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I have the following question:

Suppose, I have a finite dimensional $k$-Algebra $A$ over an arbitrary field $k$ and a finite dimensional module $M$ that is a generator-cogenerator of mod-$A$.

I'm searching for general criteria on $M$ that ensure that the Algebra $B:=End_A (M)$ is elementary and basic.

Moreover, if $B$ isn't elementary, is there a good candidate for a finite field extension $L$ of $k$ such that $B$ is elementary over $L$?

Thank you very much.

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    $\begingroup$ It will be basic and elementary iff $M/Rad(M)$ contains no isomorphic simple summands and each summand has $k$ as its endomorphism ring. I am not sure how much more there is to say. If it is just basic (meaning $M/Rad(M)$ has no isomorphic simple summands) then you have to find a splitting field for each endomorphism ring of a simple summand of $M/Rad(M)$. I am not sure of the easiest way to do this if $A$ is not a group or semigroup algebra. $\endgroup$
    – Benjamin Steinberg
    Commented Nov 4, 2013 at 15:59
  • $\begingroup$ Thank you very much for your comment. Could you give a reference for your assertions (Maybe also a reference about some representation theory of End$_A(M)$)? I am also interested in special cases for $A$, e.g. group algebras. $\endgroup$
    – Bernhard Boehmler
    Commented Nov 5, 2013 at 14:38
  • $\begingroup$ Let me write an answer because things are usually stated in the case of a projective generator and I just realized now you are not assuming this. Today I have to mark exams and some other stuff so I will try tomorrow to write something. I think what I wrote should still be true in any event. $\endgroup$
    – Benjamin Steinberg
    Commented Nov 5, 2013 at 15:33

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Assume $M$ is the direct sum of the indecomposable modules $M_i$. Then $B$ is basic iff $M$ is basic, meaning that $M_i$ is not isomorphic to $M_j$. The simple modules then are $End(M_i)/Rad(End(M_i))$ , viewing this as a module via projections. Thus the algebra is basic and additionally elementary iff all $End(M_i)/Rad(End(M_i))$ are one-dimensional. How to find such a field extension is for example described in the representation theory book by Zimmermann.

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