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.

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