Geometry of chain complexes and outer automorphisms under derived equivalence
The principle aim of the lecture is to show how geometric methods can be applied towards identifying invariants of finite dimensional algebras under derived equivalence.
We will start by sketching Rickard's seminal work on derived equivalences of module categories (completing insights of Happel and Cline-Parshall-Scott), as well as some of its applications. In particular, we will outline the status quo concerning invariants under such equivalences. The most recent item on this -- so far short -- list reflects symmetries of the algebra: namely, the identity component of the outer automorphism group. That this algebraic group is in fact an invariant was proved in joint work of M. Saorín and the speaker.
The second part of the lecture addresses the geometric underpinnings of this result, inasmuch as they are of independent interest. Roughly speaking, the main theorem in this direction links the geometry of chain complexes to Hom-groups in the derived category. A well known predecessor of this bridge, due to Voigt, concerns classical varieties of representations of an algebra A, as first studied by Gabriel and Bernstein-Gelfand-Ponomarev: If X is a point in the variety Mod_d^A of representations of fixed dimension d, and GLd.X the corresponding orbit under the conjugation action of GLd, then the vector space TX(Mod_d^A)/ TX (GLd.X) naturally embeds into Ext^1_A(X,X); here TX (-) denotes the tangent space at X. We will discuss the relevance of this result and its extension to derived categories. One of the consequences, finally, will take us back