Ph.D Thesis
Ph.D Student Adi Wolf Galois Descent and Forms of Path Algebras Department of Mathematics Full Professor Aljadeff Eli

Abstract

For a Galois extension K/k we consider the question of classifying the K/k-forms of a finite dimensional path algebra A=kΓ, i.e., find up to k-isomorphism all the k-algebras B such that A k K B k K. Here Γ is an acyclic finite quiver.

We show that when char(k)=0 the K/k-forms of A are classified by the cohomology pointed set H 1 (Gal(K/k), SΓ), where SΓ is a certain finite subgroup of permutations of vertices of Γ.

This translates the classification of K/k-forms of into a combinatorial problem. For an acyclic finite quiver Γ we define the notions of combinatorial forms and their evaluations (which are certain path type tensor algebras). We introduce a combinatorial descent for classifying the combinatorial forms, and show that the K/k-forms of are evaluations of combinatorial forms of Γ.

As a corollary, we show how to identify any hereditary finite dimensional k-algebra A with A/rad(A) commutative as an evaluation of a specific combinatorial form of an acyclic quiver Γ. Another corollary is the explicit construction of a generic object (versal torsor) for path algebras on (directed) trees.