# Are module categories over a semisimple category semisimple?

Let $(\mathcal{C}, \otimes, \oplus, I, a, r, l)$ be an abelian semisimple monoidal category, then is it necessary that all module categories over $\mathcal{C}$ be semisimple?

I can convince myself that they need to be additive and have indecomposable objects, but I cannot see semisimplicity.

Thanks

• I don't really work with module categories over monoidal categories, but shouldn't you have some 'finite-dimensionality' condition on your module category for something like this to hold? Also, you do assume that $\mathcal{C}$ is semisimple right? (it's written in the title but not in the body of your post). – Mathematician 42 Feb 13 '18 at 20:37
• I edited my question, thank. Finite dimensionality for hom-set you mean? – Fatimah Feb 13 '18 at 21:06
• Can you give me a reference where you studied these things? I'm not familiar enough with these concepts to answer straightaway. If no-one else answers, I might think about it. – Mathematician 42 Feb 13 '18 at 21:15
• Well, taking $\mathcal C:=\mathbf{finVect}_k$, wouldn't that mean that every (finitely generated, associative, unital) algebra over $k$ is semisimple? – Berci Feb 14 '18 at 0:02
• @Mathematician42 Thanks! Basically this one: arxiv.org/abs/math/0111139v1 – Fatimah Feb 22 '18 at 12:13

No, they usually are not. To elaborate on Berci's comment, if $k$ is a field and $\mathcal{C}$ is the usual monoidal category of (finite-dimensional) vector spaces over $k$, then for any $k$-algebra $A$, the category of (finitely generated) right $A$-modules is a module category over $\mathcal{C}$ (by just using the usual tensor product over $k$: if $V$ is a vector space and $M$ is an $A$-module, then $V\otimes_k M$ is again an $A$-module). This category is semisimple iff $A$ is a semisimple ring, so any non-semisimple $k$-algebra gives a counterexample (e.g., $A=k[x]$, or $A=k[x]/(x^2)$ if you want some stronger finiteness conditions).