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Take the category of (at most) countable-dimensional vector spaces over your favourite field. Then take the quotient by the Serre subcategory of finite-dimensional vector spaces. (And take a skeletal …

For a simpler, but arguably more artificial, example than Mark's, take $\mathcal{C}$ to be the category with one object and two morphisms. Then the identity functor $\mathcal{C}\to\mathcal{C}$ is "unn …

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 …

Yes, it's always zero, assuming $D(A)$ means the unbounded derived category. My complexes will be cochain complexes, and $X[1]$ will be $X$ shifted down in degree. First suppose $X$ is bounded below …

Here's a counterexample that appears in nature. Fix a prime $p$ and a field $k$ of characteristic $p$, and let $G=C_{p^{n}}$ be a cyclic group of order $p^{n}$ (where $n\geq1$ if $p$ is odd, and $n\ge …

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 …

This is probably an absurdly over-complicated answer, but ... Let $$J(X)=\left\{\sum_{x\in X}a_xx\in GX: \sum_{x\in X}a_x=0\right\}.$$ I claim that $J$ is not of the form $H\circ G\circ F$. Suppos …

Words change their meanings. The original meaning of “tilting module” is that of Happel and Ringel in the representation theory of finite dimensional algebras, which requires the projective dimension …

In the example where $\mathcal{D}$ is the category of abelian groups and $\mathcal{C}$ is the category of finite abelian groups, take $F(X)=X\otimes_\mathbb{Z}\mathbb{Q}/\mathbb{Z}$. Then the restrict …

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.

Here are a couple of "natural" constructions that produce Frobenius algebras over a field that are not necessarily symmetric. (A) The trivial extension algebra of any algebra $A$ is defined to be $A\ …

If it's considered bad form to resurrect year-old threads, then please slap my wrist (gently, please; I'm new here!) A fairly simple explicit example of a "sumpact" module that is not f.g. is as foll …

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 …

A few months after the last activity on this question, Neena Gupta gave a proof that over a field $k$ of positive characteristic, a retract of a polynomial algebra need not be a polynomial algebra: ht …