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An abelian category satisfy condition AB3 if it has small coproducts (hence cocomplete). Condition AB4 states that coproducts are exact: for any family of short exact sequences $0\to X_i \to Y_i \to Z_i \to 0$, the sequence $0\to \oplus X_i \to \oplus Y_i \to \oplus Z_i \to 0$ is exact.

What are examples of abelian categories satisfying AB3, but not AB4?

Condition AB5 says that direct limits are exact (or equivalently filtered colimits are exact).

What are examples of abelian categories satisfying AB4, but not AB5?

Alex
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1 Answers1

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There are well-known and naturally occurring examples for the dual questions, so you can just take the opposite categories.

Categories of sheaves of abelian groups are $AB3^*$ but typically not $AB4^*$, so their opposite categories are $AB3$ but typically not $AB4$.

Module categories of nonzero rings are $AB4^*$ but not $AB5^*$, so their opposite categories are $AB4$ but not $AB5$.

Jeremy Rickard
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  • In such categories (AB3 but not AB4), does homology still commute with direct sums? Obviously the standard treatment of showing this (using the exact sequence $0\rightarrow B_n\rightarrow Z_n\rightarrow H_n\rightarrow 0$) does not work in this case, so is there still a way of guaranteeing/disproving the commutativity of homology with exact sequences for such categories? – user_not_found Feb 02 '24 at 07:16
  • @moboDawn_φ No. By definition, in an abelian category which is AB3 but not AB4, there is some collection of short exact sequences (which can be regarded as complexes with zero homology) whose direct sum is not exact (i.e., has nonzero homology). Necessarily this collection will be infinite: homology commutes with finite direct sums in any abelian category. – Jeremy Rickard Feb 02 '24 at 08:41