Recent zbMATH articles in MSC 13https://zbmath.org/atom/cc/132023-11-13T18:48:18.785376ZUnknown authorWerkzeugMini-workshop: Subvarieties in projective spaces and their projections. Abstracts from the mini-workshop held November 27 -- December 3, 2022https://zbmath.org/1521.000092023-11-13T18:48:18.785376ZSummary: The major goals of this workshop are to lay paths for a systematic study of geproci (and related, e.g., projecting to almost complete intersections or full intersections) sets of points in projective spaces, study algebraic properties of their ideals (e.g. in the spirit of the Cayley-Bacharach properties), and to identify the most promising new directions for study.Algebraic structures in statistical methodology. Abstracts from the workshop held December 4--10, 2022https://zbmath.org/1521.000142023-11-13T18:48:18.785376ZSummary: Algebraic structures arise naturally in a broad variety of statistical problems, and numerous fruitful connections have been made between algebra and discrete mathematics and research on statistical methodology. The workshop took up this theme with a particular focus on algebraic approaches to graphical models, causality, axiomatic systems for independence and non-parametric models.Orders on computable ringshttps://zbmath.org/1521.030222023-11-13T18:48:18.785376Z"Wu, Huishan"https://zbmath.org/authors/?q=ai:wu.huishanSummary: The Artin-Schreier theorem says that every formally real field has orders. \textit{H. M. Friedman} et al. showed in [Ann. Pure Appl. Logic 25, 141--181 (1983; Zbl 0575.03038); addendum ibid. 28, 319--320 (1985; Zbl 0575.03039)] that the Artin-Schreier theorem is equivalent to \(\mathsf{WKL}_0\) over \(\mathsf{RCA}_0\). We first prove that the generalization of the Artin-Schreier theorem to noncommutative rings is equivalent to \(\mathsf{WKL}_0\) over \(\mathsf{RCA}_0\). In the theory of orderings on rings, following an idea of Serre, we often show the existence of orders on formally real rings by extending pre-orders to orders, where Zorn's lemma is used. We then prove that ``pre-orders on rings not necessarily commutative extend to orders'' is equivalent to \(\mathsf{WKL}_0\).The torsion-free part of the Ziegler spectrum of orders over Dedekind domainshttps://zbmath.org/1521.030872023-11-13T18:48:18.785376Z"Gregory, Lorna"https://zbmath.org/authors/?q=ai:gregory.lorna"L'Innocente, Sonia"https://zbmath.org/authors/?q=ai:linnocente.sonia"Toffalori, Carlo"https://zbmath.org/authors/?q=ai:toffalori.carloSummary: We study the \(R\)-torsion-free part of the Ziegler spectrum of an order \(\Lambda\) over a Dedekind domain \(R\). We underline and comment on the role of lattices over \(\Lambda \). We describe the torsion-free part of the spectrum when \(\Lambda\) is of finite lattice representation type.Genus two nilpotent graphs of finite commutative ringshttps://zbmath.org/1521.052172023-11-13T18:48:18.785376Z"Kalaimurugan, G."https://zbmath.org/authors/?q=ai:kalaimurugan.gnanappirakasam"Vignesh, P."https://zbmath.org/authors/?q=ai:vignesh.p"Tamizh Chelvam, T."https://zbmath.org/authors/?q=ai:tamizh-chelvam.thirugnanamConsider a finite commutative ring \(R\) with identity and denote by
\begin{itemize}
\item \(Z(R)\) the subset of the zero divisors of \(R\);
\item \(Z_N(R):=\{y \in R: \exists x \in R\setminus\{0\} \textrm{ s.t. } xy \textrm{ nilpotent in } R\}\);
\item \(I \subseteq R\) an ideal of \(R\) and
\item \(\mathrm{ann}_R(I)=\{x \in R: xI=\{0\}\}\) the annihilator ideal.
\end{itemize}
We say that an ideal \(J\) of \(R\) is essential if \(J \cap I \neq \{0\}\), for each nonzero ideal \(I\) of \(R\).
We can associate graphs to these structures:
\begin{itemize}
\item \(\Gamma(R)\): zerodivisor graph.
It is a simple and undirected graph; its vertices are the elements of \(V=Z(R) \setminus \{0\}\) and if \(x,y \in V\), there's an edge between them iff \(xy=0\).
\item \(EG(R)\): essential graph.
It is a simple and undirected graph; its vertices are the elements of \(V=Z(R) \setminus \{0\}\) and if \(x,y \in V\), there's an edge between them iff \(\mathrm{ann}_R(xy)\) is an essential ideal for \(R\).
\item \(\Gamma_N(R)\): nilpotent graph [\textit{A.-H. Li} and \textit{Q.-S. Li}, Int. J. Algebra 4, No. 5--8, 291--302 (2010; Zbl 1210.16010)].
It is a simple and undirected graph; its vertices are the elements of \(V_N=Z_N(R) \setminus \{0\}\) and if \(x,y \in V_N\), there's an edge between them iff \(xy\) is nilpotent.
\end{itemize}
Relying on some relations of [\textit{M. J. Nikmehr} et al., J. Algebra Appl. 16, No. 7, Article ID 1750132, 14 p. (2017; Zbl 1367.13005)], the paper recalls that:
\begin{itemize}
\item if \(R\) is reduced, \(\Gamma(R) \simeq EG(R) \simeq \Gamma_N(R)\);
\item if \(R\) is nonreduced, \(\Gamma(R) \subseteq EG(R) \subset \Gamma_N(R)\).
\end{itemize}
We call genus of a graph \(G\) the minimum positive integer \(g(G):=g\) such that we can embed the graph in the surface \(S_g\) without edge crossings.
The paper characterizes all finite commutative rings with identity such that their nilpotent graphs have genus two as follows
Theorem. Let \(R\) be a finite commutative ring with identity. It holds \(g(\Gamma_N(R))=2\) if and only if \(R\) is isomorphic to one of the following rings:
\begin{itemize}
\item \(\mathbb{F}_4 \times \mathbb{F}_8\)
\item \(\mathbb{F}_4 \times\mathbb{F}_9\)
\item \(\mathbb{F}_4 \times \mathbb{Z}_{11}\)
\item \(\mathbb{Z}_5 \times\mathbb{Z}_7 \)
\item \(\mathbb{Z}_2 \times\mathbb{Z}_2 \times \mathbb{F}_8\)
\item \(\mathbb{Z}_2 \times\mathbb{Z}_2 \times \mathbb{F}_9 \)
\item \(\mathbb{Z}_2 \times\mathbb{Z}_2 \times\mathbb{Z}_{11}\)
\item \(\mathbb{Z}_2 \times\mathbb{Z}_3 \times\mathbb{Z}_5 \)
\item \(\mathbb{Z}_3 \times\mathbb{Z}_3 \times \mathbb{F}_4\)
\item \(\mathbb{Z}_4 \times\mathbb{Z}_5 \)
\item \(\mathbb{Z}_2[x]/(x^2) \times\mathbb{Z}_5 \)
\item \(\mathbb{Z}_4 \times\mathbb{Z}_2 \times\mathbb{Z}_2\)
\item \(\mathbb{Z}_2[x]/(x^2) \times\mathbb{Z}_2 \times\mathbb{Z}_2 \)
\item \(\mathbb{Z}_2 \times\mathbb{Z}_2 \times\mathbb{Z}_2 \times\mathbb{Z}_3\).
\end{itemize}
Reviewer: Michela Ceria (Bari)Polynomials, \( \alpha \)-ideals, and the principal latticehttps://zbmath.org/1521.060072023-11-13T18:48:18.785376Z"Molkhasi, Ali"https://zbmath.org/authors/?q=ai:molkhasi.aliSummary: Let \(R\) be a commutative ring with an identity, \( \mathfrak{R}\) be an almost distributive lattice and \(I_\alpha(\mathfrak{R})\) be the set of all \(\alpha \)-ideals of \(\mathfrak{R}\). If \(L(R)\) is the principal lattice of \(R\), then \(R[I_\alpha(\mathfrak{R})]\) is Cohen-Macaulay. In particular, \(R[I_\alpha(\mathfrak{R})][X_1,X_2,\cdots]\) is WB-height-unmixed.Elements of high order in finite fields specified by binomialshttps://zbmath.org/1521.110782023-11-13T18:48:18.785376Z"Bovdi, V."https://zbmath.org/authors/?q=ai:bovdi.victor-a"Diene, A."https://zbmath.org/authors/?q=ai:diene.adama"Popovych, R."https://zbmath.org/authors/?q=ai:popovych.roman-o|popovych.roman-bSummary: Let \(\mathbb F_q\) be a field with \(q\) elements, where \(q\) is a power of a prime number \(p\geq 5\). For any integer \(m\geq 2\) and \(a\in \mathbb F_q^*\) such that the polynomial \(x^m-a\) is irreducible in \(\mathbb F_q[x]\), we combine two different methods to explicitly construct elements of high order in the field \(\mathbb F_q[x]/\langle x^m-a\rangle \). Namely, we find elements with multiplicative order of at least \(5^{\sqrt[3]{m/2}}\), which is better than previously obtained bound for such family of extension fields.Differential Brauer monoidshttps://zbmath.org/1521.120072023-11-13T18:48:18.785376Z"Magid, Andy R."https://zbmath.org/authors/?q=ai:magid.andy-rSummary: The differential Brauer monoid of a differential commutative ring \(R\) is defined. Its elements are the isomorphism classes of differential Azumaya \(R\) algebras with operation from tensor product subject to the relation that two such algebras are equivalent if matrix algebras over them, with entry-wise differentiation, are differentially isomorphic. The fine Bauer monoid, which is a group, is the same thing without the differential requirement. The differential Brauer monoid is then determined from the fine Brauer monoids of \(R\) and \(R^D\) and the submonoid of the Brauer monoid whose underlying Azumaya algebras are matrix rings.Correction to: ``Effective algorithm for computing Noetherian operators of zero-dimensional ideals''https://zbmath.org/1521.130012023-11-13T18:48:18.785376Z"Nabeshima, Katsusuke"https://zbmath.org/authors/?q=ai:nabeshima.katsusuke"Tajima, Shinichi"https://zbmath.org/authors/?q=ai:tajima.shinichiA formula missing in Example 2 of the pdf version of the authors' paper [ibid. 33, No. 6, 867--899 (2022; Zbl 1518.13001)] is added.A characterisation of atomicityhttps://zbmath.org/1521.130022023-11-13T18:48:18.785376Z"Tringali, Salvatore"https://zbmath.org/authors/?q=ai:tringali.salvatoreThe author studies factorability in preordered monoids (not necessarily commutative or cancellative). In particular, the author proves that an acyclic monoid is atomic if and only if it has a generating set whose elements satisfy the ACCP. Hence, an integral domain is atomic if and only the multiplicative monoid of its nonzero elements is generated by a set of elements satisfying the ascending chain condition on principal ideals. This characterization answers a question of Geroldinger and Koch.
Reviewer: Moshe Roitman (Haifa)On \(2r\)-ideals in commutative rings with zero-divisorshttps://zbmath.org/1521.130032023-11-13T18:48:18.785376Z"Alhazmy, Khaled"https://zbmath.org/authors/?q=ai:alhazmy.khaled"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.mohammedLet \(R\) be a commutative ring with identity and \(Z(R)\) its set of zero-divizors. A proper ideal \(I\) of \(R\) is said to be a uniformly \(pr\)-ideal if there exists a positive integer \(n\) such that, whenever \(x,y\in R\) with \(xy\in I\), then \(x^n\in I\) or \(y\in Z(R)\). The order of \(I\) is the smallest positive integer for which the aforementioned property holds. The goal of this paper is to study the uniformly \(pr\)-ideals with order \(\leq 2\), which are called \(2r\)-ideals. After giving several properties and characterizations of such ideals, the authors show that many known classes of ideals are \(2r\)-ideals. They also include the study of \(2r\)-ideals in polynomials rings.
Reviewer: Ali Benhissi (Monastir)\(S\)-principal ideal multiplication moduleshttps://zbmath.org/1521.130042023-11-13T18:48:18.785376Z"Aslankarayiğit Uğurlu, Emel"https://zbmath.org/authors/?q=ai:ugurlu.emel-aslankarayigit"Koç, Suat"https://zbmath.org/authors/?q=ai:koc.suat"Tekir, Ünsal"https://zbmath.org/authors/?q=ai:tekir.unsalSummary: In this paper, we study \(S\)-Principal ideal multiplication modules. Let \(A\) be a commutative ring with \(1 \neq 0\), \(S \subseteq A\) a multiplicatively closed set and \(M\) an \(A\)-module. A submodule \(N\) of \(M\) is said to be an \(S\)-\textit{multiple} of \(M\) if there exist \(s \in S\) and a principal ideal \(I\) of \(A\) such that \(sN \subseteq IM \subseteq N\). \(M\) is said to be an \(S\)-\textit{principal ideal multiplication module} if every submodule \(N\) of \(M\) is an \(S\)-multiple of \(M\). Various examples and properties of \(S\)-principal ideal multiplication modules are given. We investigate the conditions under which the trivial extension \(A\ltimes M\) is an \(S \ltimes 0\)-principal ideal ring. Also, we prove Cohen type theorem for \(S\)-principal ideal multiplication modules in terms of \(S\)-prime submodules.Companion varieties for Hesse, Hesse union dual Hesse arrangementshttps://zbmath.org/1521.130052023-11-13T18:48:18.785376Z"De Poi, Pietro"https://zbmath.org/authors/?q=ai:de-poi.pietro"Ilardi, Giovanna"https://zbmath.org/authors/?q=ai:ilardi.giovannaIn the paper under review, the authors study the so-called companion varieties for some line arrangements over the complex numbers. The starting point is the notion of unexpected hypersurfaces.
Definition. We say that a reduced set of points \(Z \subset \mathbb{P}^{N}\) admits an unexpected hypersurface of degree \(d\) if there exists a sequence of non-negative integers \(m_{1}, \dots m_{s}\) such that for all general points \(P_{1}, \dots P_{s}\) the zero dimensional subscheme \(P=m_{1}P_{1} + \dots + m_{s}P_{s}\) fails to impose independent conditions on forms of degree \(d\) vanishing along \(Z\) and the set of such forms is non-empty.
Assume now that there is a set of points \(Z \subset \mathbb{P}^{N}\) which admits a unique unexpected hypersurface \(H_{Z,P}\) of degree \(d\) and multiplicity \(m\) at a general point \(P=(a_{0}: \dots : a_{N}) \in \mathbb{P}^{N}\). Let \[F_{Z}((x_{0}:\, \dots \,: x_{N}),(a_{0}:\, \dots \,: a_{N})) = 0\] be a homogeneous polynomial equation of \(H_{Z,P}\). Let \(g_{0}, \dots , g_{M}\) be a basis of the vector space \([I(Z)]_{d}\) of homogeneous polynomials of degree \(d\) vanishing at all points of \(Z\). Under some reasonable conditions the unexpected hypersurface \(H_{Z,P}\) comes from a bi-homogeneous polynomial \(F_{Z}((x_{0}: \dots : x_{N}),(a_{0}: \dots : a_{N}))\) of bi-degree \((m,d)\). Indeed, \(F_{Z}\) can be written in a unique way as a combination \[(\star): \quad F_{Z} = h_{0}(a_{0} : \dots : a_{N})g_{0}(x_{0} : \dots : x_{N}) + \cdots + h_{M}(a_{0} : \dots : a_{N})g_{M}(x_{0} : \dots : x_{N}),\] where \(g_{0}(x_{1}: \dots : x_{N})\), \dots , \(g_{M}(x_{0} : \dots x_{N})\) are homogeneous polynomials of degree \(d\) and \(h_{0}(a_{0} : \dots : a_{N})\), \dots , \(h_{M}(a_{0}: \dots : a_{N})\) are homogeneous polynomials of degree \(m\). Therefore, there are two rational maps naturally associated with \((\star)\), namely \[\phi : \mathbb{P}^{N} \ni (x_{0} : \dots : x_{N}) \mapsto(g_{0}(x_{0}: \dots : x_{N}): \dots : g_{M}(x_{0}: \dots : x_{N})) \in \mathbb{P}^{N},\] \[\psi : \mathbb{P}^{N} \ni (a_{0} : \dots : a_{N}) \mapsto(h_{0}(a_{0}: \dots : a_{N}): \dots : h_{M}(a_{0}: \dots : a_{N})) \in \mathbb{P}^{N}.\] The images of these maps are the companion varieties. The main result of the paper under review can be formulated as follows.
Main Theorem. The image \(S\) of \(\phi\) is a smooth arithmetically Cohen-Macaulay rational surface in the case of the Hesse and the merger of the Hesse and the dual Hesse arrangements.
1) In the case of the Hesse arrangement, the surface \(S\) is of degree \(13\). More precisely, it is the plane blow up in the \(12\) points of \(Z(\mathrm{Hesse})\) (see system (2) therein for details) embedded into \(\mathbb{P}^{8}\) with the complete linear system of the quintics through \(Z(\mathrm{Hesse})\). Its ideal \(I(S)\) is generated by \(15\) quadrics.
2) In the case of the merger of the Hesse and the dual Hesse, the surface \(S\) is of degree \(43\). More precisely, it is the plane blown-up in the \(21\) points of \(Z(\mathrm{Hesse} \cup \mathrm{dualHesse})\) (see system (6) therein for details), embedded into \(\mathbb{P}^{23}\) with the complete linear system of the octics through \(Z(\mathrm{Hesse} \cup\mathrm{dualHesse})\). Its ideal \(I(S)\) is generated by \(210\) quadrics.
Reviewer: Piotr Pokora (Kraków)FMR-rings in some distinguished constructionshttps://zbmath.org/1521.130062023-11-13T18:48:18.785376Z"Ouzzaouit, Omar"https://zbmath.org/authors/?q=ai:ouzzaouit.omar"Tamoussit, Ali"https://zbmath.org/authors/?q=ai:tamoussit.aliLet \(R\) be a commutative ring with unit. Then \(R\) is called an FMR-ring if for each maximal ideal \(M\) of \(R\), the field \(R/M\) is finite. This paper is devoted to the transfer of the FMR-ring property to the trivial ring extension and to the bi-amalgamated algebra. The main results can be summarized as follows;
The case of the trivial ring extension: Let \(R\) be a commutative ring and \(E\) be an \(R\)-module. Then \(R\propto E\) is an FMR-ring if and only if so is \(R\).
The case of the bi-amalgamated algebra: Let \(f:A\to B\) and \(g:A\to C\) be two ring homomorphisms, and let \(J\) and \(J'\) be two ideals of \(B\) and \(C\), respectively, such that \(f^{-1}(J)=g^{-1}(J')\). Then \(A\bowtie^{(f,g)}(J,J')\) is an FMR-ring if and only if so are \(f(A)+J\) and \(g(A)+J'\). If, in addition, \(f\) and \(g\) are surjective, then \(A\bowtie^{(f,g)}(J,J')\) is an FMR-ring if and only if so are \(B\) and \(C\).
Reviewer: Mohamed Aqalmoun (Fès)Essentially finite generation of valuation rings in terms of classical invariantshttps://zbmath.org/1521.130072023-11-13T18:48:18.785376Z"Cutkosky, Steven Dale"https://zbmath.org/authors/?q=ai:cutkosky.steven-dale"Novacoski, Josnei"https://zbmath.org/authors/?q=ai:novacoski.josneiSummary: The main goal of this paper is to study some properties of an extension of valuations from classical invariants. More specifically, we consider a valued field \((K,\nu)\) and an extension \(\omega\) of \(\nu\) to a finite extension \(L\) of \(K\). Then we study when the valuation ring of \(\omega\) is essentially finitely generated over the valuation ring of \(\nu \). We present a necessary condition in terms of classic invariants of the extension by Hagen Knaf and show that in some particular cases, this condition is also sufficient. We also study when the corresponding extension of graded algebras is finitely generated. For this problem we present an equivalent condition (which is weaker than the one for the finite generation of the valuation rings).When does a perturbation of the equations preserve the normal cone?https://zbmath.org/1521.130082023-11-13T18:48:18.785376Z"Quy, Pham Hung"https://zbmath.org/authors/?q=ai:pham-hung-quy."Trung, Ngo Viet"https://zbmath.org/authors/?q=ai:ngo-viet-trung.Let \(I = (f_1,\ldots,f_r)\) denote an ideal of a local Noetherian ring \((R,\mathfrak{m})\). An ideal \(I' = (f_1',\ldots,f_m')\) of \(R\) is called a small pertubation of \(I\) whenever \(f_i \cong f_i' \mod \mathfrak{m}^N\) for \(i=1,\ldots,m\) and \(N \gg 0\). It is of some interest to know which properties of \(R/I\) are preserved under small pertubations. For instances when defining ideal of the singularity of an analytic space is replaced by their \(N\)-jets for some \(N \gg 0\). The paper deals with the following problem: Let \(J \subset R\) denote an arbitrary ideal. For which ideal \(I\) does there exists a number \(N\) such that \(f_i \cong f_i' \mod J^N\) for \(i=1,\ldots,m\) such that \(\operatorname{gr}_J(R/I) \cong \operatorname{gr}_J(R/I')\)? (Note that \(\operatorname{Spec} (\operatorname{gr}_J(R/I) \) is the normal cone of the blow-up of \(R/I\) along \(J\).) -- In their main result the authors provide an affirmative answer when \(f_1,\ldots,f_m\) is a \(J\)-filter regular sequence, i.e. \((f_1,\ldots,f_{i-1}) :_R f_i /(f_1,\ldots,f_{i-1})\) is of \(J\)-torsion for \(i = 1,\ldots,m\). Moreover, they prove a converse to the result by assuming that \(\operatorname{gr}_J(R/(f_1,\ldots,f_i)) \cong \operatorname{gr}_J(R/(f_1',\ldots,f_i'))\) for \(i = 1,\ldots,m\). Finally they generalize some of their results to Noetherian filtrations.
Reviewer: Peter Schenzel (Halle)Enumeration of \(\mathcal{D}\)-invariant ideals of the ring \(R_n(K,J)\)https://zbmath.org/1521.130092023-11-13T18:48:18.785376Z"Davletshin, Maksim N."https://zbmath.org/authors/?q=ai:davletshin.maksim-nikolaevichSummary: Let \(K\) be a local ring of the main ideal with a nilpotent maximal ideal \(J\). The paper is devoted to finished of solution of problem enumeration of ideals of the ring \(K\) of \(n\times n\) matrices with coefficients of \(J\) on the main diagonal and above it.Almost strongly unital ringshttps://zbmath.org/1521.130102023-11-13T18:48:18.785376Z"Oman, Greg"https://zbmath.org/authors/?q=ai:oman.greg-g"Senkoff, Evan"https://zbmath.org/authors/?q=ai:senkoff.evanAs the authors mentioned, if \(P\) is a certain property, then a mathematical structure \(\mathcal{A}\) almost has property \(P\) if \(\mathcal{A}\) does not have property \(P\), but every substructure (or quotient structure) of \(\mathcal{A}\) has property \(P\). If \(R\) is a ring (not necessarily commutative or with identity), \(S\subseteq R\) is called a subring of \(R\), if \((S,+)\) is a subgroup of \((R,+)\) and \(S\) is closed under the multiplication of \(R\). If each subring \(S\) of \(R\) has an identity, say \(1_S\) (it is possible that \(1_R\neq 1_S\)), then \(R\) is called strongly unital ring. These rings completely determined in [\textit{G. Oman} and \textit{J. Stroud}, Involve 13, No. 5, 823--828 (2020; Zbl 1479.16002)]. In fact, they proved that a nontrivial ring \(R\) is strongly unital if and only if \(R\cong F_1\times\cdots\times F_n\), where each \(F_i\) is an absolutely algebraic field (i.e., field with nonzero characteristic which is algebraic over its prime subfield).
In this article, a ring \(R\) is called almost strongly unital, if \(R\) has no identity but every proper subring of \(R\) has an identity. In Lemma 6, they prove that each commutative reduced Artinian ring has an identity. In Theorem 8, they prove that if \(R\) is a nonzero ring, then every proper subring of \(R\) has an identity if and only if either \(R\) is strongly unital or \(R\cong \frac{X\mathbb{F}_p[X]}{<X^2>}\), where \(p\) is a prime number and \(\mathbb{F}_p\) is a field with exactly \(p\) elements.
Reviewer: Alborz Azarang (Ahvaz)Analytic spread and integral closure of integrally decomposable moduleshttps://zbmath.org/1521.130112023-11-13T18:48:18.785376Z"Bivià-Ausina, Carles"https://zbmath.org/authors/?q=ai:bivia-ausina.carles"Montaño, Jonathan"https://zbmath.org/authors/?q=ai:montano.jonathanSummary: We relate the analytic spread of a module expressed as the direct sum of two submodules with the analytic spread of its components. We also study a class of submodules whose integral closure can be expressed in terms of the integral closure of its row ideals, and therefore can be obtained by means of a simple computer algebra procedure. In particular, we analyze a class of modules, not necessarily of maximal rank, whose integral closure is determined by the family of Newton polyhedra of their row ideals.Erratum to: ``Rings in which every ideal contained in the set of zero-divisors is a d-ideal''https://zbmath.org/1521.130122023-11-13T18:48:18.785376Z"Anebri, Adam"https://zbmath.org/authors/?q=ai:anebri.adam"Mahdou, Najib"https://zbmath.org/authors/?q=ai:mahdou.najib"Mimouni, Abdeslam"https://zbmath.org/authors/?q=ai:mimouni.abdeslamSummary: In this erratum, we correct a mistake in the proof of Proposition 2.7 in our paper [ibid. 37, No. 1, 45--56 (2022; Zbl 1483.13022)]. In fact the equivalence \((3)\Longleftarrow (4)\) ``\(R\) is a quasi-regular ring if and only if \(R\) is a reduced ring and every principal ideal contained in \(Z(R)\) is a 0-ideal'' does not hold as we only have \(Rx\subseteq O(S)\).Classification of 1-absorbing comultiplication modules over a pullback ringhttps://zbmath.org/1521.130132023-11-13T18:48:18.785376Z"Farzalipour, Farkhondeh"https://zbmath.org/authors/?q=ai:farzalipour.farkhonde"Ghiasvand, Peyman"https://zbmath.org/authors/?q=ai:ghiasvand.peymanSummary: One of the aims of the modern representation theory is to solve classification problems for subcategories of modules over a unitary ring \(R\). In this paper, we introduce the concept of 1-absorbing comultiplication modules and classify 1-absorbing comultiplication modules over local Dedekind domains and we study it in detail from the classification problem point of view. The main purpose of this article is to classify all those indecomposable 1-absorbing comultiplication modules with finite-dimensional top over pullback rings of two local Dedekind domains and establish a connection between the 1-absorbing comultiplication modules and the pure-injective modules over such rings. In fact, we extend the definition and results given in [17] to a more general 1-absorbing comultiplication modules case.The algebraic cohomotopy group and its propertieshttps://zbmath.org/1521.130142023-11-13T18:48:18.785376Z"Sridharan, Raja"https://zbmath.org/authors/?q=ai:sridharan.raja"Upadhyay, Sumit Kumar"https://zbmath.org/authors/?q=ai:kumar-upadhyay.sumit"Yadav, Sunil Kumar"https://zbmath.org/authors/?q=ai:yadav.sunil-kumar|kumar-yadav.sunilFor a commutative ring \(A\) with unity, a row \(\mathbf a=(a_1,\dots,a_n)\) of ring elements is called unimodular of length \(n\), if there is another row \(\mathbf b=(b_1,\dots,b_n)\) of elements in \(A\) such that their dot product \(\mathbf a\cdot \mathbf b=1\); the set of unimodular rows of \(A\) is denoted by \(\mathrm{Um}_n(A)\). The paper considers the case \(n=2\).
An equivalence relation \(\sim\) on \(\mathrm{Um}_2(A)\) is defined by either of the following two equivalent conditions: (1) There exists \((f_1(T), f_2(T))\in\mathrm{Um}_2(A[T])\) such that \((f_1(0),f_2(0))=(a,b)\) and \((f_1(1),f_2(1))=(c,d)\) (here \(A[T]\) is the polynomial ring over \(A\) with variable \(T\)); (2) There is a matrix \(\alpha\in\mathrm{SL}_2(A)\) for which there exists a matrix \(\beta(T)\in\mathrm{SL}_2(A[T])\) with \(\beta(0)=I_2\) and \(\beta(1)=\alpha\), such that \(\alpha(a, b)^t=(c, d)^t\).
The equivalence classes \(\mathrm{Um}_2(A)/_\sim = \Gamma(A)\) form a group (called the ``algebraic cohomotopy group'') under the operation \([a,b]*[c,d]=[ac+de,bc+df]\), where \(e, f\) are such that the matrix with first row \((a,e)\) and second row \((b,f)\) is in \(\mathrm{SL}_2(A)\); the unit element is \([1,0]\). This group was previously used by \textit{M. I. Krusemeyer} in [Invent. Math. 19, 15--47 (1973; Zbl 0247.14005)] (see also [\textit{M. Karoubi} and \textit{O. Villamayor}, Math. Scand. 28, 265--307 (1971; Zbl 0231.18018)]).
The authors establish some properties of the group \(\Gamma(A)\), such as the following: The group is trivial in a number of important cases, namely if \(A\) is a field, a local ring, a polynomial ring over a field, a Euclidean domain, the polynomial ring over integers, semilocal ring, ring of dimension \(0\). If \(I\) is the nil radical of \(A\), then \(\Gamma(A)\cong\Gamma(A/I)\). If \(A=A_0\oplus A_1\oplus A_2\oplus\dots\) is a positively graded ring, then \(\Gamma(A)\cong\Gamma(A_0)\), hence \(\Gamma(A)\cong\Gamma(A[X_1,\dots,X_n])\). If \(K\) is a field, then, for every \(n\), \(\Gamma( K[X_1^{\pm1},\dots, X_n^{\pm1}])=[1,0]\). One of the open problems is whether \(\Gamma(A)\) is torsion-free. The paper would benefit from some linguistic editing.
Reviewer: Radoslav M. Dimitrić (New York)Radical support for multigraded idealshttps://zbmath.org/1521.130152023-11-13T18:48:18.785376Z"Conca, A."https://zbmath.org/authors/?q=ai:conca.aldo"De Negri, E."https://zbmath.org/authors/?q=ai:de-negri.emanuela"Gorla, E."https://zbmath.org/authors/?q=ai:gorla.elisaLet \(n\) be a positive integer and \(m=(m_1, m_2,\cdots, m_n)\in\mathbb{N}^n_{>0}\). Let \(S(m)=K[x_{ij}|1\le j\le n, 1\le i\le m_j]\) be a polynomial ring over a field \(K\) endowed with the standard \(\mathbb{Z}^n\)-grading induced by setting deg\((x_{ij})=e_j\), where \(e_j\in\mathbb{Z}^n\) is the \(j\)-th standard basis vector.
\textbf{Definition.} A collection (repetitions are allowed) \(\mathcal{A}=\{A_1, A_2, \cdots, A_s\}\) of non-empty subsets of \([n]\) is a radical support with respect to a field \(K\) if for every \(m=(m_1, m_2,\cdots, m_n)\in\mathbb{N}^n_{>0}\) and for every choice of \(f_1\in S_{A_1}, \cdots, f_s\in S_{A_s}\) the ideal \(I=(f_1,\cdots, f_s)\) of \(S\) is radical. Furthermore \(\mathcal{A}\) is called a radical support if it is a radical support for every field \(K\).
In the present paper the authors give a combinatorial characterization of radical supports. Their characterization is in terms of properties of cycles in an associated labelled graph. We also show that the notion of radical support is closely related to that of Cartwright-Sturmfels ideals. In fact, any ideal generated by multigraded generators whose multidegrees form a radical support is a Cartwright-Sturmfels ideal. Conversely, a collection of degrees such that any multigraded ideal generated by elements of those degrees is Cartwright-Sturmfels is a radical support.
Reviewer: Siamak Yassemi (Tehran)Hypersurface arrangements of aCM typehttps://zbmath.org/1521.130162023-11-13T18:48:18.785376Z"Ballico, Edoardo"https://zbmath.org/authors/?q=ai:ballico.edoardo"Huh, Sukmoon"https://zbmath.org/authors/?q=ai:huh.sukmoonSummary: We investigate the arrangement of hypersurfaces on a nonsingular varieties whose associated logarithmic vector bundle is arithmetically Cohen-Macaulay (for short, aCM), and prove that the projective space is the only smooth complete intersection with Picard rank one that admits an aCM logarithmic vector bundle. We also obtain a number of results on aCM logarithmic vector bundles over several specific varieties. As an opposite situation we investigate the Torelli-type problem that the logarithmic cohomology determines the arrangement.
{{\copyright} 2022 Wiley-VCH GmbH}Some algebraic invariants of the residue class rings of the edge ideals of perfect semiregular treeshttps://zbmath.org/1521.130172023-11-13T18:48:18.785376Z"Shaukat, Bakhtawar"https://zbmath.org/authors/?q=ai:shaukat.bakhtawar"Haq, Ahtsham Ul"https://zbmath.org/authors/?q=ai:haq.ahtsham-ul"Ishaq, Muhammad"https://zbmath.org/authors/?q=ai:ishaq.muhammadSummary: Let \(S\) be a polynomial algebra over a field. If \(I\) is the edge ideal of a perfect semiregular tree, then we give precise formulas for values of depth, Stanley depth, projective dimension, regularity and Krull dimension of \(S/I\).Regularity of primes associated with polynomial parametrisationshttps://zbmath.org/1521.130182023-11-13T18:48:18.785376Z"Cioffi, Francesca"https://zbmath.org/authors/?q=ai:cioffi.francesca"Conca, Aldo"https://zbmath.org/authors/?q=ai:conca.aldoLet \(I\) be a homogeneous ideal in the polynomial ring \(K[x_1,\ldots ,x_n]\) over a field \(K\). Then, the Castelnuovo-Mumford regularity of \(I\) is defined to be \(\mathrm{reg}(I)=\max\{j-i \ | \ \beta_{i,j}(I)\ne 0\}\) where \(\beta_{i,j}(I)\)'s are the the graded Betti numbers of the ideal \(I\). Moreover, let \(J\) be the kernel of the \(K\)-algebra map
\[
\phi: K[x_1,\ldots ,x_n] \mapsto K[y_1,\ldots ,y_m]
\]
with \(\phi(x_i)=f_i\) where \(f_i\) is a homogeneous polynomial of degree \(d > 0\) for a fixed integer \(d\). The main contribution of this paper is that \(\mathrm{reg}(J)\le d^{n2^{m-1}-1}\).
Reviewer: Amir Hashemi (Isfahan)On the reduction numbers and the Castelnuovo-Mumford regularity of projective monomial curveshttps://zbmath.org/1521.130192023-11-13T18:48:18.785376Z"Lam, Tran Thi Gia"https://zbmath.org/authors/?q=ai:lam.tran-thi-giaLet \(R\) be the coordinate ring of a projective monomial curve given parametrically by a set \(M\) of monomials of degree \(d\) in two variables \(x, y\). Let \(Q=(x ,y )\) be the ideal generated by \(x ,y\) in R. It is known that \(Q\) is a minimal reduction of \(R_+\). In various samples of \(M\) the author computes the minimal reduction number \(r_Q(R_+)\) in terms of \(M\). The approach uses the fact that \(R\) is a finitely generated module over the polynomial ring \(k[x^d,y^d]\), a Noether normalization. Note that the minimal reduction number equals the Castelnuovo-Mumford regularity if \(R\) is a Cohen-Macaulay or more general a Buchsbaum ring.
Reviewer: Peter Schenzel (Halle)Canonical modules and class groups of Rees-like algebrashttps://zbmath.org/1521.130202023-11-13T18:48:18.785376Z"Mantero, Paolo"https://zbmath.org/authors/?q=ai:mantero.paolo"McCullough, Jason"https://zbmath.org/authors/?q=ai:mccullough.jason"Miller, Lance Edward"https://zbmath.org/authors/?q=ai:miller.lance-edwardLet \(S = k[x_1,\ldots,x_n]\) the polynomial ring in \(n\) variables over the field \(k\). For a homogeneous ideal \(I = (f_1,\ldots,f_m)\) the Rees-like algebra is defined by \(S[It, t^2] \subset S[t]\), where \(t\) is a variable. Let \(T = S[y_1,\ldots,y_m,z]\) be the non-standard graded polynomial ring with a natural map \(T \to S[It, t^2]\), where the grading is given by \(\deg y_i = deg f_i +1, \deg z = 2\). Let \(Q\) denote its kernel. Then there is an expression of the canonical module \(\omega_{T/Q} = \operatorname{Ext}^c_T(T/Q,T), c = \operatorname{codim} Q\). Via a concrete computation based on linkage, the authors provide an explicit, well-structured resolution of the canonical module in terms of a type of double-Koszul complex. Additionally, they give descriptions of both the divisor class group and the Picard group of a Rees-like algebra. Note that Rees-like algebra were introduced by Peeva and the second author (see [\textit{J. McCullough} and \textit{I. Peeva}, J. Am. Math. Soc. 31, No. 2, 473--496 (2018; Zbl 1390.13043)])
Reviewer: Peter Schenzel (Halle)Castelnuovo-Mumford regularity of monomial ideals arising from posetshttps://zbmath.org/1521.130212023-11-13T18:48:18.785376Z"Seyed Fakhari, S. A."https://zbmath.org/authors/?q=ai:seyed-fakhari.seyed-aminLet \(P\) be a poset and \(I\) the Stanley-Reisner ideal of the chain complex of \(P\). The author determines an upper bound for the Castelnuovo-Mumford regularity of the \(s\)th symbolic power of \(I\), \(I^{(s)}\), which depends only on the invariants of the poset \(P\). In order to do this, the author studies the behaviour of of the Stanley-Reisner ideals of chain posets when an element of the poset is deleted. This will be used to determine a constructive method for the upper bound of \(\mbox{reg}(I^{(s)})\).
Reviewer: Anda-Georgina Olteanu (Constanţa)Homotopy categories of unbounded complexes of projective moduleshttps://zbmath.org/1521.130222023-11-13T18:48:18.785376Z"Yoshino, Yuji"https://zbmath.org/authors/?q=ai:yoshino.yujiLet \(R\) be a commutative Noetherian ring. Let \(W\) and \(X\) be complexes in the derived category \(D(R)\) of chain complexes over \(R\) such that \(H_i(W)\) is finitely generated and \(H_i(W)=0\) for \(i>>0\). textit{Y. Yoshino} [Acta Math. Vietnam. 40, No. 1, 173--177 (2015; Zbl 1327.13052)] showed that \(R\mathrm{Hom}_R(W,X)=0\) if and only if \(W_L\otimes_RX=0\). In the present paper he proves an extension of the above result as follows:
{Theorem.} Let \(R\) be a commutative Noetherian ring that is generically Gorenstein (that is, \(R_{\mathfrak{p}}\) is a Gorenstein ring for every associated prime \(\mathfrak{p}\in \mathrm{Ass}(R)\)). Let \(X\) be a complex of finitely generated projective \(R\)-modules. Then, \(X\) is acyclic (that is, \(H(X)=0\)) if and only if the \(R\)-dual \(X^*\) is acyclic (that is, \(H(X^*)=0\)).
The author introduces the parallel notion of torsion-freeness and refexivity for complexes, which he calls \(^*\)torsion-free and \(^*\)reflexive complexes. A crucial point of the proof of the above Theorem is how one can relate a generic condition of the ring such as the generic Gorenstein condition with the \(^*\)torsion-free or the \(^*\)reflextive property for complexes.
{Corollary 1.} Assume that the ring \(R\) is a generically Gorenstein ring. Let \(f:X\to Y\) be a chain homomorphism between complexes of finitely generated projective modules over \(R\). Then, \(f\) is a quasi-isomorphism if and only if the \(R\)-dual \(f^*: Y^*\to X^*\) is a quasi-isomorphism.
{Corollary 2.} Assume that the ring \(R\) is a generically Gorenstein ring. Let \(M\) be a finitely generated \(R\)-module. Then the following conditions are equivalent.
\begin{itemize}
\item[(1)] \(M\) is a totally reflexive \(R\)-module.
\item[(2)] \(Ext^i_R(M, R)= 0\) for all \(i>0\).
\item[(3)] \(M\) is an infinite syzygy, that is, there is an exact sequence of infinite length of the form \(0\to M\to P_0\to P_1\to\cdots\), where each \(P_i\) is a finitely generated projective \(R\)-module.
\end{itemize}
{Corollary 3.} Under the assumption that \(R\) is a generically Gorenstein ring, we have the equality of \(G\)-dimension; \[G-dim_RM = sup\{n\in\mathbb{Z}|\mathrm{Ext}^n_R(M, R)\neq 0\},\] for a finitely generated \(R\)-module \(M\).
{Corollary 4.} Let \(R\) be a Cohen-Macaulay ring with canonical module \(\omega\). Furthermore, assume that \(R\) is a generically Gorenstein ring. If \(\mathrm{Ext}^i_R(\omega, R)=0\) for all \(i>0\), then \(R\) is Gorenstein.
{Corollary 5.} Assume that the ring \(R\) is a generically Gorenstein ring. Let \(X\) be a complex of finitely generated projective modules.
\begin{itemize}
\item[(1)] If \(H(X)\) is bounded above, that is, \(X\in D^-(R)\), then there is an isomorphism \(X^*=R\mathrm{Hom}_R(X,R)\) in the derived category \(D(R)\)
\item[(2)] If \(H(X)\) and \(H(X^*)\) are bounded above, then we have the isomorphism in the derived category: \(X=R\mathrm{Hom}_R(R\mathrm{Hom}_R(X,R), R)\).
\end{itemize}
{Corollary 6.} Assume that the ring \(R\) is a generically Gorenstein ring. Let \(X\) be a complex of finitely generated projective modules. If all the cohomology modules \(H_i(X)\) have dimension at most \(\ell\) as \(R\)-modules, then so are the modules \(H_i(X^*)\). In particular, \(X\) has cohomology modules of finite length if and only if so does \(X^*\).
Reviewer: Siamak Yassemi (Tehran)On a theorem of Gulliksen on ideals of finite projective dimensionhttps://zbmath.org/1521.130232023-11-13T18:48:18.785376Z"Alvite, Samuel"https://zbmath.org/authors/?q=ai:alvite.samuel"Barral, Nerea G."https://zbmath.org/authors/?q=ai:barral.nerea-g"Majadas, Javier"https://zbmath.org/authors/?q=ai:majadas.javierSummary: We show that a modification of the proof of a result of \textit{T. H. Gulliksen} and \textit{G. Levin} [Homology of local rings. Kingston, Ontario: Queen's University (1969; Zbl 0208.30304), Theorem 1.4.9] gives an elementary proof of the following important theorem by Avramov: if \(f : (A, \mathfrak{m}, k) \to (B, \mathfrak{n}, l)\) is a homomorphism of noetherian local rings and \(B\) is of finite flat dimension over \(A\), then the homomorphism induced in André-Quillen homology modules \(\mathrm{H}_2(A, l, l) \to \mathrm{H}_2(B, l, l)\) is injective.A colorful hochster formula and universal parameters for face ringshttps://zbmath.org/1521.130242023-11-13T18:48:18.785376Z"Adams, Ashleigh"https://zbmath.org/authors/?q=ai:adams.ashleigh"Reiner, Victor"https://zbmath.org/authors/?q=ai:reiner.victorLet \(\Delta\) be an abstract simplicial complex on a finite vertex set \(V=[n]=\{1, 2, \dots ,n\}\). Fix a field \(\mathbf{k}\) and let \(\mathbf{k}[\mathbf{x}] = \mathbf{k}[x_{1}, x_{2},\dots ,x_{n}]\) be the polynomial ring in variables indexed by the vertices \(V\). The Stanley-Reisner ring \(\mathbf{k}[\Delta]\) of \(\Delta\) over the field \(\mathbf{k}\) is the quotient \(\mathbf{k}[\mathbf{x}] / I_{\Delta}\) where \(I_{\Delta}\) is the ideal generated by all square-free monomials \(\prod_{i \in F}x_{i}\), \(F \notin \Delta\), (see [\textit{R. Fröberg}, in: Commutative algebra. Expository papers dedicated to David Eisenbud on the occasion of his 75th birthday. Cham: Springer. 317--341 (2021; Zbl 1505.13030)])
A (proper-vertex)-d-coloring of \(\Delta\) is a map \(\kappa: V=[n] \rightarrow [d]\) such that \(\kappa(i) \neq \kappa(j)\) for all edges \([i, j]\) in \(\Delta\).
Two related parts are included in the paper under review. In the first part the authors generalize Hochster's formula on resolutions of Stanley-Reisner rings (see [\textit{M. Hochster}, in: Ring Theory II, Proc. 2nd Okla. Conf. 1975, 171--223 (1977; Zbl 0351.13009)]) to a colorful version.
A universal system of parameters for face rings of simplicial posets (see [\textit{R. P. Stanley}, J. Pure Appl. Algebra 71, No. 2--3, 319--331 (1991; Zbl 0727.06009)]) is the subject of the second part of this paper. The authors show that these parameters have good properties, such as being stable under symmetries and detecting the depth of the face ring. Moreover, when resolving the face ring over these parameters, the shape is predicted, conjecturally, by the colorful Hochster formula.
Reviewer: Saïd Zarati (Tunis)Indecomposability of top local cohomology modules and connectedness of the prime divisors graphshttps://zbmath.org/1521.130252023-11-13T18:48:18.785376Z"Doustimehr, Mohammad Reza"https://zbmath.org/authors/?q=ai:doustimehr.mohammad-rezaLet \(R\) be a commutative Noetherian local ring with identity, and let \(\mathfrak a\) be an ideal of \(R\). The goal is to decompose the top local cohomology module \(\text{H}_{\mathfrak a}^d(R)\), where \(d=\dim R\), into indecomposable modules. It is assumed that \(\text{H}_{\mathfrak a}^d(R)\neq 0\) and a natural number \(t\) is given. The main result of this paper states that \(\text{H}_{\mathfrak a}^d(R)\) can be expressed as the direct sum of at least \(t\) nonzero modules if and only if the undirected graph \(\Gamma_{{\mathfrak a},R}\) has at least \(t\) connected components. This result extends the corresponding finding of \textit{M. Hochster} and \textit{C. Huneke} [Contemp. Math. 159, 197--208 (1994; Zbl 0809.13003)]. The vertex set of \(\Gamma_{{\mathfrak a},R}\) consists of all associated prime ideals \({\mathfrak p}\in \text{Ass} \ R\) such that \(\text{H}_{\mathfrak a}^d(R/{\mathfrak p}) \neq 0\), and an edge is formed between any two distinct vertices \(\mathfrak p\) and \(\mathfrak q\) if \(\text{ht}({\mathfrak p}+{\mathfrak q})=1\).
Reviewer: Kamran Divaani-Aazar (Tehran)Coartinianess of local homology modules for ideals of small dimensionhttps://zbmath.org/1521.130262023-11-13T18:48:18.785376Z"Shen, Jingwen"https://zbmath.org/authors/?q=ai:shen.jingwen"Zhang, Pinger"https://zbmath.org/authors/?q=ai:zhang.pinger"Yang, Xiaoyan"https://zbmath.org/authors/?q=ai:yang.xiaoyanLet \(R\) be a commutative noetherian ring, \(\mathfrak{a}\) an ideal of \(R\), and \(M\) an \(R\)-module. For any \(i \geq 0\), the \(i\)th local cohomology module of \(M\) with respect to \(\mathfrak{a}\) is given by \[H^{i}_{\mathfrak{a}}(M) \cong \underset{n\geq 1}\varinjlim \mathrm{Ext}^{i}_{R}\left(R/ \mathfrak{a}^{n},M\right).\] Hartshorne defines an \(R\)-module \(M\) to be \(\mathfrak{a}\)-cofinite if \(\mathrm{Supp}_{R}(M)\subseteq \mathrm{Var}(\mathfrak{a})\) and \(\mathrm{Ext}^{i}_{R}\left(R/ \mathfrak{a},M\right)\) is a finitely generated \(R\)-module for every \(i\geq 0\). Dually, for any \(i \geq 0\), the \(i\)th local homology module of \(M\) with respect to \(\mathfrak{a}\) is given by \[H^{\mathfrak{a}}_{i}(M) \cong \underset{n\geq 1}\varprojlim \mathrm{Tor}^{R}_{i}\left(R/ \mathfrak{a}^{n},M\right).\] In the same vein, an \(R\)-module \(M\) is said to be \(\mathfrak{a}\)-coartinian if \(\mathrm{Cosupp}_{R}(M)\subseteq \mathrm{Var}(\mathfrak{a})\) and \(\mathrm{Tor}^{R}_{i}\left(R/ \mathfrak{a},M\right)\) is an artinian \(R\)-module for every \(i\geq 0\).
For an \(R\)-module \(M\) with \(\mathrm{Cosupp}_{R}(M)\subseteq \mathrm{Var}(\mathfrak{a})\), the authors show that \(M\) is \(\mathfrak{a}\)-coartinian if and only if \(\mathrm{Ext}^{i}_{R}\left(R/ \mathfrak{a},M\right)\) is artinian for every \(0\leq i \leq \mathrm{cd}(\mathfrak{a},M)\), which in turn provides finitely many steps to examine \(\mathfrak{a}\)-coartinianess. They also consider the dual questions to those of Hartshorne: for which rings \(R\) and ideals \(\mathfrak{a}\) are the local homology modules \(H^{\mathfrak{a}}_{i}(M)\) \(\mathfrak{a}\)-coartinian for every artinian \(R\)-module \(M\) and every \(i\geq 0\); and whether the category of \(\mathfrak{a}\)-coartinian \(R\)-modules is an abelian subcategory of the category of \(R\)-modules. They further establish affirmative answers to these questions in the cases \(\mathrm{cd}(\mathfrak{a},R) \leq 1\) or \(\dim(R/\mathfrak{a}) \leq 1\).
Reviewer: Hossein Faridian (Clemson)Attached primes of local cohomology modules of complexeshttps://zbmath.org/1521.130272023-11-13T18:48:18.785376Z"Tri, Nguyen Minh"https://zbmath.org/authors/?q=ai:nguyen-minh-tri.|tri.nguyen-minhIn the paper under review, the author describes the attached primes of top local cohomology modules in derived categories. More precisely, let \((R, \mathfrak{m})\) be a local ring, \(\mathcal{S}\) a specialization closed subset and \(X\not\simeq 0\) an \(R\)-complex with finitely generated and bounded homology and finite dimension. Assume that \(H^d_{\mathcal{S}}(X)\neq 0\). Then \[\mathrm{Att}_RH^d_{\mathcal{S}}(X)=\{\mathfrak{p}\in \mathrm{Supp}_R(X)|cd(\mathcal{S}, R/\mathfrak{p})-inf X_{\mathfrak{p}}=d\}.\]
In addition, the author gives a generalization of the Lichtenbaum-Hartshorne vanishing theorem for complexes of \(R\)-modules.
Reviewer: Siamak Yassemi (Tehran)Classes of trivial ring extensions and amalgamations subject to pseudo-almost valuation conditionhttps://zbmath.org/1521.130282023-11-13T18:48:18.785376Z"Moutui, Moutu Abdou Salam"https://zbmath.org/authors/?q=ai:moutui.moutu-abdou-salam"Ouled Azaiez, Najib"https://zbmath.org/authors/?q=ai:ouled-azaiez.najib"Koç, Suat"https://zbmath.org/authors/?q=ai:koc.suatSummary: This paper studies the transfer of pseudo-almost valuation property (PAVR property for short) to various context of commutative ring extensions such as power series ring, trivial ring extension and amalgamation. Our work is motivated by an attempt to generate new original classes of rings satisfying this property. The obtained results are backed with new and illustrative examples arising as trivial ring extensions, amalgamations and pullback constructions.Characterizing absolutely irreducible integer-valued polynomials over discrete valuation domainshttps://zbmath.org/1521.130292023-11-13T18:48:18.785376Z"Hiebler, Moritz"https://zbmath.org/authors/?q=ai:hiebler.moritz"Nakato, Sarah"https://zbmath.org/authors/?q=ai:nakato.sarah"Rissner, Roswitha"https://zbmath.org/authors/?q=ai:rissner.roswithaFor a commutative ring \(A\) with identity, an irreducible element \(a \in A\) is called \textit{absolutely irreducible}, if for every \(k \in \mathbb N\) the element \(a^k\) has (up to associates) only the trivial factorization \(a \cdot a \cdot \ldots \cdot a\).
Let \(R\) be a discrete valuation domain with finite residue field \(R/pR\), where \(p\) is a prime element of \(R\), and let \(K\) denote the quotient field of \(R\). The authors investigate absolutely irreducible elements of the ring
\[
\text{Int}(R) = \{ F \in K[X] \mid F(R) \subseteq R \}
\]
of all integer-valued polynomials of \(K[X]\). Every irreducible \(F \in \text{Int}(R)\) can be written in the form \(F = f/p^n\) with some primitive polynomial \(f \in R[X]\) and \(n \ge 0\).
The authors prove that \(F\) is absolutely irreducible in \(\text{Int}(R)\) if and only if the \textit{fixed divisor kernel} of \(f\) is trivial (and \(f\) is not a proper power of a polynomial of \(R[X]\)). The notions and ideas for the proof are contained in Section 4 of the paper.
Furthermore, for given \(F\) as above, the authors determine explicite values \(s \in \mathbb N\) such that \(F\) is absolutely irreducible if and only if \(F^s\) has only the trivial factorization in \(\text{Int}(R)\). Finally they argue that these values for \(s\) are best possible.
Reviewer: Günter Lettl (Graz)Equality of ordinary and symbolic powers of edge ideals of weighted oriented graphshttps://zbmath.org/1521.130302023-11-13T18:48:18.785376Z"Banerjee, Arindam"https://zbmath.org/authors/?q=ai:banerjee.arindam.1"Chakraborty, Bidwan"https://zbmath.org/authors/?q=ai:chakraborty.bidwan"Das, Kanoy Kumar"https://zbmath.org/authors/?q=ai:das.kanoy-kumar"Mandal, Mousumi"https://zbmath.org/authors/?q=ai:mandal.mousumi"Selvaraja, S."https://zbmath.org/authors/?q=ai:selvaraja.sThe authors examine when the edge ideal of a weighted oriented graph is normally torsion-free, that is, when the ordinary and symbolic powers of the (weighted) edge ideal coincide for all powers. Weighted oriented graphs have been of interest to algebaists in recent years due to their connection to coding theory. The edge ideal of such an object is a monomial ideal that is generally not square-free. More precisely, let \(\mathcal D\) be a graph with \(n\) vertices and directed edges, where the directed edge from \(x\) to \(y\) is denoted \([x,y]\), and a vertex weight function \(f: \{x_1, \ldots ,x_n\} \rightarrow {\mathbb N}\). Then \(I({\mathcal D}) = (x_ix_j^{a_j} \mid [x_i,x_j] \in E({\mathcal D}))\) where \(a_j = f(x_j)\). The authors determine when \({\mathfrak m} = (x_1, \ldots , x_n)\) is a strong vertex cover of \(\mathcal D\) in terms of out neighborhoods and use this to show that for a weighted oriented complete graph, \(\mathfrak m\) is a strong vertex cover if and only if \(I({\mathcal D})^t = I({\mathcal D})^{(t)}\) for all \(t \geq 2\). The authors also characterize all strong vertex covers for weighted oriented complete bipartite graphs using sources and sinks, and show that for a weighted oriented complete bipartite graph, if every vertex of one of the bipartition sets is either a source or a sink, then \(I({\mathcal D})^t = I({\mathcal D})^{(t)}\) for all \(t \geq 2\). Further, for such graphs, if all vertices have nontrivial weights, the authors show that \(I({\mathcal D})^t = I({\mathcal D})^{(t)}\) for all \(t \geq 2\) holds if an only if either \(\mathcal D\) does not have any source vertices or if there exists one source vertex, then all vertices are either sources or sinks.
Reviewer: Susan Morey (San Marcos)Support posets of some monomial idealshttps://zbmath.org/1521.130312023-11-13T18:48:18.785376Z"Pascual-Ortigosa, Patricia"https://zbmath.org/authors/?q=ai:pascual-ortigosa.patricia"Sáenz-de-Cabezón, E."https://zbmath.org/authors/?q=ai:saenz-de-cabezon.eduardoLet \(I\subseteq k[x_1,\ldots,x_n]\) be a monomial ideal. The authors consider the support poset of \(I\). Since not any poset can be realizable as the support poset of some monomial ideal, the authors consider classes of posets for which there is at least a monomial ideal supported by them and explicitly describe these ideals. By using the Mayer-Vietoris trees, the authors determine the Betti numbers of these ideals. Moreover, for particular classes of monomial ideals, they determine combinatorial properties of the corresponding support poset. They prove, for instance, that the support poset of a series-parallel ideal is a forest (Theorem 4.5) and that converse also holds (Proposition 4.7).
Reviewer: Anda-Georgina Olteanu (Constanţa)Exchange graphs for mutation-finite non-integer quivershttps://zbmath.org/1521.130322023-11-13T18:48:18.785376Z"Felikson, Anna"https://zbmath.org/authors/?q=ai:felikson.anna"Lampe, Philipp"https://zbmath.org/authors/?q=ai:lampe.philippSummary: Skew-symmetric non-integer matrices with real entries can be viewed as quivers with non-integer arrow weights. Such quivers can be mutated following the usual rules of quiver mutation. Felikson and Tumarkin show that mutation-finite non-integer quivers admit geometric realisations by partial reflections. This allows us to define a geometric notion of seeds and thus to define the exchange graphs for mutation classes. In this paper we study exchange graphs of mutation-finite quivers. The concept of finite type generalises naturally to mutation-finite non-integer quivers. We show that for all non-integer quivers of finite type there is a well-defined notion of an exchange graph, and this notion is consistent with the classical notion of exchange graph of integer mutation types coming from cluster algebras. In particular, exchange graphs of finite type quivers are finite. We also show that exchange graphs of rank 3 affine quivers are finite modulo the action of a finite-dimensional lattice (but unlike the integer case, the rank of the lattice is higher than 1 for non-integer quivers).Generalized cluster structures related to the Drinfeld double of \(GL_n\)https://zbmath.org/1521.130332023-11-13T18:48:18.785376Z"Gekhtman, Misha"https://zbmath.org/authors/?q=ai:gekhtman.misha"Shapiro, Michael"https://zbmath.org/authors/?q=ai:shapiro.michael-z"Vainshtein, Alek"https://zbmath.org/authors/?q=ai:vainshtein.alekSummary: We prove that the regular generalized cluster structure on the Drinfeld double of \(GL_n\) constructed in [\textit{M. Gekhtman} et al., Int. Math. Res. Not. 2022, No. 6, 4181--4221 (2022; Zbl 1510.16027)] is complete and compatible with the standard Poisson-Lie structure on the double. Moreover, we show that for \(n=4\) this structure is distinct from a previously known regular generalized cluster structure on the Drinfeld double, even though they have the same compatible Poisson structure and the same collection of frozen variables. Further, we prove that the regular generalized cluster structure on band periodic matrices constructed in [Gekhtman et al., loc. cit.] possesses similar compatibility and completeness properties.A cluster structure on the coordinate ring of partial flag varietieshttps://zbmath.org/1521.130342023-11-13T18:48:18.785376Z"Kadhem, Fayadh"https://zbmath.org/authors/?q=ai:kadhem.fayadhSummary: The main goal of this paper is to show that the (multi-homogeneous) coordinate ring of a partial flag variety \(\mathbb{C} [G / P_K^-]\) contains a cluster algebra if \(G\) is any semisimple complex algebraic group. We use derivation properties and a special lifting map to prove that the cluster algebra structure \(\mathcal{A}\) of the coordinate ring \(\mathbb{C} [N_K]\) of a Schubert cell constructed by Goodearl and Yakimov can be lifted, in an explicit way, to a cluster structure \(\widehat{\mathcal{A}}\) living in the coordinate ring of the corresponding partial flag variety. Then we use a minimality condition to prove that the cluster algebra \(\widehat{\mathcal{A}}\) is equal to \(\mathbb{C} [G / P_K^-]\) after localizing some special minors.The \(h\)-polynomial and the rook polynomial of some polyominoeshttps://zbmath.org/1521.130352023-11-13T18:48:18.785376Z"Kummini, Manoj"https://zbmath.org/authors/?q=ai:kummini.manoj"Veer, Dharm"https://zbmath.org/authors/?q=ai:veer.dharmA polyomino is a finite union of unit squares with vertices at lattice points in the plane that is connected and has not finite cut-set. \textit{A. A. Qureshi} [J. Algebra 357, 279--303 (2012; Zbl 1262.13013)] associated a finitely generated graded \(\Bbbk\)-algebra \(\Bbbk[X]\) over a field \(\Bbbk.\) Let \(h(t)\) be the \(h\)-polynomial of \(\Bbbk[X].\) For \(k \in \mathbb{N},\) a \(k\)-rook configuration in \(X\) is an arrangement of \(k\) rooks in pairwise non-attacking positions. The rook polynomial \(r(t)\) of \(X\) is \(\sum_{k\in \mathbb{N}} r_k t^k\) where \(r_k\) is the number of \(k\)-rook configurations in \(X.\)
A polyomino is said to be thin if it does not contain a \(2 \times 2\) square of four unit squares. In [\textit{G. Rinaldo} and \textit{F. Romeo}, J. Algebr. Comb. 54, No. 2, 607--624 (2021; Zbl 1480.13014)] the authors proved that if \(X\) is a simple thin polyomino, then \(h(t)=r(t)\) and conjectured that this property characterizes thin polyminoes. In this paper the authors prove the conjecture in the case the vertex set of a convex polyomino is a sublattice of \(\mathbb{N}^2.\)
Reviewer: Shreedevi K. Masuti (Dharwad)Numerical semigroups with unique Apéry expansionshttps://zbmath.org/1521.130362023-11-13T18:48:18.785376Z"Pandit, Sudip"https://zbmath.org/authors/?q=ai:pandit.sudip"Saha, Joydip"https://zbmath.org/authors/?q=ai:saha.joydip"Sengupta, Indranath"https://zbmath.org/authors/?q=ai:sengupta.indranathThe authors provide a deep study of two families of numerical semigroups. The first one is \(\Gamma_4=\langle a,2a+d,3a+4d,4a+6d\rangle\), with \(a\) and \(d\) positive integers such that \(\gcd(a,d)=1\) and \(a\ge 7\). For this family of numerical semigroups, the authors compute the Apéry set with respect to \(a\), the set of pseudo-Frobenius numbers, a minimal presentation, study the syzygies of \(k[\Gamma_4]\), describe the Apéry table and tangent cone of \(k[\Gamma_4]\). In particular, they show that the elements of Apéry set of \(\Gamma_4\) with respect to \(a\) have unique expansions and that the tangent cone is Cohen-Macaulay, not Gorenstein and Buchsbaum. The authors prove that if \(a\) is a multiple of six, then \(\Gamma_4\) is symmetric, and it is not almost Gorenstein in any other case.
The other family studied is the set of numerical semigroups \(\mathfrak{G}_{n+2}=\langle a,ha+d,ha+rd,ha+r^2d,\dots,ha+r^nd\rangle\). The authors show that the elements of the Apéry set of these semigroups with respect to \(a\) have unique expansions. They also compute the Apéry table for these semigroups, and as an application prove that the tangent cone in this setting is also Cohen-Macaulay, not Gorenstein and Buchsbaum.
Reviewer: Pedro A. García Sánchez (Granada)Betti sequence of the projective closure of affine monomial curveshttps://zbmath.org/1521.130372023-11-13T18:48:18.785376Z"Saha, Joydip"https://zbmath.org/authors/?q=ai:saha.joydip"Sengupta, Indranath"https://zbmath.org/authors/?q=ai:sengupta.indranath"Srivastava, Pranjal"https://zbmath.org/authors/?q=ai:srivastava.pranjalSummary: We introduce the notion of star gluing of numerical semigroups and show that this preserves the arithmetically Cohen-Macaulay and Gorenstein properties of the projective closure. Next, we give a sufficient condition involving Gröbner basis for the matching of Betti sequences of the affine curve and its projective closure. We also study the effect of simple gluing on Betti sequences of the projective closure. Finally, we construct numerical semigroups by gluing, such that for every positive integer \(n\), the last Betti number of the corresponding affine curve and its projective closure are both \(n\).Affinoid Dixmier modules and the deformed Dixmier-Moeglin equivalencehttps://zbmath.org/1521.130382023-11-13T18:48:18.785376Z"Jones, Adam"https://zbmath.org/authors/?q=ai:jones.adam|jones.adam.1This long paper is divided into the following sections: 1. Motivation, 2. Preliminaries, 3. The Action of \(\widehat{U(\mathcal L)_K}\) on \(\widehat{D(\lambda)}\), 4. Dixmier Annihilators, 5. Locally Closed Ideals, 6. Weakly Rational Ideals, 7. Special Dixmier Annihilators.
The springboards to the paper's main results are the Dixmier-Moeglin equivalence and Iwasawa algebras, with appropriate modifications. Given a free, finitely generated \(\mathbb{Z}_p\)-Lie algebra \(\mathcal L\) (\(p\) a prime) the affinoid enveloping algebra \(\widehat{U(\mathcal L)_K}\) has been shown to be useful in the representation theory of compact \(p\)-adic Lie groups and the author aims to understand better its algebraic structure. The author defines a Dixmier module over \(\widehat{U(\mathcal L)_K}\) (a generalization of the Verma module) in order to prove that the Dixmier modules are generally irreducible and that, if \(\mathcal L\) is nilpotent, then all primitive ideals of \(\widehat{U(\mathcal L)_K}\) can be described in terms of annihilator ideals of Dixmier modules. If \(G\) is a uniform pro-\(p\) group these results will be used by the author to infer information about primitive ideals of \(KG\). The author's other main result is that the enveloping algebra \(\widehat{U(\mathcal L)_K}\) satisfies the deformed Dixmier-Moeglin equivalence in case \(\mathcal L\) is an \(\mathcal O\)-Lie lattice in \(\mathfrak g\) (\(\mathcal O\) being the valuation ring of \(K\) and \(\mathfrak g\) -- a nilpotent \(K\)-Lie algebra such that [\(\mathfrak g, \mathfrak g\)] is abelian.
Reviewer: Radoslav M. Dimitrić (New York)Anick resolution and the minimal projective resolution of \(U_q^+(C_3)\)https://zbmath.org/1521.130392023-11-13T18:48:18.785376Z"Mao, Ling Ling"https://zbmath.org/authors/?q=ai:mao.lingling"Xin, Xiao Long"https://zbmath.org/authors/?q=ai:xin.xiaolongThe notion of the minimal projective resolution of the trivial module on an algebra is very important in homological algebra as one can use it for many purposes such as computing the extensions of a given pair of modules and the global dimension. But in general, it is often very difficult to construct the minimal projective resolution for a module. Anick constructed a resolution, the so-called Anick resolution, that is larger than the minimal resolution but small enough to perform many calculations in homological algebra. One of the main ingredients of the Anick resolution is the \(n\)-chain introduced and computed by Anick using the Diamond Lemma, or equivalently, the Gröbner-Shirshov basis. The minimal projective resolutions of \(U_{q}^{+}(G_{2})\) and \(U_{q}^{+}(B_{2})\) have been computed before. In this paper, from the Anick resolution and Gröbner-Shirshov basis for quantized enveloping algebra of type \(C_{3}\), the authors compute the minimal projective resolution of the trivial module \(U_{q}^{+}(C_{3})\) and as an application, they show that the global dimension of \(U_{q}^{+}(C_{3})\) is equal to 9.
Reviewer: Hossein Faridian (Clemson)Perfectly contractile graphs and quadratic toric ringshttps://zbmath.org/1521.130402023-11-13T18:48:18.785376Z"Ohsugi, Hidefumi"https://zbmath.org/authors/?q=ai:ohsugi.hidefumi"Shibata, Kazuki"https://zbmath.org/authors/?q=ai:shibata.kazuki"Tsuchiya, Akiyoshi"https://zbmath.org/authors/?q=ai:tsuchiya.akiyoshiSummary: Perfect graphs form one of the distinguished classes of finite simple graphs. \textit{M. Chudnovsky} et al. [Ann. Math. (2) 164, No. 1, 51--229 (2006; Zbl 1112.05042)] proved that a graph is perfect if and only if it has no odd holes and no odd antiholes as induced subgraphs, which was conjectured by Berge. We consider the class \({\mathcal{A}}\) of graphs that have no odd holes, no antiholes, and no odd stretchers as induced subgraphs. In particular, every graph belonging to \({\mathcal{A}}\) is perfect. Everett and Reed conjectured that a graph belongs to \({\mathcal{A}}\) if and only if it is perfectly contractile. In the present paper, we discuss graphs belonging to \({\mathcal{A}}\) from a viewpoint of commutative algebra. In fact, we conjecture that a perfect graph \(G\) belongs to \({\mathcal{A}}\) if and only if the toric ideal of the stable set polytope of \(G\) is generated by quadratic binomials. Especially, we show that this conjecture is true for Meyniel graphs, perfectly orderable graphs, and clique separable graphs, which are perfectly contractile graphs.
{{\copyright} 2022 The Authors. \textit{Bulletin of the London Mathematical Society} is copyright {\copyright} London Mathematical Society.}Explicit minimal embedded resolutions of divisors on models of the projective linehttps://zbmath.org/1521.140082023-11-13T18:48:18.785376Z"Obus, Andrew"https://zbmath.org/authors/?q=ai:obus.andrew"Srinivasan, Padmavathi"https://zbmath.org/authors/?q=ai:srinivasan.padmavathiSummary: Let \(K\) be a discretely valued field with ring of integers \(\mathcal{O}_K\) with perfect residue field. Let \(K(x)\) be the rational function field in one variable. Let \(\mathbb{P}^1_{\mathcal{O}_K}\) be the standard smooth model of \(\mathbb{P}^1_K\) with coordinate \(x\). Let \(f(x)\in\mathcal{O}_K[x]\) be a squarefree polynomial with corresponding divisor of zeroes \(\operatorname{div}_0(f)\) on \(\mathbb{P}^1_{\mathcal{O}_K}\). We give an explicit description of the minimal embedded resolution \(\mathcal{Y}\) of the pair \((\mathbb{P}^1_{\mathcal{O}_K}, \operatorname{div}_0(f))\) by using Mac Lane's theory to write down the discrete valuations on \(K(x)\) corresponding to the irreducible components of the special fiber of \(\mathcal{Y}\).The delta invariant and fiberwise normalization for families of isolated non-normal singularitieshttps://zbmath.org/1521.140092023-11-13T18:48:18.785376Z"Greuel, Gert-Martin"https://zbmath.org/authors/?q=ai:greuel.gert-martin"Pfister, Gerhard"https://zbmath.org/authors/?q=ai:pfister.gerhardThe delta invariant, also called genus defect, is an important numerical invariant of a singular reduced curve and is therefore often considered for algebraic curves over the complex numbers, but also for curves over finite fields, e.g. in coding theory. The delta invariant has been extended to generically reduced complex analytic curves and it has been shown that it can be used to control the topology in a family of such curves by taking care of the influence of embedded points.
This invariant has been further extended to complex-analytic isolated non-normal singularities of any dimension and its behavior has been studied in connection with simultaneous normalization.
The authors prove the semicontinuity of the delta invariant in a family of schemes or analytic varieties with finitely many (not necessarily reduced) isolated non-normal singularities, in particular for families of generically reduced curves. They define and use a modified delta invariant for isolated non-normal singularities of any dimension that takes care of embedded points. Their results generalize results by Teissier and Chiang-Hsieh-Lipman for families of reduced curve singularities.
The base ring for the families can be an arbitrary principal ideal domain such that the semicontinuity result provides possible improvements for algorithms to compute the genus of a curve.
Reviewer: Vladimir P. Kostov (Nice)New elementary components of the Gorenstein locus of the Hilbert scheme of pointshttps://zbmath.org/1521.140132023-11-13T18:48:18.785376Z"Szafarczyk, Robert"https://zbmath.org/authors/?q=ai:szafarczyk.robertSummary: We construct new explicit examples of nonsmoothable Gorenstein algebras with Hilbert function \((1,n,n,1)\). This gives a new infinite family of elementary components in the Gorenstein locus of the Hilbert scheme of points and solves the cubic case of Iarrobino's conjecture.Classification and syzygies of smooth projective varieties with 2-regular structure sheafhttps://zbmath.org/1521.140362023-11-13T18:48:18.785376Z"Kwak, Sijong"https://zbmath.org/authors/?q=ai:kwak.sijong"Park, Jinhyung"https://zbmath.org/authors/?q=ai:park.jinhyungSummary: The geometric and algebraic properties of smooth projective varieties with 1-regular structure sheaf are well understood, and the complete classification of these varieties is a classical result. The aim of this paper is to study the next case: smooth projective varieties with 2-regular structure sheaf. First, we give a classification of such varieties using adjunction mappings. Next, under suitable conditions, we study the syzygies of section rings of those varieties to understand the structure of the Betti tables, and show a sharp bound for Castelnuovo-Mumford regularity.A regular interpolation problem and its applicationshttps://zbmath.org/1521.140822023-11-13T18:48:18.785376Z"Das, Nilkantha"https://zbmath.org/authors/?q=ai:das.nilkanthaLet \(k\) be an algebraically closed field of characteristic zero and \(\Phi : X \longrightarrow Y\) a regular map between affine varieties. The article under review explores the problem of interpolation of a function \(f: Y \longrightarrow k\) over \(\text{Im}(\Phi)\) by a regular function where it is known that \(f \circ \Phi: X \longrightarrow k\) is a regular function, i.e., it asks whether there exists a regular function \(g: Y \longrightarrow k\) such that \(f|_{\text{Im}(\Phi)} = g|_{\text{Im}(\Phi)}\). The author shows that the problem has an affirmative answer if \(Y\) is factorial and \(\Phi\) is almost surjective. The achieved result generalizes a result of \textit{E. Aichinger} [J. Commut. Algebra 7, No. 3, 303--315 (2015; Zbl 1330.13009)] proven for affine spaces.
As an application, the author establishes that a regular morphism from an affine variety to a factorial affine variety is biregular if and only if it is injective and almost surjective, which generalizes a result of \textit{J. Ax} [Pac. J. Math. 31, 1--7 (1969; Zbl 0194.52001)] establishing injective endomorphisms are automorphisms. The author also discusses an analytic criterion of biregularity.
Reviewer: Prosenjit Das (Thiruvananthapuram)Orbifold diagramshttps://zbmath.org/1521.160082023-11-13T18:48:18.785376Z"Baur, Karin"https://zbmath.org/authors/?q=ai:baur.karin"Pasquali, Andrea"https://zbmath.org/authors/?q=ai:pasquali.andrea"Velasco, Diego"https://zbmath.org/authors/?q=ai:velasco.diegoThe authors investigate orbifold diagrams, which are alternating strand diagrams on the disk with an orbifold point. These are collections of oriented arcs satisfying certain properties, which are defined as quotients by rotation of alternating strand diagrams on the disk. The latter have been used in the study of the coordinate ring of the Grassmannian: they give rise to clusters of the Grassmannian cluster algebras or to cluster tilting objects of the Grassmannian cluster categories.
The authors study alternating strand diagrams on the disk with an orbifold point. These are quotients by rotation of alternating strand diagrams on the disk, which are called orbifold diagrams. They associate a quiver with potential to each orbifold diagram, in such a way that its Jacobian algebra and the one associated to the covering Postnikov diagram are related by a skew-group algebra construction. Moreover, they realize this Jacobian algebra as the endomorphism algebra of a certain explicit cluster-tilting object.
Reviewer: Mee Seong Im (Annapolis)On support \(\tau\)-tilting graphs of gentle algebrashttps://zbmath.org/1521.160102023-11-13T18:48:18.785376Z"Fu, Changjian"https://zbmath.org/authors/?q=ai:fu.changjian"Geng, Shengfei"https://zbmath.org/authors/?q=ai:geng.shengfei"Liu, Pin"https://zbmath.org/authors/?q=ai:liu.pin"Zhou, Yu"https://zbmath.org/authors/?q=ai:zhou.yuSupport \(\tau\)-tilting module is the central notion of \(\tau\)-tilting theory, which generalizes the classical tilting module. This class of modules is deeply connected with representation theory, such as torsion classes, silting objects, \(t\)-structures. In contrast to tilting modules, the support \(\tau\)-tilting module can always be mutated at an arbitrary indecomposable direct summand to obtain a new support \(\tau\)-tilting module. Therefore, the support \(\tau\)-tilting modules may have a richer combinatorial structure than tilting modules.
Let \(A\) be a finite-dimensional gentle algebra over an algebraically closed field. The authors investigate the combinatorial properties of support \(\tau\)-tilting graph of \(A\). In particular, the authors prove that the support \(\tau\)-tilting graph of \(A\) is connected and has the so-called reachable-in-face property.
Reviewer: Mee Seong Im (Annapolis)Duality for multimoduleshttps://zbmath.org/1521.160252023-11-13T18:48:18.785376Z"Bertozzini, Paolo"https://zbmath.org/authors/?q=ai:bertozzini.paolo"Conti, Roberto"https://zbmath.org/authors/?q=ai:conti.roberto.1"Puttirungroj, Chatchai"https://zbmath.org/authors/?q=ai:puttirungroj.chatchaiMultimodules are a generalisation of bimodules. Much of this paper is concerned with extending the classical theory of bimodules (and the theory of multimodules over \(\mathbb{Z}\), as in Bourbaki) to a broader setting, aiming at future work studying noncommutative contravariant differential calculus. Great care is taken throughout the paper to be precise with the many fine details that underpin the basic notions, but in this review I will take the luxury of adopting far less careful notation.
Definitions.
Throughout this review, all rings are unital and associative (though the authors occasionally work in more general settings).
Let \(Z\) be a commutative ring and \(R\) a \textit{\(Z\)-central} ring, i.e., a ring equipped with a distinguished homomorphism \(\iota_R: Z\to Z(R)\).
A \textit{\(Z\)-central \(R\)-bimodule} is an \(R\)-bimodule \(M = {}_RM_R\) with a distinguished isomorphism \(\iota_M: Z\to Z(M)^0\), where \(Z(M)^0\) is defined as the subring of those \(z\in Z(R)\) satisfying \(zm = mz\) for all \(m\in M\). (It is not explicitly stated in the paper, but presumably in all cases of interest there is a relation between \(\iota_R\) and \(\iota_M\) when both exist.) Morphisms between them, \(\Phi: M\to N\), are usually taken to be \textit{\(Z\)-linear} in the sense that \(\Phi(\iota_M(z)m) = \iota_N(z)\Phi(m)\) for all \(m\in M\). A \textit{\(Z\)-central \(R\)-algebra} is a \(Z\)-central \(R\)-bimodule \(\mathcal{A}\) complete with a distributive multiplication that is compatible with the \(R\)-action on both sides.
Given two families of \(Z\)-central \(R\)-algebras \((\mathcal{A}_\alpha)_{\alpha\in A}\) and \((\mathcal{B}_\beta)_{\beta\in B}\), an \((\mathcal{A}_\alpha), (\mathcal{B}_\beta)\)-multimodule \(M = {}_{(\mathcal{A}_\alpha)} M_{(\mathcal{B}_\beta)}\) is a \(Z\)-central \(R\)-bimodule that is also a \(Z\)-central \(\mathcal{A}_\alpha, \mathcal{B}_\beta\)-bimodule for all pairs \((\alpha,\beta)\), such that every pair of left actions commutes and every pair of right actions commutes.
The first main novelty arises here. A \textit{morphism} of multimodules is a \(Z\)-linear map \(\Phi: M\to N\) satisfying appropriate linearity relations. In particular, if \(M\) is an \((\mathcal{A}_\alpha)_A, (\mathcal{B}_\beta)_B\)-multimodule and \(N\) is a \((\mathcal{C}_\gamma)_C, (\mathcal{D}_\delta)_D\)-multimodule, where \(A, B, C, D\) are indexing sets, a morphism comes with a specified function \(f: A\coprod B\to C\coprod D\), the \textit{covariance signature} of \(\Phi\). This function \(f\) encodes which parts of the morphism are covariant (specified by those indices \(\alpha\in A\) and \(\beta\in B\) which are sent to \(C\) and \(D\) respectively) which are contravariant (specified by those indices \(\alpha\in A\) and \(\beta\in B\) sent to \(D\) and \(C\) respectively. When \(R = Z\) and \(f\) is entirely covariant, the study of multimodules can be reduced to the study of bimodules (Remark 9), but it seems that arbitrary covariance signatures cannot be handled in this way in the more general setting of \(\iota_R: Z\to R\).
Basic constructions.
Continue to assume that everything mentioned is \(Z\)-central, etc.
The authors proceed to show that several natural constructions exist and behave in the expected ways:
\begin{itemize}
\item Sub-multimodules \(N\to M\) and quotient multimodules \(M\to M/N\) (Definition 12, Remark 13).
\item If \(M\) is an \((\mathcal{A}_\alpha), (\mathcal{B}_\beta)\)-multimodule and \(N\) is a \((\mathcal{C}_\gamma), (\mathcal{D}_\delta)\)-multimodule, then \(\mathrm{Hom}_Z(M,N)\) is a \((\mathcal{B}_\beta, \mathcal{C}_\gamma), (\mathcal{A}_\alpha, \mathcal{D}_\delta)\)-multimodule (Remark 15).
\item If \(M\) is an \((\mathcal{A}_\alpha), (\mathcal{B}_\beta)\)-multimodule and \(N\) is a \((\mathcal{C}_\gamma), (\mathcal{D}_\delta)\)-multimodule, then \(M\otimes_Z N\) is a \((\mathcal{A}_\alpha,\mathcal{C}_\gamma)\), \((\mathcal{B}_\beta, \mathcal{D}_\delta)\)-multimodule (Proposition 16). More general tensor products over shared actions are also considered (Remark 17), and are defined using the ``usual'' universal property with respect to appropriately balanced morphisms out of \(M\times N\).
\end{itemize}
Involutions, duals, traces, inner products.
We can now begin to understand analogues of linear-algebraic notions of duality, trace pairings and inner products in this new setting.
\begin{itemize}
\item Let \(M\) be an \((\mathcal{A}_\alpha)_A, (\mathcal{A}_\beta)_B\)-multimodule.
A multimodule morphism \(M\to M\) is an \textit{involution} (\S 4) if the corresponding covariance signature \(f: A\coprod B \to A\coprod B\) is an involution, and if -- once co- and contravariance are taken into account -- it is an involution with respect to all of the individual left and right actions (see Definition 18 for the details). It is then shown that, under appropriate conditions, if \(M\) and \(N\) admit involutions, then \(\mathrm{Hom}_Z(M,N)\) and \(M\otimes_Z N\) inherit involutions from them.
\item Let \(M\) be an \((\mathcal{A}_\alpha)_A, (\mathcal{B}_\beta)_B\)-multimodule. Fix \(I\subseteq A\) and \(J\subseteq B\).
An \textit{\((I,J)\)-dual} of \(M\) consists of a \((\mathcal{B}_\beta)_B, (\mathcal{A}_\alpha)_A\)-multimodule \(N\) and a pairing
\[
\tau: N\times M\to \bigotimes_{\alpha\in I} \mathcal{A}_\alpha \otimes_Z \bigotimes_{\beta\in J} \mathcal{B}_\beta
\]
which is universal with respect to being \((A-I, B-J)\)-balanced and \((I, J)\)-multilinear. \((I,J)\)-duals are shown to always exist, and the authors go on to discuss \(\gamma\)-conjugate duals, double duals, etc throughout \S 5. There is also some mention of the categorical features of dual pairing on multimodules in \S 5 and Appendix A.
\item Let \(M\) be an \((\mathcal{A}_\alpha)_A, (\mathcal{A}_\beta)_B\)-multimodule. Suppose that some of the \(\mathcal{A}_\xi\) are equal to other \(\mathcal{A}_\zeta\); more precisely, let \(\Gamma\in (A\coprod B)\times (A\coprod B)\) be an injective symmetric relation such that \((\xi,\zeta)\in\Gamma \implies \mathcal{A}_\xi = \mathcal{A}_\zeta\). Then a morphism \(T: M\to V\) acts like a trace with respect to \(\Gamma\) if, for all \((\xi,\zeta)\in\Gamma\), the induced actions of \(A_\xi\) and \(A_\zeta\) on \(T(M)\) are equal. For any such \(\Gamma\), it is shown that there is a universal pair \(T = T^\Gamma\) and \(V = M|\Gamma\).
\end{itemize}
Finally, with this machinery in place, the authors are able to discuss \textit{\(\Gamma\)-compatible involutions} (Definition 33), as well as \textit{left} and \textit{right \((I,J)\)-inner products} on involutive multimodules (Definition 36), and define the canonically induced \textit{Riesz maps} into appropriate dual multimodules (Theorem 39). The authors hope to go on to discuss Riesz dualities in future work.
Reviewer: William Woods (Essex)Redundancy in string cone inequalities and multiplicities in potential functions on cluster varietieshttps://zbmath.org/1521.170182023-11-13T18:48:18.785376Z"Koshevoy, Gleb"https://zbmath.org/authors/?q=ai:koshevoy.gleb-a"Schumann, Bea"https://zbmath.org/authors/?q=ai:schumann.beaSummary: We study defining inequalities of string cones via a potential function on a reduced double Bruhat cell. We give a necessary criterion for the potential function to provide a minimal set of inequalities via tropicalization and conjecture an equivalence.Free Rota-Baxter family algebras and (tri)dendriform family algebrashttps://zbmath.org/1521.170342023-11-13T18:48:18.785376Z"Zhang, Yuanyuan"https://zbmath.org/authors/?q=ai:zhang.yuanyuan.1"Gao, Xing"https://zbmath.org/authors/?q=ai:gao.xingSummary: We construct free commutative Rota-Baxter family algebras, and then free noncommutative Rota-Baxter family algebras via the method of Gröbner-Shirshov bases. We introduce the concept of dendriform (resp.~tridendriform) family algebras, and prove that Rota-Baxter family algebras of weight zero (resp.~\( \lambda)\) induce dendriform (resp.~tridendriform) family algebras. We also construct free commutative dendriform (resp.~tridendriform) family algebras.Poisson catenarity in Poisson nilpotent algebrashttps://zbmath.org/1521.170452023-11-13T18:48:18.785376Z"Goodearl, K. R."https://zbmath.org/authors/?q=ai:goodearl.kenneth-r"Launois, S."https://zbmath.org/authors/?q=ai:launois.stephaneSummary: We prove that for the iterated Poisson polynomial rings known as Poisson nilpotent algebras (or Poisson-CGL extensions), the Poisson prime spectrum is catenary, i.e., all saturated chains of inclusions of Poisson prime ideals between any two given Poisson prime ideals have the same length.Yoneda algebras of the triplet vertex operator algebrahttps://zbmath.org/1521.170712023-11-13T18:48:18.785376Z"Caradot, Antoine"https://zbmath.org/authors/?q=ai:caradot.antoine"Jiang, Cuipo"https://zbmath.org/authors/?q=ai:jiang.cuipo"Lin, Zongzhu"https://zbmath.org/authors/?q=ai:lin.zongzhuSummary: Given a vertex operator algebra \(V\), one can construct two associative algebras, the Zhu algebra \(A(V)\) and the \(C_2\)-algebra \(R(V)\). This gives rise to two abelian categories \(A(V)\)-\(\operatorname{Mod}\) and \(R(V)\)-\(\operatorname{Mod}\), in addition to the category of admissible modules of \(V\). In case \(V\) is rational and \(C_2\)-cofinite, the category of admissible \(V\)-modules and the category of all \(A(V)\)-modules are equivalent. However, when \(V\) is not rational, the connection between these two categories is unclear. The goal of this paper is to study the triplet vertex operator algebra \(\mathcal{W}(p)\), as an example to compare these three categories, in terms of abelian categories. For each of these three abelian categories, we will determine the associated Ext quiver, the Morita equivalent basic algebra, i.e., the algebra \(\operatorname{End} (\oplus_{L \in \mathrm{Irr}} P_L )^{op}\), and the Yoneda algebra \(\operatorname{Ext}^{\ast}(\oplus_{L \in \mathrm{Irr}} L, \oplus_{L \in \mathrm{Irr}} L)\). As a consequence, the category of admissible log-modules for the triplet VOA \(\mathcal{W}(p)\) has infinite global dimension, as do the Zhu algebra \(A(\mathcal{W}(p))\), and the associated graded algebra \(\operatorname{gr} A(\mathcal{W}(p))\) which is isomorphic to \(R(\mathcal{W}(p))\). We also describe the Koszul properties of the module categories of \(\mathcal{W}(p), A(\mathcal{W}(p))\) and \(\operatorname{gr} A(\mathcal{W}(p))\).Massey products for algebras over operadshttps://zbmath.org/1521.180192023-11-13T18:48:18.785376Z"Muro, Fernando"https://zbmath.org/authors/?q=ai:muro.fernandoSummary: We define a generalization of Massey products for algebras over a Koszul operad in characteristic zero, extending Massey's and Allday's and Retah's in the associative and Lie cases, respectively. We establish connections with minimal models and with Dimitrova's universal operadic cohomology class. We compute a Gerstenhaber algebra example and a hypercommutative algebra example related to the Chevalley-Eilenberg complex of the Heisenberg Lie algebra.A family of finite \(p\)-groups satisfying Carlson's depth conjecturehttps://zbmath.org/1521.200412023-11-13T18:48:18.785376Z"Garaialde Ocaña, Oihana"https://zbmath.org/authors/?q=ai:garaialde-ocana.oihana"González-Sánchez, Jon"https://zbmath.org/authors/?q=ai:gonzalez-sanchez.jon"Guerrero Sánchez, Lander"https://zbmath.org/authors/?q=ai:guerrero-sanchez.landerSummary: Let \(p>3\) be a prime number and let \(r\) be an integer with \(1<r<p-1\). For each \(r\), let moreover \(G_r\) denote the unique quotient of the maximal class pro-\(p\) group of size \(p^{r+1}\). We show that the mod-\(p\) cohomology ring of \(G_r\) has depth one and that, in turn, it satisfies the equalities in \textit{J. F. Carlson}'s depth conjecture [Math. Z. 218, No. 3, 461--468 (1995; Zbl 0837.20062)]. This is the first family of finite \(p\)-groups for which Carlson's depth conjecture has been verified besides \(p\)-groups of abelian type mod-\(p\) cohomology or extraspecial \(p\)-groups. Moreover, this computation is possible without first describing the structure of the cohomology ring.
{{\copyright} 2022 The Authors. \textit{Mathematische Nachrichten} published by Wiley-VCH GmbH.}Amenable groups of finite cohomological dimension and the zero divisor conjecturehttps://zbmath.org/1521.201132023-11-13T18:48:18.785376Z"Degrijse, Dieter"https://zbmath.org/authors/?q=ai:degrijse.dieterThe paper deals with amenable groups which are not elementary. The paper proves that every amenable group of cohomological dimension two (over the integers) is solvable focusing on Kaplansky's zero divisor conjecture and its generalizations. The paper's main results establish connections between cohomological dimension, finiteness properties, and the solvability of amenable groups under specific conditions. The strategy of the proof is partly based on the work of P. H. Kropholler.
Reviewer: Meral Tosun (İstanbul)Forcing a basis into \(\aleph_1\)-free groupshttps://zbmath.org/1521.201182023-11-13T18:48:18.785376Z"Bossaller, Daniel"https://zbmath.org/authors/?q=ai:bossaller.daniel-p"Herden, Daniel"https://zbmath.org/authors/?q=ai:herden.daniel"Pasi, Alexandra V."https://zbmath.org/authors/?q=ai:pasi.alexandra-vSummary: In this paper, we address the question of when a non-free \(\aleph_1\)-free group \(H\) can be free in a transitive cardinality-preserving model extension. Using the \(\Gamma\)-invariant, denoted \(\Gamma(H)\), we present a necessary and sufficient condition resolving this question for \(\aleph_1\)-free groups of cardinality \(\aleph_1\). Specifically, if \(\Gamma(H) = [\aleph_1]\), then \(H\) will be free in a transitive model extension if and only if \(\aleph_1\) collapses, while for \(\Gamma(H) \neq [\aleph_1]\) there exist cardinality-preserving forcings that will add a basis to \(H\). In particular, for \(\Gamma(H) \neq [\aleph_1]\), we provide a poset \((\mathcal{P}_{\mathrm{pb}}, \leqslant)\) of partial bases for adding a basis to \(H\) without collapsing \(\aleph_1\).Projective modules over the ring of pseudorational numbershttps://zbmath.org/1521.201192023-11-13T18:48:18.785376Z"Timoshenko, Egor A."https://zbmath.org/authors/?q=ai:timoshenko.egor-aleksandrovichSummary: We prove the structure theorems which give a full description of projective modules over the ring of pseudorational numbers. We construct a complete system of invariants for such modules.A note on the Łojasiewicz exponent of non-degenerate isolated hypersurface singularitieshttps://zbmath.org/1521.320332023-11-13T18:48:18.785376Z"Brzostowski, Szymon"https://zbmath.org/authors/?q=ai:brzostowski.szymonThe main result of this note is the following: the Łojasiewicz exponents of two Kouchnirenko non-degenerate holomorphic functions \(f,g : (\mathbb C^n,0) \to (\mathbb C,0)\) having an isolated singularity at zero and the same Newton diagram are equal. The proof uses Tessier's result [\textit{B. Teissier}, Invent. Math. 40, 267--292 (1977; Zbl 0446.32002)] that the Łojasiewicz exponent is constant along certain deformations. The verification that the deformations used in the proof satisfy the required conditions is given in a wider context of formal power series over any algebraically closed field.
For the entire collection see [Zbl 1429.00039].
Reviewer: Armin Rainer (Wien)The Vietoris functor and modal operators on rings of continuous functionshttps://zbmath.org/1521.540042023-11-13T18:48:18.785376Z"Bezhanishvili, G."https://zbmath.org/authors/?q=ai:bezhanishvili.guram"Carai, L."https://zbmath.org/authors/?q=ai:carai.luca"Morandi, P. J."https://zbmath.org/authors/?q=ai:morandi.patrick-jIn this paper, the authors introduce an endofunctor \(\mathcal{H}\) on the category \(\textbf{bal}\) of bounded archimedean \(\textit{l}\)-algebras and show that there is a dual adjunction between the category \(\textbf{Alg}(\mathcal{H})\) of algebras for \(\mathcal{H}\) and the category \(\textbf{Coalg}(\mathcal{V})\) of coalgebras for the Vietoris endofunctor \(\mathcal{V}\) on the category of compact Hausdorff spaces. They also introduce an endofunctor \(\mathcal{H}^u\) on the reflective subcategory of \(\textbf{bal}\) consisting of uniformly complete objects of \(\textbf{bal}\) and show that Gelfand duality lifts to a dual equivalence between \(\textbf{Alg}(\mathcal{H}^u)\) and \(\textbf{Coalg}(\mathcal{V})\). Finally, some known results are generalized.
Reviewer: Sami Lazaar (Sidi Daoued)Algebraic foundations for applied topology and data analysishttps://zbmath.org/1521.550012023-11-13T18:48:18.785376Z"Schenck, Hal"https://zbmath.org/authors/?q=ai:schenck.halThis book is a true joy to read for everyone who wants to learn about the eponymous foundations of applied topology in a data analysis context. Starting from the fundamentals of linear algebra and group theory, the author quickly manages to reach more advanced topics such as \textit{sheaves}, which may not always be part of the typical curriculum. The provided explanations and theorems are always accurate and manage to convey the proper sense of depth that would require a more detailed study of some concepts. While this book cannot replace targeted courses in, say, algebraic topology, it certainly manages to complement them and guide the reader towards additional resources if need be.
The `grand finale' of the book is arguably its discussion of persistent homology and its variants. Again, the author manages to discuss the most relevant concepts in a fashion that is not too terse but also does not burden readers with details that are unnecessary for a first understanding.
One minor issue of this book is that, due to the selection of different topics, the \textit{complexity} of the presentation varies somewhat between chapters. The reader should thus be advised to read on when getting stuck or just sample chapters of interest; there is no monotone increase in difficulty (or at least, I did not experience it).
In summary, this text book is a great companion for graduate students interested in applied topology. Through its focus on the underlying concepts, the book will remain highly-relevant for years to come; an advantage that does not apply to many other publications in data science. I can highly recommend pairing this book with the equally delightful work by \textit{R. W. Ghrist} [Elementary applied topology. [s.l.]: Createspace (2014; Zbl 1427.55001)]. Together, these two books provide a sweeping overview of an exciting nascent field, and I am sure that readers will appreciate them.
Reviewer: Bastian Rieck (Bern)Multiplicativity and nonrealizable equivariant chain complexeshttps://zbmath.org/1521.550032023-11-13T18:48:18.785376Z"Rüping, Henrik"https://zbmath.org/authors/?q=ai:ruping.henrik"Stephan, Marc"https://zbmath.org/authors/?q=ai:stephan.marcAuthors' abstract: Let \(G\) be a finite \(p\)-group and \(\mathbb{F}\) a field of characteristic \(p\). We filter the cochain complex of a free \(G\)-space with coefficients in \(\mathbb{F}\) by powers of the augmentation ideal of \(\mathbb{F}G\). We show that the cup product induces a multiplicative structure on the arising spectral sequence and compute the \(E_1\)-page as a bigraded algebra. As an application, we prove that recent counterexamples of \textit{S. B. Iyengar} and \textit{M. E. Walker} [Acta Math. 221, No. 1, 143--158 (2018; Zbl 1403.13026)] to an algebraic version of Carlsson's conjecture [\textit{G. Carlsson}, Lect. Notes Math. 1217, 79--83 (1986; Zbl 0614.57023)] can not be realized topologically.
Reviewer: Hero Saremi (Sanandaj)Realization of graded monomial ideal rings modulo torsionhttps://zbmath.org/1521.550042023-11-13T18:48:18.785376Z"So, Tseleung"https://zbmath.org/authors/?q=ai:so.tseleung"Stanley, Donald"https://zbmath.org/authors/?q=ai:stanley.donald-a|stanley.donaldA classical problem in algebraic topology asks: which commutative graded \(R\)-algebras \(A\) are isomorphic to \(H^\ast(X_A,R)\) for some space \(X_A\)? The space \(X_A\), if it exists, is called a realization of \(A\). According to \textit{J. Aguadé} [Publ., Secc. Mat., Univ. Autòn. Barc. 26, No. 2, 25--68 (1982; Zbl 0595.55001)] the problem goes back to at least Hopf, and was later explicitly stated by \textit{N. E. Steenrod} [Enseign. Math. (2) 7, 153--178 (1962; Zbl 0104.39604)]. To solve the problem in general is probably too ambitious, but many special cases have been proven. One of \textit{D. Quillen}'s motivations for his seminal work on rational homotopy theory [Ann. Math. (2) 90, 205--295 (1969; Zbl 0191.53702)] was to solve this problem over the field of rationals \(\mathbb{Q}\). \par Let \(A\) be the quotient of a graded polynomial ring \(\mathbb{Z}[Ex_1,\ldots,x_m]\otimes\Lambda[y_1,\ldots,y_n]\) by an ideal generated by monomials with leading coefficients \(1\). The authors construct a space \(X_A\) such that \(A\) is isomorphic to \(H^\ast(X_A)\) modulo torsion elements.
Reviewer: Marek Golasiński (Olsztyn)Knot theory and cluster algebrashttps://zbmath.org/1521.570012023-11-13T18:48:18.785376Z"Bazier-Matte, Véronique"https://zbmath.org/authors/?q=ai:bazier-matte.veronique"Schiffler, Ralf"https://zbmath.org/authors/?q=ai:schiffler.ralfLet \(K\) be a knot or link diagram with \(n\) crossings. The Alexander polynomial \(\Delta_K\) is an important polynomial invariant of the link which is a Laurent polynomial in one variable with integer coefficients, which can be defined in terms of homology.
On the othe hand, cluster algebras form a class of commutative rings with a set of distinguished generators called cluster variables having a remarkable combinatorial structure that can be encoded in a quiver with potential. First introduced in the context of Lie theory, they have ever since appeared in many contexts, from Poisson geometry to Teichmüller theory. The \(F\)-polynomials are central objects associated to a cluster algebra, which are the specialization of the Laurent polynomials of each cluster variable, and which can be computed from modules over the Jacobian algebra \(B\) associated with the quiver with potential.
In this article, the authors construct explicitly a quiver with potential from a link diagram, and establish a connection between knot theory and cluster algebras. The main objective of the paper is to show the direct relation between the Alexander polynomial of the link and the \(F\)-polynomials associated to that quiver with potential.
More specifically, when \(K\) is the reduced diagram of an oriented prime link without curls, the authors construct a quiver \(Q\) with a vertex for each segment of \(K\), and arrows between vertices coming from the crossings between segments in the diagram \(K\). The potential \(W\) on that quiver is then defined as the difference between \(4\)-cycles corresponding to crossing points and cycles around regions. From this quiver with potential \((Q, W)\) one can define the Jacobian algebra \(B\) associated to it.
Then the authors construct a \(B\)-module \(T = \bigoplus_{i \in K_1} T(i)\) where \(K_1\) is the set of segments of \(K\). Each \(T(i)\), with \(i\) a segment of \(K\), comes from an indecomposable representation of the quiver \(Q\), and the authors prove that the lattice of Kauffman states of a link \(K\) relative to a segment \(i\) is isomorphic to the lattice of submodules of the direct summand \(T(i)\) of the link diagram \(K\).
This last property is the key step towards the proof of the main result of the paper which is that for every segment \(i\) of \(K\), the Alexander polynomial of \(K\) is equal to the specialized \(F\)-polynomial of the \(B\)-module \(T(i)\). Here the specialization of the \(F\)-polynomial is defined explicitly from the link diagram \(K\) depending on the nature of the two crossings joined by a segment. The proof use the realization of the Alexander polynomial as a state sum, following Kauffman's approach.
As applications of this result, the authors make progress towards the unimodularity conjecture for modules of Dynkin type \(\mathbb{A}\) by associating a particular class of \(2\)-bridge links whose associated module is of the desired type, and using the known properties of the Alexander polynomial of these links. They also give explicit examples of computations of \(F\)-polynomials for the figure eight knot and the Conway knot, and recover their known Alexander polynomials with that new method.
They also conjecture that the collection of the \(T(i)\) forms a cluster in the cluster algebra whose quiver is isomorphic to the opposite of \(Q\), and that the resulting cluster automorphism is of order two.
Reviewer: Frederic Palesi (Marseille)Ulrich complexityhttps://zbmath.org/1521.680702023-11-13T18:48:18.785376Z"Bläser, Markus"https://zbmath.org/authors/?q=ai:blaser.markus"Eisenbud, David"https://zbmath.org/authors/?q=ai:eisenbud.david"Schreyer, Frank-Olaf"https://zbmath.org/authors/?q=ai:schreyer.frank-olafSummary: In this note we suggest a new measure of the complexity of polynomials, the Ulrich complexity. Valiant's conjecture on the exponential complexity of the permanent would imply exponential behavior of the Ulrich complexity as well, and this may be easier to prove. We compute some families of examples, one of which has provably exponential behavior.Computing circuit polynomials in the algebraic rigidity matroidhttps://zbmath.org/1521.682672023-11-13T18:48:18.785376Z"Malić, Goran"https://zbmath.org/authors/?q=ai:malic.goran"Streinu, Ileana"https://zbmath.org/authors/?q=ai:streinu.ileanaSummary: We present an algorithm for computing \textit{circuit polynomials} in the algebraic rigidity matroid \(\boldsymbol{\mathcal{A}}(\mathrm{CM}_n)\) associated to the Cayley-Menger ideal \(\mathrm{CM}_n\) for \(n\) points in 2D. It relies on combinatorial resultants, a new operation on graphs that captures properties of the Sylvester resultant of two polynomials in this ideal. We show that every rigidity circuit has a construction tree from \(K_4\) graphs based on this operation. Our algorithm performs an algebraic elimination guided by such a construction tree and uses classical resultants, factorization, and ideal membership. To highlight its effectiveness, we implemented the algorithm in Mathematica: it took less than 15 seconds on an example where a Gröbner basis calculation took 5 days and 6 hours. Additional speed-ups are obtained using non-\(K_4\) generators of the Cayley-Menger ideal and simple variations on our main algorithm.Perfect state transfer on bi-Cayley graphs over abelian groupshttps://zbmath.org/1521.810402023-11-13T18:48:18.785376Z"Wang, Shixin"https://zbmath.org/authors/?q=ai:wang.shixin"Feng, Tao"https://zbmath.org/authors/?q=ai:feng.tao.1Summary: The study of perfect state transfer on graphs has attracted a great deal of attention during the past ten years because of its applications to quantum information processing and quantum computation. Perfect state transfer is understood to be a rare phenomenon. This paper establishes necessary and sufficient conditions for a bi-Cayley graph having a perfect state transfer over any given finite abelian group. As corollaries, many known and new results are obtained on Cayley graphs having perfect state transfer over abelian groups, (generalized) dihedral groups, semi-dihedral groups and generalized quaternion groups. Especially, we give an example of a connected non-normal Cayley graph over a dihedral group having perfect state transfer between two distinct vertices, which was thought impossible.Differential elimination for dynamical models via projections with applications to structural identifiabilityhttps://zbmath.org/1521.930342023-11-13T18:48:18.785376Z"Dong, Ruiwen"https://zbmath.org/authors/?q=ai:dong.ruiwen"Goodbrake, Christian"https://zbmath.org/authors/?q=ai:goodbrake.christian"Harrington, Heather A."https://zbmath.org/authors/?q=ai:harrington.heather-a"Pogudin, Gleb"https://zbmath.org/authors/?q=ai:pogudin.gleb-aThe research is motivated by the differential algebra approach development to assess the structural identifiability of a dynamical models. The main contribution is employment of the conventional representation of the state-space form of the dynamical system as input-output relation to design the new formalism to generate the differential-algebraic relations between the input and output variables of a dynamical system model. The algorithm for differential elimination using the projection-based representation is proposed and applied for assessing the structural identifiability. It is implemented in Julia language and available on GitHub and validated on 10 examples including SIR and pharmacokinetics models.
Reviewer: Denis Sidorov (Irkutsk)Linear obstacles in linear systems, and ways to avoid themhttps://zbmath.org/1521.930872023-11-13T18:48:18.785376Z"Baryshnikov, Yuliy"https://zbmath.org/authors/?q=ai:baryshnikov.yu-mSummary: We describe the cohomology ring of the space of trajectories of linear control systems avoiding linear obstacles.Corrigendum to: ``\(l\)-LIPs of codes over finite chain rings''https://zbmath.org/1521.940892023-11-13T18:48:18.785376Z"Liu, Xiusheng"https://zbmath.org/authors/?q=ai:liu.xiusheng"Hu, Peng"https://zbmath.org/authors/?q=ai:hu.pengSummary: In this note, we first give an example to illustrate that the Lemma 2.22 in [ibid. 345, No. 12, Article ID 113087, 12 p. (2022; Zbl 1503.94059)] is not correct. Then we prove that a special case of the Lemma 2.22 is correct. This result ensures that the subsequent conclusions in [loc. cit.] are correct (except for a minor modification to the conditions of Theorem 3.3).