6
I. MAIN RESULTS AND EXAMPLES
We view this isomorphism as an identification when A is a T-Koszul algebra,
and in this case we let £': grT grA denote the functor given by £f(M) =
©n
0
Extp(M,T). The next theorem is the main duality result. If M is in /CT(A)
and + F2 Fi FQ M 0 is a linear fat resolution of M, then we denote
the kernel of FQ M by M 1 .
THEOREM
2.7. Let A be a Koszul algebra with respect to T.
(a) The functor £: Gr A » GrT specializes to a duality £:
/CT(A)

/CTCO,
wit/i inverse given by £' \
/CT(T)
—•
/CT(A).
(b) 7/MQ is m AT, £^en A S)\0 M0 and Mo are T-Koszul modules.
(c) If M is a T-Koszul module, then Mi[—1] and
M1[—1]
are a/so T-Koszul
modules.
(d) // M is a T-Koszul module, then
(i) £(M
1
[-l]) = (5(M))
1
[-l];
(ii) 5(M1[-1]) = (5(M))1[-1];
(iii) If F = A 0
A o
M0 is a /at module, then £(F) = £(F)Q;
(iv) £ ( M o ) = r 0
r o
( £ ( M ) ) o .
To see that the duality £:
/CT(A)
/Qr(r) specializes to the two situations
discussed in the last section, we note that if A =
AQ,
it is immediate that
JCT(A)
=
XT and £{M) = HomA(M,T) if M is in XT. Thus, Wakamatsu cotilting duality
is a special case of Theorem 2.7. When A0 is a finite product of copies of a field,
then fat modules are projective modules generated in degree 0, the category XT is
modAo, and linear fat resolutions are just linear projective resolutions. Hence, in
this case, the category
/CT(A)
is the category of Koszul modules, and the duality
described in Theorem 2.7 is Koszul duality.
The statements that fat modules and Ao-modules are T-Koszul modules cor-
respond to the fact that for Koszul algebras, projective modules and semisimple
modules are Koszul modules. The statement about Mi and M 1 corresponds to
the statement that if M is a Koszul module, then so are the 1-shift of the graded
radical of M and the first syzygy of M. Finally, using these observations, we see
that the T-Koszul duality extends the usual Koszul duality.
Our duality result implies that if A is a T-Koszul algebra and M is in
JCT(A)
then, since £{M) is in
JCT(T),
we see that £{M) is generated in degree 0. Unlike
what happens in the Koszul algebra case, £{M) being generated in degree 0 is not
sufficient for M to be in
/CT(A).
Example 4.1 below provides an example of this.
These observations motivate the following definitions.
DEFINITION
2.8. Let A be a left finite graded ring generated in degrees 0 and
1, with AQ an artin algebra, and let T be a Wakamatsu cotilting Ao-module. We
denote by GT(&) the full subcategory of gr0 A consisting of modules M such that
Ext^(T,T) ExtX(M,T) =
ExtX+1(M,T)
for all i 0. Equivalently, £T(A) consists
of the modules M in gr0 A such that £(M) is also generated in degree zero as a
module over the subalgebra of
ET(A)
generated by
ET(A)Q
and
ET(A)I.
If T is generated in degrees 0 and 1, for example if A is T-Koszul, then clearly
5 T ( A) consists of the modules M in gr0 A such that £(M) is generated in degree 0 as
a T-module. We shall see that for a T-Koszul algebra A we always have that
JCT(A)
is contained in
QT(A).
The case when we have equality is especially interesting, so
it is convenient to give the following definition.
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