On finitely presented functors, Auslander algebras and almost split sequences

This thesis consists of two parts: In Part A we study the category of finitely presented functors and use it to determine the representation type of the Auslander Algebra of Aq = K-algebra , denoted Rq (K is a field). This is possible because the category of finitely generated modules over Rq, mod Rq , is equivalent to the category of finitely presented functors from (mod Aq)ᵒᵖ to Mod k. Part A finishes with the construction of the Auslander-Reiten quiver of Rq in case q = 3. Part B deals with the construction of almost split sequences in the category mod° Δ of lattices over an R-order Δ , where R is a complete discrete rank 1 valuation ring. In the first chapter of part B we give a description of some unpublished work by J. A. Green who permitted me to include it in this thesis. This work contains a method to construct a short exact sequence 0-> N-> E-> S-> 0 in a way which gives an explicit expression for the subfunctor Im( ,g) of ( ,S) , and shows that the construction of almost split sequences can be viewed as a particular case of this problem. In the second chapter of part B we continue this work by deducing a "trace formula" which provides a practical way of dealing with a certain step of the construction of almost split sequences in mod° Δ. Then we consider the particular case where Δ is the group ring