Hidden constructions in abstract algebra, Krull Dimension, Going Up, Going Down

Thierry Coquand; Henri Lombardi

Home/Archive/arXiv

arXiv · 1712.04725

Hidden constructions in abstract algebra, Krull Dimension, Going Up, Going Down

Thierry Coquand·Henri Lombardi

View original source ↗Read source document ↗Export metadata ↓

Abstract

We present constructive versions of Krull's dimension theory for commutative rings and distributive lattices. The foundations of these constructive versions are due to Joyal, Espanõl and the authors. We show that this gives a constructive version of basic classical theorems (dimension of finitely presented algebras, Going up and Going down theorem, \ldots), and hence that we get an explicit computational content where these abstract results are used to show the existence of concrete elements. This can be seen as a partial realisation of Hilbert's program for classical abstract commutative algebra.

Explore related subjects

math.AC

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Thierry Coquand, Henri Lombardi. 2017-12-13. Hidden constructions in abstract algebra, Krull Dimension, Going Up, Going Down. https://arxiv.org/abs/1712.04725

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Generalized Additive Decompositions of Symmetric Tensors

This article addresses the Generalized Additive Decomposition (GAD) of symmetric tensors, that is, degree-$d$ forms $f \in \mathcal{S}_d$. From a geometric perspective, a GAD corresponds to representing a point on a secant of osculating varieties to the Veronese variety, providing a compact and structured description of a tensor that captures its intrinsic algebraic properties. We provide a linear algebra method for measuring the GAD size and prove that the minimal achievable size, which we call the GAD-rank of the considered tensor, coincides with the rank of suitable Catalecticant matrices, under certain regularity assumptions. We provide a new explicit description of the apolar scheme associated with a GAD as the annihilator of a polynomial-exponential series. We show that if the Castelnuovo-Mumford regularity of this scheme is sufficiently small, then both the GAD and the associated apolar scheme are minimal and unique. Leveraging these results, we develop a numerical GAD algorithm for symmetric tensors that effectively exploits the underlying algebraic structure, extending existing algebraic approaches based on eigen computation to the treatment of multiple points. We illustrate the effectiveness and numerical stability of such an algorithm through several examples, including Waring and tangential decompositions.

math.AC↗

arXiv2026

Numerical Semigroups of Sally Type II

In this paper we study numerical semigroups of Sally type of multiplicity $e$ and embedding dimension $ν\ge e-2$. We construct the minimal resolutions for these semigroup rings when they are symmetric and compute their Betti numbers. We also construct a minimal resolution for another special class of such semigroups of type $ν-1$. Finally, we propose some conjectures for the Betti numbers of families of non-symmetric Sally type semigroups in the above cases in relation to those of the corresponding Gorenstein cases of Sally type semigroups.

math.AC↗

arXiv2026

Matrix equivalence to Smith normal form: new theoretical results for multivariate polynomial matrices

This paper investigates the Smith normal form equivalence problem for multivariate polynomial matrices. Using methods from matrix theory and polynomial ideal theory, we prove that Frost and Storey's 1978 conjecture holds for a broad class of matrices: such a matrix is equivalent to its Smith normal form if and only if its reduced minors of each order generate the unit ideal. Moreover, by extending the original matrix class via automorphisms of the polynomial ring, we show that our framework applies in a substantially more general setting.

math.AC↗