Related papers: A note on co-Hopfian groups and rings

Let $k$ be an algebraically closed field of characteristic $p>0$. For a loop $\circlearrowleft$, denote its path coalgebra by $k\circlearrowleft$. In this paper, all the finite-dimensional commutative Hopf algebras over the sub coalgebras…

In this paper, we use trace methods to study the algebraic $K$-theory of rings of the form $R[x_1,\ldots, x_d]/(x_1,\ldots, x_d)^2$. We compute the relative $p$-adic $K$ groups for $R$ a perfectoid ring. In particular, we get the integral…

Let $H$ be a finite-dimensional Hopf algebra over an algebraically closed field of characteristic 0. If $H$ is not semisimple and $\dim(H)=2n$ for some odd integer $n$, then $H$ or $H^*$ is not unimodular. Using this result, we prove that…

The article gives the second part of the treatise on Regular Algebraic $K$-theory (Sections V & VI) of the author. Regular algebraic $K$-theory for groups is a homology theory for discrete groups closely connected to (but different from)…

We prove that every group ring of a non-abelian locally free group which is the union of an ascending sequence of free groups is primitive. In particular, every group ring of a countable non-abelian locally free group is primitive. In…

It is proved that given any prime ideal $\mathfrak{p}$ of height at least 2 in a countable commutative noetherian ring $A$, there are uncountably many more dualizable objects in the $\mathfrak{p}$-local $\mathfrak{p}$-torsion stratum of the…

Let $\alpha$ be a non-zero algebraic number. Let $K$ be the Galois closure of $\mathbb{Q}(\alpha)$ with Galois group $G$ and $\bar{\mathbb{Q}}$ be the algebraic closure of $\mathbb{Q}$. In this article, among the other results, we prove the…

In this note we introduce pro-Hall $R$-groups as inverse limits of Hall $R$-groups and show that for the binomial closure $S^{bin}$ of any ring $S$ discriminated by $\mathbb{Z}_p$, the free pro-Hall $S^{bin}$-group $\mathbb{F}(A,S^{bin})$…

Let $G$ be the simple algebraic group $\mathrm{SL}_2$ defined over an algebraically closed field $k$ of characteristic $p > 0$. Using results of A. Parker, we develop a method which gives, for any $q \in \mathbb{N}$, a closed form…

We prove finite generation of the cohomology ring of any finite dimensional pointed Hopf algebra, having abelian group of grouplike elements, under some mild restrictions on the group order. The proof uses the recent classification by…

We show that every central simple algebra A over a field k is Brauer equivalent to a quotient of a finite dimensional Hopf algebra over the same field (that is- A is Hopf Schur). If the characteristic of the field is zero, or if the algebra…

Let K be a nontrivial knot in the 3-sphere with the exterior E(K), and u in G(K), the fundamental group of E(K), a slope element represented by an essential simple closed curve on the boundary of E(K). Since the normal closure of u in G(K)…

Let $\Bbbk$ be a field of characteristic $p>0$, $V$ a finite-dimensional $\Bbbk$-vector-space, and $G$ a finite $p$-group acting $\Bbbk$-linearly on $V$. Let $S = \Sym V^*$. We show that $S^G$ is a polynomial ring if and only if the…

Let $\mathbb{K}$ be an algebraically closed field of characteristic zero and $\mathbb{K}[x,y]$ the polynomial ring. The group $\text{SL}_{2}\left(\mathbb{K}[x,y]\right)$ of all matrices with determinant equal to $1$ over $\mathbb{K}[x,y]$…

In a paper from 2002, Bruns and Gubeladze conjectured that graded algebra retracts of polytopal algebras over a field $k$ are again polytopal algebras. Motivated by this conjecture, we prove that graded algebra retracts of Stanley-Reisner…

Let $\Bbbk$ be an algebraically closed field of characteristic $p>0$. We study the general structures of $p^n$-dimensional Hopf algebras over $\Bbbk$ with $p^{n-1}$ group-like elements or a primitive element generating a…

We show that the hermitian K-theory space of a commutative ring R can be identified, up to A^1-homotopy, with the group completion of the groupoid of oriented finite Gorenstein R-algebras, i.e., finite locally free R-algebras with…

Let $n=2g+2$ be a positive even integer, $f(x)$ a degree $n$ complex polynomial without multiple roots and $C_f: y^2=f(x)$ the corresponding genus $g$ hyperelliptic curve over the field $\C$ of complex numbers. Let a $(g-1)$-dimensional…

We investigate a generalization of Hopf algebra $\mathfrak{sl}_{q}(2)$ by weakening the invertibility of the generator $K$, i.e. exchanging its invertibility $KK^{-1}=1$ to the regularity $K\overline{K}K=K$. This leads to a weak Hopf…

Let p be a prime. We prove that if a finite group G has non-abelian Sylow p-subgroups, and the class size of every p-element in G is coprime to p; then G contains a simple group as a subquotient which exhibits the same property. In addition…