CHAPTER I
Main results and examples
In order to get an understanding for the background for our theory of T-Koszul
algebras, we first give an account of some of the main results in the two classical
cases. We then state some of the central results of this paper. We want to quickly
get to a variety of examples, so we postpone the proofs of these results to the next
chapter.
1. Two classical cases
In this section we recall relevant properties for cotilting modules and Koszul
algebras, in particular with respect to existence of dualities for subcategories.
Tilting theory has played a central role in the representation theory of artin
algebras [4, 15, 17, 22]. One of the aspects of the theory deals with equivalence or
duality between appropriate subcategories of module categories. We here consider
artin algebras, even though there are analogous results in more general contexts.
Let A be an artin algebra. Let T be selforthogonal in mod A, the category of
finitely generated (left) A-modules. We say T is a tilting module if pdAT oc and
there is an exact sequence 0 A To ^ Ti —•••— Tn ^ 0 with the T; in
add T. Dually, T is a cotilting module if idAT oc and there is an exact sequence
0 - Tn - Ti T0 -• D(A) -• 0 with the T* in addT, where D(A) is an
injective generator.
We have the following important properties for the full subcategory ^ T asso-
ciated with a cotilting module T.
THEOREM
1.1. If T is a cotilting module, ^T has the following properties.
(a)
±T
is resolving.
(b) ^T is functorially finite.
(c) ^T has almost split sequences.
Let T = EndA(T) for a cotilting A-module T. Then T is a cotilting module
over T, the algebra A is isomorphic to EndA(T), and the module theories for mod A
and mod T are closely connected in the following way.
THEOREM
1.2. If T is a cotilting module, the functor HomA( ,T): mod A —•
rnodT induces a duality Homr( , T): j^T p~T.
Note that D(A) is clearly a cotilting module. In this case j^D(A) = mod A, the
algebra T is isomorphic to
Aop,
the subcategory r-D(A) is mod
Aop,
and we recover
the duality D = HomA( ,D(A)): mod A - mod Aop.
In the general case T is an injective object in ±T in the sense that Ext^C , T) =
(0) for i 0 and C in ^T. In addition we have the following.
l
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