Related papers: On an example of Nagarajan

For $K$ an infinite field of characteristic other than two, consider the action of the special orthogonal group $\operatorname{SO}_t(K)$ on a polynomial ring via copies of the regular representation. When $K$ has characteristic zero,…

We present two noncommutative algebras over a field of characteristic zero that each posses a family of actions by cyclic groups of order $2n$, represented in $n \times n$ matrices, requiring generators of degree $3n$.

A power series ring over non-Noetherian rings can fail to be flat over the base ring, and its dimension can be infinite, even when the dimension of the base ring is finite. We study the case when the base ring has Krull dimension 0, and…

Working over an algebraically closed base field $k$ of characteristic 2, the ring of invariants $R^G$ is studied, where $G$ is the orthogonal group O(n) or the special orthogonal group SO(n), acting naturally on the coordinate ring $R$ of…

We answer a question of Abhyankar by constructing an algebraic closure of the field of power series over a field of positive characteristic, using "generalized power series". (The corresponding construction in characteristic 0 dates back to…

Let $k$ be a perfect field of characteristic $p>0$, $k(t)_{per}$ the perfect closure of $k(t)$ and $A$ a $k$-algebra. We characterize whether the ring $A\otimes_k k(t)_{per}$ is noetherian or not. As a consequence, we prove that the ring…

Let $k$ be a field of characteristic zero, let $R$ be the ring of formal power series in $n$ variables over $k$ and let $D(R,k)$ be the ring of $k-$linear differential operators in $R$. If $M$ is a finitely generated $D(R,k)-$module then…

Let R* be an ideal-adic completion of a Noetherian integral domain R and let L be a subfield of the total quotient ring of R* such that L contains R. Let A denote the intersection of L with R*. The integral domain A sometimes inherits nice…

Let $k$ be a field and $G \subseteq Gl_n(k)$ be a finite group with $|G|^{-1} \in k$. Let $G$ act linearly on $A = k[X_1, \ldots, X_n]$ and let $A^G$ be the ring of invariant's. Suppose there does not exist any non-trivial one-dimensional…

In this paper we exhibit an example of a three-dimensional regular local domain (A, n) having a height-two prime ideal P with the property that the extension PA^ of P to the n-adic completion A^ of A is not integrally closed. We use a…

We show that if X is a toric scheme over a regular ring containing a field then the direct limit of the K-groups of X taken over any infinite sequence of nontrivial dilations is homotopy invariant. This theorem was known in characteristic…

We extend and generalize the results of Scheiderer (2006) on the representation of polynomials nonnegative on two-dimensional basic closed semialgebraic sets. Our extension covers some situations where the defining polynomials do not…

We investigate actions of cyclic groups on polynomial rings with two variables, and the minimal free resolution of the corresponding invariant ring. In particular, we fully classify several cases, including the case the defining ideal has…

In characteristic two and dimension two, wild Z/2Z-actions on k[[u,v]] ramified precisely at the origin were classified by Artin, who showed in particular that they induce hypersurface singularities. We introduce in this article a new class…

Over fields of characteristic two, we construct an infinite ascending chain of GL-stable ideals in the coordinate ring of infinite skew-symmetric matrices. This construction provides the first known example of a non-noetherian GL-algebra,…

Using Fourier-Mukai transformations, we prove some results about the ring of unipotent vector bundles on elliptic curves in positive characteristics. This ring was determined by Atiyah in characteristic zero, who showed that it is a…

In this paper, we study the ring of invariants under the action of SL(m,K)\times SL(n,K) and SL(m,K)\times SL(n,K)\times SL(2,K) on the 3-dimensional array of indeterminates of form m\times n\times 2, where K is an infinite field. And we…

We determine the rings of invariants in the symmetric algebra on the dual of a vector space V over the field of two elements, for the group G of orthogonal transformations preserving a non-singular quadratic form on V. The invariant ring is…

Miyanishi proved that the ring of invariants of any $\mathbb{G}_a$ action on $\mathbb{A}^3$ is $\mathbb{A}^2$, when the field $k$ has zero characteristic. However, it is not known if this result holds when $k$ has positive characteristic.…

Combining a characterisation by B\'elair, Kaplan, Scanlon and Wagner of certain $\rm NIP$ valued fields of characteristic $p$ with Dickson's construction of cyclic algebras, we provide examples of noncommutative $\rm NIP$ division ring of…