* Re: injectivity
@ 1997-05-29 17:33 categories
0 siblings, 0 replies; 3+ messages in thread
From: categories @ 1997-05-29 17:33 UTC (permalink / raw)
To: categories
Date: Wed, 28 May 1997 11:22:00 +0200
From: Dr. Reinhard B/rger (Prof. Dr. Pumpl^nn) <Reinhard.Boerger@FernUni-Hagen.de>
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
^ permalink raw reply [flat|nested] 3+ messages in thread
* Re: injectivity
@ 1997-05-16 1:17 categories
0 siblings, 0 replies; 3+ messages in thread
From: categories @ 1997-05-16 1:17 UTC (permalink / raw)
To: categories
Date: Tue, 13 May 1997 17:52:00 -0400
From: Michael Barr <barr@triples.math.mcgill.ca>
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
^ permalink raw reply [flat|nested] 3+ messages in thread
* injectivity
@ 1997-05-09 16:11 categories
0 siblings, 0 replies; 3+ messages in thread
From: categories @ 1997-05-09 16:11 UTC (permalink / raw)
To: categories
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
^ permalink raw reply [flat|nested] 3+ messages in thread
end of thread, other threads:[~1997-05-29 17:33 UTC | newest]
Thread overview: 3+ messages (download: mbox.gz / follow: Atom feed)
-- links below jump to the message on this page --
1997-05-29 17:33 injectivity categories
-- strict thread matches above, loose matches on Subject: below --
1997-05-16 1:17 injectivity categories
1997-05-09 16:11 injectivity categories
This is a public inbox, see mirroring instructions
for how to clone and mirror all data and code used for this inbox;
as well as URLs for NNTP newsgroup(s).