We find conditions on the local cohomology modules of multi-Rees algebras of admissible filtrations which enable us to predict joint reduction numbers. As a consequence, we are able to prove a generalization of a result of Reid, Roberts and Vitulli in the setting of analytically unramified local rings for completeness of power products of complete ideals.

We obtain a characterisation of the monomial ideals $I\subseteq \mathbb{C}[x_{1},\dots ,x_{n}]$ of finite colength that satisfy the condition $e(I)={\mathcal{L}}_{0}^{(1)}(I)\cdots {\mathcal{L}}_{0}^{(n)}(I)$, where ${\mathcal{L}}_{0}^{(1)}(I),\dots ,{\mathcal{L}}_{0}^{(n)}(I)$ is the sequence of mixed Łojasiewicz exponents of $I$ and $e(I)$ is the Samuel multiplicity of $I$. These are the monomial ideals whose integral closure admits a reduction generated by homogeneous polynomials.

Let $k$ be a field and $\mathbb{V}$ the affine threefold in $\mathbb{A}^4_k$ defined by $x^m y=F(x, z, t)$, $m \ge 2$. In this paper, we show that $\mathbb{V} \cong \mathbb{A}^3_k$ if and only if $f(z, t): = F(0, z, t)$ is a coordinate of $k[z, t]$. In particular, when $k$ is a field of positive characteristic and $f$ defines a non-trivial line in the affine plane $\mathbb{A}^2_k$ (we shall call such a $\mathbb{V}$ as an Asanuma threefold), then $\mathbb{V}\ncong \mathbb{A}^3_k$ although $\mathbb{V} \times \mathbb{A}^1_k \cong \mathbb{A}^4_k$, thereby providing a family of counter-examples to Zariski’s cancellation conjecture for the affine 3-space in positive characteristic. Our main result also proves a special case of the embedding conjecture of Abhyankar–Sathaye in arbitrary characteristic.

Computational Galois theory, in particular the problem of computing the Galois group of a given polynomial, is a very old problem. Currently, the best algorithmic solution is Stauduhar’s method. Computationally, one of the key challenges in the application of Stauduhar’s method is to find, for a given pair of groups $H<G$ , a $G$ -relative $H$ -invariant, that is a multivariate polynomial $F$ that is $H$ -invariant, but not $G$ -invariant. While generic, theoretical methods are known to find such $F$ , in general they yield impractical answers. We give a general method for computing invariants of large degree which improves on previous known methods, as well as various special invariants that are derived from the structure of the groups. We then apply our new invariants to the task of computing the Galois groups of polynomials over the rational numbers, resulting in the first practical degree independent algorithm.

We prove that ${ \mathbb{Z} }_{{p}^{n} } $ and ${ \mathbb{Z} }_{p} [t] / ({t}^{n} )$ are polynomially equivalent if and only if $n\leq 2$ or ${p}^{n} = 8$. For the proof, employing Bernoulli numbers, we explicitly provide the polynomials which compute the carry-on part for the addition and multiplication in base $p$. As a corollary, we characterize finite rings of ${p}^{2} $ elements up to polynomial equivalence.

The aim of this paper is to give a classification theorem for commutative torsion filial rings.

Let X be a smooth projective variety of dimension n in Pr, and let π:X→Pn+c be a general linear projection, with c>0. In this paper we bound the scheme-theoretic complexity of the fibers of π. In his famous work on stable mappings, Mather extended the classical results by showing that the number of distinct points in the fiber is bounded by B:=n/c+1, and that, when n is not too large, the degree of the fiber (taking the scheme structure into account) is also bounded by B. A result of Lazarsfeld shows that this fails dramatically for n≫0. We describe a new invariant of the scheme-theoretic fiber that agrees with the degree in many cases and is always bounded by B. We deduce, for example, that if we write a fiber as the disjoint union of schemes Y′ and Y′′ such that Y′ is the union of the locally complete intersection components of Y, then deg Y′+deg Y′′red≤B. Our method also gives a sharp bound on the subvariety of Pr swept out by the l-secant lines of X for any positive integer l, and we discuss a corresponding bound for highly secant linear spaces of higher dimension. These results extend Ran’s ‘dimension +2 secant lemma’.

Let k be an algebraically closed field of positive characteristic p. We consider which finite groups G have the property that every faithful action of G on a connected smooth projective curve over k lifts to characteristic zero. Oort conjectured that cyclic groups have this property. We show that if a cyclic-by-p group G has this property, then G must be either cyclic or dihedral, with the exception of A4 in characteristic two. This proves one direction of a strong form of the Oort conjecture.

This work is concerned with the question of when a von Neumann regular ring is expressible as a directed union of Artinian subrings.

For i = 1,…, n let ai be a homogeneous polynomial of degree ri(>0) in the graded polynomial ring R[x1, …, xm], or R[x] for short, where R is a commutative ring with unity and x1, …, xm are indeterminates of degree 1. Let of degree - 1 be a formal inverse of xj and let U denote the graded R[x]-module In [2, §2] we introduced a graded complex of r-modules.

Suppose D is an integral domain with quotient field K and that L is an extension field of K. We show in Theorem 4 that if the complete integral closure of D is an intersection of Archimedean valuation domains on K, then the complete integral closure of D in L is an intersection of Archimedean valuation domains on L; this answers a question raised by Gilmer and Heinzer in 1965.

Let A be a subring of a commutative ring B. If the natural mapping from the prime spectrum of B to the prime spectrum of A is injective (respectively bijective) then the pair (A, B) is said to have the injective (respectively bijective) Spec-map. We give necessary and sufficient conditions for a pair of rings A and B graded by a free abelian group to have the injective (respectively bijective) Spec-map. For this we first deal with the polynomial case. Let l be a field and k a subfield. Then the pair of polynomial rings (k[X], l[X]) has the injective Spec-map if and only if l is a purely inseparable extension of k.

Differentially simple local noetherian Q -algebras are shown to be always (a certain type of) subrings of formal power series rings. The result is established as an illustration of a general theory of differential filtrations and differential completions.

The concept of a permutation polynomial function over a commutative ring with 1 can be generalized to multiplace functions in two different ways, yielding the notion of a k-ary permutation polynomial function (k > 1, k ∈ N) and the notion of a strict k-ary permutation polynomial function respectively. It is shown that in the case of a residue class ring Zm of the integers these two notions coincide if and only if m is squarefree.

The construction, for a module M over a commutative ring A (with identity) and a multiplicatively closed subset S of A, of the module of fractions S-1M is, of course, one of the most basic ideas in commutative algebra. The purpose of this note is to present a generalization which constructs, for a positive integer n and what is called a triangular subset U of An = A × A × … × A (n factors), a module U-n M of generalized fractions, a typical element of which has the form

where m∈ M and (u1, … un)∈U.

Let L/K be a quadratic extension of algebraic number fields, and D a central L-division algebra of finite L-dimension d2. If - is an involution (i.e., a ring antiautomorphism of period two) of D, we write S(-) for the set of - symmetric elements of D: