On Modular Vector Invariant Fields

Let [Formula: see text] be a finite field of any characteristic and [Formula: see text] be the general linear group over [Formula: see text]. Suppose W denotes the standard representation of [Formula: see text], and [Formula: see text] acts diagonally on the direct sum of W and its dual space W∗. Let G be any subgroup of [Formula: see text]. Suppose the invariant field [Formula: see text], where [Formula: see text] in [Formula: see text] are homogeneous invariant polynomials. We prove that there exist homogeneous polynomials [Formula: see text] in the invariant ring [Formula: see text] such that the invariant field [Formula: see text] is generated by [Formula: see text] over [Formula: see text].

Relations between modular invariants of a vector and a covector in dimension two

We exhibit a set of generating relations for the modular invariant ring of a vector and a covector for the two-dimensional general linear group over a finite field.

Generating sequences and semigroups of valuations on 2 dimensional normal local rings

In this thesis we develop a method for constructing generating sequences for valuations dominating the ring of a two dimensional quotient singularity. Suppose that K is an algebraically closed field of characteristic zero, K[X, Y] is a polynomial ring over K and v is a rational rank 1 valuation of the field K(X, Y) which dominates K[X, Y](X,Y) . Given a finite Abelian group H acting diagonally on K[X, Y], and a generating sequence of v in K[X, Y] whose members are eigenfunctions for the action of H, we compute a generating sequence for the invariant ring K[X, Y]H. We use this to compute the semigroup SK[X,Y ]H (v) of values of elements of K[X, Y]H. We further determine when SK[X,Y ]H (v) is a finitely generated SK[X,Y ]H (v)-module.

Rings whose finitely generated right ideals are quasi-projective

An invariant ring A is arithmetical if and only if every finitely generated ideal M of the ring A is a quasi-projective A-module and every endomorphism of this module may be extended to an endomorphism of the module A