A new family of algebras whose representation schemes are smooth (Q332222)

The setting of the article is the following: \(A\) is a finitely generated associative \(k\)-algebra, \(k\) algebraically closed. Let \(B\) be any ring, and let \(M_n(B)\) be the ring of matrices with entries in \(B\). It is a known fact that the functor \(R:\mathcal C_k\rightarrow\mathrm {Sets}\) from the category of commutative algebras to sets given by \(R(B)=\mathrm {Hom}_{\mathcal N_k}(A,M_n(B))\), where \(\mathcal N_k\) is the category of associative \(k\)-algebras, is represented by a commutative algebra \(V_n(A)\), and \(\mathrm {Rep}^n(A)=\mathrm {Spec}(V_n(A))\) is \textit{the scheme of linear representations} of dimension \(n\) of \(A\).NEWLINENEWLINEDue to Grothendieck, formally smoothness is a generalization of free algebras, and it is known that when \(A\) is formally free, then \(\mathrm {Rep}^n(A)\) is smooth. If \(A\) is finite-dimensional, then it is formally smooth if and only if it is hereditary, and so \(\mathrm {Rep}^n(A)\) is smooth for all \(n\) if and only if \(A\) is hereditary.NEWLINENEWLINEThis is more complicated for infinite-dimensional algebras. Then there exist hereditary algebras that are not formally smooth. The main goal of the present article is to find sufficient and necessary conditions for \(\mathrm {Rep}^n_A\) to be smooth.NEWLINENEWLINEThere is a well known obstruction theory for \(A\)-modules \(M\in\mathrm {Rep}^n_A(k)\). This says that an infinitesimal deformation of \(M\) can be lifted formally (inductively by small morphisms) if and only if the element \(o(M)=0\) in the obstruction space \(H\subseteq\mathrm {Ext}^2_A(M,M)\).NEWLINENEWLINEIf \(\mathrm {Ext}^2_A(M,M)=0\) for all \(M\in\mathrm {Rep}^n_A(k),\;n\geq 1,\) \(A\) is called \textit{finitely unobstructed}.NEWLINENEWLINEBy analysing the local geometry of \(\mathrm {Rep}^n_A\) using upper semicontinuity of certain dimension functions arising from the bar resolution of \(A\), \textit{C. Geiss} and \textit{J. A. de la Peña} [Manuscr. Math. 88, No. 2, 191--208 (1995; Zbl 0851.16011)] proved that when \(A\) is finite dimensional, finitely unobstructedness implies that \(\mathrm {Rep}^n_A\) is smooth.NEWLINENEWLINEThis article study the smoothness problem via the adjunction \(\mathrm {Hom}_{\mathcal C_k}(V_n(A),B)\overset\cong\rightarrow\mathrm {Hom}_{\mathcal N_k}(A,M_n(B))\). This allows the use of the Harrison cohomology of \(V_n(A)\) instead of Hochshild cohomology of \(A\); the Harrison cohomology of a commutative \(k\)-algebra is the symmetric part of its Hochschild cohomology, and an affine ting \(R\) is regular if and only if its second Harrison cohomology vanishes.NEWLINENEWLINEThe article's main result extends known results on smoothness to infinite-dimensional finitely generated algebras. It says that for \(A\) a finitely generated \(k\)-algebra, \(f:V_n(A)\rightarrow k\) a \(k\)-algebra map, and \(\rho:A\rightarrow M_n(k)\) the algebra map corresponding to \(f\) through the above adjunction, there exists a linear embedding of \(\mathrm {Harr}^2(V_n(A), {}_fk)\) into \(H^2(A,{}_\rho M_n(k)_\rho)\). In particular, this gives that \(M\in\mathrm {Rep}^n_A\) is a regular point whenever \(\mathrm {Ext}^2_A(M,M)=0\).NEWLINENEWLINEThe embedding given in the main result is not an isomorphism in general, and the article contains a counter example using 2-Calabi Yau algebras.NEWLINENEWLINEThe necessary definitions and preliminaries are included and makes the article self contained. This includes the representability and obstruction theory in the case of representations. Several explicit examples on finitely obstructed algebras are given, discussing the case of hereditary algebras and formally smooth algebras. The relationship between the deformation theory of \(M\in\mathrm {Rep}^n_A(k)\) and the deformation theory of \(V_n(A)\) as used in algebraic geometry is established.NEWLINENEWLINEThis article is self contained and explicit, and it is a very nice introduction to the base concepts of moduli spaces constructed by deformation theory. It strengthen the link between representation theory and algebraic geometry in a way that can be be easily followed in both directions.