Smooth morphisms of module varieties


Grzegorz Zwara


Abstract: Let k be an algebraically closed field. For any finite dimensional k-algebra B, one defines module varieties \mo_B^d(k), d\geq 1. Each \mo_B^d(k) is a sum of connected components \mo_B^\ud(k) consisting of orbits of the modules with the dimension vector \ud=(d_1,...,d_r), where \sum d_i.\dim_kS_i=d and \{S_i\ is a complete set of pairwise nonisomorphic simple B-modules.

Any homomorphism \phi:A\to B of finite dimensional k-algebras induces regular morphisms \phi^\{(d)}:\mo_B^d(k)\to\mo_A^d(k) of module varieties, for all d\geq 1. We denote by \phi^\{(\ud)} the morphism \phi^\{(d)} restricted to \mo_B^\ud(k). We are interested in homomorphisms \phi such that morphisms \phi^\{(d)} have nice geometric properties. The main result is as follows.

Theorem. Assume that the morphism \phi^\{(\ud)}:\mo_B^\ud(k)\to\mo_A^d(k) preserves codimensions of Gl_d(k)-orbits for any dimension vector \ud, where d=\sum d_i\cdot\dim_kS_i. Then each morphism \phi^\{(\ud)}:\mo_B^\ud(k)\to\ov\{\im\phi^\{(\ud)}} is smooth.

As an application we will show that the geometry of modules from a homogeneous tube is closely related to the geometry of nilpotent matrices.



created Mon Mar 20 10:54:56 PST 2000