Ph.D Thesis | |

Ph.D Student | Wolf Adi |
---|---|

Subject | Galois Descent and Forms of Path Algebras |

Department | Department of Mathematics |

Supervisor | PROF. Eli Aljadeff |

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 *

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),
S_{Γ})*, where

This translates
the classification of *K/k*-forms of *kΓ* 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 *kΓ* 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.