Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Hi,

in 1971 M.Auslander showed that the representation dimension of $A$ is $\neq 1$ for every Artin algebra $A$.

Does anybody have a reference paper or book proving this? Is the proof easy and / or does it need many prerequisites?

Thanks for the help.

share|improve this question
add comment

1 Answer

up vote 4 down vote accepted

First of all, you have to assume that $A$ is non-semi-simple. For a semi-simple Artin algebra, the representation dimension is defined to be 1.

For a non-semi-simple algebra, the representation dimension is, by definition, the smallest $d$ such that there exists $M$ an $A$-module which is both a generator and a co-generator, and such that the global dimension of the endomorphism ring of $M$ is $d$.

To show the representation dimension of $M$ is not 1, we need to show that the endomorphism ring of $M$ is not hereditary.

Since $M$ is a generator and a co-generator, it contains all the projective indecomposables and all the injective decomposables as direct summands.

If $A$ is non-semi-simple, then it has a projective indecomposable module $P$ which is not simple. Let $Q$ be another projective which has a non-zero map to $P$. Suppose $P$ is the projective cover of the simple $S$, and let $I$ be its injective hull. Then the composition of the maps from $Q$ to $P$ and from $P$ to $I$ is zero. This shows that there are relations among the elements the endomorphism ring of $M$. It follows that the endomorphism ring is not hereditary.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.