Some Studies Of Extension Algebra A(C, B)

Representation theory of algebra is a new branch of algebra developedin the seventies of 20th century. Tilting theory is an important subject inthe study of representation theory of finite dimensional algebra. Let C andB be two finite dimensional basic algebras over a field k. we suppose that Cand B are given by a quiver Q_C=(Q₀,Q₁)with relations{ρ_i|i∈I_C}and aquiverΓ_B=(Γ₀,Γ₁)with relations{ρj|j∈I_B}respectively.We assume thatQ₀=Γ₀ and define a new k-algebra A given by the quiver (?)=(Q₀,Q₁∪Γ₁)with relations{ρ_i|i∈I_C}∪{ρ_j|j∈I_B}∪{βα|α∈Q₁,β∈Γ₁}．Then A isa finite dimensional k-algebra．We call A the extension of C and B,denotedby A(C,B)．The main focus of this dissertation gives some properties of theextension algebra by means of homological algebra and tilting theories．Wediscuss relations between tilting C-modules and tilting A-modules by meansof functors．We give a necessary and sufficient condition on M(?)_C A is a tiltingA-module．And then,we discuss different equivalence in the same subcategoryof A-modules．If M₁(?)_C A and M₂(?)_C A are tilting A-modules,we prove thatthey induce the same torsion theory in mod A if and only if M₁ and M₂ inducethe same torsion theory in modC．...