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I thought that I had shown that the category of $\mathcal{A}$ of all torsion abelian groups is a Grothendieck category:

  1. All coproducts exist, they are just the coproducts of abelian groups;
  2. The colimits are also just the colimits in the category of abelian groups, since the cokernel of a morphism of torsion groups is also a torsion group;
  3. There is a generator.

The third point is the least obvious. Let $G=\bigoplus_{n\in \mathbb{N}}\mathbb{Z}/n\mathbb{Z}$ and $A$ be an arbitrary torsion abelian group. The morphism $\bigoplus_{0\neq a\in A}\mathbb{Z}\to A$, which takes the 1 of the summand corresponding to $a$ to $a,$ is an epimorphism in the category of abelian groups. But since $A$ is torsion, this filters through the map $\bigoplus_{0\neq a\in A}\mathbb{Z}/\text{ord}(a)\mathbb{Z}\to A$, which then naturally extends to a morphism $\bigoplus_{0\neq a\in A}G\to A.$

However, one of the comments at Complete but not cocomplete category says:

For example the category of torsion abelian groups is not grothendieck (there is no progenerator).

Although I do agree that there is no progenerator, the axioms of a Grothendieck category seemingly do not require its existence, only the existence of a not necessarily projective generator.

So in the end, is the category of torsion abelian groups actually Grothendieck?

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  • $\begingroup$ You have not shown the "Grothendieck" condition AB5: that direct unions preserve finite intersections, or equivalently that filtered colimits are exact. $\endgroup$ Jul 7, 2022 at 19:27
  • $\begingroup$ @VladimirSotirov I thought that's what I did in 2: the colimits in this category are just the colimits in the category of all abelian groups, and those are exact. $\endgroup$ Jul 7, 2022 at 21:25

1 Answer 1

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I think your observations are correct, and hold more generally.

Given a (not necessarily commutative) ring $A$, a torsion class is a full subcategory of (left) $A$-modules closed under quotients, coproducts, and extensions. In particular it is closed under colimits. A hereditary torsion class is one that is also closed under subobjects. In particular, the fact the filtered colimits of $A$-modules are exact implies filtered colimits of such torsion modules are exact. Finally, any hereditary torsion class is generated by the cyclic $A$-modules that are in the class (see chapter VI, Proposition 3.6. in Bo Stenstrom's book Rings of Quotients - An Introduction to Methods of Ring Theory). Thus hereditary torsion classes should be Grothendieck categories.

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