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I think I have an example. Fix a chain of fields $k_\alpha$ indexed by ordinals $\alpha$, where $k_\alpha\subset k_\beta$ is an infinite field extension for all pairs $\alpha<\beta$ of ordinals. Fir …

The category of countable abelian groups is an essentially small abelian category, and has enough projectives and injectives (the countable free abelian groups and the countable divisible groups respe …

I was recently reminded about this old question on math.stackexchange. Let $\operatorname{Mod}R$ be the category of (right) modules for a ring $R$. The questioner mistakenly thought that the Freyd-Mit …

The category of finite abelian $p$-groups (where $p$ is your favourite prime) is an abelian category with no proper nonzero Serre subcategories, but not every short exact sequence splits.

In Conjecture $1_{\infty}$ of Simson, Daniel, On large indecomposable modules, endo-wild representation type and right pure semisimple rings., Algebra Discrete Math. 2003, No. 2, 93-118 (2003). ZBL106 …

If $A$ is a $k$-algebra, and $M$,$N$ are finite-dimensional $A$-modules, then $$\operatorname{Ext}^i_A(M,N)\cong\operatorname{Tor}^A_i(M,N^*)^*$$ (where $*$ denotes $k$-dual). So $\operatorname{Ext}^ …

This is Proposition 4.2 in Buchsbaum’s Satellites and universal functors, Annals of Mathematics 71(2), pp. 199–209 (1960). Well, to be precise, that is the dual result (for contravariant functors). Bu …

Here's an example of a full exact embedding of the module category of a non-Noetherian ring $S$ into that of a Noetherian ring $R$, preserving all direct sums and direct products. So this gives an exa …

Here's a simpler, but less consequential, example. Take the category of "eventually constant" functors from ordinals (considered as a category with a single morphism $\alpha\to\beta$ when $\alpha\leq …

Here's another example for Question 2 that I encountered in nature. In the derived category of modules for a ring, pick one module $M_i$ for each $i\in\mathbb{Z}$, and let $A_i=M_i[i]$. Then the natur …

It's not true, without boundedness conditions, that a left exact functor always preserves quasi-isomorphisms between complexes of $F$-acyclic objects. As alluded to in the question, a chain map is a q …

No, not in general. For example, let $R=k[x]/(x^2)$ for a field $k$ and let $X$ be the object $$\dots\stackrel{x}{\to}R\stackrel{x}{\to}R\stackrel{x}{\to}R\to0\to0\to\dots$$ of the derived category o …

I don't think (3) implies (1). For example, the opposite category of the category of abelian groups satisfies (3), but is not AB5.

The statement about simple Lie algebras is not true. A (finitely generated right) module $P$ for a ring $R$ is stably free if $P\oplus R^m\cong R^n$ for some integers $m,n$. Suppose $R$ has a non-f …

This is Proposition 3.1.2 in the thesis of Parra, "Hearts of $t $-structures which are Grothendieck or module categories".