Publication

Reduction modulo p of Noether’s problem

2025

Abstract

Let R be a complete valuation ring of mixed characteristic (0, p) with algebraically closed fraction field K and residue field k. Let X/R be a smooth projective morphism. We show that if X k is stably rational, then H 3 K et (X K , ‫)ޚ‬ is torsion-free. The proof uses the integral p-adic Hodge theory of Bhatt, Morrow and Scholze and the study of differential forms in positive characteristic. We then apply this result to study the Noether problem for finite p-groups.

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2025

Abstract

Official source

Related concepts (32)

Residue field

In mathematics, the residue field is a basic construction in commutative algebra. If R is a commutative ring and m is a maximal ideal, then the residue field is the quotient ring k = R/m, which is a field. Frequently, R is a local ring and m is then its unique maximal ideal. This construction is applied in algebraic geometry, where to every point x of a scheme X one associates its residue field k(x). One can say a little loosely that the residue field of a point of an abstract algebraic variety is the 'natural domain' for the coordinates of the point.

Valuation (algebra)

In algebra (in particular in algebraic geometry or algebraic number theory), a valuation is a function on a field that provides a measure of the size or multiplicity of elements of the field. It generalizes to commutative algebra the notion of size inherent in consideration of the degree of a pole or multiplicity of a zero in complex analysis, the degree of divisibility of a number by a prime number in number theory, and the geometrical concept of contact between two algebraic or analytic varieties in algebraic geometry.

Characteristic (algebra)

In mathematics, the characteristic of a ring R, often denoted char(R), is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0). If this sum never reaches the additive identity the ring is said to have characteristic zero. That is, char(R) is the smallest positive number n such that: if such a number n exists, and 0 otherwise.

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Reduction modulo p of Noether’s problem

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