On the deformation theory of finite dimensional algebras (Q1911213)

We are concerned with finite dimensional algebras \(A\) for which some of the low cohomology groups \(H^i(A)\) (\(i=1,2,3\)) vanish. In section 1, we relate the groups \(H^i(A)\) with local properties of the scheme \(\text{alg}_d\) of \(d\)-dimensional \(k\)-algebra structures (for \(d=\dim_k A\)). For example, we show that if \(H^3(A)=0\), then the point \(\alpha\) in \(\text{alg}_d\) corresponding to \(A\) is smooth. As a consequence, if \(H^1(A)=0\) and \(H^3(A)=0\), then there is an open neighbourhood \(\mathcal U\) of \(\alpha\) in \(\text{alg}_d\) such that for every \(\beta\) in \(\mathcal U\), the set of points corresponding to algebras isomorphic to \(B\) has dimension \(\dim_k H^2(A)\). Therefore, in this situation the different concepts of rigidity considered in the literature coincide. In section 2 we concentrate on strongly simply connected tame algebras \(A\) of polynomial growth. For these algebras \(H^1(A)=0\) and \(H^3(A)=0\). We calculate \(\dim_k H^2(A)\) and therefore we find ridigity conditions. We construct explicitly a geometric \(\dim_k H^2(A)\)-parametric family of strongly simply connected algebras of polynomial growth.