Derived equivalences of selfinjective algebras preserve singularities.
Let \(A\) and \(B\) be finite dimensional self-injective algebras over an algebraically closed field \(k\). Then, one defines an algebraic variety \(\text{mod}(A,d)\) of \(d\)-dimensional \(A\)-modules as the variety of possible \(A\)-module structures on a fixed \(d\)-dimensional vector space. The base change group \(\text{Gl}_d(k)\) acts on \(\text{mod}(A,d)\) and a module \(M\) is said to be a degeneration of a module \(N\) if \(M\) belongs to the Zariski closure \(\overline O_N=\overline{\text{Gl}_d(k)\cdot N}\) of the \(\text{Gl}_d(k)\)-orbit of \(N\). Now, any two pointed algebraic varieties \((X_1,x_1)\) and \((X_2,x_2)\) are said to be smoothly equivalent if there is a third \((Z,z_0)\) and smooth morphisms \(\xi_i\colon(Z,z_0)\to(X_i,x_i)\) for \(i=1\) and \(i=2\). The equivalence class of \((X,x)\) is denoted \(\text{Sing}(X,x)\). The main result of the paper under review is the following statement. Suppose there is an \(A\)-\(B\)-bimodule \(M\) and a \(B\)-\(A\)-bimodule \(N\) so that \(M\otimes_BN\) is isomorphic to the direct sum of \(A\) and a projective \(A\)-\(A\)-bimodule, and likewise for \(N\otimes_AM\). If the \(A\)-module \(V\) degenerates to \(W\), then \(\text{Sing}(\overline O_V,W)=\text{Sing}(\overline O_{M\otimes_AV},M\otimes_AW)\). As an application the authors show that the closures of the orbits of the module varieties of a Brauer tree algebra are Cohen-Macaulay, normal, and, if the characteristic of \(k\) is \(0\), have rational singularities. The proof of the main theorem is a reduction to a statement on Grothendieck groups, and then the authors apply an earlier result of the second author. The existence of such bimodules \(M\) and \(N\) is implied by an equivalence between the derived categories of the two algebras.
scientific article; zbMATH DE number 2018264