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) =
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 .
2.7. Let A be a Koszul algebra with respect to T.
(a) The functor £: Gr A » GrT specializes to a duality £:

wit/i inverse given by £' \
(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
are a/so T-Koszul
(d) // M is a T-Koszul module, then
(i) £(M
[-l]) = (5(M))
(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 £:
/Qr(r) specializes to the two situations
discussed in the last section, we note that if A =
it is immediate that
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
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
then, since £{M) is in
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
Example 4.1 below provides an example of this.
These observations motivate the following definitions.
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) =
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
generated by
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
is contained in
The case when we have equality is especially interesting, so
it is convenient to give the following definition.
Previous Page Next Page