If a tensor product of modules is semi-simple, are the tensor factors semi-simple ? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T12:36:59Z http://mathoverflow.net/feeds/question/22707 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/22707/if-a-tensor-product-of-modules-is-semi-simple-are-the-tensor-factors-semi-simple If a tensor product of modules is semi-simple, are the tensor factors semi-simple ? GD 2010-04-27T11:06:35Z 2010-07-11T23:01:07Z <p>Suppose $M$ and $M'$ are two $R$-moules (I am most interested in the case of $R$ a DVR). If $M\otimes M'$ is a semi-simple module (i.e., every submodule is a direct summand) then is it true that the tensor factors $M$ and $M'$ are semi-simple ?</p> <p>I.e., if this is not true then it would be nice to see a counter-example.</p> http://mathoverflow.net/questions/22707/if-a-tensor-product-of-modules-is-semi-simple-are-the-tensor-factors-semi-simple/22709#22709 Answer by Angelo for If a tensor product of modules is semi-simple, are the tensor factors semi-simple ? Angelo 2010-04-27T11:19:35Z 2010-04-27T11:19:35Z <p>No; if $R$ is a DVR with uniformizing paramenter $t$, the $R$-module $(R/tR) \otimes (R/t^2R) = R/tR$ is semisimple, but $R/t^2R$ is not.</p> http://mathoverflow.net/questions/22707/if-a-tensor-product-of-modules-is-semi-simple-are-the-tensor-factors-semi-simple/31454#31454 Answer by Akhil Mathew for If a tensor product of modules is semi-simple, are the tensor factors semi-simple ? Akhil Mathew 2010-07-11T19:22:32Z 2010-07-11T19:22:32Z <p>Another example: Let $A,B$ be two non-semisimple modules over a ring whose tensor product is zero (e.g. $R = \mathbb{Z}, A = B = \mathbb{Q}/\mathbb{Z}$); this was discussed elsewhere.</p> http://mathoverflow.net/questions/22707/if-a-tensor-product-of-modules-is-semi-simple-are-the-tensor-factors-semi-simple/31477#31477 Answer by George McNinch for If a tensor product of modules is semi-simple, are the tensor factors semi-simple ? George McNinch 2010-07-11T23:01:07Z 2010-07-11T23:01:07Z <p>I can't resist providing the following comment and references, even though it is likely not really relevant to the (already old by now) original post.</p> <p><em>Quick background:</em> Let $k$ be a field of char. $p>0$ and let $G$ be a(ny) group. Serre proved [Invent. Math. 116 (1994), no. 1-3, 513--530] that if $V$ and $W$ are finite dimensional, semisimple $kG$-modules and $$(\dim V - 1) + (\dim W -1) &lt; p$$ then $V \otimes_k W$ is a semisimple $kG$-module. (The argument is quite nice - one reduces to alg. closed $k$, and replaces $G$ by a certain algebraic group over $k$ whose identity component is reductive. One then has to argue that $[G:G^0]$ has order prime to $p$, so one is reduced to consideration of connected reductive $G$. And that case is handled by some "weight combinatorics" and the linkage principle.)</p> <p>In a subsequent paper [Semisimplicity and tensor products of group representations: converse theorems. With an appendix by Walter Feit. J. Algebra 194 (1997), no. 2, 496--520] Serre proved some "converse theorems". For example, he shows that $$V \otimes_k W \quad \text{semisimple} \implies V \ \text{semisimple if \dim W \not \equiv 0 \pmod{p}}$$ Examples (due to Feit, and included in the paper) show that one can't get rid of the assumption on $\dim W$.</p>