I have given some thought to this question. I do not have a complete answer, but no one else has posted anything, so I will give what I have. First off, the functor category [I,Vect_k] is an AB5 category with a projective generator and hence a module category. In the particular case that I is the orbits of a group, finite or not, it is just k[G] modules. Now if k is finite, then k[G] is semisimple, whence all modules are injective, unless char(k) | #(G), the so-called modular case. In that case, I haven't worked out the details, but I think the tensor product of finite-dimensional injectives is injective. The argument uses duality in k. In fact, the category is self dual (a *-autonomous category). On the other hand, I think it unlikely that this is true for infinite dimensional spaces, but I do not have a counter-example. There are categories, for instance Ab, in which the tensor product of injectives is injective. The reason for Ab is that every injective is a direct sum of indecomposable injectives and the only non-zero tensor product of indecomposable injectives is Q tensor Q = Q. ================================================
From cat-dist@mailserv.mta.ca Fri May 9 12:19:50 1997 Received: from Math.McGill.CA (Gauss.Math.McGill.CA [132.206.150.3]) by triples.math.mcgill.ca (8.6.8/8.6.6) with SMTP id MAA04846; Fri, 9 May 1997 12:19:46 -0400 Received: from mailserv.mta.ca ([138.73.102.50]) by Math.McGill.CA (4.1/SMI-4.1) id AA21821; Fri, 9 May 97 12:24:36 EDT Received: by mailserv.mta.ca; id AA20359; Fri, 9 May 1997 13:11:56 -0300 Date: Fri, 9 May 1997 13:11:56 -0300 (ADT) From: categories <cat-dist@mta.ca> To: categories <categories@mta.ca> Subject: injectivity Message-Id: <Pine.OSF.3.90.970509131147.20454A-100000@mailserv.mta.ca> Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO
Date: Fri, 9 May 1997 15:30:53 +0200 (MET DST) From: Marek Golasinski <marek@mat.uni.torun.pl> Dear Colleagues, Let $Vect_k$ be the category of vector spaces over a field $k$ and $I$ a small category. Consider the category $I-Vect_k$ of all covariant functors from $I$ to $Vect_k$. For two object $F,F'$ of the category $I-Vect_k$ consider their tensor product $F\otimes F'$ such that $(F\otimes F')(i)=F(i)\otimes F'(i)$ for all $i\in I$ and in the obvious way on the morphisms of $I$. 1) Is it true that this tensor product $F\otimes F'$ is injective provided that $F$ and $F'$ are injective? I am really intersted in its particular case. Namely, let $G$ be a finite group and $O(G)$ the finite associated category of canonical orbits. Objects of $O(G)$ are given by the finite $G$-sets $G/H$ for all subgroups $H\subsetq G$ and morphisms by eqivariant maps. 2) What about preserving the injectivity by the above defined tensor product in the functor category $O(G)-Vect_k$? If that is not true for $I=O(G)$ then I would greatly appreciate getting a counterexample. Many thanks in advance for your kind attention on the problem above. With my best regards, Marek Golasinski