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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 …

No, not always. In Positselski, Leonid; Schnürer, Olaf M., Unbounded derived categories of small and big modules: is the natural functor fully faithful?, J. Pure Appl. Algebra 225, No. 11, Article ID …

Interpreting the various matrices as maps between free modules in the usual way, the question becomes: If $M$ is a submodule of $R^m$, and $\alpha,\beta: R^k\to M$ are epimorphisms, then must $\alpha$ …

This is Theorem 30.29 in Curtis, Charles W.; Reiner, Irving, Methods of representation theory, with applications to finite groups and orders. Vol. I, Pure and Applied Mathematics. A Wiley-Interscience …

I don't know the group of smallest order for which the conjecture has not been verified. But certainly it is known to be true for all groups of order less than 200. There are general results that deal …

In almost all cases I know of where people have proved derived equivalences between blocks of finite groups, the proof hasn't really gone that way (i.e., finding a virtual bimodule and refining it to …

Such rings were called "$F$-semiperfect", and more recently (thanks to rschweib for the information) "semiregular". One characterization is that these are the rings $R$ such that $\bar{R}=R/J(R)$, the …

I think the algebra you describe is actually the algebra denoted by $\hat{A}$ in the paper you reference. $\bar{A}$ is an uncompleted version. Neither algebra is an Artin algebra, or even an artinian …

The two notions are the same. Clearly the first implies the second. Assume that $\sigma^{(m)}$ is isomorphic to $\sigma$. So there is some $a\in GL_n(k)$ such that $a\sigma^{(m)}(g)a^{-1}=\sigma(g)$ f …

Representations of $B$ (or at least an equivalent category) are studied in the literature under the name of "cohomological Mackey functors". Theorem 1.1 of Bouc, Serge; Stancu, Radu; Webb, Peter, On t …

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 …

No. Let $R=\mathbb{Z}\times\mathbb{Z}$, let $P$ and $Q$ be the projective modules $\mathbb{Z}\times0$ and $0\times\mathbb{Z}$, and let $$P_\bullet=\dots\longrightarrow0\longrightarrow P\otimes_\mathb …

As I said in comments, there is a fair amount of literature to be found by Googling "infinite socle series". More specifically, a module $M$ for which (in the notation of the question) $\overline{\te …

Without more conditions it's not true. Take the Nakayama algebra with two simples and indecomposable projectives $$P(1)=\matrix{1\\2\\1}\hspace{1cm}\text{and}\hspace{1cm}P(2)=\matrix{2\\1}$$ Then th …

No. Let $G=C_2\times C_2$, generated by elements $g$ and $h$, and let $K$ be any finite field of characteristic $2$. Let $V$ be a finite dimensional $K$-vector space of dimension greater than one wi …