This question is about proposition 1.8.1 in Grothendieck's tohoku paper. (I am reading out of Michael Barr's translation).

The proposition says that if $\mathcal{A}$ is an abelian category which is cocomplete and and$$\sum (A_i \cap B) = B \cap \sum A_i$$ for all objects $A$, all subobjects $B$ of $A$ and lattices $\{A_i \}$ of subobjects of $A$ then filtered colimits are exact.

Grothendieck leaves the proof to the reader. Could anyone share some insight on how to prove the above claim?

Yesterday I asked this question on math stackexchange and Zhen Lin helped me sort out the converse, but I am still stuck with the proposition and no one seems interested in the question.

I hope that this question is suitable for mathoverflow.

  • 1
    $\begingroup$ Dear Daniel, you can find a proof of this fact in H. Schubert, Kategorien I, Springer (1970), Satz 14.6.5. (I hope you know some German.) $\endgroup$ – Fred Rohrer Aug 10 '11 at 4:05
  • $\begingroup$ For a proof written in english you can look in Stemstrom's book. $\endgroup$ – Simone Virili Nov 16 '12 at 23:40

It is clear that colimits are right exact, since colimits commute with colimits in general. Thus it remains to show that filtered colimits preserve monomorphisms. Let $A_i \subseteq B_i$ be a filtered system of monomorphisms. It has to be shown that $\text{colim} A_i \to \text{colim} B_i$ is a monomorphism. Let $K$ be the kernel. Then $K \cap A_i = 0$ by assumption. But then $K = \text{colim} K \cap A_i = 0$, where the first equality follows easily from the assumption.


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