THEOREM 1.12. Let A be a Koszul algebra.
(a) A is quadratic.
(b) £ ( A ) ~ A
2. Statements of main results
In this section we state some of the main results of this paper, and give the
definitions necessary for this.
Throughout this section, A = Ao+Ai H will be a positively Z-graded algebra
generated in degrees 0 and 1. Let gr0 A denote the full subcategory of gr(A) whose
objects consist of graded modules generated in degree 0. We will freely view the
category of Ao-modules and its various subcategories as embedded in gr0 A by
viewing a Ao-module MQ as the graded module ^ JV$ with Ni = (0) if i ^ 0 and
No = M0.
We make the following assumptions throughout the section unless otherwise
stated. First we will always assume that Ao is an artin algebra and that A is left
finite. By T we will denote a Wakamatsu cotilting Ao-module.
Let T = ®n
ExtJ(r, T). We define £: gr A - gr T by
£(M) = ©
n 0
where £{M) has a natural graded T-module structure via the Yoneda product. We
note that if A = AQ, then £, when restricted to XT, is the Wakamatsu cotilting
duality described in the previous section. Next we note that if AQ is a product of
n copies of a field fc, then the only Wakamatsu Ao-cotilting module is T = Ao up
to multiplicity of direct summands. If A is a Koszul algebra, then the functor £
restricts to the Koszul duality on the category of Koszul modules /C(A). In this
section, we define a general setting where Wakamatsu cotilting duality and Koszul
duality fit in as special cases.
We shall define and introduce two conditions on A which will be important for
our definition of an algebra which is Koszul with respect to T.
DEFINITION 2.1. Let y be a subcategory of mod A0.
(a) We say that a Ao-bimodule B satisfies the ^-condition with respect to y if
B®Aoyc y.
(b) A Ao-bimodule B satisfies the TOT-condition with respect to y, if
{B,y) = (0) for each i 1.
(c) When T is a Wakamatsu cotilting Ao-module, the graded ring A satisfies
the 0-condition and/or the Tor-condition with respect to T, if A as a Ao-
bimodule satisfies the corresponding conditions. That is, each A* satisfies
the corresponding condition with respect to XT-
When both the (g-condition and the Tor-condition hold, we say that the ®-
Tor-conditions are satisfied.
These conditions may seem artificial, but in Section 4 of Chapter III, they
will be motivated by showing that conditions like 0-Tor-conditions are a necessary
consequence of the existence of a duality theory.
Note that in both Wakamatsu cotilting theory and Koszul algebra theory, the
0-Tor-conditions are trivially satisfied, and these conditions will be part of our
definition of an algebra being Koszul with respect to T. In Chapter III we give
a more general setting for a duality theory to hold, but we postpone the general
rather technical conditions for clarity of exposition.
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