Related papers: Euler class group of Laurent polynomial ring : loc…

We provide a class of commutative Noetherian domains $R$ of dimension $d$ such that every finitely generated projective $R$-module $P$ of rank $d$ splits off a free summand of rank one. On this class, we also show that $P$ is cancellative.…

Suppose that $G$ is a finite group and $k$ is a field of characteristic $p >0$. Let $\mathcal{M}$ be the thick tensor ideal of finitely generated modules whose support variety is in a fixed subvariety $V$ of the projectivized prime ideal…

Let $R$ be an algebra over a ring $\Bbbk$, $T$ an $R$-algebra, $M$ a finitely generated projective $R$-module, and $N$ a $T$-module. Let $G$ be a linearly reductive group scheme over $\Bbbk$ equipped with a representation…

The purpose of this paper and its sequel, is to introduce a new class of modules over a commutative ring $R$, called $\mathbb{P}$-radical modules (modules $M$ satisfying the prime radical condition "$(\sqrt[p]{{\cal{P}}M}:M)={\cal{P}}$" for…

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 $\mathbb{K}[x,y]$…

Let $R$ be a finite unital commutative ring. We introduce a new class of finite groups, which we call hereditary groups over $R$. Our main result states that if $G$ is a hereditary group over $R$ then a unital algebra isomorphism between…

We classify, up to isomorphism, the $\mathbb{Z}_pG$-modules of rank $1$ (i.e., the quotients of $\mathbb{Z}_pG$) for $G$ cyclic of order $p$, where $\mathbb{Z}_p$ is the ring of $p$-adic integers. This allows us in particular to determine…

For a unimodular totally disconnected locally compact group $G$ we introduce and study an analogue of the Hattori-Stallings rank $\tilde{\rho}(P)\in\mathbf{h}_G$ for a finitely generated projective rational discrete left $\mathbb…

We define a class $\mathcal{U}$ of solvable groups of finite abelian section rank which includes all such groups that are virtually torsion-free as well as those that are finitely generated. Assume that $G$ is a group in $\mathcal{U}$ and…

A finite group $G$ is called a Schur group, if any Schur ring over $G$ is the transitivity module of a point stabilizer in a subgroup of $\sym(G)$ that contains all right translations. We complete a classification of abelian $2$-groups by…

Let G be a simply connected simple algebraic group over an algebraically closed field K of characteristic p>0 with root system R, and let ${\mathfrak g}={\cal L}(G)$ be its restricted Lie algebra. Let V be a finite dimensional ${\mathfrak…

Let R be a connected noetherian commutative ring, and let G be a simply connected reductive group over R of isotropic rank ge 2. The elementary subgroup E(R) of G(R) is the subgroup generated by the R-points U_P^+(R) and U_P^-(R) of the…

We construct elementary subgroups of all reductive groups of the local isotropic rank $\geq 2$ over rings and prove their basic properties. In particular, our results may be applied to the automorphism groups of any finitely generated…

In this paper we give elementary conditions completely characterising when the theory of modules of a Pr\"ufer domain is decidable. Using these results, we show that the theory of modules of the ring of integer valued polynomials is…

Let G be a finite p-group which does not contain a rank two elementary abelian p-group as a direct factor. Then the ideal of essential classes in the mod-p cohomology ring of G is a Cohen-Macaulay module whose Krull dimension is the p-rank…

It is well known that the ring radical theory can be approached via language of modules. In this work, we present some generalizations of classical results from module theory, in the two-sided and graded sense. Let $\mathsf{G}$ be a group,…

We prove that for a countable, commutative ring $R$, the class of countable $R$-modules either has only countably many isomorphism types, or else it is Borel complete. The machinery gives a succinct proof of the Borel completeness of TFAB,…

A group of order $p^n$ ($p$ prime) has an indecomposable polynomial invariant of degree at least $p^{n-1}$ if and only if the group has a cyclic subgroup of index at most $p$ or it is isomorphic to one of two particular groups of small…

If $\mathfrak{g} = \mathfrak{g}_{\overline{0}} \oplus \mathfrak{g}_{\overline{1}}$ is a Lie superalgebra over an algebraically closed field $k$ of characteristic 0, the notion of an endotrivial module has recently been extended to…

We consider the group G of R-automorphisms of the polynomial ring R[x] especially in the case where R is the ring of integers modulo n. We describe conjugacy classes in G, and in the case where n = 4, we describe more explicitly the…