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I made the exact same question on MSE some days ago and there wasn't any response whatsoever so far, so thought probably I have to ask here to get an answer. So the question goes as follows:

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?

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    $\begingroup$ Roughly speaking, endo-trivial module arise naturally if you consider the stable module category for $\mathbb{K}G$ for a finite group $G$. Endotrivial modules become invertible (under the product induced by tensor product) in the stable module category. As to the last question, the Krull-Scmidt theorem holds for finite dimensional $\mathbb{K}G$-modules. Furthermore, if $U$ is non-projective, then $U \otimes U^{\ast}$ has at least one non-projective indecomposable summand. Hence an endotrivial module can have at most one non-projective indecomposable summand. $\endgroup$
    – Geoff Robinson
    Commented Jun 18, 2017 at 12:16
  • $\begingroup$ Thank you very much for your answer @GeoffRobinson! I have two questions to make, firstly could you please be more analytic with the conclusion of your comment, "...can have at most one non-projective indecomposable summand", because it isn't quite clear to me the at most one part. And regarding the aim of their study, what you imply is that because are occurring naturally in a certain setup we want to understand through their structure collectively, the representation theory of $G$, right? $\endgroup$
    – mayer_vietoris
    Commented Jun 18, 2017 at 13:46
  • $\begingroup$ Typos corrected: If $U,V$ are different non-projective indecomposable summands of $M,$ then $(U \otimes U^{\ast}) \oplus (V \otimes V^{\ast})$ is a summand of $M \otimes M^{\ast}$, and has at least two different non-projective indecomposable summands. But if $M$ is endotrivial, then $\mathbb{K}$ should be the unique non-projective indecomposable summand of $M \otimes M^{\ast}.$ As to second question- I gave one reason for interest. The stable module category is part of the representation theory of $\mathbb{K}G.$ $\endgroup$
    – Geoff Robinson
    Commented Jun 18, 2017 at 14:15
  • $\begingroup$ I suppose I should have said that ( at least when $p$ divides $|G|$) an endotrivial module clearly has at least one non-projective indecomposable summand, so has exactly one – Geoff Robinson 15 mins ago $\endgroup$
    – Geoff Robinson
    Commented Jun 18, 2017 at 16:18
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    $\begingroup$ Thank you very much for your replies, were really helpful. You can merge them up to write an answer if you want. I would like to make one more question, just in case you're aware of. Do you know any reference were there is a proof/explanation, why $T(G)= \mathbb{Z}_2$, where $G$ is a cyclic of order $p$ group, and $T(G)$ the group of iso. clasees of e-t modules? $\endgroup$
    – mayer_vietoris
    Commented Jun 18, 2017 at 19:02

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A good reference for this and related questions is Endo-Permutation modules over p-groups I and its sequel, that contain a lot of the background results. (Here, for example, is where Dade computes T(G) for G a cyclic or abelian p-group.)

A more recent reference, whose references cover a lot of more recent work on the subject, is Torsion-free endotrivial modules, which contains a statement and a reference to a proof of the result about splitting off a unique (up to isomorphism) indecomposable summand that you ask about.

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