Recent zbMATH articles in MSC 13Chttps://zbmath.org/atom/cc/13C2024-02-28T19:32:02.718555ZWerkzeugOn the matroidal path idealshttps://zbmath.org/1527.050532024-02-28T19:32:02.718555Z"Khashyarmanesh, Kazem"https://zbmath.org/authors/?q=ai:khashyarmanesh.kazem"Nasernejad, Mehrdad"https://zbmath.org/authors/?q=ai:nasernejad.mehrdad"Qureshi, Ayesha Asloob"https://zbmath.org/authors/?q=ai:qureshi.ayesha-asloobSummary: In this paper, we prove that the set of all paths of a fixed length in a complete multipartite graph is the bases of a matroid. Moreover, we discuss the Cohen-Macaulayness and depth of powers of \(t\)-path ideals of a complete multipartite graph.An extension of the Beauville-Laszlo descent theoremhttps://zbmath.org/1527.130102024-02-28T19:32:02.718555Z"Banerjee, Abhishek"https://zbmath.org/authors/?q=ai:banerjee.abhishek.1|banerjee.abhishekSummary: For a \(k\)-algebra \(A\), the category \(\mathcal{C}_A\) of \(A\)-modules taking values in a \(k\)-linear abelian category \(\mathcal{C}\) was introduced by Popescu. The algebraic properties of \(\mathcal{C}_A\), including Grothendieck's theory of flat descent, were developed by Artin and Zhang. In this note, we show that Beauville-Laszlo descent also holds in the category \(\mathcal{C}_A\).Commutative rings whose proper ideals are pure-semisimplehttps://zbmath.org/1527.130112024-02-28T19:32:02.718555Z"Baghdari, S."https://zbmath.org/authors/?q=ai:baghdari.samaneh"Behboodi, M."https://zbmath.org/authors/?q=ai:behboodi.mahmood"Moradzadeh-Dehkordi, A."https://zbmath.org/authors/?q=ai:moradzadeh-dehkordi.aliRecall that for a left \(R\)-module \(M\), \(\sigma(M)\) denotes the smallest Grothendieck subcategory of \(R\)-Mod containing \(M\), i.e., \(\sigma[M]\) consists of all left \(R\)-module \(N\) which is isomorphic to a factor module of \(M^{(I)}\) for some index set \(I\). Recall that an \(R\)-module \(M\) is called pure-semisimple if every module in the category \(\sigma[M]\) is a direct sum of finitely generated (and indecomposable) modules. A theorem from commutative algebra due to Köthe, Cohen-Kaplansky and Griffith states that a commutative ring \(R\) is pure-semisimple if and only if every \(R\)-module is a direct sum of cyclic modules, and the latter holds true if and only if, \(R\) is an Artinian principal ideal ring. Consequently, every (or finitely generated, cyclic) ideal of \(R\) is pure-semisimple if and only if \(R\) is an Artinian principal ideal ring. Therefore, a natural question is the following: Whether the same is true if one only assumes that every proper ideal of \(R\) is pure-semisimple? The goal of this paper is to answer this question. The structure of such rings is completely described as either an Artinian principal ideal ring or a local rings \(R\) with a maximal ideal \(M = Rx \oplus T\) in which \(Rx\) is Artinian uniserial while \(T\) is semisimple. Also, they give several characterizations for commutative rings whose proper principal (finitely generated) ideals are pure-semisimple.
Reviewer: Tongsuo Wu (Shanghai)Unimodular rows over monoid extensions of overrings of polynomial ringshttps://zbmath.org/1527.130122024-02-28T19:32:02.718555Z"Mathew, Maria A."https://zbmath.org/authors/?q=ai:mathew.maria-a"Keshari, Manoj K."https://zbmath.org/authors/?q=ai:keshari.manoj-kumarSummary: Let \(R\) be a commutative Noetherian ring of dimension \(d\) and \(M\) a commutative cancellative torsion-free seminormal monoid. Then: (1) Let \(A\) be a ring of type \(R[d, m, n]\) and \(P\) be a projective \(A[M]\)-module of rank \(r \geq\max\{ 2,d + 1\}\). Then the action of \(E(A[M] \oplus P)\) on \((A[M] \oplus P)\) is transitive and (2) Assume \((R, m, K)\) is a regular local ring containing a field \(k\) such that either \(\mathrm{char}\,k = 0\) or \(\mathrm{char}\,k = p\) and \(\mathrm{tr}\)-\(\mathrm{deg}\, K / \mathbb{F}_p \geq 1\). Let \(A\) be a ring of type \(R[d, m, n]^* \) and \(f \in R\) be a regular parameter. Then all finitely generated projective modules over \(A[M], A[M]_f\) and \(A[M]\otimes_R R(T)\) are free. When \(M\) is free both results are due to \textit{M. K. Keshari} and \textit{S. A. Lokhande} [J. Pure Appl. Algebra 218, No. 6, 1003--1011 (2014; Zbl 1283.13008)].Baer submodules of modules over commutative ringshttps://zbmath.org/1527.130132024-02-28T19:32:02.718555Z"Anebri, Adam"https://zbmath.org/authors/?q=ai:anebri.adam"Kim, Hwankoo"https://zbmath.org/authors/?q=ai:kim.hwankoo"Mahdou, Najib"https://zbmath.org/authors/?q=ai:mahdou.najibLet \(R\) be a commutative ring with non-zero unit and \(M\) be an \(R\)-module. A submodule \(N\) of \(M\) is called a \(d\)-submodule if \(\mathrm{Ann}_R(m)\subseteq \mathrm{Ann}_R(m')\) for some \(m\in N\) and \(m'\in M\) implies that \(m'\in N\) and is called a \(fd\)-submodule if \(\mathrm{Ann}_R(F)\subseteq \mathrm{Ann}_R(m')\) for a finite subset \(F\subseteq N\) and \(m'\in M\) implies that \(m'\in N\). The paper is dedicated to the study of these classes of submodules: characterizations, properties and especially many nice examples. These classes of submodules are also used to characterize some classes of modules: von Neumann modules, reduced modules etc.
Reviewer: Cristodor-Paul Ionescu (Bucureşti)Polymatroidal ideals and linear resolutionhttps://zbmath.org/1527.130142024-02-28T19:32:02.718555Z"Bandari, Somayeh"https://zbmath.org/authors/?q=ai:bandari.somayehLet \(R=K[x_1, \ldots, x_n]\) be a polynomial ring over a field \(K\) and \(\mathfrak{m} = (x_1, \ldots, x_n)\). In addition, for a monomial ideal \(I\) of \(R\), the \textit{monomial localization} of \(I\) with respect to a monomial prime ideal \(\mathfrak{p}\) is the monomial ideal \(I( \mathfrak{p})\) which can be obtained from \(I\) by substituting the variables \(x_i \notin \mathfrak{p}\) by \(1\). Let \(I \subset R\) be a monomial ideal generated in a single degree. Then the ideal \(I\) is \textit{polymatroidal} if for any two elements \(u,v\in \mathcal{G}(I)\) such that \(\deg_{x_i}(u) > \deg_{x_i}(v)\) there exists an index \(j\) with \(\deg_{x_j}(u) < \deg_{x_j}(v)\) such that \(x_j(u/x_i)\in I\). Now, suppose that \(I\) is a polymatroidal ideal of \(R\).
According to [\textit{J. Herzog} et al., J. Algebr. Comb. 37, No. 2, 289--312 (2013; Zbl 1258.13014)], it is well-known that a monomial localization of a polymatroidal ideal is again polymatroidal. In this direction, \textit{S. Bandari} and \textit{J. Herzog} [Eur. J. Comb. 34, No. 4, 752--763 (2013; Zbl 1273.13035)] have made a conjecture which says that a monomial ideal \(I\) is polymatroidal if and only if \(I(\mathfrak{p})\) has a linear resolution for all monomial prime ideals \(\mathfrak{p}\).
In particular, they proved that this conjecture is true in the following cases: \begin{itemize} \item \(I\) is generated in degree \(2\) ; \item \(I\) contains at least \(n-1\) pure powers; \item \(I\) is monomial ideal in at most three variables; \item \(I\) has no embedded prime ideal and either \(|\mathrm{Ass}(R/I)|\leq 3\) or \(\mathrm{height} (I)= n-1\). \end{itemize}
In this paper, the author focuses on the following statement: ( \(\dagger\) ) Let \(I\) be a monomial ideal with linear resolution such that \(I\mathfrak{m}\) is polymatroidal. Then \(I\) is polymatroidal. Particularly, it should be noted that if Bandari-Herzog's conjecture is true, then the statement \((\dagger)\) holds. This paper has two main results. First, the author established that the statement \((\dagger)\) is true in the following cases: \begin{itemize} \item \(I \mathfrak{m}\) is polymatroidal with strong exchange property ; \item \(I\) is a monomial ideal in at most \(4\) variables. \end{itemize}
Secondly, the author proves that the first homological shift ideal of a polymatroidal ideal is again polymatroidal in the following proposition:
{Proposition 2.13.} Let \(I \subset R\) be a polymatroidal ideal. Then \(\mathrm{HS}_1(I)=(I\mathfrak{m})^{\leq \textbf{a}},\) where \(\textbf{a}=(a_1, \ldots, a_n)\) and \(a_i=\max\{\deg_{x_i}(u) \mid u\in \mathcal{G}(I)\}\). In particular, \(\mathrm{HS}_1(I)\) is polymatroidal.
Reviewer: Mehrdad Nasernejad (Lens)On virtually Cohen-Macaulay simplicial complexeshttps://zbmath.org/1527.130192024-02-28T19:32:02.718555Z"Kenshur, Nathan"https://zbmath.org/authors/?q=ai:kenshur.nathan"Lin, Feiyang"https://zbmath.org/authors/?q=ai:lin.feiyang"McNally, Sean"https://zbmath.org/authors/?q=ai:mcnally.sean"Xu, Zixuan"https://zbmath.org/authors/?q=ai:xu.zixuan"Yu, Teresa"https://zbmath.org/authors/?q=ai:yu.teresaSummary: We examine virtual resolutions of Stanley-Reisner ideals for a product of projective spaces by providing sufficient conditions for a simplicial complex to be virtually Cohen-Macaulay (to have a virtual resolution with length equal to its codimension). In particular, we show that all balanced simplicial complexes are virtually Cohen-Macaulay.Cohen-Macaulay binomial edge ideals of small graphshttps://zbmath.org/1527.130262024-02-28T19:32:02.718555Z"Bolognini, Davide"https://zbmath.org/authors/?q=ai:bolognini.davide"Macchia, Antonio"https://zbmath.org/authors/?q=ai:macchia.antonio"Rinaldo, Giancarlo"https://zbmath.org/authors/?q=ai:rinaldo.giancarlo"Strazzanti, Francesco"https://zbmath.org/authors/?q=ai:strazzanti.francescoThe main aim of this work is to explore a graph-theoretical characterization of Cohen-Macaulay binomial edge ideals, a class of binomial ideals defined starting from simple graphs. In fact, for a given simple finite graph \(G=(V(G), E(G))\) with \(m=|V(G)|\), we define the \textit{binomial edge ideal} of \(G\) as \(J_G:=(x_iy_j - x_jy_i \mid \{i,j\} \in E(G)) \subset R=K[x_1, \ldots, x_m, y_1, \ldots, y_m].\) This definition has been presented in [\textit{J. Herzog} et al., Adv. Appl. Math. 45, No. 3, 317--333 (2010; Zbl 1196.13018)] and [\textit{M. Ohtani}, Commun. Algebra 39, No. 3, 905--917 (2011; Zbl 1225.13028)].
To understand the main results of this paper, we require to review some definitions. For a subset \(S \subset V(G)\), we denote by \(G\setminus S\) the subgraph induced by \(G\) on the vertices \(V(G)\setminus S\) and by \(c_G(S)\) the number of connected components of \(G \setminus S\). A subset \(S \subset V(G)\) is called a \textit{cut-point set}, or simply \textit{cut set}, of \(G\) if either \(S = \emptyset\) or \(c_G(S) > c_G(S \setminus \{i\})\) for every \(i\in S\). In prticular, we denote by \(\mathcal{C}(G)\) the collection of cut sets of \(G\). In addition, a graph \(G\) is called \textit{accessible} if \(J_G\) is unmixed and \(\mathcal{C}(G)\) is an \textit{accessible set system}, i.e., for every non-empty \(S \in \mathcal{C}(G)\) there exists \(s\in S\) such that \(S\setminus \{s\}\in \mathcal{C}(G)\). Furthermore, a \textit{block}, or biconnected graph, is a graph that does not have cut vertices and \textit{adding a whisker} to a vertex \(v\) of a graph means attaching a pendant edge \(\{v, f\}\), where \(f\) is a new vertex. In particular, \(f\) is a \textit{free vertex}, which means that it belongs to a unique maximal clique. Historically, the study of Cohen-Macaulay binomial edge ideals concentrated on the search of classes of graphs and of constructions preserving this property. Specially, in [\textit{D. Bolognini} et al., J. Algebr. Comb. 55, No. 4, 1139--1170 (2022; Zbl 1496.13036)], the authors presented the following conjecture in terms of the structure of the cut sets of the associated graph. \textbf{Conjecture.} Let \(G\) be a graph. Then \(R/J_G\) is Cohen-Macaulay if and only if \(G\) is accessible. This conjecture holds for chordal, bipartite, and traceable graphs. On the other hand, for every graph \(G\), the following implications hold:
\begin{align*}
J_G \text{ strongly unmixed } &\Rightarrow R/J_G \text{ Cohen-Macaulay }\\
& \Rightarrow R/J_G \text{ satisfies Serre's condition } (S_2) \Rightarrow G \text{ accessible}. \tag{\(\dagger\)}
\end{align*}
The authors of this paper tried to provide both theoretical and computational evidence for conjecture above by proving that it holds for new classes of graphs.
We finally summarize the main results of this paper in the following theorems.
\textbf{Theorem. } Let \(G\) be one of the following:
\begin{itemize}
\item[1.] a block with \(n\) vertices and \(k \geq n-2\) whiskers;
\item[2.] a block with whiskers, where the vertices of the block are at most \(11\);
\item[3.] a graph with up to \(15\) vertices.
\end{itemize}
Then the conditions in (\(\dagger\)) are all equivalent. In particular, conjecture above holds for all the graphs above and in these cases the Cohen-Macaulayness of \(R/J_G\) does not depend on the field.
\textbf{Theorem. } Let \(B\) be a block with \(n\) vertices and \(\overline{B}\) be the graph obtained by adding \(k > 0\) whiskers to \(B\). Assume that \(\overline{B}\) is accessible and satisfies one of the following properties:
\begin{itemize}
\item[1.] \(B\) contains a free vertex;
\item[2.] \(B\) has a vertex of degree at most two;
\item[3.] \(\overline{B}\) has \(k\leq 3\) whiskers;
\item[4.] there is a cut vertex \(v\) of \(\overline{B}\) such that \(|N_B(v)| \geq \lfloor\frac{n+r}{2}\rfloor-1\), where \(r\) is the number of cut vertices adjacent to \(v\) plus one;
\item[5.] \(\overline{B}\) has \(k = 4\) whiskers and the induced subgraph on the cut vertices of \(\overline{B}\) is a block;
\item[6.] \(\overline{B}\) has \(k \geq n - 2\) whiskers.
\end{itemize}
Then there exists a cut vertex of \(\overline{B}\) for which \(J_{\overline{B}\setminus \{v\}}\) is unmixed.
Reviewer: Mehrdad Nasernejad (Lens)Upper topology and its relation with the projective moduleshttps://zbmath.org/1527.140032024-02-28T19:32:02.718555Z"Tarizadeh, Abolfazl"https://zbmath.org/authors/?q=ai:tarizadeh.abolfazlThis article presents some new results on projective modules and on the upper topology of an ordinal number. Then it is shown that the rank map of a locally finite type projective module is continuous with respect to the upper topology.
Reviewer: Chen Sheng (Harbin)