Theorem. The representation dimension of any artin algebra is finite

Let A be an artin algebra. Consider an A-modules M of the form M = _AA \oplus D(A_AA)\oplus M', where M' is an arbitrary finite length module and its endomorphism ring E = End(M). The representation dimension of A is the infimum of the global dimension of all such algebras E.

The representation dimension was introduced in order to reduce questions about modules to questions dealing with modules of small finite projective dimension. It was introduced by M.Auslander in his Queen Mary notes (1971) and his question whether the rep dim is finite remained open for 30 years! But this is only one aspect of the paper. Indeed, the question has to be put into the broader context of relating artin algebras to quasi-hereditary ones (the quasi-hereditary algebras have been introduced by Cline-Parshall-Scott in order to deal with highest weight categories as they arise in the rep theory of semisimple algebraic groups and Lie algebras, but they have turned out to be somewhat technical, however very useful algebraic objects - since also many other artin algebras which arise in nature turn out to be quasi-hereditary.) Of particular interest is the fact that quasi-hereditary algebras are of finite global dimension and that they allow an inductive construction of the corresponding derived category. Now Iyama shows the following:

Theorem Given any module X of finite length over any artin algebra, then there is a module Y (explicitly constructed) such that the endomorphism ring of X\oplus Y is quasi-hereditary.

This is a very basic result, showing in some sense that such a module category is "locally like a quasi-hereditary one".

See also Igusa-Todorov where the case of rep dim A = 3 is discussed. Recall: Let A be an artin algebra.

• rep dim A = 0 if and only if A is semisimple.
• Never rep dim A = 1.
• rep dim A = 2 if and only if A is representation finite and not semisimple.
• No example is known with rep dim A greater or equal to 4.