4 I. MAIN RESULTS AND EXAMPLES

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

0

ExtJ(r, T). We define £: gr A - • gr T by

£(M) = ©

n 0

ExtX(M,T),

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

Toif0

{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.