Journal of Algebra and Its Applications最新文献

Let $m$ be an odd positive integer and $D_m({𝒜})$ be the $m$-periodic derived category of a finitary hereditary Abelian category ${𝒜}$. In this note, we prove that there is an embedding of algebras from the derived Hall algebra of $D_m({𝒜})$ defined by Xu–Chen [Hall algebras of odd periodic triangulated categories, Algebr. Represent. Theory16(3) (2013) 673–687] to the extended derived Hall algebra of $D_m({𝒜})$ defined in [H. Zhang, Periodic derived Hall algebras of hereditary Abelian categories, preprint (2023), arXiv:2303.02912v2]. This homomorphism is given on basis elements, rather than just on generating elements.

We study Abelian groups whose endomorphism rings are V-rings. Let $G$ be a non-reduced Abelian group, We prove that $\mathrm{\text{End}}(G)$ is a V-ring on either side if and only if $G=B\oplus {\mathbb{Q}}^n$ where $B$ is a tame elementary Abelian group. We observe that a reduced group whose endomorphism is a V-ring, is an sp-group. Recognizing that $\mathrm{\text{End}}(G)$ is also an sp-group of ${\prod }_{p\in \mathbb{P}}\mathrm{\text{End}}(G_p)$, we show that $\mathrm{\text{End}}(G)/\oplus \mathrm{\text{End}}(G_p)$ is a V-ring if and only if $\mathrm{\text{End}}(G)$ is a V-ring.

We investigate the question of how to compute the cotensor product, and more generally the derived cotensor (i.e. Cotor) groups, of a tensor product of comodules. In particular, we determine the conditions under which there is a Künneth formula for Cotor. We show that there is a simple Künneth theorem for Cotor groups if and only if an appropriate coefficient comodule has trivial coaction. This result is an application of a spectral sequence we construct for computing Cotor of a tensor product of comodules. Finally, for certain families of nontrivial comodules which are especially topologically natural, we work out necessary and sufficient conditions for the existence of a Künneth formula for the $0$th Cotor group, i.e. the cotensor product. We give topological applications in the form of consequences for the $E_2$-term of the Adams spectral sequence of a smash product of spectra, and the Hurewicz image of a smash product of spectra.

Let $G$ be a graph with vertex set $V(G)$, $f$ a permutation of $V(G)$. Define ${\delta }_f(x,y)=|d(x,y)-d(f(x),f(y))|$ and ${\delta }_f(G)=\sum {\delta }_f(x,y)$, where the sum is taken over all unordered pairs $x$, $y$ of distinct vertices of $G$. Let $\pi (G)$ denote the smallest positive value of ${\delta }_f(G)$ among all permutations $f$ of $V(G)$. A permutation $f$ with ${\delta }_f(G)=\pi (G)$ is called a near automorphism of