8
$\begingroup$

Let $R$ be a ring (not necessarily commutative).

Let $A$ be a left $R$-module. When does the functor $\text{Hom}(A,-)$ preserve direct sums - in the category of left $R$-modules? For example, this is certainly true when $A$ is free and finitely generated (EDIT: or finitely generated in general, as suggested by Pierre-Yves).

Do we always need the finitely generated condition?

  • 0
    Dear Evariste: It seems clear to me that direct sums are preserved if $A$ is finitely generated. I'd be tempted to say that the converse holds. I'll think about it. Many users know this kind of things much better than I, and I hope (for you and for everybody) they they'll answer your question.2011-11-02
  • 0
    Answered here: http://mathoverflow.net/questions/59282/sums-compact-objects-f-g-objects-in-categories-of-modules2011-12-19

1 Answers 1

7

There have been two previous incorrect versions of the answer. I apologize for them, and thank Mariano for his friendly and efficient help. Thank you also to Evariste!

Here are the claims:

(1) If $A$ is finitely generated, then the functor $\text{Hom}_R(A,?)$ preserves direct sums.

(2) If there is an increasing sequence $(A_i)_{i\ge1}$ of proper submodules whose union is $A$, then the functor $\text{Hom}_R(A,?)$ does not preserve all the direct sums.

I don't know if such a sequence $(A_i)_{i\ge1}$ exists whenever $A$ is not finitely generated. (And I'm very anxious to know if this is true.)

Proof of (1). We have a natural $\mathbb Z$-linear injection
$$ F:\bigoplus_i\ \text{Hom}_R(A,B_i)\to\text{Hom}_R(A,\oplus_i\ B_i). $$ Moreover $g\in\text{Hom}_R(A,\oplus_i\ B_i)$ is in the image of $F$ if and only if $g(A)$ is contained into the sum of finitely many $B_i$.

In particular $F$ is bijective if $A$ is finitely generated.

Proof of (2). The natural $R$-linear map from $A$ into $\oplus\,A/A_i$ is not in the image of $F$.

EDIT. Here are three references:

http://ncatlab.org/nlab/show/coproduct-preserving+representable

Preservation of direct sums and finite generation

https://mathoverflow.net/questions/59282/sums-compact-objects-f-g-objects-in-categories-of-modules/81333#81333

  • 0
    Ok, you convinced me. Thanks!2011-11-02
  • 0
    Dear Evariste: Thanks a lot! Unfortunately, I fear that there a mistake in my post. More precisely, I think: (1) The part from "Assume $A$ is **not** finitely generated" is incorrect. (2) The main statement is correct. While trying to fix (1), I'm leaving the post as is for the moment. I'm really sorry...2011-11-02
  • 0
    Hm... The most important thing for me was the question whether finitely generated is sufficient, as I really didn't know what the answer should be. So that's fine. I'll think again about what happens for non-finitely generated modules.2011-11-02
  • 0
    Dear @Evariste: I hope it's correct now. Thanks again for having asked such an interesting question and for being so positive!2011-11-02
  • 0
    Dear Pierre: in your updated construction, it may well be the case that an element $b\in A$ does not belong to infinitely many of the $A_a$'s, so that the map you propose does not really take values in the direct sum.2011-11-02
  • 0
    What you want is an increasing sequence $(A_i)_{i\geq1}$ of proper submodules whose union is $A$. I'm pretty sure you can always get such a thing in a non-finitely generated module.2011-11-02
  • 0
    Dear @Mariano: [I wrote this before seeing your last comment. Things seem to converge:] You're right! (I had just realized that). Right now it seems to me that it suffices to take a countably generated quotient, but I've already mad so many mistakes... (I mean: on this question; I'm not talking about the one you corrected the other day... I'm getting more and more indebted to you...)2011-11-02
  • 0
    Dear @Mariano: Your formulation "an increasing sequence $(A_i)_{i\ge1}$ of proper submodules whose union is $A$" is outstanding. I'm looking for a precise argument showing that such exists if $A$ is not f. g. If you find an argument to this effect before me (which is very likely), and if you're kind enough to post it, I'll restrict my answer to the f. g. case, and refer to yours.2011-11-02
  • 0
    Excuse me, in the answer above, is $F$ a $\mathbb{Z}$-liner map or an $R$-linear map? $A$ is finitely generated means $R$-finitely generated? Thanks a lot!2012-06-06
  • 1
    Dear @Peter: $F$ is $\mathbb Z$-linear, and "finitely generated" means "finitely generated over $R$".2012-06-06
  • 0
    Dear @Pierre-Yves Gaillard: Thank you very much! I will check that by myself.2012-06-06