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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 …

No. The projective model structure on chain complexes of modules over a ring is an abelian model category, and the homotopy category is the derived category, which is never abelian unless the ring is …

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 …

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\ …

The motivation for the definition was an attempt to explain structurally the phenomenon of an "isotypy". This makes sense for arbitrary blocks, but let's stick to principal blocks for simplicity. Sup …

Inspired by the reference Julian gave in his comment, here's an explicit example of a two-parameter family of indecomposable representations. First, I'll describe a one-parameter family of indecompos …

Yes. In fact, $V$ has an infinite dimensional semisimple quotient, which is decomposable since any simple $\mathbb{F}_pG$-module is a quotient of $\mathbb{F}_pG$ and so is finite-dimensional. Let $V' …

As Geoff thought, the answer is contained in James and Kerber (it's Theorem 2.5.13 in "The Representation Theory of the Symmetric Group", Encyclopedia of Mathematics and its Applications vol. 16, 1981 …

If $C$ is a right module then $C^*$ is a left module. In Landrock, $\text{soc}(R)$ is the right socle. As he proves, it is a two-sided ideal, but it may not be semisimple as a left module (it is not …

The condition on Brauer characters is not sufficient. Let $G$ be a $p$-group, $\pi$ any nontrivial representation over $\mathbb{F}_p$, and $\sigma$ the trivial representation over $\mathbb{Q}_p$ of t …

Here's one way of constructing counterexamples for finite groups. Suppose $M$ is a periodic $kG$-module with period $p$: i.e., the $p$th syzygy $\Omega^pM$ is isomorphic to $M$, but $\Omega M\not\con …

In Nüsken, M. "Unique tensor factorization of algebras", Math Ann. (1999) 315-341 this is proved for $K$ of characteristic zero. As far as I know, it's still open in positive characteristic, although …

In a very short paper put on the arXiv recently, García, Margolis and del Río give examples of nonisomorphic finite $2$-groups $G$ and $H$ with $\mathbb{F}_2G\cong\mathbb{F}_2H$, thus solving the modu …

There seems to be a classification of representations for the example you mention (and more generally for representations of $C_p$ over $\mathbb{Z}/p^s\mathbb{Z}$) in V. S. Drobotenko, E. S. Drobo …

Let $K$ be an algebraic closure of $k$. The following lemma must surely be well-known, but I haven't found an explicit reference, so I'll include a proof at the end of this post. Lemma. If $S$ is …