Noetherian-like properties in polynomial and power series rings

There are many Noetherian-like rings. Among them, we are interested in SFT-rings, piecewise Noetherian rings, and rings with Noetherian prime spectrum. Some of them are stable under polynomial extensions but none of them are stable under power series extensions. We give partial answers to some open questions related with stabilities of such rings. In particular, we show that any mixed extensions $R[X_1??[X_n?$ over a zero-dimensional SFT ring R are also SFT-rings, and that if R is an SFT-domain such that $R/P$ is integrally closed for each prime ideal P of R, then $R[X]$ is an SFT-ring. We also give a direct proof that if R is an SFT Prüfer domain, then $R[X_1,?,X_n]$ is an SFT-ring. Finally, we show that the power series extension $R?X?$ over a Prüfer domain R is piecewise Noetherian if and only if R is Noetherian.     

Lengths of involutions in finite Coxeter groups

Let W be a finite Coxeter group and X a subset of W. The length polynomial $L_{W,X}(t)$ is defined by $L_{W,X}(t)={\sum }_{x\in X}{t}^{?(x)}$, where ? is the length function on W. If $X=\{x\in W:x^2=1\}$ then we call $L_{W,X}(t)$ the involution length polynomial of W. In this article we derive expressions for the length polynomial where X is any conjugacy class of involutions, and the involution length polynomial, in any finite Coxeter group W. In particular, these results correct errors in [11] for the involution length polynomials of Coxeter groups of type $B_n$ and $D_n$. Moreover, we give a counterexample to a unimodality conjecture stated in [11].     

A formula about W-operator and its application to Hurwitz number

$W$-operators are differential operators on the polynomial ring. Mironov, Morosov and Natanzon construct the generalized Hurwitz numbers. They use the $W$-operator to prove a formula for the generating function of the generalized Hurwitz numbers. A special example of the $W$-operator is the cut-and-join operator. Goulden and Jackson use the cut-and-join operator to calculate the simple Hurwitz number. In this paper, we study the relation between $W$-operator $W\left(\left[d\right]\right)$ and the central elements $K_{1^{n?d}d}$ in $?S_n$. Based on the relation we find, we give another proof about a differential equation of the generating function of $d$-Hurwitz number.     

A1-invariance of non-stable K1-functors in the equicharacteristic case

We apply the techniques developed by I. Panin for the proof of the equicharacteristic case of the Serre–Grothendieck conjecture for isotropic reductive groups (Panin et al., 2015; Panin, 2019) to obtain similar injectivity and ${\mathbb{A}}^1$-invariance theorems for non-stable $K_1$-functors associated to isotropic reductive groups. Namely, let $G$ be a reductive group over a commutative ring $R$. We say that $G$ has isotropic rank $\ge n$, if every non-trivial normal semisimple $R$-subgroup of $G$ contains ${\left({\mathbf{G}}_{m,R}\right)}^n$. We show that if $G$ has isotropic rank $\ge 2$ and $R$ is a regular domain containing a field, then $K_1^G\left(R\left[x\right]\right)=K_1^G\left(R\right)$, where $K_1^G\left(R\right)=G\left(R\right)/E\left(R\right)$ is the corresponding non-stable $K_1$-functor, also called the Whitehead group of $G$. If $R$ is, moreover, local, then we show that $K_1^G\left(R\right)\to K_1^G\left(K\right)$ is injective, where $K$ is the field of fractions of $R$.     

Cancellation of projective modules over polynomial extensions over a two-dimensional ring

Let R be a commutative Noetherian ring of dimension two with $1/2\in R$ and let $A=R[X_1,?,X_n]$. Let P be a projective A-module of rank 2. In this article, we prove that P is cancellative if ${\wedge }^2(P)\oplus A$ is cancellative.     

Stable-like fluctuations of Biggins’ martingales

Let ${\left(W_n\left(\theta \right)\right)}_{n\in {\mathbb{N}}_0}$ be Biggins’ martingale associated with a supercritical branching random walk, and let $W\left(\theta \right)$ be its almost sure limit. Under a natural condition for the offspring point process in the branching random walk, we show that if the law of $W_1\left(\theta \right)$ belongs to the domain of normal attraction of an $\alpha$-stable distribution for some $\alpha \in (1,2)$, then, as $n\to \infty$, there is weak convergence of the tail process ${(W\left(\theta \right)?W_{n?k}\left(\theta \right))}_{k\in {\mathbb{N}}_0}$, properly normalized, to a random scale multiple of a stationary autoregressive process of order one with $\alpha$-stable marginals.     

Applications and homological properties of local rings with decomposable maximal ideals

We construct a local Cohen–Macaulay ring R with a prime ideal $\mathfrak{p}\in \mathrm{Spec}(R)$ such that R satisfies the uniform Auslander condition (UAC), but the localization $R_{\mathfrak{p}}$ does not satisfy Auslander's condition (AC). Given any positive integer n, we also construct a local Cohen–Macaulay ring R with a prime ideal $\mathfrak{p}\in \mathrm{Spec}(R)$ such that R has exactly two non-isomorphic semidualizing modules, but the localization $R_{\mathfrak{p}}$ has $2^n$ non-isomorphic semidualizing modules. Each of these examples is constructed as a fiber product of two local rings over their common residue field. Additionally, we characterize the non-trivial Cohen–Macaulay fiber products of finite Cohen–Macaulay type.     

Almost slender rings and compact rings

Classical results concerning slenderness for commutative integral domains are generalized to commutative rings with zero divisors. This is done by extending the methods from the domain case and bringing them in connection with results on the linear topologies associated to non-discrete Hausdorff filtrations. In many cases a weakened notion “almost slenderness” of slenderness is appropriate for rings with zero divisors. Special results for countable rings are extended to rings said to be of “bounded type” (including countable rings, ‘small’ rings, and, for instance, rings that are countably generated as algebras over an Artinian ring).More precisely, for a ring R of bounded type it is proved that R is slender if R is reduced and has no simple ideals, or if R is Noetherian and has no simple ideals; moreover, R is almost slender if R is not perfect (in the sense of H. Bass). We use our methods to study various special classes of rings, for instance von Neumann regular rings and valuation rings. Among other results we show that the following two rings are slender: the ring of Puiseux series over a field and the von Neumann regular ring $k^{\mathbb{N}}/k^{(\mathbb{N})}$ over a von Neumann regular ring k.For a Noetherian ring R we prove that R is a finite product of local complete rings iff R satisfies one of several (equivalent) conditions of algebraic compactness. A 1-dimensional Noetherian ring is outside this ‘compact’ class precisely when it is almost slender. For the rings of classical algebraic geometry we prove that a localization of an algebra finitely generated over a field is either Artinian or almost slender. Finally, we show that a Noetherian ring R is a finite product of local complete rings with finite residue fields exactly when there exists a map of R-algebras $R^{\mathbb{N}}\to R$ vanishing on $R^{(\mathbb{N})}$.     

Algebraic jump loci for rank and Betti numbers over Laurent polynomial rings

Let C be a chain complex of finitely generated free modules over a commutative Laurent polynomial ring $L_s$ in s indeterminates. Given a group homomorphism $p:{\mathbb{Z}}^s?{\mathbb{Z}}^t$ we let $p_{!}(C)=C{?}_{L_s}{L}_t$ denote the resulting induced complex over the Laurent polynomial ring $L_t$ in t indeterminates. We prove that the Betti number jump loci, that is, the sets of those homomorphisms p such that $b_k(p_{!}(C))>b_k(C)$, have a surprisingly simple structure. We allow non-unital commutative rings of coefficients, and work with a notion of Betti numbers that generalises both the usual one for integral domains, and the analogous concept involving McCoy ranks in case of unital commutative rings.     

Approximation in higher-order Sobolev spaces and Hodge systems

Let $d\ge 2$ be an integer, $1\le l\le d?1$ and φ be a differential l-form on ${\mathbb{R}}^d$ with ${\dot{W}}^{1,d}$ coefficients. It was proved by Bourgain and Brezis ([5, Theorem 5]) that there exists a differential l-form ψ on ${\mathbb{R}}^d$ with coefficients in $L^{\infty }\cap {\dot{W}}^{1,d}$ such that $d\phi =d\psi$. In the same work, Bourgain and Brezis also left as an open problem the extension of this result to the case of differential forms with coefficients in the higher order space ${\dot{W}}^{2,d/2}$ or more generally in the fractional Sobolev spaces ${\dot{W}}^{s,p}$ with $sp=d$. We give a positive answer to this question, provided that $d?\kappa \le l\le d?1$, where κ is the largest positive integer such that $\kappa <\min ?(p,d)$. The proof relies on an approximation result (interesting in its own right) for functions in ${\dot{W}}^{s,p}$ by functions in ${\dot{W}}^{s,p}\cap L^{\infty }$, even though ${\dot{W}}^{s,p}$ does not embed into $L^{\infty }$ in this critical case. The proofs rely on some techniques due to Bourgain and Brezis but the context of higher order and/or fractional Sobolev spaces creates various difficulties and requires new ideas and methods.     

Exterior powers and Tor-persistence

A commutative Noetherian ring R is said to be Tor-persistent if, for any finitely generated R-module M, the vanishing of ${\mathrm{Tor}}_i^R(M,M)$ for $i?0$ implies M has finite projective dimension. An open question of Avramov, et al. asks whether any such R is Tor-persistent. In this work, we exploit properties of exterior powers of modules and complexes to provide several partial answers to this question; in particular, we show that every local ring $(R,\mathfrak{m})$ with ${\mathfrak{m}}^3=0$ is Tor-persistent. As a consequence of our methods, we provide a new proof of the Tachikawa Conjecture for positively graded rings over a field of characteristic different from 2.     

A note on Hadwiger’s conjecture for W5-free graphs with independence number two

The Hadwiger number of a graph $G$, denoted $h\left(G\right)$, is the largest integer $t$ such that $G$ contains $K_t$ as a minor. A famous conjecture due to Hadwiger in 1943 states that for every graph $G$, $h\left(G\right)\ge \chi \left(G\right)$, where $\chi \left(G\right)$ denotes the chromatic number of $G$. Let $\alpha \left(G\right)$ denote the independence number of $G$. A graph is $H$-free if it does not contain the graph $H$ as an induced subgraph. In 2003, Plummer, Stiebitz and Toft proved that $h\left(G\right)\ge \chi \left(G\right)$ for all $H$-free graphs $G$ with $\alpha \left(G\right)\le 2$, where $H$ is any graph on four vertices with $\alpha \left(H\right)\le 2$, $H=C_5$, or $H$ is a particular graph on seven vertices. In 2010, Kriesell subsequently generalized the statement to include all forbidden subgraphs $H$ on five vertices with $\alpha \left(H\right)\le 2$. In this note, we prove that $h\left(G\right)\ge \chi \left(G\right)$ for all $W_5$-free graphs $G$ with $\alpha \left(G\right)\le 2$, where $W_5$ denotes the wheel on six vertices.     

L-orthogonality in Daugavet centers and narrow operators

We study the presence of L-orthogonal elements in connection with Daugavet centers and narrow operators. We prove that if $\mathrm{dens}(Y)?{\omega }_1$ and $G:X?Y$ is a Daugavet center with separable range then, for every non-empty $w^{?}$-open subset W of $B_{X^{??}}$, it follows that $G^{??}(W)$ contains some L-orthogonal to Y. In the context of narrow operators, we show that if X is separable and $T:X?Y$ is a narrow operator, then given $y\in B_X$ and any non-empty $w^{?}$-open subset W of $B_{X^{??}}$ then W contains some L-orthogonal u so that $T^{??}(u)=T(y)$. In the particular case that $T^{?}(Y^{?})$ is separable, we extend the previous result to $\mathrm{dens}(X)={\omega }_1$. Finally, we prove that none of the previous results holds in larger density characters (in particular, a counterexample is shown for ${\omega }_2$ under the assumption $2^c={\omega }_2$).