The concept of finitely presented functor was introduced by Auslander. Proposition 3.1 of Auslander & Reiten provides a way of dealing with the category of finitely presented functors, that seems concrete and easy to use, at least in some examples. The study of this category, using this particular line of thought, is the main purpose of this work. In §1 I recall some basic definitions and give the required notation. In §2 I state the theorem of Auslander & Reiten referred to above and give a new proof of this result. The first part of this proof is an immediate consequence of the theory developed by Green. In §3 I state and prove an unpublished theorem by J. A. Green and I introduce a new category
I
such that the category of finitely presented functors. mmod
A
, is equivalent to a quotient category
I
/
J
, where
J
is an ideal of
I
. In §4 I give some examples of properties of mmod
A
, stated and proved in terms of the category
I
, by using the equivalence of categories referred to in §3. In §5 I consider the particular case where
A
=
A
q
=
k
-alg <
z
:
z
q
= 0>, apply the results of previous sections to study mmod
A
q
and make conclusions about the representation type of the Auslander algebra of
A
q
.
Sheaves of noncommutative algebras and the Beilinson-Bernstein equivalence of categories
