of \(a\) and the vector space \(\Hom(T,\text{Ext}^2_{a\text{-bimod}}(a,a))\). Next the authors consider the total Ext group \[ \text{Ext}^\bullet_{a\text{-bimod}}(a,a)=\bigoplus_{i\geq 0} \text{Ext}^i_{a\text{-bimod}}(a,a) \] which is a graded vector space with an associative algebra structure given by Yoneda product. Another fundamental result due to \textit{M. Gerstenhaber} [Ann. Math. (2) 78, 267-288 (1963; Zbl 0131.27302)] says that the Yoneda product on \(\text{Ext}^\bullet_{a\text{-bimod}}(a,a)\) is graded commutative. This allows the conclusion, when combining it with the above result, that any \(\mathcal O\)-deformation of \(a\) gives rise to a graded algebra homomorphism \[ \text{deform}\colon\text{Sym}(T[-2])\to \text{Ext}^{2\bullet}_{a\text{-bimod}}(a,a) \] where \(\text{Sym}(T[-2])\) denotes the commutative graded algebra freely generated by the vector space \(T\) placed in degree \(2\). The main result of this paper is concerned with the problem of lifting this morphism to the level of derived categories. To do so, the authors need to consider infinite order formal deformations of \(a\), which are given as follows. Let \(\mathcal O\) be a formally smooth local \(k\)-algebra with maximal ideal \(\mathfrak m\) such that \(\mathcal O/\mathfrak m=k\) and assume that \(\mathcal O\) is complete in the \(\mathfrak m\)-adic topology. The finite dimensional \(k\)-vector space \(T:=(\mathfrak m/\mathfrak m^2)^*\) may be viewed as the tangent space to \(\text{Spec}(\mathcal O)\) at the base point and there is a canonical isomorphism \(\mathcal O/\mathfrak m^2=k\oplus T^*\). An infinite order formal \(\mathcal O\)-deformation of \(a\) consists of a complete topological \(\mathcal O\)-algebra \(A\) such that \(A/\mathfrak m^nA\) is a free \(\mathcal O/\mathfrak m^n\)-module for all \(n\), together with an algebra isomorphism \(\psi\colon a\to A/\mathfrak mA\). Reducing such an infinite order deformation modulo \(\mathfrak m^2\), one obtains a first order \(\mathcal O/\mathfrak m^2\)-deformation of \(a\). The main result of the paper is as follows: Any infinite order formal \(\mathcal O\)-deformation \((A,\psi)\) of \(a\) provides a canonical lift of the graded algebra morphism \(\text{deform}\) above, which is associated to the corresponding first order \(\mathcal O/\mathfrak m^2\)-deformation, to a dg-algebra morphism \[ \mathfrak{Deform}\colon\text{Sym}(T[-2])\to\text{RHom}_{a\text{-bimod}}(a,a)= \text{REnd}_{a\otimes_ka^{\text{op}}}(a). \] In particular, this means that one can map a basis of the vector space \[ \text{deform}(T[-2])\subset\text{Ext}^2_{a\text{-bimod}}(a,a) \] to a set of pairwise commuting elements in \(\text{RHom}_{a\text{-bimod}}(a,a)\). This commutativity statement implicitly involves a particular DG algebra model for \(\text{RHom}_{a\text{-bimod}}(a,a)\). The model the authors use as well as their construction of the morphism \(\mathfrak{Deform}\) both involve the full infinite order deformation \((A,\psi)\), i.e. the full \(\mathcal O\)-algebra structure on \(A\), and not only the corresponding first order deformation resulting from \(A/\mathfrak m^2A\).