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Not necessarily. For example, let $k$ be a field, and $R$ the $k$-algebra $k\oplus V$, where $V$ is an infinite dimensional square-zero ideal. Let $C$ be the category of $R$-modules and $T$ the Serre …

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}^ …

I posted the following example of a Grothendieck category with no simple objects in answer to a question on math.stackexchange about seven and a half years ago. I seem to have said at the time that it …

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 …

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 …

Here's a simple counterexample for (2). Let $\mathcal{A}$ be the category of sequences of linear maps $U\stackrel{\alpha}{\to}V\stackrel{\beta}{\to}W$ between vector spaces over a field $k$ such that …

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 …

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.

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 answer to Question 1 is "yes". I'll identify $\mathsf{A}'$ with a Serre subcategory of $\mathsf{A}$ via $i_*$, and the quotient $\mathsf{A}/\mathsf{A}'$ with $\mathsf{A}''$ via $j^*$. Then for a …

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 …

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 …

Let $k$ be a field, and let $A$ be the $3$-dimensional commutative $k$-algebra $k[x,y]/(x^2,xy,y^2)$. Then in the category of $A$-modules there is a unique indecomposable injective, namely the dual $D …

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 …

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