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Let $I$ be a directed category and let $A$ be the category of $R$-modules ($R$ any ring). I'm trying to understand why the direct limit functor $$ \varinjlim_{I}: A^I \to A $$ is exact. Since it has a right adjoint it's sufficient to show that it preserves monomorphisms. Let $\alpha: F \to G$ be a monomorphism in $A^I$. The proofs in the literature I know of (Weibel: "Introduction to homological algebra", 2.6.15 or Eisenbud: "Commutative Algebra", A6.4) seem to use the following as definition for $\alpha$ being mono:

$\alpha(i): F(i) \to G(i)$ is a monomorphism in $A$ for each $i \in obj(I).$ $\hspace{27pt}$ $\hspace{27pt}$ $(*)$

It's easy to see that $(*)$ implies that $\alpha$ is mono in the usual sense (e.g. if $\beta: H \to F$ is a homomorphism in $A^I$ such that $\alpha \beta = 0$ then $\beta = 0$).

Does anyone know if $(*)$ is equivalent to this definition of a monomorphism ?

I was only able to settle the following special case:

Let $A$ be an abelian category and $I$ a small category such that for all $i, j \in obj(I)$:

  • $Hom_I(i,i) = \lbrace id_i \rbrace $
  • $Hom_I(i,j) \neq \emptyset \implies Hom_I(j,i) = \emptyset \hspace{5pt} (i \neq j)$
Then $\alpha$ above is mono iff $(*)$ holds.

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Showing that taking direct limit is exact is exercise 19 in chapter 2 of Atiyah-Macdonald (there's a hint there). –  Łukasz Grabowski Jan 10 '11 at 13:03
    
@Lukasz: Thanks for the reference. But they also take (*) as definition for being mono. –  Ralph Jan 10 '11 at 13:18
    
@Ralph: To show that mono implies (*) at a given i \in I consider a system over I which is non-zero only at i. –  Łukasz Grabowski Jan 10 '11 at 13:34
    
The point Sergio is making in his answer is a very good one -- it concerns your initial motivation that you want to show that filtered colimits of $R$-modules are exact. This is a special property of module categories and it is important to notice that the dual is wrong. Filtered limits of short exact sequences need not be exact, see en.wikipedia.org/wiki/Mittag-Leffler_condition –  Theo Buehler Jan 10 '11 at 14:40

3 Answers 3

The answer is yes if $A=R\mbox{-Mod}$ has pullbacks (which I'm pretty sure it does). See Tom Leinster's answer to a similar question.

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Good point . –  Martin Brandenburg Jan 10 '11 at 13:55
    
Thanks Finn, that's what I was looking for. –  Ralph Jan 10 '11 at 14:14

An Abelian category where filtrant colimts are exats is called a (abelian) Grothendieck category, no all Abelan categories are Grothendieck. Of course R-Mod is Grothendieck, because the forgetful functors U: R-Mod -> Set create limits (then preserve and reflexct Monomorphisms) and U create filtrant colimits, just because these (in Set) are coherent with finite products and algebraic structures.

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$A^I$ is an abelian category. Namely, the direct sums, kernels and cokernels may be constructed pointwise. In particular, $\alpha$ is mono iff $\ker(\alpha)=0$. But a functor $I \to A$ vanishes iff it vanishes pointwise. Thus $\alpha$ mono iff all $\alpha(i)$ mono.

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Why the downvote? –  Martin Brandenburg Mar 15 '12 at 8:32

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