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
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
Michael Barr mentions the example Ab. There is even an easier reason why tensor products of injectives in Ab are injective and it even injectivity of one factor suffices: Injecitive abelian groups coincide with divisible abelian groups and a tensor product is divisible if one factor is. This holds in a more general sitution, e.g. for modules over a principal ideal domain. It might be worthwile to look for a general (categorical) reason for this phenomenon. Greetings Reinhard Boerger
participants (3)
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Dr. Reinhard B/rger -
Marek Golasinski -
Michael Barr