Vrftjkry

Recently I came across the notion of Endo-Trivial modules (out of brevity's sake e-t), and was a surprise for me that there is a huge (and rather complicated) theory behind them. I recall that an e-t module is a finitely-generated $mathbb{K}G$-module over a field of positive characteristic $mathbb{K}$ and say $G$ a $p$-group, such that $M^{*} otimes M cong mathbb{K} oplus textit{(proj)}$, where $(proj)$ states for some projective $mathbb{K}G$-module. Apparently there should be a kind of modular represenation theoretic argument to study those objects, however isn't clear to me. Do you know what's the initiative behind their study?

Also, there is a decomposition of those modules always, namely
$$M = M_0 oplus textit{(proj)},$$
for some indecomposable submodule $M_0$, and some projective. However this isn't clear either. The latter should be some kind of version of Krull-Schmidt theorem, since $M$ is a f.g module over an Artinian Ring (and therefore of finite length), hence a decomposition into indecomposables exists and is unique (up to isomorphism). However the theorem doesn't mention anything about projectivity for the indecomposables, so I can't come up with a better idea unfortunately.

Could you please help me out?

Endotrivial modules were defined in the 70's, they are the building blocks of endo-permutation modules and endo-permutation modules are important because for a $p$-nilpotent group $G$ the sources of the irreducible modules are the endo-permutation modules. When I say source here I'm talking about the Green correspondence which defines what a source and vertex is for a module. The idea, broadly, is that the modular representation theory of $G$ is wild, you cannot classify all the indecomposable modules, so you instead look for a smaller class of modules that you can classify and hope that this smaller class has some relation to the larger category so that the classification can enable you to prove boarder results. If you want to learn more about the Green correspondence there is a great book by Alperin called "Local Representation Theory" which will get you to some of the easier cases very quickly and without much background.

As for why endotrivial modules are a direct sum of an indecomposable and a projective, I can't give you the exact argument but I can tell you it's not going to follow from Krull-Schmidt or anything that is similarly elementary. The study of endotrivial modules is surprisingly hard given that they have such an elementary definition. The proofs tend to use very difficult cohomilogical arguments. Dade's original arguments about endotrivial modules relied on proving that the endotrivial condition is actually a local condition on the cohomilogical variety of the module, using Quillen's stratification theorem for a description of that variety. More recently the cohomology variety description can be simplified to what's called a rank variety description (google Jon Carlson to find tons of talks and papers about this) and I believe the argument you're looking for is related to the fact that the variety of an indecomposable module is itself indecomposable, but as I said I can't remember the exact details.

There is an easy(ish) way to prove that endotrivial modules are the sum of an indecomposable and a projective module. The statement is that a module $M$ is projective if and only if $Motimes M^*$ is projective. Suppose that this is true. Then if an endotrivial module $M$ had two separate indecomposable summands that were not projective, say $A$ and $B$, then you would see both $Aotimes A^*$ and $Botimes B^*$ as summands of $Motimes M^*$, so $M$ is not endotrivial.

Thus it suffices to prove the claim. This follows from two subfacts. The first is that if $M$ is projective and $N$ is any module, then $Motimes N$ is projective. If you believe this for $M$ free, then write $M$ as a summand of a free module and you are done.

The second fact is that, for any (absolutely) indecomposable module $M$, $M$ is a summand of $Motimes M^*otimes M$. This isn't very easy, but it's a case of writing down a specific map and seeing that it works. I believe this first appears in two papers of Benson-Carlson ($pnmid dim(M)$) and Auslander-Carlson ($pmid dim(M)$).

Putting these things together gives you the result, you don't need to invoke the support variety (which easily shows that $Motimes M^*$ cannot be projective if $M$ is not).