ACKNOWLEDGMEN T xi
both special cases. This will be discussed at the end of the paper. But note
that the example of T being a cotilting module can be viewed as a special case
also of the second version, since Ext\ (T,T) = (0) for i 0, and consequently
ExtA (T,T) = EndA(T). Note also that the functor £ = @i0Ext\( , T) coin-
cides with the functor HomA( , T) on the subcategory j^T. In fact, here we are
at the heart of the motivation for our paper, namely to develop a (contravariant)
correspondence via Ext-algebras, containing both the Koszul and cotilting case,
including the associated dualities, as special cases.
Our basic setting is the following. We assume that A = Ao -f Ai H where Ao
is an artin algebra and the A^ are finitely generated (left) Ao-modules, and T is a
finitely generated cotilting module over Ao, actually more generally a Wakamatsu
cotilting module (See page 2 for the definition). Then the pair (T,T) will have the
same properties. We want to identify a class of rings A such that we have a desired
correspondence (and such that Y runs through the same class of rings) along with
a duality theory for subcategories of the graded A-modules generated in degree
zero and of the graded T-modules generated in degree zero, induced by the functor
5 = 0i oExtX( ,T).
For identifying candidates for subcategories generalizing the category of Koszul
modules for Koszul algebras, we take two different approaches. One approach fol-
lows closely the classical approach, after making some initial assumptions obviously
satisfied for the two cases we want to generalize. For the second approach we in-
vestigate more closely under which operations the category of Koszul modules and
the category ^T associated with a cotilting module T are closed. Then we require
our category of Koszul modules to satisfy these properties, having in mind that we
would like to have a good duality theory.
The paper is organized as follows. Chapter I starts with a brief discussion of the
classical cases and then states the main results and examples of our first approach
to T-Koszul algebras. Chapter II and III are devoted to the different approaches
to the definition of a T-Koszul algebra and a corresponding category of T-Koszul
modules. Some further results, along with open questions, are discussed in Chapter
IV.
Koszul algebras, together with dualities between Koszul modules, are discussed
in [12, 13, 14], and for DG-algebras in [16], along with dualities between associated
derived categories in [12, 16]. There is another generalization of a graded ring
A = Ao + Ai + being Koszul, due to Woodcock [21]. He assumes that A is
projective as a left and as a right Ao-module, i.e. each A^ is projective as a left and
as a right Ao-module. This is not necessarily the case in our setting. Moreover
he has no restrictions on Ao- His point of view is considering the Koszul complex
as the defining property he wants to generalize. However, assuming that A is a
left finite graded ring generated in degrees 0 and 1, with AQ an artin algebra, and
choosing T =
D(AQ P ) ,
one can show that A is a T-Koszul algebra in the sense of
this paper. A generalization of Koszul in a direction different from ours is given in
[11], to a categorical setting.
Acknowledgment
The first and the last author want to thank NSF, NSA and the Maurice Auslan-
der Fund of the American Scandinavian Foundation for support. The first author
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