Recent zbMATH articles in MSC 13https://zbmath.org/atom/cc/132022-09-13T20:28:31.338867ZWerkzeugA structure-oriented construction of the classical number domains. With a view to order structures, algebraic and topological structureshttps://zbmath.org/1491.000022022-09-13T20:28:31.338867Z"Maurer, Christian"https://zbmath.org/authors/?q=ai:maurer.christianPublisher's description: Dieses Buch entwickelt systematisch die Konstruktion der klassischen Zahlenbereiche mit Blick auf die wichtigsten mathematischen Strukturen: Ordnungsstrukturen, algebraische Strukturen und topologische Strukturen. Kurze Zusammenfassungen je Kapitel/Abschnitt erleichtern die Übersicht und das Verinnerlichen der Inhalte. Das Buch bietet einen weit vernetzten Überblick über die fachwissenschaftlichen Grundlagen und deren zentrale Zusammenhänge; damit dient es Studierenden im Fach- und insbesondere im Lehramtsstudium Mathematik für Grundschule und Sekundarstufe I als wertvolle Ergänzung und Begleitung während der ersten Semester. Lehrende finden hier eine Alternative zum klassischen Einstieg ins Studium. Darüber hinaus ist das Buch auch für Quereinsteiger -- etwa Lehrkräfte anderer Fächer -- zur berufsbegleitenden Weiterbildung geeignet.The relation between polynomial calculus, Sherali-Adams, and sum-of-squares proofshttps://zbmath.org/1491.030772022-09-13T20:28:31.338867Z"Berkholz, Christoph"https://zbmath.org/authors/?q=ai:berkholz.christophSummary: We relate different approaches for proving the unsatisfiability of a system of real polynomial equations over Boolean variables. On the one hand, there are the static proof systems Sherali-Adams and sum-of-squares (a.k.a. Lasserre), which are based on linear and semi-definite programming relaxations. On the other hand, we consider polynomial calculus, which is a dynamic algebraic proof system that models Gröbner basis computations.
Our first result is that sum-of-squares simulates polynomial calculus: any polynomial calculus refutation of degree \(d\) can be transformed into a sum-of-squares refutation of degree \(2d\) and only polynomial increase in size. In contrast, our second result shows that this is not the case for Sherali-Adams: there are systems of polynomial equations that have polynomial calculus refutations of degree 3 and polynomial size, but require Sherali-Adams refutations of large degree \(\Omega(\sqrt{n}/\log n)\) and exponential size.
A corollary of our first result is that the proof systems Positivstellensatz and Positivstellensatz Calculus, which have been separated over non-Boolean polynomials, simulate each other in the presence of Boolean axioms.
For the entire collection see [Zbl 1381.68010].Inverse relations and reciprocity laws involving partial Bell polynomials and related extensionshttps://zbmath.org/1491.050302022-09-13T20:28:31.338867Z"Schreiber, Alfred"https://zbmath.org/authors/?q=ai:schreiber.alfredSummary: The objective of this paper is, mainly, twofold: Firstly, to develop an algebraic setting for dealing
with Bell polynomials and related extensions. Secondly, based on the author's previous work on multivariate
Stirling polynomials [Discrete Math. 338, No. 12, 2462--2484 (2015; Zbl 1321.11030)], to present a number of new results related to different types of inverse relationships, among these (1) the use of multivariable Lah polynomials for characterizing self-orthogonal families of
polynomials that can be represented by Bell polynomials, (2) the introduction of `generalized Lagrange inversion polynomials' that invert functions characterized in a specific way by sequences of constants, (3) a general
reciprocity theorem according to which, in particular, the partial Bell polynomials \(B_{n,k}\) and their orthogonal
companions \(A_{n,k}\) belong to one single class of Stirling polynomials: \(A_{n,k} = (-1)^{n-k}B_{-k,-n}\). Moreover, of some
numerical statements (such as Stirling inversion, Schlömilch-Schläfli formulas) generalized polynomial versions
are established. A number of well-known theorems (Jabotinsky, Mullin-Rota, Melzak, Comtet) are given new
proofs.A graph associated to a commutative semiringhttps://zbmath.org/1491.050612022-09-13T20:28:31.338867Z"Atani, Shahabaddin Ebrahimi"https://zbmath.org/authors/?q=ai:ebrahimi-atani.shahabaddin"Khoramdel, Mehdi"https://zbmath.org/authors/?q=ai:khoramdel.mehdi"Hesari, Saboura Dolati Pish"https://zbmath.org/authors/?q=ai:dolati-pishhesari.sabouraSummary: Let \(R\) be a commutative finite semiring with nonzero identity and \(H\) be an arbitrary multiplicatively closed subset \(R\). The generalized identity-summand graph of \(R\) is the (simple) graph \(G_H (R)\) with all elements of \(R\) as the vertices, and two distinct vertices \(x\) and \(y\) are adjacent if and only if \(x + y \in H\). In this paper, we study some basic properties of \(G_H (R)\). Moreover, we characterize the planarity, chromatic number, clique number and independence number of \(G_H (R)\).A new view toward vertex decomposable graphshttps://zbmath.org/1491.052052022-09-13T20:28:31.338867Z"Guo, Jin"https://zbmath.org/authors/?q=ai:guo.jin"Li, Meiyan"https://zbmath.org/authors/?q=ai:li.meiyan"Wu, Tongsuo"https://zbmath.org/authors/?q=ai:wu.tongsuoSummary: In this paper, we bring a new view about closed neighbourhood to show the vertex decomposability of graphs. Making use of the characterization of hereditary vertex decomposable graphs, we introduce a class of vertex decomposable graphs, which include some well-known classic vertex decomposable graphs such as clique-whiskered graphs and Cameron-Walker graphs.A variant of the proof of van der Waerden's theorem by Furstenberghttps://zbmath.org/1491.110152022-09-13T20:28:31.338867Z"Eyidoğan, Sadık"https://zbmath.org/authors/?q=ai:eyidogan.sadik"Özkurt, Ali Arslan"https://zbmath.org/authors/?q=ai:ozkurt.ali-arslanSummary: Let \(R\) be a commutative ring with identity. In this paper, for a given monotone decreasing positive sequence and an increasing sequence of subsets of \(R\), we will define a metric on \(R\) using them. Then, we will use this kind of metric to obtain a variant of the proof of van der Waerden's theorem by \textit{H. Furstenberg} [Recurrence in ergodic theory and combinatorial number theory. Princeton, New Jersey: Princeton University Press (1981; Zbl 0459.28023)].Systems of polynomials with at least one positive real zerohttps://zbmath.org/1491.120012022-09-13T20:28:31.338867Z"Wang, Jie"https://zbmath.org/authors/?q=ai:wang.jie|wang.jie.2|wang.jie.1|wang.jie.3|wang.jie.4Rees algebras of ideals of star configurationshttps://zbmath.org/1491.130012022-09-13T20:28:31.338867Z"Costantini, A."https://zbmath.org/authors/?q=ai:costantini.andrea|costantini.antonio|costantini.alessandra"Drabkin, B."https://zbmath.org/authors/?q=ai:drabkin.benjamin"Guerrieri, L."https://zbmath.org/authors/?q=ai:guerrieri.lorenzoGiven a collection of distinct hypersurfaces in \(\mathbb P^n\), assume that any \(j\) meet in codimension \(j\) or else the intersection is empty. (We say that then they meet properly.) A star configuration of codimension \(c\) is the union of the codimension \(c\) complete intersections cut out by all possible choices of \(c\) of the hypersurfaces. These have been extensively studied from different points of view, and in particular much is known about their homogeneous ideals, Hilbert functions, free resolutions and symbolic powers. When all the hypersurfaces have degree 1, the configurations are called linear star configurations. If \(I = (g_1,\dots,g_\mu)\) is an ideal in a Noetherian ring \(R\), the Rees algebra of \(I\) is the subalgebra \(\mathcal R(I) := R[It] = R[g_1 t,\dots,g_\mu t] \subseteq R[t]\) of the polynomial ring \(R[t]\). These algebras have also been extensively studied. In this paper the authors study Rees algebras of star configurations. Specifically, they ask when the ideal of the star configuration is of fiber type and when it is of linear type. They characterize when it is of linear type and they give sufficient conditions for it to be of fiber type. They also give results for linear star configurations. The approach is more algebraic than previous studies have been.
Reviewer: Juan C. Migliore (Notre Dame)Epsilon multiplicity for Noetherian graded algebrashttps://zbmath.org/1491.130022022-09-13T20:28:31.338867Z"Das, Suprajo"https://zbmath.org/authors/?q=ai:das.suprajoSummary: The notion of epsilon multiplicity was originally defined by \textit{B. Ulrich} and \textit{J. Validashti} [Math. Proc. Camb. Philos. Soc. 151, No. 1, 95--102 (2011; Zbl 1220.13006)] as a limsup, and they used it to detect integral dependence of modules. It is important to know if it can be realized as a limit. In this article, we show that the relative epsilon multiplicity of reduced Noetherian graded algebras over an excellent local ring exists as a limit. An important special case of \textit{S. D. Cutkosky}'s result [Adv. Math. 264, 55--113 (2014; Zbl 1350.13032), Theorem 6.3] concerning epsilon multiplicity is obtained as a corollary of our main theorem. We also produce a multigraded generalization of a result due to \textit{H. Dao} and \textit{J. Montaño} [Trans. Am. Math. Soc. 371, No. 5, 3483--3503 (2019; Zbl 1409.13036)] about monomial ideals.On commutative Gelfand ringshttps://zbmath.org/1491.130032022-09-13T20:28:31.338867Z"Aliabad, Ali Rezaie"https://zbmath.org/authors/?q=ai:aliabad.ali-rezaie"Badie, Mehdi"https://zbmath.org/authors/?q=ai:badie.mehdi"Nazari, Sajad"https://zbmath.org/authors/?q=ai:nazari.sajadSummary: By studying and using the quasi-pure part concept, we improve some statements and show that some assumptions in some articles are superfluous. We give some characterizations of Gelfand rings. For example: we prove that \(R\) is Gelfand if and only if \(m \left(\sum_{\alpha \in A} I_\alpha \right) = \sum_{ \alpha \in A} m(I_\alpha)\), for each family \(\{I_\alpha \}_{\alpha \in A}\) of ideals of \(R\), in addition if \(R\) is semiprimitive and \(\operatorname{Max}(R) \subseteq Y \subseteq\operatorname{Spec}(R)\), we show that \(R\) is a Gelfand ring if and only if \(Y\) is normal. We prove that if \(R\) is reduced ring, then \(R\) is a von Neumann regular ring if and only if \(\operatorname{Spec}(R)\) is regular. It has been shown that if \(R\) is a Gelfand ring, then \(\operatorname{Max}(R)\) is a quotient of \(\operatorname{Spec}(R)\), and sometimes \(h_M(a)\)'s behave like the zerosets of the space of maximal ideal. Finally, it has been proven that \(Z (\operatorname{Max}(C(X))) = \{h_M(f) : f \in C(X)\}\) if and only if \(\{h_M(f) : f \in C(X)\}\) is closed under countable intersection if and only if \(X\) is pseudocompact.ZPI property in amalgamated duplication ringhttps://zbmath.org/1491.130042022-09-13T20:28:31.338867Z"Hamed, Ahmed"https://zbmath.org/authors/?q=ai:hamed.ahmed"Malek, Achraf"https://zbmath.org/authors/?q=ai:malek.achrafSummary: Let \(A\) be a commutative ring. We say that \(A\) is a ZPI ring if every proper ideal of \(A\) is a finite product of prime ideals [\textit{H. S. Butts} and \textit{R. W. Gilmer jun.}, Can. J. Math. 18, 1183--1195 (1966; Zbl 0144.02703)]. In this paper, we study when the amalgamated duplication of \(A\) along an ideal \(I\), \(A \bowtie I\) to be a ZPI ring. We show that if \(I\) is an idempotent ideal of \(A\), then \(A\) is a ZPI ring if and only if \(A \bowtie I\) is a ZPI ring.\textit{SG}-projective ideals in one dimensional Noetherian domainshttps://zbmath.org/1491.130052022-09-13T20:28:31.338867Z"Hu, Kui"https://zbmath.org/authors/?q=ai:hu.kui"Lim, Jung Wook"https://zbmath.org/authors/?q=ai:lim.jung-wook"Zhou, De Chuan"https://zbmath.org/authors/?q=ai:zhou.dechuanSummary: In this paper, we show that every \textit{SG}-projective ideal of a one dimensional Noetherian domain is 2-generated. We also prove that an integral domain is an NWF domain if and only if it is weakly \textit{SG}-hereditary.On strongly \(J\)-Noetherian ringshttps://zbmath.org/1491.130062022-09-13T20:28:31.338867Z"Rostami, Esmaeil"https://zbmath.org/authors/?q=ai:rostami.esmaeilThe derived sequence of a pre-Jaffard familyhttps://zbmath.org/1491.130072022-09-13T20:28:31.338867Z"Spirito, Dario"https://zbmath.org/authors/?q=ai:spirito.darioThe author introduces and studies pre-Jaffard and weakly Jaffard families of overrings of a commutative integral domain \(D\). The concept of pre-Jaffard family is obtained by relaxing the locally finite property of a Jaffard family to compactness in the Zariski topology. A weak Jaffard family is pre-Jaffard.
Starting with a pre-Jaffard family \(\Theta\) of \(D\), the author defines recursively for each ordinal \(\alpha\) an overring \(T_{\alpha}\) of the domain \(D\) and a weakly Jaffard family \(\Theta_{\alpha}\) of \(T_{\alpha}\). The family of the overrings \(\{T_\alpha\}\) is called the derived sequence with respect to \(\Theta\). The minimal ordinal \(\alpha\) such that \(T_{\alpha}=T_{\alpha^{\prime}}\) for all \(\alpha^{\prime}>\alpha\), which is proved to exist, is called the Jaffard degree of the family \(\Theta\). Quoting from the authors' abstract, the derived sequence `allows to decompose stable semistar operations and singular length functions in more cases than what is allowed by Jaffard families.'
The author applies his general results to the one-dimensional case. If the domain \(D\) is one-dimensional, then \(\{\Theta := D_{M} \mid M\in \text{Max}(D)\}\) is a pre-Jaffard family of \(D\). In this case the derived sequence is described purely topologically, and the Jaffard degree of \(\Theta\) is equal to the Cantor-Bendixson rank of \(\text{Max}(D)^{\text{inv}}\), where `inv' means the inverse topology.
Reviewer: Moshe Roitman (Haifa)The exact annihilating-ideal graph of a commutative ringhttps://zbmath.org/1491.130082022-09-13T20:28:31.338867Z"Visweswaran, Subramanian"https://zbmath.org/authors/?q=ai:visweswaran.subramanian"Lalchandani, Premkumar T."https://zbmath.org/authors/?q=ai:lalchandani.premkumar-tGiven a commutative ring \(R\) with identity, an ideal \(I\) of \(R\) is said to be an exact annihilating ideal if there exists a non-zero ideal \(J\) such that \(\mathrm{Ann}(I) = J\) and \(\mathrm{Ann}(J) = I\). Here \(\mathrm{Ann}(I)\) denotes the annihilator of \(I\) in \(R\), and similarly for \(\mathrm{Ann}(J)\). Let \(\mathbb{EA}(R)^*\) denote the set of all nonzero exact annihilating ideals of \(R\). \textit{P. T. Lalchandani} [J. Algebra Relat. Top. 5, No. 1, 27--33 (2017; Zbl 1390.13009)] introduced the exact annihilating-ideal graph of \(R\), denoted by \(\mathbb{EAG}(R)\), whose vertex set is \(\mathbb{EA}(R)^*\) and distinct vertices \(I\) and \(J\) are adjacent if and only if \(\mathrm{Ann}(I) = J\) and \(\mathrm{Ann}(J) = I\).
The paper under review is devoted to the study of the exact annihilating-ideal graphs of commutative rings. The authors first discuss some basic properties of exact annihilating ideals for special rings, such as special principal ideal rings, principal ideal domains and von Neumann regular rings which are not fields. Then the authors investigate the graph structures of \(\mathbb{EAG}(R)\) for special principal ideal rings and reduced rings which admit only a finite number of minimal prime ideals.
Reviewer: Houyi Yu (Chongqing)Valued fields with finitely many defect extensions of prime degreehttps://zbmath.org/1491.130092022-09-13T20:28:31.338867Z"Kuhlmann, Franz-Viktor"https://zbmath.org/authors/?q=ai:kuhlmann.franz-viktorThis paper proves results about Artin-Schreier extensions, independent defect extensions
and Kummer extensions of valued fields.
An Artin-Schreier extension is an extension $K(\theta)|K$, where the field $K$ has characteristic $p>0$ and $\theta^p-\theta\in K$.
A valued field $(K,v)$ is said to be semitame if either its residue field $Kv$ has characteristic $0$ or $\operatorname{char}Kv=p>0$, its value group $vK$ is $p$-divisible and the homomorphism $x\mapsto x^p$ is surjective from $\mathcal{O}/p\mathcal{O}$ onto $\mathcal{O}/p\mathcal{O}$, where $\mathcal{O}$ denotes the valuation ring of the completion of $(K,v)$ with respect to the topology defined by the valuation.
Assume that $\operatorname{char}K=p>0$. The author proves that if $(K,v)$ is not dense in its perfect hull, then it admits infinitely many distinct Artin-Schreier extensions. He deduces that if $K$ admits only finitely many Artin-Schreier extensions, then $(K,v)$ is semitame and $Kv$ is perfect.
Now, let $(L|K,v)$ be a Galois extension of degree $p$, where $\operatorname{char}Kv=p$, $vL=vK$ and $Lv=Kv$.
Let $\sigma$ be a generator of $\mathrm{Gal}(L|K)$ and $\Sigma=\left\{v\left(\frac{\sigma f-f}{f}\right) : f\in L\backslash \{0\}\right\}$. One can prove that the set $\Sigma$ is independent of the choice of the generator $\sigma$. Then we say that $(L|K,v)$ has independent defect if there exists a proper convex subgroup $H$ of $vK$ such that $\Sigma=\{\alpha \in vK : \alpha >H\}$. Otherwise, we say that $(L|K,v)$ has dependent defect.
In the case where $\operatorname{char}K=p>0$, the author proves that if $(K,v)$ admits an Artin-Schreier extension with dependent defect, then it admits infinitely many Artin-Schreier extensions with dependent defect. We turn to Kummer extensions, that is extensions $(K(\eta)|K,v)$ where $\operatorname{char}K=0<p=\operatorname{char}Kv$ and $\eta^p\in K$. If this holds, then we can assume that $v(\eta-1)>0$. We say that this extension has super-independent defect if there exists $\alpha$ in the divisible hull of $vK$ such that $v(\eta-K)\leq \alpha<\frac{vp}{p}$.
Otherwise we say that this extension has super-dependent defect. The author assumes that
$vK$ has no nontrivial proper convex subgroup and $K$ contains a primitive $p$-th root of $1$. He proves that if $(K,v)$ admits an extension of degree $p$ with super-dependent defect,
then it admits infinitely many extensions of degree $p$ with super-dependent defect.
The last part of this paper is dedicated to fill a gap in a previous paper, that is to prove an assertion which proof had been forgotten.
Reviewer: Gerard Leloup (Le Mans)Extensions of valuation domains and going-uphttps://zbmath.org/1491.130102022-09-13T20:28:31.338867Z"Sarussi, Shai"https://zbmath.org/authors/?q=ai:sarussi.shaiSummary: Suppose \(F\) is a field with valuation \(v\) and valuation domain \(O_v\), \(E/F\) is a finite-dimensional field extension, and \(R\) is an \(O_v\)-subalgebra of \(E\) such that \(F \cdot R = E\) and \(R \cap F = O_v\). It is known that \(R\) satisfies LO, INC, GD and SGB over \(O_v\); it is also known that under certain conditions \(R\) satisfies GU over \(O_v\). In this paper, we present a necessary and sufficient condition for the existence of such \(R\) that does not satisfy GU over \(O_v\). We also present an explicit example of such \(R\) that does not satisfy GU over \(O_v\).Dynamic evaluation of integrity and the computational content of Krull's lemmahttps://zbmath.org/1491.130112022-09-13T20:28:31.338867Z"Schuster, Peter"https://zbmath.org/authors/?q=ai:schuster.peter-michael"Wessel, Daniel"https://zbmath.org/authors/?q=ai:wessel.daniel"Yengui, Ihsen"https://zbmath.org/authors/?q=ai:yengui.ihsenIn the paper under review, the authors suggest a constructive procedure to determine the Krull dimension of a commutative ring. This procedure has been mostly developed in a previous paper by \textit{G. Kemper} and \textit{I. Yengui} [J. Algebra 557, 278--288 (2020; Zbl 1440.13112)]. Such paper deals with a constructive characterisation of the more general valuative dimension of a domain and contains only one non-constructive step: a reduction from the general case to the integral case. In the paper under review, the authors present a constructive argument for this reduction step via a dynamical solution.
Reviewer: Paolo Lella (Trento)Associated prime ideals of equivariant coinvariant algebras, Steenrod operations, and Krull's Going Down Theoremhttps://zbmath.org/1491.130122022-09-13T20:28:31.338867Z"Smith, Larry"https://zbmath.org/authors/?q=ai:smith.larry|smith.larry.1In this note the author shows that the associated prime ideals of the equivariant coinvariant algebra \(\mathbb{F}[V]\otimes_{\mathbb{F}[V]^G}\mathbb{F}[V]\) must be either a minimal prime ideal or the maximal ideal, where \(G\) is a finite group, \(\mathbb{F}\) is a finite field, and \(V=\mathbb{F}^n,\) is the representation space, with the invariant ring denoted by \(\mathbb{F}[V]^G.\)
The most important part of the note is that the main result quoted above is proved by using the Steenrod operations, \(\mathcal{P}^*.\) Moreover, there are similar interesting results obtained, namely;
\begin{itemize}
\item The Going Down Theorem holds for the equivariant coinvariant algebra from either tensor factors.
\item If \(A\) is a Noetherian unstable \(\mathcal{P}^*\)-algebra generated by its linear forms and \(\mathfrak{p}\subset A\) is a minimal prime ideal, then \(A/\mathfrak{p}\) is a polynomial algebra on \(\textrm{co-ht}(\mathfrak{p})\) linear generators.
\item If \(A\) is a finitely generated standard graded \(\mathcal{P}^*\)-algebra, then the \(\mathcal{P}^*\)-invariant prime ideals of \(A\) are generated by the linear forms.
\item If \(\mathfrak{p}\subset\mathbb{F}[V]\) is a prime ideal and \(\mathfrak{P}\subset\mathbb{F}[V]\otimes_{\mathbb{F}[V]^G}\mathbb{F}[V]\) a prime ideal lying over it, then \(\mathrm{ht}(\mathfrak{p})=\mathrm{ht}(\mathfrak{P}),\textrm{co-ht}(\mathfrak{p})=\textrm{co-ht}(\mathfrak{P}),\) and \(\mathrm{ht}(\mathfrak{p})+\textrm{co-ht}(\mathfrak{p})=\mathrm{ht}(\mathfrak{P})+\textrm{co-ht}(\mathfrak{P})=n. \)
\end{itemize}
The note ends with an appendix containing two lemmas; \textit{Derivation lemma} and \textit{\(\mathcal{P}^*\)-derivation lemma.}
Reviewer: Ugur Madran (Eqaila)Correction to: ``Classification of non-local rings with genus two zero-divisor graphs''https://zbmath.org/1491.130132022-09-13T20:28:31.338867Z"Asir, T."https://zbmath.org/authors/?q=ai:asir.t"Mano, K."https://zbmath.org/authors/?q=ai:mano.karuppiah"Tamizh Chelvam, T."https://zbmath.org/authors/?q=ai:tamizh-chelvam.thirugnanamCorrection of some errors in the statement and proof of Theorem 4 in the first two authors' paper [ibid. 24, No. 1, 237--245 (2020; Zbl 1436.13016)].\(S\)-2-absorbing second submoduleshttps://zbmath.org/1491.130142022-09-13T20:28:31.338867Z"Farshadifar, Faranak"https://zbmath.org/authors/?q=ai:farshadifar.faranakSummary: Let \(R\) be a commutative ring with identity, \(S\) be a multiplicatively closed subset of \(R\), and let \(M\) be an \(R\)-module. In this paper, we introduce and investigate some properties of the notion of \(S\)-2-absorbing second submodules of \(M\) as a generalization of \(S\)-second submodules and strongly 2-absorbing second submodules of \(M\). Also, we obtain some results concerning \(S\)-2-absorbing submodules of \(M\).On the Cohen-Macaulayness of bracket powers of generalized mixed product idealshttps://zbmath.org/1491.130152022-09-13T20:28:31.338867Z"Moghimipor, Roya"https://zbmath.org/authors/?q=ai:moghimipor.royaAuthor's abstract: Let \(L\) be the generalized mixed product ideal induced by a monomial ideal \(I\). For every integer \(k\geq 1\), we denote the kth bracket power of \(L\) by \(L[k]\). We study algebraic properties of \(L[k]\), and show that \(L[k]\) is Cohen-Macaulay if \(I\) is Cohen-Macaulay.
Reviewer: Amir Mafi (Sanandaj and Tehran)Vanishing of Ext modules over Cohen-Macaulay ringshttps://zbmath.org/1491.130162022-09-13T20:28:31.338867Z"Kimura, Kaito"https://zbmath.org/authors/?q=ai:kimura.kaito"Otake, Yuya"https://zbmath.org/authors/?q=ai:otake.yuya"Takahashi, Ryo"https://zbmath.org/authors/?q=ai:takahashi.ryo.2|takahashi.ryo|takahashi.ryo.3Summary: In this article, we give criteria for projectivity of modules in terms of vanishing of Ext modules. One of the applications shows that the Auslander-Reiten conjecture holds for Cohen-Macaulay normal rings.
For the entire collection see [Zbl 1489.16003].Ladder determinantal rings over normal domainshttps://zbmath.org/1491.130172022-09-13T20:28:31.338867Z"Sather-Wagstaff, Sean K."https://zbmath.org/authors/?q=ai:sather-wagstaff.sean"Se, Tony"https://zbmath.org/authors/?q=ai:se.tony"Spiroff, Sandra"https://zbmath.org/authors/?q=ai:spiroff.sandraAuthors' abstract: We explicitly describe the divisor class groups and semidualizing modules for ladder determinantal rings with coefficients in an arbitrary normal domain for arbitrary ladders, not necessarily connected, and all sizes of minors.
Reviewer: S. A. Seyed Fakhari (Tehran)Splitting the conormal module for licci idealshttps://zbmath.org/1491.130182022-09-13T20:28:31.338867Z"Johnson, Mark R."https://zbmath.org/authors/?q=ai:johnson.mark-rSummary: For a licci ideal in a power series ring over a field, it is shown that its conormal module has a free summand precisely when the ideal is a hypersurface section. Using results of \textit{B. Ulrich} [J. Pure Appl. Algebra 39, 165--175 (1986; Zbl 0575.13013)], in the Gorenstein case one can show, up to deformation, that the conormal module is indecomposable.Lyubeznik numbers of almost complete intersection and linked idealshttps://zbmath.org/1491.130192022-09-13T20:28:31.338867Z"Nadi, Parvaneh"https://zbmath.org/authors/?q=ai:nadi.parvaneh"Rahmati, Farhad"https://zbmath.org/authors/?q=ai:rahmati.farhadLet \(A\) be a local commutative Noetherian ring containing a field \(k\). Associated to \(A\) are some invariants called the ``Lyubeznik numbers'' of \(A\), \(\lambda_{i,j}(A)\), defined in terms of certain local cohomology: \(\lambda_{i,j}(A) = \dim_k \hbox{Ext}_R^i (k, H_I^{n-j}(R))\). The authors are interested in describing the Lyubeznik numbers using linkage. Earlier, the authors jointly with Eghbali initiated a study in this direction. The present paper continues this work. Specifically, they study properties of these numbers for almost complete intersection ideals and for squarefree monomial ideals which are linked in two steps.
Reviewer: Juan C. Migliore (Notre Dame)Hilbert's syzygy theorem for monomial idealshttps://zbmath.org/1491.130202022-09-13T20:28:31.338867Z"Alesandroni, Guillermo"https://zbmath.org/authors/?q=ai:alesandroni.guillermoSummary: We give a new proof of Hilbert's Syzygy Theorem for monomial ideals. In addition, we prove the following. If \(S=k[x_1, \dots, x_n]\) is a polynomial ring over a field, \(M\) is a squarefree monomial ideal in \(S\), and each minimal generator of \(M\) has degree larger than \(i\), then \(\mathrm{pd}(S/M)\leq n - i\).Virtual criterion for generalized Eagon-Northcott complexeshttps://zbmath.org/1491.130212022-09-13T20:28:31.338867Z"Booms-Peot, Caitlyn"https://zbmath.org/authors/?q=ai:booms-peot.caitlyn"Cobb, John"https://zbmath.org/authors/?q=ai:cobb.johnSummary: Given any map of finitely generated free modules, \textit{D. A. Buchsbaum} and \textit{D. Eisenbud} defined a family of generalized Eagon-Northcott complexes associated to it [Adv. Math. 18, 245--301 (1975; Zbl 0336.13007)]. We give sufficient criterion for these complexes to be virtual resolutions, thus adding to the known examples of virtual resolutions, particularly those not coming from minimal free resolutions.The quantitative behavior of asymptotic syzygies for Hirzebruch surfaceshttps://zbmath.org/1491.130222022-09-13T20:28:31.338867Z"Bruce, Juliette"https://zbmath.org/authors/?q=ai:bruce.julietteIn the paper under review, the author studies the so-called Ein, Erman, and Lazarsfeld's normality heuristic [\textit{L. Ein} et al., J. Reine Angew. Math. 702, 55--75 (2015; Zbl 1338.13023)] for the asymptotic linear syzygies of Hirzebruch surfaces embedded by \(\mathcal{O}(d,2)\). More specifically, let \(X\) be projective variety of dimension \(n\) over an arbitrary field \(\mathbb{K}\). Given a sequence of very ample line bundles \(\{L_{d}\}_{d\in \mathbb{N}}\), one wants to study how the graded Betti numbers of \(X\) behave asymptotically with respect to \(L_{d}\) if \(d \gg 0\), that is, one is interested in the syzygies of the section ring \[R(X;L_{d}) :=\bigoplus_{k \in \mathbb{Z}}H^{0}(X,kL_{d})\] as a module over \(S = \mathrm{Sym} \, H^{0}(X,L_{d})\). Considering the graded minimal free resolution \[0 \rightarrow F_{r_{d}} \rightarrow \cdots \quad \cdots \rightarrow F_{1} \rightarrow F_{0} \rightarrow R(X;L_{d}) \rightarrow 0,\] let \[K_{p,q}(X;L_{d}) := \mathrm{span}_{\mathbb{K}} \langle\text{minimal generators of } \, F_{p} \, \text{ of degree } \, (p+q)\rangle\] be the finite dimensional \(\mathbb{K}\)-vector space of minimal syzygies of homological degree \(p\) and degree \(p+q\). We write \(k_{p,q}(X;L_{d})\) for \(\mathrm{dim} \, K_{p,q}(X;L_{d})\), and then form the Betti table of \((X;L_{d})\) by placing \(k_{p,q}(X;L_{d})\) in the \((q,p)\)-th spot.
In this setup, we can state Ein, Erman, and Lazarsfeld's heuristic as follows: if \(\{L_{d}\}_{d\in \mathbb{N}}\) is a sequence of line bundles growing in positivity, then for any \(q \in [1, ..., n]\) there exists a function \(F_{q}(d)\), depending on \(X\), such that if \(\{p_{d}\}_{d\in \mathbb{N}}\) is a sequence of non-negative integers such that \[(\star): \quad \quad \mathrm{lim}_{d \rightarrow \infty}\bigg(p_{d} - (r_{d}/2 + a\sqrt{r_{d}}/2)\bigg) = 0,\] where \(a \in \mathbb{R}\) is a fixed constant, then \[F_{q}(d) \cdot k_{p_{d},q}(X;L_{d}) \rightarrow e^{-a^{2}/2}.\]
Now we can formulate the main results of the paper.
Denote by \(\mathbb{F}_{t}\) the Hirzebruch surface embedded by the line bundle \(\mathcal{O}_{\mathbb{F}_{t}}(d,2)\).
Theorem A. If \(\{p_{d}\}_{d \in \mathbb{N}}\) is a sequence of non-negative integers satisfying \((\star)\) for some real number \(a \in \mathbb{R}\), then \[\frac{3\sqrt{2\pi}}{2^{r_{d}}\sqrt{r_{d}}} \cdot k_{p_{d},1}(\mathbb{F}_{t}, \mathcal{O}_{\mathbb{F}_{t}}(d,2)) = e^{-a^{2}/2}\bigg(1 + O\bigg(\frac{1}{\sqrt{r_{d}}}\bigg)\bigg).\]
Theorem B. There does not exist a function \(F_{2}(d)\) such that if \(\{p_{d}\}_{d\in \mathbb{N}}\) is a sequence of non-negative integers satisfying \((\star)\) for some real number \(a \in \mathbb{R}\), then \[F_{2}(d)\cdot k_{p_{d},2}(\mathbb{F}_{t},\mathcal{O}_{\mathbb{F}_{t}}(d,2)) = e^{-a^{2}/2}\bigg(1 + O\bigg(\frac{1}{\sqrt{r_{d}}}\bigg)\bigg).\]
Reviewer: Piotr Pokora (Kraków)On \textit{w-FI}-flat and \textit{w-FI}-injective moduleshttps://zbmath.org/1491.130232022-09-13T20:28:31.338867Z"Almahdi, Fuad Ali Ahmed"https://zbmath.org/authors/?q=ai:almahdi.fuad-ali-ahmed"Bouba, El Mehdi"https://zbmath.org/authors/?q=ai:bouba.el-mehdi"Tamekkante, Mohammed"https://zbmath.org/authors/?q=ai:tamekkante.mohammedAs we all know, flat modules and injective module are important in the study of homological algebra. In this vein, the authors introduce and study the concepts of \(w\)-FI-flat modules and \(w\)-FI-injective modules, which are in the context of the \(w\)-operation theory of FI-flat modules and FI-injective modules [\textit{L. Mao} and \textit{N. Ding}, J. Algebra 309, No. 1, 367--385 (2007; Zbl 1117.16002)]. The introduced modules are studied and compared with other classical modules. Among other things, they show that every module has a \(w\)-FI-flat cover. A couple of examples are also provided to illustrate the results.
Reviewer: Hwankoo Kim (Asan)Regularity relative to a hereditary torsion theory for modules over a commutative ringhttps://zbmath.org/1491.130242022-09-13T20:28:31.338867Z"Qiao, Lei"https://zbmath.org/authors/?q=ai:qiao.lei|qiao.lei.1|qiao.lei.2"Zuo, Kai"https://zbmath.org/authors/?q=ai:zuo.kaiThis paper is concerned with \(w\)-regularity for \(w\)-coherent rings and \(w\)-Noetherian rings. Recall that a ring \(R\) is called \(w\)-Noetherian if every ideal of \(R\) is of \(w\)-finite type, and is called \(w\)-coherent if every \(w\)-finite type ideal is of \(w\)-finitely presented type. An \(R\)-module \(M\) is said to be \(w\)-projective if Ext\(^1_R(L(M), N)\) is a \(GV\)-torsion module for any torsionfree \(w\)-module \(N\), where \(L(M) = (M/tor_{GV})_w\). Let \(\mathscr{P}^{\dag}_w\) to denote the class of \(GV\)-torsionfree \(R\)-modules \(N\) with the property that Ext\(^k_R(M, N) = 0\) for all \(w\)-projective \(R\)-modules \(M\) and for all integers
\(k \geq 1.\) Then an \(R\)-module \(M\) is said to be a weak \(w\)-projective module if Ext\(^1_R(M, N) = 0\) for all \(N\in \mathscr{P}^{\dag}_w\). Using weak \(w\)-projective modules, Wang and Qiao define the weak \(w\)-projective dimension w.\(w\)-pd\(_R(M)\) of an \(R\)-module \(M\) to be the shortest length of weak \(w\)-projective
\(w\)-resolutions of \(M\), and define the global weak \(w\)-projective dimension gl.w.\(w\)-dim\((R)\) of a ring \(R\) to be supremum of weak \(w\)-projective dimensions of all \(R\)-modules.
The authors call a ring \(R\) \(w\)-regular if every finitely generated ideal of \(R\) has finite weak \(w\)-projective dimension. The authors show that the \(w\)-coherent \(w\)-regular domains are exactly the Prüfer \(v\)-multiplication domains and that an integral domain is \(w\)-Noetherian and \(w\)-regular if and only if it is a Krull domain. The authors also prove the \(w\)-analogue of the global version of the Serre-Auslander-Buchsbaum Theorem. Among other things, the authors show that every \(w\)-Noetherian \(w\)-regular ring is the direct sum of a finite number of Krull domains. Finally, the authors obtain that the global weak \(w\)-projective dimension of a \(w\)-Noetherian
ring is 0, 1, or \(\infty\).
The article is well written, and the related results are very exciting.
Reviewer: Xiaolei Zhang (Zibo)On the stable under specialization sets and cofiniteness of local cohomologyhttps://zbmath.org/1491.130252022-09-13T20:28:31.338867Z"Aghapournahr, Moharram"https://zbmath.org/authors/?q=ai:aghapournahr.moharram"Hatamkhani, Marziye"https://zbmath.org/authors/?q=ai:hatamkhani.marziyeThe paper under review deals with a generalization of the concept of cofiniteness defined in [\textit{R. Hartshorne}, Invent. Math. 9, 145--164 (1970; Zbl 0196.24301)] with a view using the so-called stable under specialization defined in [\textit{K. Divaani-Aazar} et al., J. Algebra Appl. 18, No. 1, Article ID 1950015, 22 p. (2019; Zbl 1454.13025)].
Reviewer: Majid Eghbali (Tehran)Integral domains issued from associated primeshttps://zbmath.org/1491.130262022-09-13T20:28:31.338867Z"Kim, Hwankoo"https://zbmath.org/authors/?q=ai:kim.hwankoo"Tamoussit, Ali"https://zbmath.org/authors/?q=ai:tamoussit.aliAll rings considered in this paper are commutative with unity.
Recall first that a prime ideal \(P\) of an integral domain \(D\) is called an associated prime ideal of a principal ideal \(aD\) of \(D\) if \(P\) is minimal over \((aD:bD)\) for some \(b\in D\backslash aD.\) For a property \(\mathcal{X}\) of an integral domain \(D,\) the authors called \(D\) an MZ-\(\mathcal{X}\) domain (for Mott-Zafrullah) if \(D_{P}\) has the property \(\mathcal{X}\) for each associated prime ideal \(P\) of a principal ideal of \(D.\) Several types of MZ-\( \mathcal{X}\) domains were considered. For instance, the authors proved that for an integral domain \(D,\) \(D\) is an MZ-\(\mathcal{X}\) domain if and only if \(D_{S}\) is an MZ-\(\mathcal{X}\) domain for any multiplicative set \(S\) of \(D\) which is also equivalent to the fact that \(D\) is a \(t\)-locally MZ-\(\mathcal{X }\) domain. Special cases of the property \(\mathcal{X}\) were also examined, giving rise to some characterizations of almost Dedekind domains and \(t\) -almost Dedekind domains. If \(\mathcal{X}\) is one of the following properties: almost valuation, valuation, DVR, Jaffard or Noetherian and if \(D \) is an integral domain then \(D[X]\) (respectively, \(D[[X]])\) is an MZ-\( \mathcal{X}\) domain if and only if so is \(D[X,X^{-1}]\) (respectively, \( D[[X]][X^{-1}]).\)
Moreover, the transfer of the properties MZ-Noetherian, MZ-Mori, MZ-almost valuation from the integral domain \(D\) to the Serre's conjecture ring \( D\langle X\rangle \) and the Nagata ring \(D(X)\) was investigated.
The case of the power series ring over MZ-Noetherian domains (respectively, MZ-valuation domains, MZ-DVR domains) was also studied.
Finally, the authors introduced integral domains satisfying the ascending chain condition (ACC) on associated prime ideals. They proved among others that an integral domain \(D\) satisfies the ACC on associated prime ideals if and only if the ring \(D[X]\) (respectively, \(D\langle X\rangle ,\) \(D(X))\) does.
Reviewer: Sana Hizem (Monastir)On length densitieshttps://zbmath.org/1491.130272022-09-13T20:28:31.338867Z"Chapman, Scott T."https://zbmath.org/authors/?q=ai:chapman.scott-thomas"O'Neill, Christopher"https://zbmath.org/authors/?q=ai:oneill.christopher"Ponomarenko, Vadim"https://zbmath.org/authors/?q=ai:ponomarenko.vadimThe set of lengths of an element \(x\) in a commutative monoid \(M\) and the elasticity for elements of monoid \(M\) along with the elasticity of monoid \(M\) itself have been well studied in the literature, particularly for Krull monoid, numerical monoid, Puiseux monoids and arithmetic congruence monoids. This paper discusses various new notions such as a length density \(\operatorname{LD}(x)\), asymptotic length density \(\overline{\operatorname{LD}}(x)\) of an element \(x\) and length density \(\operatorname{LD}(M)\) of a monoid \(M\). Analogous to the acceptance of elasticity of \(M\), the authors define the acceptance of length density. The length density of \(M\) is accepted if there exists \(x\in M\) such that \(\operatorname{LD}(x) = \operatorname{LD}(M)\).
The paper is organised as follows: Introduction includes a brief discussion of all the crucial definitions and notions which are required for the rest of the article. Section 2 discusses the basic properties of length density. In particularly, bounds are calculated for \(\operatorname{LD}(x)\) and \(\operatorname{LD}(M)\)s, and several examples of monoids have been given for which these bounds are met. These results further give examples of monoids for which length density is (is not) accepted. Section 3 answers the problem of the existence of a monoid having any irrational number in \((0, 1)\) as a length density, which is a similar result to the existence of a monoid having irrational elasticity in \((0, 1)\). However, this paper presents a new construction to prove the result for the length density case. Section 3 further computes the length density of block monoids and discusses several other examples. The last section of the article deals with the sufficient condition which guarantees the existence of asymptotic length density, this condition is particularly satisfied by a finitely generated monoid, \(C\)-monoid and Krull monoids with finite divisor class group. Authors also construct an example of a monoid which lacks asymptotic length density.
The article is well readable and has many good results with much information about the literature.
Reviewer: Nitin Bisht (Indore)Split absolutely irreducible integer-valued polynomials over discrete valuation domainshttps://zbmath.org/1491.130282022-09-13T20:28:31.338867Z"Frisch, Sophie"https://zbmath.org/authors/?q=ai:frisch.sophie"Nakato, Sarah"https://zbmath.org/authors/?q=ai:nakato.sarah"Rissner, Roswitha"https://zbmath.org/authors/?q=ai:rissner.roswithaLet \((R,M)\) be a DVR with quotient field \(K\) and with finite residue field. The main object of study of this paper is the absolutely irreducible elements (i.e. irreducible elements all of whose powers have a unique factorization) of the ring of integer-valued polynomials Int\((R):=\{f\in K[X]\ \vert\ f(R)\subseteq R\}.\) The main result is Theorem 2 that completely characterizes absolutely irreducible polynomials of Int\((R)\) that split over \(K.\) Applications are given.
Reviewer: Cristodor-Paul Ionescu (Bucureşti)Matchings and squarefree powers of edge idealshttps://zbmath.org/1491.130292022-09-13T20:28:31.338867Z"Erey, Nursel"https://zbmath.org/authors/?q=ai:erey.nursel"Herzog, Jürgen"https://zbmath.org/authors/?q=ai:herzog.jurgen"Hibi, Takayuki"https://zbmath.org/authors/?q=ai:hibi.takayuki"Saeedi Madani, Sara"https://zbmath.org/authors/?q=ai:madani.sara-saeediLet \(G=(V,E)\) be a simple graph on a vertex set \(V\). The edge ideal of \(G\) is the squarefree monomial ideal
\[
I(G)=(\textbf{x}_e \colon \; e \in E),
\]
where \(\textbf{x}_e=x_i x_j\) for \(e=\{i,j\}\). The \(k\)th squarefree power of \(G\), denoted by \(I(G)^{[k]}\), is the squarefree monomial ideal generated by all squarefree monomials in \(I(G)^k\), the \(k\)th ordinary power of \(I(G)\). The paper under review gives an upper bound for the regularity of the squarefree powers of edge ideal of a graph \(G\) and investigates when such powers are linearly related or have linear resolution.
A finite set of edges \(M = \{e_1,\ldots, e_k\} \subseteq E\) is called a matching of \(G\) if \(e_i \cap e_j = \varnothing\) for \(1 \leq i < j \leq k\). Given a matching \(M\) of \(G\), let \(u_M=\prod_{e \in M} \textbf{x}_e\). It turns out that
\[
I(G)^{[k]}=\left(u_M \colon \; M \text{ is a matching in } G \text{ and } |M|=k \right).
\]
In the study of squarefree powers of edge ideals, there are three important invariants of a graph \(G\) which are introduced as follows:
\begin{itemize}
\item \textbf{Matching number.} The matching number of \(G\), denoted by \(\nu(G)\), is the maximum cardinality of a matching of \(G\). Clearly, \(I(G)=I(G)^{[1]}\) and \(I(G)^{[k]}=0\) for \(k>\nu(G)\). It is shown in [\textit{M. Bigdeli} et al., Commun. Algebra 46, No. 3, 1080--1095 (2018; Zbl 1428.13032), Theorem 5.1] that \(I(G)^{[\nu(G)]}\) has linear quotients.
\item \textbf{Induced matching number.} An induced matching of \(G\) is a matching \(M = \{e_1,\ldots, e_k\}\) of \(G\) such that the induced subgraph of \(G\) on \(\cup_{i=1}^k e_i\) has exactly \(k\) edges. the number \(\nu_1(G)\) stands for the induced matching number of \(G\) which is the maximum size of an induced matching in \(G\).
\item \textbf{Restricted matching number.} A restricted matching of \(G\) is a matching in which there exists an edge which provides a gap in \(G\) with any other edge in the matching. The maximum size of a restricted matching in \(G\) is denoted by \(\nu_0(G)\).
\end{itemize}
One may observe that
\[
\nu_1(G) \leq \nu_0(G) \leq \nu(G).
\]
One of the main results of the paper under review is that for a graph \(G\), and for all \(1 \leq k \leq \nu_1(G)\) one has the inequality
\[
\mathrm{reg}(I(G)^{[k]}) \geq k+\nu_1(G).
\]
On the other hand, the authors show that \(\mathrm{reg}(I(G)^{[k]}) \leq k+\nu(G)\) for \(k=2\). This inequality is already known for \(k=1\) [\textit{H. T. Hà} and \textit{A. Van Tuyl}, J. Algebr. Comb. 27, No. 2, 215--245 (2008; Zbl 1147.05051), Theorem 1.5] and \(k=\nu(G)\) [\textit{M. Bigdeli} et al., Commun. Algebra 46, No. 3, 1080--1095 (2018; Zbl 1428.13032), Theorem 5.1].
A homogeneous ideal \(I\) in the polynomial ring \(S=\mathbb{K}[x_1, \ldots, x_n]\) is said to be linearly related, if the first syzygy module of \(I\) is generated by linear relations. The other main result of this paper states that if \(G\) is a graph and \(I(G)^{[k]}\) is linearly related then so is \(I(G)^{[k+1]}\). It follows that there exists a smallest integer \(\lambda(I(G))\) for which \(I(G)^{[k]}\) is linearly related for all \(k \geq \lambda(I(G))\). It is known [\textit{M. Bigdeli} et al., Commun. Algebra 46, No. 3, 1080--1095 (2018; Zbl 1428.13032), Lemma 5.2] that \(\lambda(I(G)) \geq \nu_0(G)\) and there are some examples with strict inequality. The authors show that \(I(G)^{[\nu_0(G)]}\) is linearly related if \(\nu_0(G) \leq 2\).
A squarefree monomial ideal \(I\) is said to satisfy the squarefree Ratliff property, if \(I^{[k]} \colon I = I^{[k]}\) for all \(k \geq 2\). The last main result of the paper under review states that any nonzero squarefree monomial ideal satisfies the squarefree Ratliff property.
Reviewer: Ali Akbar Yazdan Pour (Zanjan)Very well-covered graphs and local cohomology of their residue rings by the edge idealshttps://zbmath.org/1491.130302022-09-13T20:28:31.338867Z"Kimura, K."https://zbmath.org/authors/?q=ai:kimura.kyouko"Pournaki, M. R."https://zbmath.org/authors/?q=ai:pournaki.mohammad-reza"Terai, N."https://zbmath.org/authors/?q=ai:terai.naoki"Yassemi, S."https://zbmath.org/authors/?q=ai:yassemi.siamakLet \(G\) be a simple graph without isolated vertices and let \(I(G)\) denote its edge ideal. Recall that \(G\) is called very well-covered if \(I(G)\) is unmixed and the number of vertices of \(G\) is equal to \(2\operatorname{ht} I(G)\).
The main result of this article is a structure theorem on very well-covered graphs, based on Cohen-Macaulay very well-covered graphs. The latter were characterized in [\textit{M. Crupi} et al., Nagoya Math. J. 201, 117--131 (2011; Zbl 1227.05218)]. In the present article, the authors show that if \(G\) is very well-covered, then there exists a very well-covered Cohen-Macaulay graph \(H\), such that \(G\) can be obtained from \(H\) by replacing the edges in a certain subset of edges of \(H\), \(\{e_1,\dots,e_{d}\}\), with complete bipartite graphs \(K_{n_1,n_1},\dots, K_{n_d,n_d}\) (Theorem 3.5). The authors then use this result to obtain formulas for the Hilbert series of the local cohomology modules of the residue ring of the edge ideal (Theorem 4.4), and also for the regularity (Theorem 5.1) and depth (Theorem 5.3) of this ring. The two last results compare to Theorem 4.12 in [\textit{M. Mahmoudi} et al., J. Pure Appl. Algebra 215, No. 10, 2473--2480 (2011; Zbl 1227.13017)] and Theorem 1.1 in [\textit{K. Kimura} et al., Nagoya Math. J. 230, 160--179 (2018; Zbl 1411.13018)], respectively.
Reviewer: Jorge Neves (Coimbra)Frobenius-Witt differentials and regularityhttps://zbmath.org/1491.130312022-09-13T20:28:31.338867Z"Saito, Takeshi"https://zbmath.org/authors/?q=ai:saito.takeshiIn [J. Algebra 524, 110--123 (2019; Zbl 1408.13054)], the authors introduced the module of total \(p\)-differentials for a ring over \(\mathbb{Z}/p^{2}\). The paper under review studies a similar construction for a ring over \(\mathbb{Z}_{(p)}\). Given a (commutative) ring \(R\), an \(R\)-module \(M\) and a prime number \(p\), the paper defines a Frobenius-Witt derivation (or a FW-derivation) from \(A\) to \(M\) as a mapping \(w:A\rightarrow M\) such that \(w(a+b) = w(a) + w(b) - P(a, b)w(p)\) and \(w(ab) = b^{p}w(a)+a^{p}w(b)\) where \(P = \sum_{i=1}^{p-1}{\frac{(p-1)!}{i!(p-i)!}}X^{i}Y^{p-i}\in\mathbb{Z}[X, Y]\). It is shown that there exists a universal pair of an \(A\)-module \(F\Omega_{A}^{1}\) and an FW-derivation \(w:A\rightarrow F\Omega_{A}^{1}\) called the \textit{module of} FW-\textit{differentials} of \(A\) and the \textit{universal} FW-\textit{derivation}, respectively. The main result of the paper under review is a proof of a regularity criterion in this case. It says that under a suitable finitness condition, a Noetherian local ring \(A\) with residue field \(k\) of characteristic \(p\) is regular if and only if the \(A/pA\)-module of FW-differentials \(F\Omega_{A}^{1}\) is free of rank \(d+r\) where \(d = \dim A\) and \([k:k^{p}] = p^{r}\). The construction of \(F\Omega_{A}^{1}\) is sheafified and the author obtains a sheaf of FW-differentials \(F\Omega_{X}^{1}\) on a scheme \(X\). It is used in [\textit{T. Saito}, Algebra Number Theory 16, No. 2, 335--368 (2022; Zbl 07516272)] to define the cotangent bundle and the microsupport of an étale sheaf in mixed characteristic. The last part of the paper under review studies the relation of \(F\Omega_{X}^{1}\) with \(\mathcal{H}_{1}\) of cotangent complexes.
Reviewer: Alexander B. Levin (Washington)On exponential morphisms over commutative ringshttps://zbmath.org/1491.130322022-09-13T20:28:31.338867Z"El Kahoui, M'hammed"https://zbmath.org/authors/?q=ai:el-kahoui.mhammed"Hammi, Aziza"https://zbmath.org/authors/?q=ai:hammi.azizaSummary: We give elementary and self-contained proofs of two results concerning exponential morphisms on polynomial rings.Homological aspects of derivation modules and critical case of the Herzog-Vasconcelos conjecturehttps://zbmath.org/1491.130332022-09-13T20:28:31.338867Z"Jorge-Pérez, Victor H."https://zbmath.org/authors/?q=ai:jorge-perez.victor-h"Miranda-Neto, Cleto B."https://zbmath.org/authors/?q=ai:miranda-neto.cleto-bLet \(R\) be a finitely generated algebra over a field \(k\) of characteristic zero and let \(\mathrm{Der}_{k}(R)\) be the \(R\)-module of \(k\)-linear derivations of \(R\). The Zariski-Lipman conjecture asserts that if \(\mathrm{Der}_{k}(R)\) is a free \(R\)-module, then \(R\) is a polynomial ring over \(k\). This has been proven in several special cases, for instance by \textit{M. Hochster} [J. Algebra 47, 411--424 (1977; Zbl 0401.13006)] in the graded case. J. Herzog and W. Vasconcelos suggested to replace the freeness of \(\mathrm{Der}_{k}(R)\) by the seemingly weaker condition of having finite projective dimension. Thus, if separated from the Zariski-Lipman conjecture, they ask whether \(\mathrm{Der}_{k}(R)\) is free if its projective dimension is finite. This question has been answered affirmatively for graded complete intersections with isolated singularity (see [\textit{J. Herzog} and \textit{A. Martsinkovsky}, Comment. Math. Helv. 68, No. 3, 365--384 (1993; Zbl 0799.14016)]) and for some other types of rings.
The paper under review investigates the impact of imposing the finiteness of the projective dimension (and, more generally, of the Gorenstein dimension) of \(\mathrm{Der}_{k}(R)\) in the case when \(k\) is a unital commutative ring and \(R\) a local Noetherian \(k\)-algebra. The main problem considered in the paper is whether or when the above-mentioned key hypothesis ensures the freeness of \(\mathrm{Der}_{k}(R)\). The authors found two independent types of such conditions. First, a certain condition on the depth of the tensor product \(\mathrm{Der}_{k}(R)\otimes_{R}U\) where \(U\) is a torsionless \(R\)-module. Second, a (co)homological vanishing condition of the form \(\mathrm{Ext}_{R}^{i}(M, N) = 0\) or \(\mathrm{Tor}_{i}^{R}(M, N) = 0\) for some suitable modules \(M\) and \(N\). Both results are obtained with the assumption that the depth of \(R\) does not exceed \(3\). In the last part of the paper, the authors apply the obtained results to the critical case \(\operatorname{depth}R = 3\).
Reviewer: Alexander B. Levin (Washington)Factorization of some polynomials over finite local commutative rings and applications to certain self-dual and LCD codeshttps://zbmath.org/1491.130342022-09-13T20:28:31.338867Z"Köse, Şeyda"https://zbmath.org/authors/?q=ai:kose.seyda"Özbudak, Ferruh"https://zbmath.org/authors/?q=ai:ozbudak.ferruhAn important class of codes is quasi-twisted (QT) codes over fnite felds. They include cyclic codes, quasi-cyclic codes and constacyclic codes as special subclasses. These codes have been studied in detail recently. For example, there are structure results using Chinese Remainder Theorem, as in the case of quasi-cyclic codes over fnite felds. Self-dual codes over fnite felds are very interesting with many connections to other areas in mathematics. Linear codes with Complementary Duals (LCD) are related to self-dual codes as a kind of opposite extreme. They have very important applications to cryptography. The study of LCD codes dates back at least to a paper of \textit{J. L. Massey} [Discrete Math. 106/107, 337--342 (1992; Zbl 0754.94009)].
Double circulant codes form a special subclass of quasi-cyclic codes. Similarly doublenegacirculant codes form a special subclass of quasi-twisted codes. In [\textit{A. Alahmadi} et al., Discrete Appl. Math. 222, 205--212 (2017; Zbl 1437.94085); Des. Codes Cryptography 86, No. 6, 1257--1265 (2018; Zbl 1387.94126)] double circulant and double negacirculant codes over fnite felds were studied from the perspective of enumeration, which have asymptotic goodness consequences.
In this paper the authors determine the unique factorization of some polynomials over a fnite local commutative ring with identity explicitly. This solves and generalizes the main conjecture of Qian, Shi and Solé in [\textit{L. Qian} et al., Cryptogr. Commun. 11, No. 4, 717--734 (2019; Zbl 1459.94170)]. They also give some applications to enumeration of certain generalized double circulant self-dual and linear complementary dual (LCD) codes over some fnite rings together with an application in asymptotic coding theory.
Reviewer: Cenap Özel (İzmir)Solving degree, last fall degree, and related invariantshttps://zbmath.org/1491.130352022-09-13T20:28:31.338867Z"Caminata, Alessio"https://zbmath.org/authors/?q=ai:caminata.alessio"Gorla, Elisa"https://zbmath.org/authors/?q=ai:gorla.elisaSummary: In this paper we study and relate several invariants connected to the solving degree of a polynomial system. This provides a rigorous framework for estimating the complexity of solving a system of polynomial equations via Gröbner bases methods. Our main results include a connection between the solving degree and the last fall degree and one between the degree of regularity and the Castelnuovo-Mumford regularity.Two algorithms for computing the general component of jet scheme and applicationshttps://zbmath.org/1491.130362022-09-13T20:28:31.338867Z"Cañón, Mario Morán"https://zbmath.org/authors/?q=ai:canon.mario-moran"Sebag, Julien"https://zbmath.org/authors/?q=ai:sebag.julienLet \(X\) be an integral variety over a perfect field \(k\), \(\mathcal L_m(X)\) its jet scheme of level \(m\in \mathbb N\) and \(\mathcal L_{\infty}(X)\) its arc scheme. The general component \(\mathcal G_m(X)\) of \(\mathcal L_m(X)\) is the Zariski closure of \(\mathcal L_m(\mathrm{Reg}(X))\).
If \(X\) is smooth on \(k\) then the geometry and topology of \(\mathcal L_m(X)\) are well understood. In this paper the authors consider the case \(X\) is not smooth and study some properties of the general component \(\mathcal G_m(X)\) by means of a smooth birational model of \(X\).
Indeed, under the further hypothesis that \(X\) is affine embedded in \(\mathbb A^N_k\), the authors prove that a birational model of \(X\) provides a description of \(\mathcal G_m(X)\) that gives rice to an algorithm which computes a Groebner basis of the defining ideal of \(\mathcal G_m(X)\) in \(\mathbb A^N_k\) as a subscheme of \(\mathcal L_m(X)\) (Algorithm~2). The authors also extend to arbitrary integral varieties over perfect fields over arbitrary characteristic another algorithm ''already introduced in the Ph.D. Thesis of Kpognon'' (see also [\textit{K. Kpognon} and \textit{J. Sebag}, Commun. Algebra 45, No. 5, 2195--2221 (2017; Zbl 1376.14018)]) ``for the study of arc scheme associated with integral affine plane curves in characteristic zero'' (Algorithm~1). Several examples and comments to the implementation of the algorithms, which is available in SageMath, are provided in Sections~6 and~7.
The given results are applied for further studies of plane curves, concerning differential operators logarithmic along an affine plane curve and the rationality of a motivic power series that is introduced by the authors and ``which encodes the geometry of all \(\mathcal G_m(X)\)'' (Sections 8 and 9).
Reviewer: Francesca Cioffi (Napoli)A generalization of the Boulier-Buchberger criterion for the computation of characteristic sets of differential idealshttps://zbmath.org/1491.130372022-09-13T20:28:31.338867Z"Hashemi, Amir"https://zbmath.org/authors/?q=ai:hashemi.amir"Ollivier, François"https://zbmath.org/authors/?q=ai:ollivier.francoisIn [\textit{F. Boulier} et al., Appl. Algebra Eng. Commun. Comput. 20, No. 1, 73--121 (2009; Zbl 1185.12003)] the authors presented a differential analog of the Buchberger criterion for the computation of Gröbner bases of polynomial ideals given in [\textit{B. Buchberger}, Lect. Notes Comput. Sci. 72, 3--21 (1979; Zbl 0417.68029)]. The original differential version of the criterion (that detects useless reductions of \(S\)-polynomials in the computation of characteristic sets of differential ideals) requires linear polynomials. Later on, this result was extended in [\textit{A. Hashemi} and \textit{Z. Touraji}, Lect. Notes Comput. Sci. 8592, 466--471 (2014; Zbl 1434.13001)] to the case of products of linear factors (however, this paper does not contain a complete proof of the result).
The paper under review presents a more general statement where the linear factors are not assumed to depend on the same differential indeterminate, but only on some differential polynomial.
Reviewer: Alexander B. Levin (Washington)Degree upper bounds for involutive baseshttps://zbmath.org/1491.130382022-09-13T20:28:31.338867Z"Hashemi, Amir"https://zbmath.org/authors/?q=ai:hashemi.amir"Parnian, Hossein"https://zbmath.org/authors/?q=ai:parnian.hossein"Seiler, Werner M."https://zbmath.org/authors/?q=ai:seiler.werner-mAuthors' abstract: The aim of this paper is to investigate upper bounds for the maximum degree of the elements of any minimal Janet basis of an ideal generated by a set of homogeneous polynomials. The presented bounds depend on the number of variables and the maximum degree of the generating set of the ideal. For this purpose, by giving a deeper analysis of the method due to \textit{T. W. Dubé} [SIAM J. Comput. 19, No. 4, 750--773 (1990; Zbl 0697.68051)], we improve (and correct) his bound on the degrees of the elements of a reduced Gröbner basis. By giving a simple proof, it is shown that this new bound is valid for Pommaret bases, as well. Furthermore, based on Dubé's method, and by introducing two new notions of genericity, so-called J-stable position and prime position, we show that Dubé's (new) bound holds also for the maximum degree of polynomials in any minimal Janet basis of a homogeneous ideal in any of these positions. Finally, we study the introduced generic positions by proposing deterministic algorithms to transform any given homogeneous ideal into these positions.
Reviewer: Michael F. Singer (Raleigh)Local dynamics of non-invertible maps near normal surface singularitieshttps://zbmath.org/1491.140552022-09-13T20:28:31.338867Z"Gignac, William"https://zbmath.org/authors/?q=ai:gignac.william"Ruggiero, Matteo"https://zbmath.org/authors/?q=ai:ruggiero.matteoIn the study of the dynamics of a dominant non-invertible holomorphic map germ \(f:(\mathbb{C}^2,0) \to (\mathbb{C}^2,0)\), a successful approach consists in investigating the dynamics of \(f\) on modifications of \((\mathbb{C}^2,0)\). Here a modification \(\pi:X_\pi \to (\mathbb{C}^2,0)\) is a proper holomorphic map that is an isomorphism over \(\mathbb{C}^2 \setminus \{0\}\), and one studies then the dynamics of the induced (meromorphic) map \(f_\pi:X_\pi \dashrightarrow X_\pi\) on the exceptional set \(\pi^{-1}\{0\}\). In this memoir the authors generalize many results to the singular case, namely replacing \((\mathbb{C}^2,0)\) by the germ of a normal surface singularity \((X,x_0)\).
A first main result is that the problematic situation of indeterminacy points for infinitely many powers \(f_\pi^n\) cannot occur (except in the very special case of a finite germ at a cusp singularity). Namely, for any modification \(\pi:X_\pi \to (X,x_0)\) one can find a modification \(\pi':X_{\pi'} \to (X,x_0) \) dominating \(\pi\) such that, if \(E\) is an exceptional divisor of \(\pi'\), then \(f_{\pi'}^n (E)\) is an indeterminacy point of the lift \(f_{\pi'}: X_{\pi'} \dashrightarrow X_{\pi'}\) for at most finitely many \(n\). Moreover \(X_{\pi'}\) can be chosen to have at most cyclic quotient singularities.
As in the smooth case, the strategy to prove this is analyzing the dynamics on a space that encodes all such modifications simultaneously, namely a suitable space \(\mathcal V_X\) of centered, rank one semivaluations on the local ring \(\mathcal O_{X,x_0}\), with the induced \(f_*:\mathcal V_X \to \mathcal V_X\). The result is roughly as follows: there is a subset \(S\subset \mathcal V_X\), homeomorphic to either a point, a closed interval or a circle, such that \(f_*(S) =S\) and for any quasimonomial valuation \(v\in \mathcal V_X\) we have that \(f_*^n(v) \to S\) as \(n \to \infty\). Its proof is the core of the paper, with main technical tool the construction of a suitable distance on \(\mathcal V_X\) and the study of its non-expanding properties.
Further, the authors derive three applications. The first is an `asymptotic functoriality' result, partially controlling the fact that the pull-back on the group of exceptional divisors of a modification \(\pi\) is in general not functorial. The second treats the sequence of attraction rates of a quasimonomial \(v\in \mathcal V_X\): it eventually satisfies an integral linear recursion relation (with a similar exception as before). The third says that the first dynamical degree of \(f\) is a quadratic integer.
It is also worth mentioning that, since the used techniques are valuative (rather than complex analytic), all results are in fact valid over an arbitrary field of characteristic zero, and that some results are even valid in positive characteristic.
Reviewer: Wim Veys (Leuven)Finite torsors over strongly \(F\)-regular singularitieshttps://zbmath.org/1491.140672022-09-13T20:28:31.338867Z"Carvajal-Rojas, Javier"https://zbmath.org/authors/?q=ai:carvajal-rojas.javierSummary: We investigate finite torsors over big opens of spectra of strongly \(F\)-regular germs that do not extend to torsors over the whole spectrum. Let \((R,\mathfrak{m},\mathfrak{k},K)\) be a strongly \(F\)-regular \(\mathfrak{k}\)-germ where \(\mathfrak{k}\) is an algebraically closed field of characteristic \(p>0\). We prove the existence of a finite local cover \(R\subset R^*\) so that \(R^*\) is a strongly \(F\)-regular \(\mathfrak{k}\)-germ and: for all finite algebraic groups \(G/\mathfrak{k}\) with solvable neutral component, every \(G\)-torsor over a big open of \(\mathrm{Spec}R^*\) extends to a \(G\)-torsor everywhere. To achieve this, we obtain a generalized transformation rule for the \(F\)-signature under finite local extensions. Such formula is used to show that the torsion of \(\mathrm{Cl}R\) is bounded by \(1/s(R)\). By taking cones, we conclude that the Picard group of globally \(F\)-regular varieties is torsion-free. Likewise, this shows that canonical covers of \(\mathbb{Q}\)-Gorenstein strongly \(F\)-regular singularities are strongly \(F\)-regular.Counting isolated points outside the image of a polynomial maphttps://zbmath.org/1491.140812022-09-13T20:28:31.338867Z"El Hilany, Boulos"https://zbmath.org/authors/?q=ai:el-hilany.boulosSummary: We consider a generic family of polynomial maps \(f := (f_1, f_2): \mathbb{C}^2 \rightarrow \mathbb{C}^2\) with given supports of polynomials, and degree deg \(f := \max(\operatorname{deg} f_1, \operatorname{deg} f_2)\). We show that the (non-) properness of maps \(f\) in this family depends uniquely on the pair of supports, and that the set of isolated points in \(\mathbb{C}^2 \setminus f (\mathbb{C}^2)\) has a size of at most 6 deg \(f\). This improves an existing upper bound \((\operatorname{deg} f - 1)^2\) proven by Jelonek. Moreover, for each \(n \in \mathbb{N} \), we construct a dominant map \(f\) as above, with \(\operatorname{deg} f = 2 n + 2\), and having \(2n\) isolated points in \(\mathbb{C}^2 \setminus f (\mathbb{C}^2)\). Our proofs are constructive and can be adapted to a method for computing isolated missing points of \(f\). As a byproduct, we describe those points in terms of singularities of the bifurcation set of \(f\).Generalizations of Samuel's criteria for a ring to be a unique factorization domainhttps://zbmath.org/1491.140832022-09-13T20:28:31.338867Z"Daigle, Daniel"https://zbmath.org/authors/?q=ai:daigle.daniel"Freudenburg, Gene"https://zbmath.org/authors/?q=ai:freudenburg.gene"Nagamine, Takanori"https://zbmath.org/authors/?q=ai:nagamine.takanoriGiven a unique factorization domain (UFD), the authors study conditions on ring extensions to be UFDs as well. Their results include inter alia generalizations of criteria given by \textit{P. Samuel} [Lectures on unique factorization domains. Notes by M. Pavman Murthy. Bombay: Tata Institute of Fundamental Research (1965; Zbl 0184.06601)] and \textit{S. Mori} [Jpn. J. Math., New Ser. 3, 223--238 (1977; Zbl 0393.13003)]. As application they construct \(\mathbb{Z}\)-graded non-noetherian rational UFDs of dimension \(3\) over an arbitrary field \(k\). Moreover, they show that a certain class of rings defined by trinomial relations, which appear i.a. as Cox rings of varieties with torus ation of complexity one, are UFDs.
Reviewer: Milena Wrobel (Oldenburg)On the existence of \(B\)-root subgroups on affine spherical varietieshttps://zbmath.org/1491.140842022-09-13T20:28:31.338867Z"Avdeev, R. S."https://zbmath.org/authors/?q=ai:avdeev.roman-s"Zhgoon, V. S."https://zbmath.org/authors/?q=ai:zhgoon.vladimir-sSummary: Let \(X\) be an irreducible affine algebraic variety that is spherical with respect to an action of a connected reductive group \(G\). In this paper, we provide sufficient conditions, formulated in terms of weight combinatorics, for the existence of one-parameter additive actions on \(X\) normalized by a Borel subgroup \(B \subset G\). As an application, we prove that every \(G\)-stable prime divisor in \(X\) can be connected with an open \(G\)-orbit by means of a suitable \(B\)-normalized one-parameter additive action.
See also [\textit{I. Arzhantsev} and the first author, Sel. Math., New Ser. 28, No. 3, Paper No. 60, 37 p. (2022; Zbl 07523717)].Schemes of modules over gentle algebras and laminations of surfaceshttps://zbmath.org/1491.160212022-09-13T20:28:31.338867Z"Geiß, Christof"https://zbmath.org/authors/?q=ai:geiss.christof"Labardini-Fragoso, Daniel"https://zbmath.org/authors/?q=ai:labardini-fragoso.daniel"Schröer, Jan"https://zbmath.org/authors/?q=ai:schroer.janThe class of gentle algebras was defined by \textit{I. Assem} and \textit{A. Skowroński} [Math. Z. 195, 269--290 (1987; Zbl 0601.16022)]. Module categories over gentle algebras can be described combinatorially [\textit{B. Wald} and \textit{J. Waschbüsch}, J. Algebra 95, 480--500 (1985; Zbl 0567.16017); \textit{M. C. R. Butler} and \textit{C. M. Ringel}, Commun. Algebra 15, 145--179 (1987; Zbl 0612.16013)]. The class of Jacobian algebras associated to triangulations of unpunctured marked surfaces is a special class of gentle algebras. In this paper, the authors study some geometric properties of the representation theory of gentle algebras. First, they classify the irreducible components of the affine schemes of modules over gentle algebras and describe all smooth points of these schemes. They show that most of irreducible components are generically reduced and if a gentle algebra \(A\) has no loops, then each irreducible component is generically reduced. For the class of Jacobian algebras associated to triangulations of unpunctured marked surfaces they give a bijection between the set of generically \(\tau\)-reduced decorated irreducible components and the set of laminations of the surface, where a lamination of an unpunctured marked surface \((S,M)\) is a set of homotopy classes of curves and loops in \((S,M)\), which do not intersect each other, together with a positive integer attached to each class [\textit{G. Musiker} et al., Compos. Math. 149, No. 2, 217--263 (2013; Zbl 1263.13024)]. This bijection has some application to cluster algebras were defined by \textit{S. Fomin} and \textit{A. Zelevinsky} [J. Am. Math. Soc. 15, No. 2, 497--529 (2002; Zbl 1021.16017)]. \textit{C. Geiß} et al. [J. Am. Math. Soc. 25, No. 1, 21--76 (2012; Zbl 1236.13020)] proved that the generic Caldero-Chapoton functions form a basis, called the generic basis, of the coefficient-free upper cluster algebra \(U_{(S,M)}\) associated with an unpunctured marked surface \((S,M)\). By using the above bijection, the authors prove that the generic basis coincides with Musiker-Schiffler-Williams' bangle basis [\textit{G. Musiker} et al., Compos. Math. 149, No. 2, 217--263 (2013; Zbl 1263.13024)] of the coefficient-free cluster algebra \(A_{(S,M)}\) associated with an unpunctured marked surface \((S,M)\).
Reviewer: Alireza Nasr-Isfahani (Isfahan)On BBW parabolics for simple classical Lie superalgebrashttps://zbmath.org/1491.170142022-09-13T20:28:31.338867Z"Grantcharov, Dimitar"https://zbmath.org/authors/?q=ai:grantcharov.dimitar"Grantcharov, Nikolay"https://zbmath.org/authors/?q=ai:grantcharov.nikolay"Nakano, Daniel K."https://zbmath.org/authors/?q=ai:nakano.daniel-k"Wu, Jerry"https://zbmath.org/authors/?q=ai:wu.jerry-chun-tehLet \(\mathfrak{g} = \mathfrak{g}_{\overline{0}} \oplus \mathfrak{g}_{\overline{1}}\) be a classical simple Lie superalgebra over \(\mathbb{C}\). In the paper under review, the authors introduce a class of parabolic subalgebras, called \textit{BBW parabolic subalgebras}, which are shown to have some special cohomological properties. For the construction of these BBW parabolic subalgebras, the authors use \textit{detecting subalgebras} obtained by using the stable action of \(G_{\overline{0}}\) on \(\mathfrak{g}_{\overline{1}}\), where \(\mathfrak{g} = \mathrm{Lie} G\). Detecting subalgebras are first introduced and studied in [\textit{B. D. Boe} et al., Trans. Am. Math. Soc. 362, No. 12, 6551--6590 (2010; Zbl 1253.17012)]. More precisely, the BBW parabolic subalgebra \(\mathfrak{b}\) is generated by the negative Borel subalgebra for \(\mathfrak{g}_{\overline{0}}\) and the detecting subalgebra \(\mathfrak{f}\). Even though \(\mathfrak{b}\) is a parabolic subalgebra and technically is not a Borel subalgebra, the authors view \(\mathfrak{b}\) as being analogous to a Borel subalgebra for a complex simple Lie algebra, and the detecting subalgebra \(\mathfrak{f}\) like a maximal torus.
The main application of the existence and properties of the BBW type parabolic subalgebras is the proof of a conjecture by Boe, Kujawa and Nakano, which concerns the equality of several support varieties for modules in the category \(\mathcal{F}_{(\mathfrak{g}, \mathfrak{g}_{\overline{0}})}\) of finite-dimensional \(\mathfrak{g}\)-modules which are completely reducible over \(\mathfrak{g}_{\overline{0}}\) (Theorem 1.2.1).
Another application is given in Section 4 and is connected to the sheaf cohomology for \(G/B\) where \(\mathfrak{b} = \mathrm{Lie} B\). In particular, let \(R^j\mathrm{ind}^G_B\mathbb{C}\) for \(j \geq 0\) denote the higher sheaf cohomology groups of \(G/B\) for the trivial line bundle. Then the authors consider the Poincaré series
\[
p_{G,B}(t) = \sum_{i=0}^{\infty} \left( \dim R^i\mathrm{ind}^G_B\mathbb{C} \right) t^i
\]
and give a complete computation, using data from the classical BBW (Bott-Borel-Weil) theorem, for all classical simple Lie superalgebras except when \(\mathfrak{g} = \mathfrak{p}(n)\) (Theorem 4.10.1). In particular, it is shown that for \(\mathfrak{g} \neq \mathfrak{p}(n)\) the Poincaré polynomial \(p_{G,B}(t)\) is related to the Poincaré polynomial for a finite reflection group \(W_{\overline{1}}\). Section 5 is devoted to investigating the situation for \(\mathfrak{g} = \mathfrak{p}(n)\). For \(\mathfrak{p}(2)\) and \(\mathfrak{p}(3)\) it is shown that \(p_{G,B}(t)\) is governed by the BBW theorem. However, for \(\mathfrak{p}(4)\) this is not the case and some open questions are presented by the authors at the end of Section 5. The proof of Theorem 1.2.1 is given in Section 6. Section 7 contains several tables which provide reference for the construction of BBW parabolic subalgebras and for the relationship between the various Poincaré series.
Reviewer: Elitza Hristova (Sofia)An introduction to abstract algebra. Sets, groups, rings, and fieldshttps://zbmath.org/1491.200012022-09-13T20:28:31.338867Z"Weintraub, Steven H."https://zbmath.org/authors/?q=ai:weintraub.steven-hPublisher's description: This book is a textbook for a semester-long or year-long introductory course in abstract algebra at the upper undergraduate or beginning graduate level.
It treats set theory, group theory, ring and ideal theory, and field theory (including Galois theory), and culminates with a treatment of Dedekind rings, including rings of algebraic integers.
In addition to treating standard topics, it contains material not often dealt with in books at this level. It provides a fresh perspective on the subjects it covers, with, in particular, distinctive treatments of factorization theory in integral domains and of Galois theory.
As an introduction, it presupposes no prior knowledge of abstract algebra, but provides a well-motivated, clear, and rigorous treatment of the subject, illustrated by many examples. Written with an eye toward number theory, it contains numerous applications to number theory (including proofs of Fermat's theorem on sums of two squares and of the Law of Quadratic Reciprocity) and serves as an excellent basis for further study in algebra in general and number theory in particular.
Each of its chapters concludes with a variety of exercises ranging from the straightforward to the challenging in order to reinforce students' knowledge of the subject. Some of these are particular examples that illustrate the theory while others are general results that develop the theory further.Tracial moment problems on hypercubeshttps://zbmath.org/1491.440062022-09-13T20:28:31.338867Z"Le, Cong Trinh"https://zbmath.org/authors/?q=ai:le-cong-trinh.Summary: In this paper we introduce the \textit{tracial \(K\)-moment problem} and the \textit{sequential matrix-valued \(K\)-moment problem} and show the equivalence of the solvability of these problems. Using a Haviland's theorem for matrix polynomials, we solve these \(K\)-moment problems for the case where \(K\) is the hypercube \([-1,1]^n\).The spectrum of a compact element in a locally convex algebrahttps://zbmath.org/1491.460352022-09-13T20:28:31.338867Z"Babalola, Victor Adekola"https://zbmath.org/authors/?q=ai:babalola.victor-adekola"Bassey, Unanowo Nyong"https://zbmath.org/authors/?q=ai:bassey.unanowo-nyongSummary: This paper, in the main, obtains the Riesz-Schauder-Leray type of results for compact elements in a locally convex algebra. By way of application, results on existence of idempotents in locally convex algebras are also obtained.
For the entire collection see [Zbl 1477.46002].Locally convex quasi *-algebras and their derivationshttps://zbmath.org/1491.460372022-09-13T20:28:31.338867Z"Sokopo, Bulelani"https://zbmath.org/authors/?q=ai:sokopo.bulelaniSummary: In this note. a collection of results on locally convex quasi *-algebras and derivations of these structures is presented. Some generalisations and extensions of results obtained for Banach quasi *-algebras are given. In addition, by introducing a new notion of well-behavedness for derivations of locally convex quasi *-algebras further results are obtained, which generalise well-known results from the case of \(C^*\)-algebras.
For the entire collection see [Zbl 1477.46002].Picard-Borel algebrashttps://zbmath.org/1491.460442022-09-13T20:28:31.338867Z"Esterle, Jean"https://zbmath.org/authors/?q=ai:esterle.jeanSummary: A Picard-Borel algebra is a commutative, unital, complex algebra \(A\) such that every family \((u_\lambda)_{\lambda\in\Lambda}\) of invertible elements of \(A\), which are pairwise linearly independent, is linearly independent. A Picard-Borel algebra is said to be nontrivial, if \(u\notin\mathbb{C}1\) for some invertible element \(u\in A\). \par The algebra \(\mathbb{C}[X]\) of complex polynomials is an obvious example of trivial Picard-Borel algebra, and results from the celebrated 1897 paper ``Sur les zéros des fonctions entières'' by
\textit{É.~Borel} [Acta Math. 20, 357--396 (1897; JFM 28.0360.01)]
show that the algebra \(\mathcal{H}(\mathbb{C})\) of entire functions on \(\mathbb{C}\) is a Picard-Borel algebra. The main result of the paper shows that those Picard-Borel algebras, which are Fréchet algebras, are integral domains.
For the entire collection see [Zbl 1477.46002].On the hit problem for the Steenrod algebra in the generic degree and its applicationshttps://zbmath.org/1491.550142022-09-13T20:28:31.338867Z"Tin, Nguyen Khac"https://zbmath.org/authors/?q=ai:tin.nguyen-khacLet \(\mathcal{A}\) be the mod \(2\) Steenrod algebra and \(\mathbf{P}_n:=\mathbb{F}_{2}[x_{1},x_{2},\dots,x_{n}]\)
be the graded polynomial algebra over the field \(\mathbb{F}_2\) in \(n\) variables \(x_i\) of degree \(1.\) \(\mathbf{P}_n\) is isomorphic to the mod \(2\) cohomology algebra of the product of \(n\) copies of the infinite dimensional real projective space \(\mathbb{R}P^{\infty},\) and, as such, has a natural module structure over \(\mathcal{A}.\) The paper investigates the Peterson hit problem of finding
a minimal set of generators for \({\mathbf P}_n\) as a module over \(\mathcal{A}\) or, equivalently, of finding a vector space basis for \(\mathbb{F}_2 \otimes_{\mathcal{A}}\mathbf{P}_n\) in each degree \(d.\)
The author considers the case \(n=5,\) \( d = 5(2^s-1)+11\cdot2^{s+1},\) and develops a result of \textit{Đặng Võ Phúc} for the case \(s=0\) [J. Korean Math. Soc. 58, No. 3, 643--702 (2021; Zbl 1485.55019)] and
asserts that if \( s \geq 0,\) then the dimensions of \(\mathbb{F}_2 \otimes_{\mathcal{A}} \mathbf{P}_n\) are:
\[\dim \left(\mathbb{F}_2 \otimes_{\mathcal{A}}\mathbf{P}_n \right)_d = \left\{\begin{array}{lll}
965 & \mbox{ if } s=0 \\
3053 & \mbox{ if } s\geq 1.
\end{array}
\right. \]
The author then applies a result of \textit{N. Sum} [Adv. Math. 274, 432--489 (2015; Zbl 1367.55010)] to show that if \(d=5(2^r-1)+49\cdot2^{r},\) then
\[
\dim \left (\mathbb{F}_2 \otimes_{\mathcal{A}}\mathbf{P}_6 \right)_d=\begin{array}{lll}
192339 & \mbox{ if }r>4.
\end{array}
\]
Reviewer: Mbakiso Fix Mothebe (Gaborone)Nine equilibrium points of four point charges on the planehttps://zbmath.org/1491.780022022-09-13T20:28:31.338867Z"Lee, Tsung-Lin"https://zbmath.org/authors/?q=ai:lee.tsung-lin"Tsai, Ya-Lun"https://zbmath.org/authors/?q=ai:tsai.ya-lunSummary: We find a specific configuration for four point charges on the plane and show, with charge values in a small region, there are nine equilibrium points on the same plane, which reaches the claimed upper bound of Maxwell's conjecture. A procedure of computing bifurcation curves is presented for assisting in locating the region in the parameter space yielding the nine equilibrium points.