# Commutative Algebra

## New submissions

[ total of 10 entries: 1-10 ]
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### New submissions for Tue, 20 Mar 18

[1]
Title: Restricting homology to hypersurfaces
Comments: 17 pages. To appear in "Geometric and topological aspects of the representation theory of finite groups", to be published by Springer in the series titled "Proceedings in Mathematics"
Subjects: Commutative Algebra (math.AC); Group Theory (math.GR)

This paper concerns the homological properties of a module $M$ over a commutative noetherian ring $R$ relative to a presentation $R\cong P/I$, where $P$ is local ring. It is proved that the Betti sequence of $M$ with respect to $P/(f)$ for a regular element $f$ in $I$ depends only on the class of $f$ in $I/\mathfrak{n} I$, where $\mathfrak{n}$ is the maximal ideal of $P$. Applications to the theory of supports sets in local algebra and in the modular representation theory of elementary abelian groups are presented.

[2]
Title: On a question of Swan
Authors: Dorin Popescu
Subjects: Commutative Algebra (math.AC); Algebraic Geometry (math.AG); K-Theory and Homology (math.KT)

We show that a regular local ring is a filtered inductive limit of regular local rings, essentially of finite type over $\bf Z$.

### Cross-lists for Tue, 20 Mar 18

[3]  arXiv:1803.06411 (cross-list from math.AG) [pdf, ps, other]
Title: The 21 reducible polars of Klein's quartic
Comments: 28 pages, contains Singular's script
Subjects: Algebraic Geometry (math.AG); Commutative Algebra (math.AC); Combinatorics (math.CO)

We describe the singularities and related properties of the arrangement of 21 reducible polars of Klein's quartic, containing Klein's well-known arrangement of $21$ lines.

[4]  arXiv:1803.06774 (cross-list from math-ph) [pdf, ps, other]
Title: Toda type equations over multi-dimensional lattices
Subjects: Mathematical Physics (math-ph); Commutative Algebra (math.AC); Exactly Solvable and Integrable Systems (nlin.SI)

We introduce a class of recursions defined over the $d$-dimensional integer lattice. The discrete equations we study are interpreted as higher dimensional extensions to the discrete Toda lattice equation. We shall prove that the equations satisfy the coprimeness property, which is one of integrability detectors analogous to the singularity confinement test. While the degree of their iterates grows exponentially, they exhibit pseudo-integrable nature in terms of the coprimeness property. We also prove that the equations can be expressed as mutations of a seed in the sense of the Laurent phenomenon algebra.

[5]  arXiv:1803.06879 (cross-list from math.CO) [pdf, other]
Title: The tree of numerical semigroups with low multiplicity
Subjects: Combinatorics (math.CO); Commutative Algebra (math.AC)

We show that the number of numerical semigroups with multiplicity three, four or five and fixed genus is increasing as a function in the genus. To this end we use the Kunz polytope for these multiplicities. Counting numerical semigroups with fixed multiplicity and genus is then an integer partition problem with some extra conditions (those of membership to the Kunz polytope). For the particular case of multiplicity four, we are able to prove that the number of numerical semigroups with multiplicity four and genus $g$ is the number of partitions $x+y+z=g+6$ with $0<x\le y\le z$, $x\neq 1$, $y\neq 2$ and $z\neq 3$.

### Replacements for Tue, 20 Mar 18

[6]  arXiv:1607.03972 (replaced) [pdf, ps, other]
Title: $F$-singularities under generic linkage
Subjects: Commutative Algebra (math.AC); Algebraic Geometry (math.AG)
[7]  arXiv:1701.07735 (replaced) [pdf, ps, other]
Title: Notes on finitely generated flat modules
Subjects: Commutative Algebra (math.AC)
[8]  arXiv:1703.08281 (replaced) [pdf, ps, other]
Title: Big Cohen-Macaulay algebras and the vanishing conjecture for maps of Tor in mixed characteristic
Title: On the Noether number of $p$-groups