Search Results

It's not in general true for the category of modules for a ring. For example, let $R=\mathbb{C}[x]/(x^2)$, let $X=R\oplus R$ be the free module on two generators, and let $Y_1$ and $Y_2$ be the submo …

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 …

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 …

For $G$ a finite group, $H^3(G,\mathbb{Z})$ is isomorphic to the Schur multiplier, and you’ll find lots of examples using that as a search term (also, “Schur-trivial” is sometimes used to mean “having …

I suspect that this is one of those things that was essentially well-known to many people before anybody wrote it down, so it will be hard to pin down the first person to prove it. But the main idea …

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 …

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 think it's less confusing if you swap the roles of 0 and 1, as then the basic operation you're using to generate the entries of $\alpha\times\beta$ is addition mod 2. Then, if you write $\alpha_i$ …

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 you probably want the following paper. Auslander, Maurice, On the dimension of modules and algebras. III. Global dimension, Nagoya Math. J. 9, 67-77 (1955). ZBL0067.27103.

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 …

Yes. The full subcategory consisting of those $M$ for which this natural map is an isomorphism (for all $N$) is a thick subcategory by a five lemma argument, and the thick subcategory generated by $A …

In the representation theory of finite dimensional algebras, at least, it's called a "trivial extension algebra" (although that sometimes refers to the special case where $N$ is the vector space dual …

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 …

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 …