Recent zbMATH articles in MSC 13https://zbmath.org/atom/cc/132021-11-25T18:46:10.358925ZWerkzeugLooking for a new version of Gordon's identitieshttps://zbmath.org/1472.050132021-11-25T18:46:10.358925Z"Afsharijoo, Pooneh"https://zbmath.org/authors/?q=ai:afsharijoo.poonehSummary: We give a commutative algebra viewpoint on Andrews recursive formula for the partitions appearing in Gordon's identities, which are a generalization of Rogers-Ramanujan identities. Using this approach and differential ideals, we conjecture a family of partition identities which extend Gordon's identities. This family is indexed by \(r\geq 2\). We prove the conjecture for \(r=2\) and \(r=3\).Möbius and coboundary polynomials for matroidshttps://zbmath.org/1472.051622021-11-25T18:46:10.358925Z"Johnsen, Trygve"https://zbmath.org/authors/?q=ai:johnsen.trygve"Verdure, Hugues"https://zbmath.org/authors/?q=ai:verdure.huguesSummary: We study how some coefficients of two-variable coboundary polynomials can be derived from Betti numbers of Stanley-Reisner rings. We also explain how the connection with these Stanley-Reisner rings forces the coefficients of the two-variable coboundary polynomials and Möbius polynomials to satisfy certain universal equations.Specialization method in Krull dimension two and Euler system theory over normal deformation ringshttps://zbmath.org/1472.111352021-11-25T18:46:10.358925Z"Ochiai, Tadashi"https://zbmath.org/authors/?q=ai:ochiai.tadashi"Shimomoto, Kazuma"https://zbmath.org/authors/?q=ai:shimomoto.kazumaSummary: The aim of this article is to establish the specialization method on characteristic ideals for finitely generated torsion modules over a complete local normal domain \(R\) that is module-finite over \(\mathcal{O}[[x_1,\ldots,x_d]]\), where \(\mathcal{O}\) is the ring of integers of a finite extension of the field of \(p\)-adic integers \(\mathbb{Q}_p\). The specialization method is a technique that recovers the information on the characteristic ideal \({\text{char}}_R (M)\) from \(\operatorname{char}_{R/I}(M/IM)\), where \(I\) varies in a certain family of nonzero principal ideals of \(R\). As applications, we prove Euler system bound over Cohen-Macaulay normal domains by combining the main results in [the authors, Nagoya Math. J. 218, 125--173 (2015; Zbl 1325.13023)] and then we prove one of divisibilities of the Iwasawa main conjecture for two-variable Hida deformations generalizing the main theorem obtained in
[the first author, Compos. Math. 142, No. 5, 1157--1200 (2006; Zbl 1112.11051)].New bounds and an efficient algorithm for sparse difference resultantshttps://zbmath.org/1472.120052021-11-25T18:46:10.358925Z"Yuan, Chun-Ming"https://zbmath.org/authors/?q=ai:yuan.chunming"Zhang, Zhi-Yong"https://zbmath.org/authors/?q=ai:zhang.zhiyongSummary: The sparse difference resultant introduced in [\textit{Wei Li} et al., J. Symb. Comput. 68, Part 1, 169--203 (2015; Zbl 1328.65266)] is a basic concept in difference elimination theory. In this paper, we show that the sparse difference resultant of a generic Laurent transformally essential system can be computed via the sparse resultant of a simple algebraic system arising from the difference system. Moreover, new order bounds of sparse difference resultant are found. Then we propose an efficient algorithm to compute sparse difference resultant which is the quotient of two determinants whose elements are the coefficients of the polynomials in the algebraic system. The complexity of the algorithm is analyzed and experimental results show the efficiency of the algorithm.On graded 1-absorbing prime idealshttps://zbmath.org/1472.130012021-11-25T18:46:10.358925Z"Abu-Dawwas, Rashid"https://zbmath.org/authors/?q=ai:abu-dawwas.rashid"Yıldız, Eda"https://zbmath.org/authors/?q=ai:yildiz.eda"Tekir, Ünsal"https://zbmath.org/authors/?q=ai:tekir.unsal"Koç, Suat"https://zbmath.org/authors/?q=ai:koc.suatLet \(R\) be a commutative ring graded by a group. A graded 1-absorbing prime ideal of \(R\) is defined by the authors as being a proper graded ideal \(P\) such that for any non-invertible homogeneous elements \(x,y,z\in P\), either \(xy\in P\) or \(z\in P\). Several basic results and equivalent characterizations of such ideals are presented. It is showed that the graded prime ideal of a graded 1-absorbing prime ideal is graded prime. It is proved that if every nonzero proper graded ideal of \(R\) is graded 1-absorbing prime, then every homogeneous element \(x\) of \(R\) is \(\pi\)-regular, i.e., there exist a positive integer \(n\) and \(y\in R\) such that \(x^{2n}y=x^n\). A graded version of the prime avoidance theorem is presented.Sets of arithmetical invariants in transfer Krull monoidshttps://zbmath.org/1472.130022021-11-25T18:46:10.358925Z"Geroldinger, Alfred"https://zbmath.org/authors/?q=ai:geroldinger.alfred"Zhong, Qinghai"https://zbmath.org/authors/?q=ai:zhong.qinghaiA commutative monoid is Mori if it satisfies the ascending chain condition on divisorial ideals and a commutative Krull monoid is a completely integrally closed Mori monoid. A transfer Krull monoid is a monoid that allows a weak transfer homomorphism to a commutative Krull monoid, equivalently, to a monoid of zero-sum sequences. Clearly, commutative Krull monoids are transfer Krull monoids, but transfer Krull monoids need not be commutative nor completely integrally closed nor Mori.
In this paper under review, the authors studid arithmetical invariants of transfer Krull monoids. Among other things, they showed that (i) every transfer Krull monoid \(H\) is fully elastic (i.e., every rational number between \(1\) and the elasticity of \(H\) can be realized as the elasticity of an element in \(H\)) and (ii) if a commutative Krull monoid has sufficiently many prime divisors in all classes, then all sets of invariants under consideration (e.g., catenary degrees, minimal relations, distances, and tame degrees) are intervals. All these results were proved by a series of technical lemmas.A note on \(w\)-split moduleshttps://zbmath.org/1472.130032021-11-25T18:46:10.358925Z"Almahdi, Fuad Ali Ahmed"https://zbmath.org/authors/?q=ai:almahdi.fuad-ali-ahmed"Assaad, Refat Abdelmawla Khaled"https://zbmath.org/authors/?q=ai:assaad.refat-abdelmawla-khaledLet \(R\) be a commutative ring with identity. Then an ideal \(J\) of \(R\) is said to be a \textit{Glaz-Vasconcelos ideal} (a GV-ideal for short) if J is finitely generated and the natural homomorphism \(\varphi:R\rightarrow \text{Hom}_R(J,R)\) is an isomorphism. An \(R\)-module \(M\) is called GV-\textit{torsion} if for each \(x\in M\), there exists a GV-ideal \(J\) with \(Jx=0\).
The notion of \(w\)-split modules was first introduced in [\textit{F. Wang} and \textit{L. Qiao}, Commun. Algebra 48, No. 8, 3415--3428 (2020; Zbl 1446.13009)]. An \(R\)-module \(M\) is \(w\)-\textit{split} if and only if \(\text{Ext}^1_R(M,N)\) is GV-torsion for all \(R\)-modules \(N\). In this paper, the authors characterize some classical commutative rings in terms of \(w\)-split modules. For example, they show that \(R\) is semisimple if and only if every \(R\)-module is \(w\)-split, and that \(R\) is Von Neumann regular if and only if every finitely presented \(R\)-module is \(w\)-split. Moreover, they also introduce the \(w\)-split dimension of \(R\)-modules. Finally, the relation between projective dimension and \(w\)-split dimension are given.Infinite prime avoidancehttps://zbmath.org/1472.130042021-11-25T18:46:10.358925Z"Chen, Justin"https://zbmath.org/authors/?q=ai:chen.justinLet \(R\) be a ring, and let \(\Lambda\) be a subset of the set of all prime ideals of \(R\). The set \(\Lambda\) satisfies prime avoidance if whenever \(I\subseteq \bigcup_{\mathfrak{p}\in \Lambda}\mathfrak{p}\) for some ideal \(I\) of \(R\), then \(I\subseteq \mathfrak{p}\) for some \(\mathfrak{p}\in \Lambda\). The author investigates prime avoidance for arbitrary set of prime ideals in a commutative ring, and establishes conditions under which prime avoidance is satisfied. Several examples and counterexamples are constructed to illustrate the prime avoidance phenomena.On the integral domains characterized by a Bézout property on intersections of principal idealshttps://zbmath.org/1472.130052021-11-25T18:46:10.358925Z"Guerrieri, Lorenzo"https://zbmath.org/authors/?q=ai:guerrieri.lorenzo"Loper, K. Alan"https://zbmath.org/authors/?q=ai:loper.k-alanThe authors study two classes of integral domains: Bézout intersection domains and strong Bézout intersection domains . An integral domain \(D\) is called:
\(\bullet\) Bézout intersection domain (BID) if for any finite collection of pairwise incomparable elements \(a_{1}, a_{2}, \ldots, a_{n}\in D\), the ideal \[I = \left(a_{1}\right)\cap \left(a_{2}\right)\cap \ldots \cap \left(a_{n}\right)\] is either principal or is not finitely generated.
\(\bullet\) Strong Bézout intersection domain (SBID) if for any finite collection of pairwise incomparable elements \(a_{1}, a_{2}, \ldots, a_{n}\in D\) with \(n\geq 2\), the ideal \[I = \left(a_{1}\right)\cap \left(a_{2}\right)\cap \ldots \cap \left(a_{n}\right)\] is not finitely generated.
In Section 2, several proprieties of BID and SBID are established. In Theorem 2.5, it is shown that the classical \(k+\mathfrak{m}\) construction produces a wide class of examples of SBID. In Theorem 2.7, the notions of BID and SBID are studied in pullback diagrams of type square. A construction of a Krull domain which is also a BID is given in Theorem 2.14.
In Section 3, an operation denoted \(\xi\) is defined on the class of ideals. The operation \(\xi\) is a variant of the classical \(w\) operation , which is itself a variant of the classical \(t\) classical operation. In Theorem 3.8, the operations \(\xi\), \(w\), \(t\) are used to gives equivalent conditions for a domain to be an SBID.
In Section 4, several constructions of strong Bézout intersection domains are given. In Example 4.9, two examples of completely integrally closed SBID are described. Other examples of SBID domains are given in Proposition 4.10 and Theorem 4.12.The ideal theory of intersections of prime divisors dominating a normal Noetherian local domain of dimension twohttps://zbmath.org/1472.130062021-11-25T18:46:10.358925Z"Heinzer, William"https://zbmath.org/authors/?q=ai:heinzer.william-j"Olberding, Bruce"https://zbmath.org/authors/?q=ai:olberding.bruce-mLet \(R\) be a normal Noetherian local domain of Krull dimension two and let \(F\) denote its quotient field. In the present paper, the authors examine intersections of rank one discrete valuation rings that birationally dominate \(R\). More specifically, recall that a valuation overring \(V\) of \(R\) whose maximal ideal contains the maximal ideal of \(R\) and whose residue field has transcendence degree 1 over the residue field of \(R\) is called a prime divisor that dominates \(R\). In other words, the prime divisors that dominate \(R\) are precisely the overrings of \(R\) that arise as the localization of the integral closure of a finitely generated \(R\)-subalgebra of \(F\) at a height one prime ideal that contains the maximal ideal of \(R\).
This article is devoted to the study of the intersection of prime divisors that dominate \(R\). It is known that the intersection of finitely many such rings is a PID, while the intersection of all prime divisors that dominate \(R\) is simply the domain \(R\) itself and so, in general, is quite far from being a PID. In particular, the authors focus on the intersection of prime divisors that dominate \(R\) and are ``within some fixed number of steps away from'' \(R\), where the steps here involve the number of normalized local quadratic transforms needed to reach the prime divisor.
One of the main results shows that if a collection of such prime divisors is taken below a certain ``level'', then the intersection is an almost Dedekind domain (i.e., an integral domain such that each localization at a maximal ideal is a DVR) having the property that every nonzero ideal can be represented uniquely as an irredundant intersection of powers of maximal ideals. Note that, in general, for arbitrary almost Dedekind domains, every ideal is an intersection of primary ideals but, because this intersection is typically infinite, may not be able to be refined to an irredundant intersection and so such a decomposition turns out to be unhandy to be useful.Of limit key polynomialshttps://zbmath.org/1472.130072021-11-25T18:46:10.358925Z"Alberich-Carramiñana, Maria"https://zbmath.org/authors/?q=ai:alberich-carraminana.maria"Boix, Alberto F. F."https://zbmath.org/authors/?q=ai:boix.alberto-f"Fernández, Julio"https://zbmath.org/authors/?q=ai:fernandez.julio"Guàrdia, Jordi"https://zbmath.org/authors/?q=ai:guardia.jordi"Nart, Enric"https://zbmath.org/authors/?q=ai:nart.enric"Roé, Joaquim"https://zbmath.org/authors/?q=ai:roe.joaquimLet \(K\) be a field and \(v\) a valuation on the polynomial ring \(K[x]\), with value group \(\Gamma_v\). For each \(\gamma\in \Gamma_v\), we have the following abelian groups \(\mathcal{P}_{\gamma}^+=\{ g\in K[x]; \mu(g)>\gamma\}\subset\mathcal{P}_{\gamma}=\{ g\in K[x]; \mu(g)\geq\gamma\}\). The graded algebra \(gr_v(K[x])=\oplus_{\gamma\in\Gamma_v}\mathcal{P}_{\gamma}/ \mathcal{P}_{\gamma}^+\) is an integral domain. A MacLane-Vaquie (MLV) key polynomial for \(v\) is a monic polynomial \(\phi\in K[X]\) whose initial term generates a prime ideal in \(gr_v(K[x])\), which cannot be generated by the initial term of a polynomial of smaller degree. The abstract key polynomials for \(v\) are defined in a technical way. In the paper under review, the authors try to find relations between the MLV key polynomials for valuations \(\mu\leq v\) and the abstract key polynomials for \(v\).Resurgence and Castelnuovo-Mumford regularity of certain monomial curves in \(\mathbb{A}^3\)https://zbmath.org/1472.130082021-11-25T18:46:10.358925Z"D'Cruz, Clare"https://zbmath.org/authors/?q=ai:dcruz.clareAuthor's abstract: Let \(p\) be the defining ideal of the monomial curve \(\mathcal{C}(2q +1, 2q +1+m, 2q +1+2m)\) in the affine space \(\mathbb{A}^3_k\) parameterised by \((x^{2q+1}, x^{2q+1+m}, x^{2q+1+2m})\), where \(gcd(2q + 1, m) =1\). In this paper we compute the resurgence of \(p\), the Waldschmidt constant of \(p\) and the Castelnuovo-Mumford regularity of the symbolic powers of \(p\).Results on the Hilbert coefficients and reduction numbershttps://zbmath.org/1472.130092021-11-25T18:46:10.358925Z"Mafi, Amir"https://zbmath.org/authors/?q=ai:mafi.amir"Naderi, Dler"https://zbmath.org/authors/?q=ai:naderi.dlerSummary: Let \((R,\mathfrak{m})\) be a \(d\)-dimensional Cohen-Macaulay local ring, \(I\) an \(\mathfrak{m}\)-primary ideal and \(J\) a minimal reduction of \(I\). In this paper we study the independence of reduction ideals and the behavior of the higher Hilbert coefficients. In addition, we also give some examples.Generalized \(F\)-signatures of Hibi ringshttps://zbmath.org/1472.130102021-11-25T18:46:10.358925Z"Higashitani, Akihiro"https://zbmath.org/authors/?q=ai:higashitani.akihiro"Nakajima, Yusuke"https://zbmath.org/authors/?q=ai:nakajima.yusukeLet \(R\) be a \(d\)-dimensional Noetherian ring of characteristic \(p>0\). \(R\) is said to have FFRT (finite \(F\)-representation type) if there is a finite set of isomorphism classes of finitely generated indecomposable modules \(\{M_0, \ldots, M_n\}\) such that for any \(e \in \mathbb{N}\) there are \(c_{i,e} \geq 0\), such that \[R^{1/p^e} \cong M_0^{\oplus c_{0,e}}\oplus M_1^{\oplus c_{1,e}}\oplus \cdots \oplus M_n^{\oplus c_{n,e}}.\] The generalized \(F\)-signature of \(M_i\) with respect to \(R\) is \(s(M_i,R):=\underset{e \rightarrow \infty}\lim\displaystyle\frac{c_{i,e}}{p^{ed}}\).
A Hibi ring is a special type of toric ring defined via a poset. For toric rings \(R\) of characteristic \(p\), it is known that \(R\) has FFRT and the indecomposable modules of \(R\) are the conical divisors of \(R\). The goal of this nice paper is to determine the generalized \(F\)-signatures for the conical divisors of a Hibi ring.
The main theorem determines the generalized \(F\)-signature for a conical divisor of a Segre product of polynomial rings of dimension \(d\), which is a Hibi ring, in terms of the number of elements of the symmetric group on a set of \(d\) elements which certain descent properties. The authors claim that the methods used to prove this result can also be used to determine the generalized \(F\)-signature for a conical divisor for other Hibi rings; their running example of a Hibi ring which is not a Segre product provides an illustration of this claim.Fundamental results on \(s\)-closureshttps://zbmath.org/1472.130112021-11-25T18:46:10.358925Z"Taylor, William D."https://zbmath.org/authors/?q=ai:taylor.william-dLet \(R\) be a commutative noetherian ring of prime characteristic \(p >0\). For an ideal \(I\) of \(R\), let \(I^{[p^e]}=(f^{p^e}\mid f \in I)\) be the \({p^e}\)-th Frobenius power of \(I\). Introduced by the author [J. Algebra 509, 212--239 (2018; Zbl 1406.13022)], the weak \(s\)-closure \(I^{\{s \}}\) (\(s \in \mathbb{R}\)) of the ideal \(I\) is the set of all elements \(x \in R\) such that there exists \(c \in R\), not in any minimal prime ideal of \(R\), such that \(cx^{p^e} \in I^{\lceil sp^e \rceil}+I^{[p^e]}\) for all \(e \gg 0\). The \(s\)-closure \(I^{\text{cl}_s}\) of \(I\) is the ideal obtained through the iteration of the weak \(s\)-closure until the chain of ideals stabilizes. It should be noted that for \(s=0\) one recovers the well-known tight closure \(I^{*}\) of the ideal \(I\).
This paper further develops the theory of \(s\)-closures. As stated by the author, ``the three main goals of this paper are to understand the structure of the \(s\)-closure in the graded case, identify situations in which \(I^{\{s \}}=I^{\text{cl}_s}\), and to compare the \(s\)-closures for different values of \(s\) using a generalization of the Briançon-Skoda theorem.'' In particular, it is proved that \(I^{\{s \}}=I^{\text{cl}_s}\) for monomial ideals in polynomial rings.Minimal cellular resolutions of the edge ideals of forestshttps://zbmath.org/1472.130122021-11-25T18:46:10.358925Z"Barile, Margherita"https://zbmath.org/authors/?q=ai:barile.margherita"Macchia, Antonio"https://zbmath.org/authors/?q=ai:macchia.antonioThe authors consider edge ideals of forests and give an explicit construction of a cellular resolution that is minimal. Their minimal resolution is based on the Lyubeznik resolution and on the discrete Morse theory. They present a procedure for selecting the admissible symbols which generate the free modules of the resolution. In the end of the paper, they compute the graded Betti numbers and the projective dimension.On perfect co-annihilating-ideal graph of a commutative Artinian ringhttps://zbmath.org/1472.130132021-11-25T18:46:10.358925Z"Mirghadim, S. M. Saadat"https://zbmath.org/authors/?q=ai:mirghadim.s-m-saadat"Nikmehr, M. J."https://zbmath.org/authors/?q=ai:nikmehr.mohammad-javad"Nikandish, R."https://zbmath.org/authors/?q=ai:nikandish.rezaLet \(R\) be a commutative ring with identity \(1\neq 0.\) The co-annihilating-ideal graph of \(R,\) denoted by \(A_R,\) is the simple undirected graph whose vertex set is the set of all nonzero proper ideals of \(R\) and two distinct vertices \(I\) and \(J\) are adjacent whenever \(Ann(I) \cap Ann(J) = (0).\) In this paper, authors characterize all Artinian rings for which both of the graphs \(A_R\) and its complement \(\overline{A}_R\) are chordal. Moreover authors obtained all Artinian commutative rings \(R\) for which \(A_R\) and its complement \(\overline{A}_R\) are perfect. In particular, it is proved that, for a commutative Artinian ring \(R,\) the graph \(\overline{A}_R\) is a perfect graph if and only if \(|Max(R)|\leq 4.\)On distance Laplacian spectrum of zero divisor graphs of the ring \(\mathbb{Z}_n \)https://zbmath.org/1472.130142021-11-25T18:46:10.358925Z"Pirzada, S."https://zbmath.org/authors/?q=ai:pirzada.shariefuddin"Rather, B. A."https://zbmath.org/authors/?q=ai:rather.bilal-a"Chishti, T. A."https://zbmath.org/authors/?q=ai:chishti.tariq-aLet \(R\) be a finite commutative ring with identity \(1\neq 0.\) The zero-divisor graph \(\Gamma(R)\) is the simple undirected graph with the set of all non-zero zero-divisors \(Z^*(R)\) as the vertex set and two distinct vertices \(x\) and \(y\) are adjacent if \(x.y=0\) in \(R.\) Zero-divisor graphs of commutative rings are well studied in several aspects of graph theory for the past three decades. For a graph \(G,\) let \(A(G)\) be the adjacency matrix \(G.\) Let \(Deg(G)\) be the diagonal matrix of vertex degrees of vertices in \(G.\) The matrices \(L(G) = Deg(G) - A(G)\) and \(Q(G) = Deg(G) + A(G)\) are respectively the Laplacian and the signless Laplacian matrices and these matrices are real symmetric and positive semi-definite. It is assumed that \(0=\lambda_n\leq \lambda_{n-1}\leq\cdots\leq \lambda_1\) are the Laplacian eigenvalues of \(L(G).\) In this paper, authors obtained the distance Laplacian spectrum of the zero divisor graphs \(\Gamma(\mathbb{Z}_n\)) for different values of \(n\in \{pq, p^2q, (pq)^2, p^z~\text{for some}~ z\geq 2\}\) where \(p\) and \(q(p < q)\) are distinct primes. Further it is proved that the zero-divisor graph \(\Gamma(\mathbb{Z}_n)\) is distance Laplacian integral if and only if \(n\) is prime power or product of two distinct primes.Characterization of cubic Galois extensionshttps://zbmath.org/1472.130152021-11-25T18:46:10.358925Z"Wagner, Heidrun"https://zbmath.org/authors/?q=ai:wagner.heidrun(no abstract)Monomial generators of complete planar idealshttps://zbmath.org/1472.130162021-11-25T18:46:10.358925Z"Alberich-Carramiñana, Maria"https://zbmath.org/authors/?q=ai:alberich-carraminana.maria"Àlvarez Montaner, Josep"https://zbmath.org/authors/?q=ai:alvarez-montaner.josep"Blanco, Guillem"https://zbmath.org/authors/?q=ai:blanco.guillemLet \((X,O)\) be a germ of smooth complex surface and \(\mathcal{O}_{X,O}\) the ring of germs of holomorphic functions in a neighbourhood of \(O\), and let \(\mathfrak{m}\) be the maximal ideal at \(O\). Let \(\pi:X'\rightarrow X\) be a proper birational morphism that can be achieved as a sequence of blow-ups along a set of points. Given an effective \(\mathbb{Z}\)-divisor \(D\) in \(X'\) we may consider its associated ideal \(\pi_*\mathcal{O}_{X'}(-D)\), whose stalk at \(O\) is denoted as \(H_D\). This type of ideals are complete ideals of \(\mathcal{O}_{X,O}\). Among the class of divisors defining the same complete ideal, we may find a unique maximal representative, which has the property of being antinef. Zariski showed that the above correspondence is an isomorphism of semigroups between the set of complete \(\mathfrak{m}\)-primary ideals and the set of antinef divisors with exceptional support.
In the present paper the authors make this correspondence explicit computationally: given a proper birational morphism \(\pi:X'\rightarrow X\) and an antinef divisor \(D\) in \(X'\), they provide an algorithm that gives a system of generators of the ideal \(H_D\). This algorithm also captures the topological type of \(D\).
Applying the algorithm, the authors provide a method to compute the integral closure of any ideal \(\mathfrak{a}\subseteq \mathcal{O}_{X,O}\). They apply these results to planar ideals, multiplier ideals and a familiy of complete ideals described by valuative conditions given by the interesection multiplicity of the elements of \(\mathcal{O}_{X,O}\) with a fixed germ of plane curve.
The algorithms developed in the paper have been implemented in the computer algebra system \verb|Magma|.Isomorphism classes of commutative algebras generated by idempotentshttps://zbmath.org/1472.130172021-11-25T18:46:10.358925Z"Inoue, Kazuyo"https://zbmath.org/authors/?q=ai:inoue.kazuyo"Kawai, Hideyasu"https://zbmath.org/authors/?q=ai:kawai.hideyasu"Onoda, Nobuharu"https://zbmath.org/authors/?q=ai:onoda.nobuharuModules over discrete valuation domains. IIIhttps://zbmath.org/1472.130182021-11-25T18:46:10.358925Z"Krylov, P. A."https://zbmath.org/authors/?q=ai:krylov.piotr-a"Tuganbaev, A. A."https://zbmath.org/authors/?q=ai:tuganbaev.askar-aSummary: This review paper is a continuation of two previous review papers devoted to properties of modules over discrete valuation domains. The first part of this work was published in [the authors, J. Math. Sci., New York 145, No. 4, 4997--5117 (2007; Zbl 1178.13012); translation from Itogi Nauki Tekh., Ser. Sovrem. Mat. Prilozh., Temat. Obz. 121 (2006); J. Math. Sci., New York 151, No. 5, 3255--3371 (2008; Zbl 1229.13018); translation from Itogi Nauki Tekh., Ser. Sovrem. Mat. Prilozh., Temat. Obz. 122 (2006)]. In this review paper, we preserve the numeration of chapters and sections of parts I and II. The first part consists of Chapter 9 ``Appendix,'' Secs. 40-42. In Sec. 40, we consider \(p\)-adic torsion-free modules with isomorphic automorphism groups. Section 41 is devoted to torsion-free modules over a complete discrete valuation domain with isomorphic radicals of their endomorphism rings. The volume of the paper does not allow us to provide the proofs of all the results that appeared after publication of the previous parts and directly related to the issues under consideration in it. In the final Sec. 42, we describe some of these new results.Algebraic properties of edge ideals of some vertex-weighted oriented cyclic graphshttps://zbmath.org/1472.130192021-11-25T18:46:10.358925Z"Wang, Hong"https://zbmath.org/authors/?q=ai:wang.hong.7|wang.hong|wang.hong.2|wang.hong.5|wang.hong.1|wang.hong.3|wang.hong.4"Zhu, Guangjun"https://zbmath.org/authors/?q=ai:zhu.guangjun"Xu, Li"https://zbmath.org/authors/?q=ai:xu.li"Zhang, Jiaqi"https://zbmath.org/authors/?q=ai:zhang.jiaqiLet \(\mathcal{D}=(V(\mathcal{D}),E(\mathcal{D}),\omega)\) be a vertex-weighted oriented graph, i.e., a digraph with no loop, multiple or bidirected edge, which is equipped with a weight function \(\omega\) defined from \(V(\mathcal{D})=\{1,\ldots,n\}\) to the positive integers. Let \(\mathbb{K}\) be a field and \(S=\mathbb{K}[x_1,\ldots,x_n]\) the polynomial ring over \(\mathbb{K}\). Then, the edge ideal associated with \(\mathcal{D}\) is the monomial ideal in \(S\) generated by all the monomials of the form \(x_{i}x_{j}^{\omega(j)}\), where \((i,j)\in E(\mathcal{D})\) is a directed edge of \(\mathcal{D}\). \par This class of ideals was introduced in [\textit{C. Paulsen} et al., J. Algebra Appl. 12, No. 5, 24p (2013; Zbl 1266.05048)], as a generalization of the well-studied edge ideal of simple graphs initiated by \textit{A. Simis} et al. [J. Algebra. 167, No. 2, 389--416 (1994; Zbl 0816.13003)]. \par The authors of the paper under review aim to study some homological invariants of \(I(\mathcal{D})\). More precisely, by using methods of Betti splitting and polarization of monomial ideals, they present some characteristic-free formulas for the regularity and the projective dimension of \(I(\mathcal{D})\) for some special classes of vertex-weighted oriented graphs \(\mathcal{D}\).SFI-injective and SFI-flat moduleshttps://zbmath.org/1472.130202021-11-25T18:46:10.358925Z"Al-Shehri, Nada Hassan"https://zbmath.org/authors/?q=ai:al-shehri.nada-hassan"Ouarghi, Khalid"https://zbmath.org/authors/?q=ai:ouarghi.khalidThroughout \(R\) is always a commutative ring with identity and all modules are unitary. Recall that a module \(M\) is called FP-injective if \(\mathrm{Ext}^1_R (N, M) = 0\) for any finitely presented module \(N\). \textit{L. Mao} and \textit{N. Ding} [J. Algebra 309, No. 1, 367--385 (2007; Zbl 1117.16002)] introduced and studied FI-injective modules and FI-flat modules, where a module \(M\) (resp., \(N\)) is called FI-injective (resp., FI-flat) if \(\mathrm{Ext}^1_R(G, M) = 0\) (resp. \(\mathrm{Tor}_1^R(N, G) = 0\)) for any FP-injective module \(G\). \textit{W. Li} et al. [Commun. Algebra 45, No. 9, 3816--3824 (2017; Zbl 1390.18032)] introduced the strongly FP-injective module as follows: A module \(C\) is called strongly FP-injective if \(\mathrm{Ext}^i_R(N, C)=0\) for any finitely presented module \(N\) and any integer \(i \geq 1\). Naturally the paper under review introduced the SFI-injective modules and SFI-flat modules by replacing `FP-injective module' by `strongly FP-injective module' in the definition of FI-injective modules and FI-flat modules respectively. Among other results, it is shown that a ring \(R\) is Quasi-Frobenius (resp., IF) if and only if every \(R\)-module is SFI-injective (resp., SFI-flat). An example is given to show that SFI-injective (resp., SFI-flat) is not necessarily injective (resp., flat).On the radius of the category of extensions of matrix factorizationshttps://zbmath.org/1472.130212021-11-25T18:46:10.358925Z"Shimada, Kaori"https://zbmath.org/authors/?q=ai:shimada.kaori"Takahashi, Ryo"https://zbmath.org/authors/?q=ai:takahashi.ryoSummary: Let \(S\) be a commutative noetherian ring. The extensions of matrix factorizations of non-zerodivisors \(x_1, \ldots, x_n\) of \(S\) form a full subcategory of finitely generated modules over the quotient ring \(S /(x_1 \cdots x_n)\). In this paper, we investigate the radius (in the sense of Dao and Takahashi) of this full subcategory. As an application, we obtain an upper bound of the dimension (in the sense of Rouquier) of the singularity category of a local hypersurface of dimension one, which refines a recent result of Kawasaki, Nakamura and Shimada.Symbolic powers of codimension two Cohen-Macaulay idealshttps://zbmath.org/1472.130222021-11-25T18:46:10.358925Z"Cooper, Susan"https://zbmath.org/authors/?q=ai:cooper.susan-marie"Fatabbi, Giuliana"https://zbmath.org/authors/?q=ai:fatabbi.giuliana"Guardo, Elena"https://zbmath.org/authors/?q=ai:guardo.elena"Lorenzini, Anna"https://zbmath.org/authors/?q=ai:lorenzini.anna"Migliore, Juan"https://zbmath.org/authors/?q=ai:migliore.juan-carlos"Nagel, Uwe"https://zbmath.org/authors/?q=ai:nagel.uwe"Seceleanu, Alexandra"https://zbmath.org/authors/?q=ai:seceleanu.alexandra"Szpond, Justyna"https://zbmath.org/authors/?q=ai:szpond.justyna"Tuyl, Adam Van"https://zbmath.org/authors/?q=ai:van-tuyl.adamLet \(X\) be a codimension two arithmetically Cohen-Macaulay scheme in \(\mathbb{P}^n\) and \(I_X\) its defining ideal. The authors consider the problem of equality between the ordinary and symbolic powers of \(I_X\), that is \(I_X^m=I_X^{(m)}\) for all \(m\geq1\). They survey known results about these equality, and they extend some of these results. They give necessary and sufficient conditions for the above equality, in terms of the number of generators of \(I_X\) for the case of codimension three arithmetically Gorenstein schemes that are locally complete intersection. They also consider the importance of the hypothesis in the presented characterization by dropping some of these hypotheses and analyzing what happens. In the end of the paper, they consider arithmetically Cohen-Macaulay set of points in \(\mathbb{P}_1\times\mathbb{P}_1\) and give new short proofs of known results.On the minimal free resolution of symbolic powers of cover ideals of graphshttps://zbmath.org/1472.130232021-11-25T18:46:10.358925Z"Fakhari, S. A. Seyed"https://zbmath.org/authors/?q=ai:seyed-fakhari.seyed-aminAuthor's abstract: For any graph \(G\), assume that \(J(G)\) is the cover ideal of \(G\). Let \(J(G)^{(k)}\) denote the \(k\)th symbolic power of \(J(G)\). We characterize all graphs \(G\) with the property that \(J(G)^{(k)}\) has a linear resolution for some (equivalently, for all) integer \(k\geq 2\). Moreover, it is shown that for any graph \(G\), the sequence
\((reg(J(G)^{(k)})_{k=1}^{\infty}\) is nondecreasing. Furthermore, we compute the largest degree of minimal generators of \(J(G)^{(k)}\) when \(G\) is either an unmixed of a claw-free graph.Green-Lazarsfeld condition for toric edge ideals of bipartite graphshttps://zbmath.org/1472.130242021-11-25T18:46:10.358925Z"Greif, Zachary"https://zbmath.org/authors/?q=ai:greif.zachary"McCullough, Jason"https://zbmath.org/authors/?q=ai:mccullough.jasonThe authors consider toric ideals of bipartite graphs and study the Green-Lazarsfeld condition \(\mathbf{N}_p\), for \(p\geq1\). Their characterizations are given in terms of the combinatorics of the given graph or its complementary graph. They also consider the graded Betti numbers of the toric ideal and give necessary sufficient conditions for not vanishing these numbers.Explicit Pieri inclusionshttps://zbmath.org/1472.130252021-11-25T18:46:10.358925Z"Hunziker, Markus"https://zbmath.org/authors/?q=ai:hunziker.markus"Miller, John A."https://zbmath.org/authors/?q=ai:miller.john-a"Sepanski, Mark"https://zbmath.org/authors/?q=ai:sepanski.mark-rSummary: By the Pieri rule, the tensor product of an exterior power and a finite-dimensional irreducible representation of a general linear group has a multiplicity-free decomposition. The embeddings of the constituents are called Pieri inclusions and were first studied by \textit{J. Weyman} [Schur functors and resolutions of minors. Brandeis University, Waltham USA (PhD Thesis) (1980)] and described explicitly by \textit{P. J. Olver} [``Differential hyperforms I'', University of Minnesota, Mathematics Report, 82--101 (1980), \url{https://www-users.cse.umn.edu/~olver/a_/hyper.pdf}]. More recently, these maps have appeared in the work of \textit{D. Eisenbud} et al. [Ann. Inst. Fourier 61, No. 3, 905--926 (2011; Zbl 1239.13023)] and of \textit{S. V. Sam} [J. Softw. Algebra Geom. 1, 5--10 (2009; Zbl 1311.13039)] and Weyman to compute pure free resolutions for classical groups.
In this paper, we give a new closed form, non-recursive description of Pieri inclusions. For partitions with a bounded number of distinct parts, the resulting algorithm has polynomial time complexity whereas the previously known algorithm has exponential time complexity.Homological invariants of powers of fiber productshttps://zbmath.org/1472.130262021-11-25T18:46:10.358925Z"Nguyen, Hop D."https://zbmath.org/authors/?q=ai:nguyen.dang-hop"Vu, Thanh"https://zbmath.org/authors/?q=ai:vu.thanhIn the present paper, authors studied asymptotic behaviers of powers of certain homogeneous ideals in a polynomial ring.
Let \(R\) (resp.\ \(S\)) be a positive-dimensional polynomial ring over a field~\(k\) with homogeneous maximal ideal \(\mathfrak m\) (resp.\ \(\mathfrak n\)) and \(I \subset \mathfrak m^2\) (resp.\ \(J \subset \mathfrak n^2\)) a homogeneous ideal of \(R\) (resp.\ \(S\)). The fiber product of \(R/I\) and \(S/J\) over \(k\) is
\[
R/I \times_k S/J = \{(f,g) \in R/I \times S/J \mid f \bmod \mathfrak m/I = g \bmod \mathfrak n/J\}.
\]
Let \(T = R \otimes_k S\) and \(F = IT + JT + \mathfrak m \mathfrak nT\). Then \(R/I \times_k S/J \cong R \otimes_k S/ (I, J, \mathfrak m \mathfrak n)\). We considered powers of \(F\).
The first result concerns the Castelnuovo-Mumford regularity. We gave an equality
\[
\operatorname{reg}_T F^s = \max\{\operatorname{reg}_R \mathfrak m^{s-i} I^i + s - i, \operatorname{reg}_S \mathfrak n^{s-i} J^i + s-i \mid i = 1, \dots, s\}
\]
for \(s \geq 1\) if (\#) \(\operatorname{char} k = 0\), or (\#\#) \(I\) and \(J\) are monomial ideals, or (\#\#\#) \(I^t\) and \(J^t\) are componentwise linear for \(i = 1\), \dots, \(s\).
The second one shows \(\operatorname{depth} F^s = 1\) for \(s \geq 2\) if (\#) or (\#\#).
The third result gives inequalities
\[
\operatorname{ld}_T F^s \geq \max\{ \operatorname{ld}_R I^s, \operatorname{ld}_S J^s\}
\]
for \(s \geq 1\), and
\[
\operatorname{ld}_T F^s \leq \max\{ \operatorname{ld}_R \mathfrak m^{s-i} I^i, \operatorname{ld}_S \mathfrak n^{s-i} J^i \mid i = 1, \dots, s\}
\]
for \(s \geq 1\) if (\#) or (\#\#) or (\#\#). Here, \(\operatorname{ld}_T F\) denotes the linearly defect of \(F\), introduced by \textit{J. Herzog} and \textit{S. Iyengar} [J. Pure Appl. Algebra 201, No. 1--3, 154--188 (2005; Zbl 1106.13011)].Virtual resolutions of monomial ideals on toric varietieshttps://zbmath.org/1472.130272021-11-25T18:46:10.358925Z"Yang, Jay"https://zbmath.org/authors/?q=ai:yang.jayGiven a smooth toric variety \(X=X(\Sigma)\) and a \(\mathrm{Pic}(X)\)-graded module \(M\), then a free complex \(F\) of graded \(\mathrm{k}[\Sigma]\)-modules is a virtual resolution of \(M\) if the corresponding complex \(\widetilde{F}\) of vector bundles on \(X\) is a resolution of \(\widetilde{M}\). In the paper under review, the author uses cellular resolutions of monomial ideals to prove an analog of Hilbert's syzygy theorem for virtual resolutions of monomial ideals on smooth toric varieties.Invariant Hochschild cohomology of smooth functionshttps://zbmath.org/1472.130282021-11-25T18:46:10.358925Z"Miaskiwskyi, Lukas"https://zbmath.org/authors/?q=ai:miaskiwskyi.lukasGroup actions have been extensively studied in several branches of mathematics. A group can act on a set with or without underlying algebraic properties. An interesting aspect of the theory of group actions is the study of invariant subspaces, subalgebras, submanifolds e.t.c. which are subsets left ``unchanged'' by the action of the group.
The Hochschild cohomology of an associative algebra is a ring of special interest because of its use in deformation theory. Whenever a Lie group acts on a smooth manifold \(M\), there is an induced action of the Lie group on the Hochschild cohomology of the smooth manifold \(M\). The interplay of Hochschild (co-)homology and group actions haven by widely studied for instance in [\textit{J. L. Brylinski}, Algebras associated with group actions and their homology, Brown Preprint (1987)] and [\textit{J. Block} and \textit{E. Getzler}, Ann. Sci. Éc. Norm. Supér. (4) 27, No. 4, 493--527 (1994; Zbl 0849.55008)].
Let \(\mathbb{K}\) be a field, \(A\) an associative \(\mathbb{K}\)-algebra and \(\mathcal{N}\) an \(A\)-bimodule. The space of Hochschild cochains is defined to be \[\textbf{HC}^{\bullet}(A,\mathcal{N}) = \bigoplus_{n=0}^{\infty}\textbf{HC}^{n}(A,\mathcal{N})\] where \(\textbf{HC}^{n}(A,\mathcal{N})\) consists of maps \(\phi:\stackrel{n-times}{A\times A\times \cdots \times A}\rightarrow \mathcal{N}\) satisfying \(d^{*}\phi=0\), where \(d^*\) is the differential. The Hochschild cohomology of \(A\) with coefficients in \(\mathcal{N}\) is obtain by taking the cohomology of the cochain complex \(\cdots\rightarrow\textbf{HC}^{n-1}(A,\mathcal{N})\xrightarrow{d_n^{*}} \textbf{HC}^{n}(A,\mathcal{N})\rightarrow\cdots\) and putting the pieces together i.e. \[\textbf{HH}^{\bullet}(A,\mathcal{N}) = \bigoplus_{n=0}^{\infty}\textbf{HH}^{n}(A,\mathcal{N}),\] where \( \textbf{HH}^{n}(A,\mathcal{N}) : = \frac{\ker d_{n+1}^{*}}{ \Im d_n^{*}}.\) Suppose that a group \(G\) acts on \(\textbf{HC}^{n}(A,\mathcal{N})\) and consider the space of \textit{invariant cochains}: \[\textbf{HC}^{\bullet}_G(A,\mathcal{N}):=\{\phi\in\textbf{HC}^{\bullet}(A,\mathcal{N})\;| g\cdot\phi = \phi \;\forall g\in G\},\] the paper under review considers two subspaces namely;
\begin{enumerate}
\item[(i).] the \textbf{cohomology of invariant cochains} given by \[ \textbf{HH}^{\bullet}_G(A,\mathcal{N}) : = \frac{\ker d^{*}|_{\textbf{HC}^{\bullet}_G(A,\mathcal{N})}}{ \Im d^{*}|_{\textbf{HC}^{\bullet}_G(A,\mathcal{N})} }\] obtained from the cochain complex \( \cdots\rightarrow\textbf{HC}^{n-1}_G(A,\mathcal{N})\xrightarrow{d_n^{*}} \textbf{HC}^{n}_G(A,\mathcal{N})\xrightarrow{d_{n+1}^{*}} \textbf{HC}^{n+1}_G(A,\mathcal{N})\rightarrow\cdots\) and
\item[(ii).] the \textbf{space of invariant classes} \[ \textbf{HH}^{\bullet}(A,\mathcal{N})^G : = \{ [\phi]\in\textbf{HH}^{\bullet}(A,\mathcal{N})\;|\; g[\phi]=[\phi]\}.\]
\end{enumerate}
The main result of the article under review was to establish that whenever \(A\) is the space of smooth functions on a manifold and the group action is ``proper'', these two subspaces are isomorphic i.e. \(\textbf{HH}^{\bullet}_G(A,\mathcal{N})\cong\textbf{HH}^{\bullet}(A,\mathcal{N})^G\). The author also draws connections between this notion of invariant Hochschild cohomology and invariant multivector fields on a smooth manifold.Corrigendum to: ``Strength conditions, small subalgebras, and Stillman bounds in degree \(\leq 4\)''https://zbmath.org/1472.130292021-11-25T18:46:10.358925Z"Ananyan, Tigran"https://zbmath.org/authors/?q=ai:ananyan.tigran"Hochster, Melvin"https://zbmath.org/authors/?q=ai:hochster.melvinSummary: The statement and proof of a proposition, which appeared in [the authors, ibid. 373, No. 7, 4757--4806 (2020; Zbl 1452.13017)], about the locus where strength of a form is at most \(k\) are corrected: the locus is constructible but not known to be closed. Needed corrections are made in the proof of another result about strength: the statement of that result is not changed.Local cohomology modules and their propertieshttps://zbmath.org/1472.130302021-11-25T18:46:10.358925Z"Azami, J."https://zbmath.org/authors/?q=ai:azami.jafar"Hasanzad, M."https://zbmath.org/authors/?q=ai:hasanzad.masoumehLet \(R\) be a commutative 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 \[H^{i}_{\mathfrak{a}}(M) \cong \underset{n\geq 1}{\mathrm{varinjlim}} \mathrm{Ext}^{i}_{R}\left(\frac{R}{\mathfrak{a}^{n}},M\right).\] Moreover, the \(\mathfrak{a}\)-transform of \(M\) is given by \[D_{\mathfrak{a}}(M) \cong \underset{n\geq 1}{\mathrm{varinjlim}}\mathrm{Hom}_{R}(\mathfrak{a}^{n},M).\] If in particular, \((R,\mathfrak{m},k)\) is noetherian local, and \(M\) is non-zero finitely generated of Krull dimension \(n > 0\), then \(M\) is said to be a generalized Cohen-Macaulay \(R\)-module if \(H^{i}_{\mathfrak{m}}(M)\) is finitely generated for every \(i \neq n\). In case \(R\) is complete and \(M\) is generalized Cohen-Macaulay of Krull dimension \(n\geq 2\), the authors show that \[\mathrm{Hom}_{R}\left(H_{\mathfrak{m}}^{n}\left(\mathrm{Hom}_{R}\left(H_{\mathfrak{m}}^{n}\left(D_{\mathfrak{m}}\left(M\right)\right),E_{R}(k)\right)\right),E_{R}(k)\right) \cong D_{\mathfrak{m}}(M).\] Then they obtain some results on the finiteness of the Bass numbers, cofiniteness, and cominimaxness of the local cohomology modules with respect to proper ideals.Gotzmann monomials in four variableshttps://zbmath.org/1472.130312021-11-25T18:46:10.358925Z"Bonanzinga, Vittoria"https://zbmath.org/authors/?q=ai:bonanzinga.vittoria"Eliahou, Shalom"https://zbmath.org/authors/?q=ai:eliahou.shalomA monomial ideal, \(J\subset K[x_1,\dots,x_n]\), is called Borel-stable if, for every monomial \(v\in J\) and for every index \(j\), \[ x_j\mid v \implies x_i({v}/{x_j}) \in J,\text{ for all }1\leq i \leq j. \] If \(u\) is a monomial, the authors denote by \(B(u)\) the smallest Borel stable monomial ideal that contains \(u\) and call \(u\) a Gotzmann monomial if the Hilbert function of \(B(u)\) attains Macaulay's lower bound for a certain degree and, hence, on [\textit{G. Gotzmann}, Math. Z. 158, 61--70 (1978; Zbl 0352.13009) ]. Their goal is to classify all Gotzmann monomials in \textit{four} variables. All monomials in one or two variables are Gotzmann monomials; the case of three variables is the contained in Proposition 8 of [\textit{S. Murai}, Ill. J. Math. 51, No. 3, 843--852 (2007; Zbl 1155.13012) ].
The main result of the article (Theorem 7.7) states that a monomial in four variables \(x_1^ax_2^bx_3^cx_4^t\) is a Gotzmann monomial if and only if \[ t\geq \binom{\binom{b}{2}}{2} + \frac{b+4}{3}\binom{b}{2} + (b+1)\binom{c+1}{2}+\binom{c+1}{3}-c. \]The support of local cohomology moduleshttps://zbmath.org/1472.130322021-11-25T18:46:10.358925Z"Katzman, Mordechai"https://zbmath.org/authors/?q=ai:katzman.mordechai"Zhang, Wenliang"https://zbmath.org/authors/?q=ai:zhang.wenliangSummary: We describe the support of \(F\)-finite \(F\)-modules over polynomial rings \(R\) of prime characteristic. Our description yields an algorithm to compute the support of such modules; the complexity of our algorithm is also analysed. To the best of our knowledge, this is the first algorithm to avoid extensive use of Gröbner bases and hence of substantial practical value. We also use the idea behind this algorithm to prove that the support of \(H^j_I(S)\) is Zariski closed for each ideal \(I\) of \(S\) where \(R\) is noetherian commutative ring of prime characteristic with finitely many isolated singular points and \(S=R/gR (g\in R)\).Irreducibility and factorizations in monoid ringshttps://zbmath.org/1472.130332021-11-25T18:46:10.358925Z"Gotti, Felix"https://zbmath.org/authors/?q=ai:gotti.felixFrom the author's abstract: For an integral domain \(R\) and a commutative cancellative monoid \(M\), the ring consisting of all polynomial expressions with coefficients in \(R\) and exponents in \(M\) is called the monoid ring of \(M\) over \(R\). An integral domain \(R\) is called \textit{atomic} if every nonzero nonunit element can be written as a product of irreducibles. In the study of the atomicity of integral domains, the building blocks are the irreducible elements. Thus, tools to prove irreducibility are crucial to study atomicity. In the first part of the paper under review, the author extends Gauss's Lemma and Eisenstein's Criterion from polynomial rings to monoid rings. An integral domain \(R\) is called \textit{half-factorial} (or an \textit{HFD}) if any two factorizations of a nonzero nonunit element of \(R\) have the same number of irreducible elements (counting repetitions). In the second part of the paper, the author determines which monoid algebras with nonnegative rational exponents are Dedekind domains, Euclidean domains, PIDs, UFDs, and HFDs. As a side result, the author characterizes the submonoids of \((\mathbb{Q}_{\geq 0},+)\) satisfying a dual notion of half-factoriality known as other-half-factoriality, where \((\mathbb{Q}_{\geq 0}, +)\) is the additive monoid of the nonnegative rational numbers.
This paper is well written.
For the entire collection see [Zbl 1446.20006].The extension property in the category of direct sum of cyclic torsion-free modules over a BFDhttps://zbmath.org/1472.130342021-11-25T18:46:10.358925Z"Abdelalim, Seddik"https://zbmath.org/authors/?q=ai:abdelalim.seddik"Chaichaa, Abdelhak"https://zbmath.org/authors/?q=ai:chaichaa.abdelhak"El Garn, Mostafa"https://zbmath.org/authors/?q=ai:el-garn.mostafaLet \(M\) be a direct sum of cyclic torsion-free modules over an integral bounded factorization domain \(A\). Let \(\alpha\) be an automorphism of the \(A\)-module \(M\). The aim of the paper is to show that \(\alpha\) satisfies the extension property, i.e., for any monomorphism \(\lambda :M\rightarrow N\) of \(A\)-modules, there exists an automorphism \(\overline{\alpha}\) of \(N\) such that \(\overline{\alpha}\lambda=\lambda \alpha\), if and only if \(\alpha\) is the multiplication by an invertible element of \(A\).
For the entire collection see [Zbl 1433.16001].The socle module of a monomial idealhttps://zbmath.org/1472.130352021-11-25T18:46:10.358925Z"Chu, Lizhong"https://zbmath.org/authors/?q=ai:chu.lizhong"Herzog, Jürgen"https://zbmath.org/authors/?q=ai:herzog.jurgen"Lu, Dancheng"https://zbmath.org/authors/?q=ai:lu.danchengAuthors' abstract: For any ideal \(I\) in a Noetherian local ring or any graded ideal \(I\) in a standard graded \(K\)-algebra over a field \(K\), we introduce the socle module Soc\((I)\), whose graded components give us the socle of the powers of \(I\). It is observed that Soc\((I)\) is a finitely generated module over the fiber cone of \(I\). In the case that \(S\) is the polynomial ring and all powers of \(I\subseteq S\) have linear resolution, we define the module Soc\(^*(I)\), which is a module over the Rees ring of \(I\). For the edge ideal of a graph and for classes of polymatroidal ideals, we study the module structure of their socle modules.On quasi-equigenerated and Freiman cover ideals of graphshttps://zbmath.org/1472.130362021-11-25T18:46:10.358925Z"Drabkin, Benjamin"https://zbmath.org/authors/?q=ai:drabkin.benjamin"Guerrieri, Lorenzo"https://zbmath.org/authors/?q=ai:guerrieri.lorenzoThe present paper addresses the problem of computing the number of generators of the powers of a homogeneous ideal \(I\) in a polynomial ring \(R=\mathbf{k}[x_1,\dots,x_n]\). If all the generators of \(I\) have the same degree with respect to a standard (resp. non-standard) \(\mathbf{N}\)-grading then we say that \(I\) is equigenerated (resp. quasi-equigenerated). \textit{J. Herzog} et al. show in [Int. J. Algebra Comput. 29, No. 5, 827--847 (2019; Zbl 1423.13105)] that the number of generators of \(I^2\) is bigger than or equal to \(l(I)\mu(I)-\binom{l(I)}{2}\), where \(\mu(I)\) is the number of generators of \(I\) and \(l(I)\) is the analytic spread of \(I\). Other bounds for any powers of \(I\) are given by \textit{J. Herzog} and \textit{G. Zhu} [Commun. Algebra 47, No. 1, 407--423 (2019; Zbl 1410.13007)]. These bounds are consequence of a theorem by Freiman in additive number theory and therefore in the case that the bound for \(I^2\) is met, \(I\) is called a Freiman ideal, i.e. \(I\) is an equigenerated monomial ideal such that \(\mu(I^2)=l(I)\mu(I)-\binom{l(I)}{2}\).
The authors study Freiman ideals among cover ideals of graphs, and show that in general, graphs that are close enough to be complete have Freiman cover ideal. He also characterize Freiman cover ideals among pairs of complete graphs sharing a vertex, circulant graphs and whiskered graphs.On the depth and Stanley depth of the integral closure of powers of monomial idealshttps://zbmath.org/1472.130372021-11-25T18:46:10.358925Z"Seyed Fakhari, S. A."https://zbmath.org/authors/?q=ai:seyed-fakhari.seyed-aminLet $M$ be a finitely generated $\mathbb{Z}^n$-graded $S$-module, where $S=\mathbb{K}[x_1,\dots,x_n]$ and $\mathbb{K}$ is a field. For any homogenuous element $u\in M$, the $\mathbb{K}$-subspace $u\mathbb{K}[Z]$ is called a \textit{Stanley space} of dimension $|Z|$, if it is a free $\mathbb{K}[Z]$-module. A decomposition $\mathcal{D}$ of $M$ as a finite direct sum of Stanley spaces is called a \textit{Stanley decomposition} of $M$. The minimum dimension of a Stanley space in $\mathcal{D}$ is called the Stanley depth of $\mathcal{D}$ and is denoted by $\mathrm{sdepth}(\mathcal{D})$. The \textit{Stanley depth} of $M$ is defined as: $$\mathrm{sdepth}(M):=\max \{\mathrm{sdepth}(\mathcal{D}) |\mathcal{D}\text{ is a Stanley decomposition of }M\}.$$ This depth is defined to be $\infty$ for the zero module. The inequality $\mathrm{depth}(M)\leq\mathrm{sdepth}(M)$ is known as \textit{Stanley's inequality}. Let $G$ be a graph with edge ideal $I(G)$, $k\gg 0$ and $p$ be the number of its bipartite connected components. In the present paper it is shown that:
\begin{enumerate}
\item[(i)] $\mathrm{sdepth}(S/I(G))\geq p$; In particular, $S/\overline{I(G)^k}$ satisfies Stanley inequality.
\item[(ii)] $\mathrm{sdepth}(\overline{I(G)^k})\geq p+1$. In particular, $\overline{I(G)^k}$ satisfies Stanley inequality.
\item[(iii)] If $G$ is a connected bipartite graph whose girth is $g$ and $k\leq g/2+1$, then $\mathrm{sdepth}(\overline{I(G)^k})\geq 2$.
\end{enumerate}
Let $I\subset S$ be a nonzero monomial ideal with the analytic spread $l(I)$. Among other results, it is proved that the sequence $\{\mathrm{depth}(I^k/I^{k+1})\}^\infty_{k=0}$ converges to $n-l(I)$. Finally, it is shown that if $I$ is an integrally closed monomial ideal, then $\mathrm{depth}(S/I^m)\leq\mathrm{depth}(S/I)$ and $\mathrm{Ass}(S/I)\subseteq\mathrm{Ass}(S/I^m)$, for every integer $m\geq 1$.Projective dimension and regularity of edge ideals of some weighted oriented graphshttps://zbmath.org/1472.130382021-11-25T18:46:10.358925Z"Zhu, Guangjun"https://zbmath.org/authors/?q=ai:zhu.guangjun"Xu, Li"https://zbmath.org/authors/?q=ai:xu.li"Wang, Hong"https://zbmath.org/authors/?q=ai:wang.hong.7|wang.hong|wang.hong.2"Tang, Zhongming"https://zbmath.org/authors/?q=ai:tang.zhongmingA vertex-weighted oriented graph is a triplet \(D=(V(D),E(D),w)\), where \(V(D)=\{x_1,\dots,x_n\}\) is the vertex set, \(E(D)\) is the edge set and \(w:V(D)\rightarrow\mathbb{N}\) is a weight function. The edge ideal of \(D\), denoted by \(I(D)\), is the ideal of the polynomial ring \(S=k[x_1,x_2,\dots,x_n]\) which is generated by monomials \(\{x_ix_j^{w_j}|x_ix_j\in E(D)\}\). In the present article, the authors consider edge ideals of vertex-weighted oriented graphs and they provide some formulas for the projective dimension and the regularity of edge ideals associated to special vertex weighted rooted graphs. In fact, it is shown that if \(D\) satisfies one of the conditions
\begin{enumerate}
\item[(i)] \(D\) is a weighted oriented star graph;
\item[(ii)] \(D\) is a weighted oriented rooted forest such that \(w(x)\geq 2\) if \(d(x)\neq 1\);
\item[(iii)] \(D\) is a weighted oriented rooted cycle such that \(w(x)\geq 2\) for every vertex \(x\in V(D)\);
\end{enumerate}
then the projective dimension and regularity of the edge ideal is calculated by the following formulas: \(\mathrm{pd}(I(D)=|E(D)|-1;\ \ \mathrm{reg}(I(D))=\sum_{x\in V(D)}w(x)-|E(D)|+1\).Quantum Grothendieck rings as quantum cluster algebrashttps://zbmath.org/1472.130392021-11-25T18:46:10.358925Z"Bittmann, Léa"https://zbmath.org/authors/?q=ai:bittmann.lea\textit{D. Hernandez} and \textit{B. Leclerc} [Duke Math. J. 154, No. 2, 265--341 (2010; Zbl 1284.17010)] first realized that the Grothendieck ring of a certain monoidal subcategory \(\mathcal{C}_1\) of the category \(\mathcal{C}\) of finite-dimensional \(U_q(L\mathfrak{g})\)-modules had the structure of a cluster algebra. They thus proved that the Grothendieck ring of a certain monoidal subcategory \(\mathcal{O}^+_\mathbb{Z}\) of the category \(\mathcal{O}\) had a cluster algebra structure of infinite rank, for which one can take as initial seed the classes of the positive prefundamental representations. That is, the category \(\mathcal{O}^+_{\mathbb{Z}}\) contains the finite-dimensional representations and the positive prefundamental representations whose spectral parameter satisfy an integrality condition. Moreover, certain exchange relations, such as the Baxter relation, coming from cluster mutations appear naturally.
In order to construct of quantum Grothendieck ring for the category \(\mathcal{O}\) of representations of the quantum loop algebra introduced by \textit{D. Hernandez} and \textit{M. Jimbo} [Compos. Math. 148, No. 5, 1593--1623 (2012; Zbl 1266.17010)], previous approaches were no longer applicable. The geometrical approach of \textit{H. Nakajima} [Ann. Math. (2) 160, No. 3, 1057--1097 (2005; Zbl 1140.17015)] and \textit{M. Varagnolo} and \textit{E. Vasserot} [Prog. Math. 210, 345--365 (2003; Zbl 1162.17307)] (in which the \(t\)-graduation naturally comes from the graduation of cohomological complexes) requires a geometric interpretation of the objects in the category \(\mathcal{O}\), which is yet to be found. The more algebraic approach consisting of realizing the (quantum) Grothendieck ring as an invariant under a sort of Weyl symmetry, which allowed \textit{D. Hernandez} [Adv. Math. 187, No. 1, 1--52 (2004; Zbl 1098.17009)] to define a quantum Grothendieck ring of finite-dimensional representations in non-simply laced types, is again no longer relevant for the category \(\mathcal{O}\). However, only the cluster algebra approach yields results in this context.
The author constructs a quantum Grothendieck ring for a certain monoidal subcategory of the category \(\mathcal{O}\) (Theorem 5.2.1, page 180). She uses the cluster algebra structure of the Grothendieck ring of this category to define the quantum Grothendieck ring as a quantum cluster algebra. When the underlying simple Lie algebra is of type \(A\), she proves that this quantum Grothendieck ring contains the quantum Grothendieck ring of the category of finite-dimensional representations of the associated quantum affine algebra (Theorem 8.1.1, page 193).Building maximal green sequences via component preserving mutationshttps://zbmath.org/1472.130402021-11-25T18:46:10.358925Z"Bucher, Eric"https://zbmath.org/authors/?q=ai:bucher.eric"Machacek, John"https://zbmath.org/authors/?q=ai:machacek.john-m"Runburg, Evan"https://zbmath.org/authors/?q=ai:runburg.evan"Yeck, Abe"https://zbmath.org/authors/?q=ai:yeck.abe"Zwede, Ethan"https://zbmath.org/authors/?q=ai:zwede.ethanSummary: We introduce a new method for producing both maximal green and reddening sequences of quivers. The method, called component preserving mutations, generalizes the notion of direct sums of quivers and can be used as a tool to both recover known reddening sequences as well as find reddening sequences that were previously unknown. We use the method to produce and recover maximal green sequences for many bipartite recurrent quivers that show up in the study of periodicity of \(T\)-systems and \(Y\)-systems. Additionally, we show how our method relates to the dominance phenomenon recently considered by Reading. Given a maximal green sequence produced by our method, this relation to dominance gives a maximal green sequence for infinitely many other quivers. Other applications of this new methodology are explored including computing of quantum dilogarithm identities and determining minimal length maximal green sequences.Connected sums of graded Artinian Gorenstein algebras and Lefschetz propertieshttps://zbmath.org/1472.130412021-11-25T18:46:10.358925Z"Iarrobino, Anthony"https://zbmath.org/authors/?q=ai:iarrobino.anthony-a"McDaniel, Chris"https://zbmath.org/authors/?q=ai:mcdaniel.chris"Seceleanu, Alexandra"https://zbmath.org/authors/?q=ai:seceleanu.alexandraLet \(A\) and \(B\) be graded Artinian Gorenstein (AG) algebras with the same socle degree, \(d\). Let \(T\) be an AG algebra of socle degree \(k<d\). Suppose that there are surjective maps \(\pi_A : A \rightarrow T\) and \(\pi_B: B \rightarrow T\). The connected sum algebra \(A \#_T B\) is a certain quotient of the fibered product \(A \times_T B\). The connected sum of two AG algebras is again an AG algebra. In this paper the authors first give two alternative descriptions of this construction, including a careful study of how it relates to Macaulay-Matlis duality. They also show that if \(A\) and \(B\) are graded AG algebras satisfying the strong Lefschetz property (SLP) then over \(T = \mathbb F\) (the ground field), the connected sum also has the SLP. This is not true for a general choice of \(T\). However, they also show that connected sums do retain the WLP to some extent.Properties of the resolutions of almost Gorenstein algebrashttps://zbmath.org/1472.130422021-11-25T18:46:10.358925Z"Zappalà, Giuseppe"https://zbmath.org/authors/?q=ai:zappala.giuseppeAlmost Gorenstein Artinian algebras are rings of the form \(R/I\) with \(I=J+ (f)\) where \(R\) is a polynomial ring and \(J\) a Gorenstein ideal of \(R\). The author studies properties of the minimal free resolutions of almost Gorenstein algebras, describing the second term \(F_2\) of the resolution.
The second part of the article is focused on almost complete intersection algebra in codimension 3, ie Artinian algebra of the form \(k[x,y,z]/I\) with \(I\) minimally generated by four polynomials. For such algebras, the author provides characterizations of the minimal free resolution and of Betti numbers.Ideals modulo a primehttps://zbmath.org/1472.130432021-11-25T18:46:10.358925Z"Abbott, John"https://zbmath.org/authors/?q=ai:abbott.john-a"Bigatti, Anna Maria"https://zbmath.org/authors/?q=ai:bigatti.anna-maria"Robbiano, Lorenzo"https://zbmath.org/authors/?q=ai:robbiano.lorenzoThe present paper deals with the problem of reducing an ideal modulo \(p\), i.e. relating an ideal \(I\) in the polynomial ring \(\mathbb{Q}[x_1,\dots,x_n]\) to a corresponding ideal in \(\mathbb{F}_p[x_1,\dots,x_n]\) where \(p\) is a prime number.
The authors define a notion of \(\sigma\)-good prime, where \(\sigma\) is a term ordering and relate it to other similar notions in the literature. Furthermore, the paper introduces a new invariant called universal denominator, which is independent of the term ordering and allows to show that all but finitely many primes are good for \(I\) (see Definiton 2.4).
The methods in the paper make it easy to detect bad primes, a key feature in modular methods (Theorem 4.1 and Corollary 4.2).
The paper includes practical applications to modular computations of Gröbner bases and also includes examples of computations using the computer algebra systems \verb|CoCoA| and \verb|SINGULAR|.Saturations of subalgebras, SAGBI bases, and U-invariantshttps://zbmath.org/1472.130442021-11-25T18:46:10.358925Z"Bigatti, Anna Maria"https://zbmath.org/authors/?q=ai:bigatti.anna-maria"Robbiano, Lorenzo"https://zbmath.org/authors/?q=ai:robbiano.lorenzoLet \(R=K[x_1,\dots ,x_n]\) and \(F\) be a (not necessarily finite) subset of \(R\). Then the subalgebra of \(R\) generated by \(F\) is denoted \(K[F]\). Similar to the notion of Grobner bases for ideals of \(R\), we can define the notion of SAGBI Gröbner basis for \(K[F]\) (see e.g. the paper of the second author and \textit{M. Sweedler} [Lect. Notes Math. 1430, 61--87 (1990; Zbl 0725.13013)] which is regarded as a pioneer work).
Let \(S\) be a \(K\)-subalgebra of the polynomial ring \(R\) , and let \(0 \ne g\in S\). We denote the set \(\bigcup_{i=0}^\infty \{ f \in R \ | \ g^i f \in S\}\) by \(S : g^\infty\).
The problem that the authors address in this paper is as follows: Given polynomials \(g_1,\dots, g_r \in R\), let \(S= K[g_1,\dots, g_r]\) and \(0\ne g \in S\). The problem is to compute a set of generators for \(S : g^\infty\). In the first part of the paper, an algorithm has been presented to compute a set of generators for \(S : g^\infty\) which terminates if and only if it is finitely generated.
In the second part of the paper, the authors consider the case that \(S\) is graded. They show that two operations of computing a SAGBI basis for \(S\) and a set of generators for \(S : g^\infty\) commute and this leads to nice algorithms for computing with \(S : g^\infty\).A comparison of unrestricted dynamic Gröbner basis algorithmshttps://zbmath.org/1472.130452021-11-25T18:46:10.358925Z"Langeloh, Gabriel Mattos"https://zbmath.org/authors/?q=ai:langeloh.gabriel-mattosA Gröbner Basis algorithm is dynamic when it may change the monomial ordering during the computation. Dynamic Gröbner Basis algorithms often yield smaller output bases and ocasionally faster running times than the traditional ones.
The present paper proposes three new dynamic unrestricted Gröbner Basis algorithms based on the concept of neighbourhoods for monomial orders. These algorithms are experimentally compared to other dynamic algorithms. The results show that the proposed algorithms produce small bases containing polynomials of low degree, with little overhead.Coisotropic hypersurfaces in Grassmannianshttps://zbmath.org/1472.130462021-11-25T18:46:10.358925Z"Kohn, Kathlén"https://zbmath.org/authors/?q=ai:kohn.kathlenThis paper studies the so-called higher associated hypersurfaces of a projective variety via the notion of coisotropy. For a \(k\)-dimensional projective variety \(X\) in \(\mathbb{P}^n\), the \(i\)-th associated hypersurface of \(X\) consists of (the Zariski closure of) all \((n-k-1+i)\)-dimensional linear spaces in \(\mathbb{P}^n\) that meet \(X\) at a smooth point non-transversely, which is a subvariety of a Grassmannian. Historically, the cases \(i = 0\) and \(i=1\) have been studied as the Chow and Hurwitz form of \(X\), respectively.
A main result of this paper is a new and direct proof of a characterization (due originally to Gel'fand, Kapranov and Zelevinsky) of such hypersurfaces in the Grassmannian. Namely, a hypersurface in the Grassmannian is the associated hypersurface of some (irreducible) projective variety iff it is coisotropic, i.e. every normal space at a smooth point of the hypersurface is spanned by rank 1 homomorphisms. Since the notion of coisotropy does not depend on the underlying projective variety, this provides an intrinsic description of all higher associated hypersurfaces (hence the term coisotropic hypersurfaces).
In addition, many other results on coisotropic hypersurfaces are given: e.g. the coisotropic hypersurfaces of the projective dual of \(X\) are the reverse of those of \(X\), and the degrees of these are precisely the polar degrees of \(X\). It is also shown that hyperdeterminants are precisely the coisotropic hypersurfaces associated to Segre varieties. Finally, equations for the Cayley variety of all coisotropic forms of a given degree are given, inside Grassmannians of lines. The author has also written a Macaulay2 package to explicitly realize computation of coisotropic hypersurfaces.Systems of polynomial equations, higher-order tensor decompositions, and multidimensional harmonic retrieval: a unifying framework. Part II: The block term decompositionhttps://zbmath.org/1472.130472021-11-25T18:46:10.358925Z"Vanderstukken, Jeroen"https://zbmath.org/authors/?q=ai:vanderstukken.jeroen"Kürschner, Patrick"https://zbmath.org/authors/?q=ai:kurschner.patrick"Domanov, Ignat"https://zbmath.org/authors/?q=ai:domanov.ignat"De Lathauwer, Lieven"https://zbmath.org/authors/?q=ai:de-lathauwer.lievenFormal moduli problems and formal derived stackshttps://zbmath.org/1472.140042021-11-25T18:46:10.358925Z"Calaque, Damien"https://zbmath.org/authors/?q=ai:calaque.damien"Grivaux, Julien"https://zbmath.org/authors/?q=ai:grivaux.julienSummary: This paper presents a survey on formal moduli problems. It starts with an introduction to pointed formal moduli problems and a sketch of proof of a Theorem (independently proven by \textit{J. P. Pridham} [Adv. Math. 224, No. 3, 772--826 (2010; Zbl 1195.14012)] and \textit{J. Lurie} [``Derived algebraic geometry X: formal moduli problems'', (2011)]) which gives a precise mathematical formulation for Drinfeld's derived deformation theory philosophy. This theorem provides a correspondence between formal moduli problems and differential graded Lie algebras. The second part deals with Lurie's general theory of deformation contexts, which we present in a slightly different way than the original paper, emphasizing the (more symmetric) notion of Koszul duality contexts and morphisms thereof. In the third part, we explain how to apply this machinery to the case of non-split formal moduli problems under a given derived affine scheme; this situation has been dealt with recently by \textit{J. Nuiten} [Adv. Math. 354, Article ID 106750, 63 p. (2019; Zbl 1433.14007)], and requires to replace differential graded Lie algebras with differential graded Lie algebroids. In the last part, we globalize this to the more general setting of formal thickenings of derived stacks, and suggest an alternative approach to results of \textit{D. Gaitsgory} and \textit{N. Rozenblyum} [A study in derived algebraic geometry. Volume I: Correspondences and duality. Providence, RI: American Mathematical Society (AMS) (2017; Zbl 1408.14001)].
For the entire collection see [Zbl 1471.14005].Nonlinear traceshttps://zbmath.org/1472.140122021-11-25T18:46:10.358925Z"Ben-Zvi, David"https://zbmath.org/authors/?q=ai:ben-zvi.david"Nadler, David"https://zbmath.org/authors/?q=ai:nadler.davidSummary: We combine the theory of traces in homotopical algebra with sheaf theory in derived algebraic geometry to deduce general fixed point and character formulas. The formalism of dimension (or Hochschild homology) of a dualizable object in the context of higher algebra provides a unifying framework for classical notions such as Euler characteristics, Chern characters, and characters of group representations. Moreover, the simple functoriality properties of dimensions clarify celebrated identities and extend them to new contexts. \par We observe that it is advantageous to calculate dimensions, traces and their functoriality directly in the nonlinear geometric setting of correspondence categories, where they are directly identified with (derived versions of) loop spaces, fixed point loci and loop maps, respectively. This results in universal nonlinear versions of Grothendieck-Riemann-Roch theorems, Atiyah-Bott-Lefschetz trace formulas, and Frobenius-Weyl character formulas. We can then linearize by applying sheaf theories, such as the theories of ind-coherent sheaves and \(\mathcal{D}\)-modules constructed by \textit{D. Gaitsgory} and \textit{N. Rozenblyum} [Contemp. Math. 610, 139--251 (2014; Zbl 1316.14006)]. This recovers the familiar classical identities, in families and without any smoothness or transversality assumptions. On the other hand, the formalism also applies to higher categorical settings not captured within a linear framework, such as characters of group actions on categories.
For the entire collection see [Zbl 1471.14005].Syzygies of the apolar ideals of the determinant and permanenthttps://zbmath.org/1472.140582021-11-25T18:46:10.358925Z"Alper, Jarod"https://zbmath.org/authors/?q=ai:alper.jarod"Rowlands, Rowan"https://zbmath.org/authors/?q=ai:rowlands.rowanGiven a polynomial \(f\in \mathbb K[y_1,\ldots,y_k]\) one defines its apolar ideal \(f^{\bot}\) as \[f^{\bot}=\{g\in\mathbb K[y_1,\ldots,y_k] : \partial g(f)=0\}.\] Recall that to a monomial \(y^{\alpha}=y_1^{\alpha_1}\ldots y_k^{\alpha_k}\) one associates a differential operator \[\frac{\partial}{\partial y^{\alpha}}=\frac{\partial}{\partial y_1^{\alpha_1}\cdots\partial y_k^{\alpha_k}}\] and extends this definition linearly to all polynomials.
Thus for \(g=\sum c_{\alpha}y^{\alpha}\) one associates a differential operator \[\partial g= \sum c_{\alpha}\frac{\partial}{\partial y^{\alpha}}.\] The authors of the paper under review are interested in apolar ideals of two specific polynomials \(\mathrm{def}_n\) and \(\mathrm{perm}_n\) which are elements of the ring \(\mathbb K[x_{11},\ldots, x_{1n},x_{21},\ldots,x_{nn}]\) defines as \[\mathrm{def}_n=\sum_{\sigma\in S_n} \mathrm{sgn}(\sigma) x_{1\sigma(1)}\ldots x_{n\sigma(n)}\] and \[\mathrm{perm}_n=\sum_{\sigma\in S_n} x_{1\sigma(1)}\ldots x_{n\sigma(n)}.\] [\textit{S. M. Shafiei}, J. Commut. Algebra 7, No. 1, 89--123 (2015; Zbl 1364.13024)] showed that the ideals \((\mathrm{def}_n)^{\bot}\) and \((\mathrm{perm}_n)^{\bot}\) are generated by quadrics. She provided an explicit minimal set of generators. The authors extend this study to the first syzygies. They show that the first syzygies of \((\mathrm{def}_n)^{\bot}\) are linear except in characteristic two, where both polynomials and hence their apolar ideals coincide. Thus \((\mathrm{def}_n)^{\bot}\) satisfies at lest the \(N_3\) property of \textit{M. L. Green} [J. Differ. Geom. 19, 125--167, 168--171 (1984; Zbl 0559.14008)].
On the other hand syzygies of \((\mathrm{perm}_n)^{\bot}\) require also quadratic generators, in arbitrary characteristic. Thus one can distinguish both polynomials by properties of their minimal graded free resolution.
The paper is clearly written and all arguments are kept pretty effective, even if some of them are quite involved.Computing the equisingularity type of a pseudo-irreducible polynomialhttps://zbmath.org/1472.140692021-11-25T18:46:10.358925Z"Poteaux, Adrien"https://zbmath.org/authors/?q=ai:poteaux.adrien"Weimann, Martin"https://zbmath.org/authors/?q=ai:weimann.martinIn the paper under review, the authors characterize a class of germs of plane curve singularities, containing irreducible ones, whose equisingularity type can be computed in an expected quasi-linear time with respect to the discriminant valuation of a Weierstrass equation.Corrigendum to: ``The Gerstenhaber problem for commuting triples of matrices is `decidable' ''https://zbmath.org/1472.150242021-11-25T18:46:10.358925Z"O'Meara, Kevin C."https://zbmath.org/authors/?q=ai:omeara.kevin-cSummary: I correct a slip in an argument in the above paper [the author, ibid. 48, No. 2, 453--466 (2020; Zbl 1436.15018)]. This does not affect the
main result: the Gerstenhaber Problem is Turing computable for all fields.Some properties of Serre subcategories in the graded local cohomology moduleshttps://zbmath.org/1472.160212021-11-25T18:46:10.358925Z"Hassani, Feysal"https://zbmath.org/authors/?q=ai:hassani.feysal"Rasuli, Rasul"https://zbmath.org/authors/?q=ai:rasuli.rasulSummary: Let \(R = \oplus_{n \geq 0} R_n\) be a standard homogeneous Noetherian ring with local base ring \((R_0, \mathfrak{m}_0)\) and let \(M\) be a finitely generated graded $R$-module. Let \(H_{R_+}^i(M)\) be the \(i\)th local cohomology module of \(M\) with respect to \(R_+ = \oplus_{n > 0} R_n\). Let \(\mathcal{S}\) be a Serre subcategory of the category of \(R\)-modules and let \(i\) be a nonnegative integer. In this paper, if \(\dim R_0 \leq 1\), then we investigate some conditions under which the \(R\)-modules \(R_0 / \mathfrak{m}_0 \otimes_{R_0} H_{R_+}^i(M), \Gamma_{\mathfrak{m}_0 R}(H_{R_+}^i(M))\) and \(H_{\mathfrak{m}_0 R}^1(H_{R_+}^i(M))\) are in \(\mathcal{S}\) for all \(i \geq 0\). Also, we prove that if \(\dim R_0 \leq 2\), then the graded \(R\)-module \(H_{\mathfrak{m}_0}^1(H_{R_+}^i(M))\) is in \(\mathcal{S}\) for all \(i \geq 0\). Finally, we prove that if \(n\) is the biggest integer such that \(H_{\mathfrak{a}}^i(M) \notin \mathcal{S}\), then \(H_{R_+}^i(M) / \mathfrak{m}_0 H_{R_+}^i(M) \in \mathcal{S}\) for all \(i \geq n\).Rings additively generated by tripotents and nilpotentshttps://zbmath.org/1472.160352021-11-25T18:46:10.358925Z"Chen, Huanyin"https://zbmath.org/authors/?q=ai:chen.huanyin"Abdolyousefi, Marjan Sheibani"https://zbmath.org/authors/?q=ai:sheibani-abdolyousefi.marjanIn this paper the authors investigate connections between rings with the property that every element is the sum of two idempotents and a nilpotent that commute (i.e., strongly 2-nil-clean rings) and rings such that every element is the sum of two orthogonal idempotents and a unit (i.e., feebly clean rings). In Theorem 2.5, it is proved that a ring \(R\) is strongly 2-nil-clean rings if and only if (1) \(R\) is feebly clean, (2) \(J(R)\) is nil, and (3) \(U(R/J(R))\) has exponent \(\leq 2\). Moreover, connections with exchange rings are also provided in the last section of the paper. It is proved in Theorem 3.3 that in the above theorem the condition (1) can be replaced by (1') R is weakly exchange.Rings in which idempotents generate maximal or minimal idealshttps://zbmath.org/1472.160362021-11-25T18:46:10.358925Z"Dube, Themba"https://zbmath.org/authors/?q=ai:dube.themba"Ghirati, Mojtaba"https://zbmath.org/authors/?q=ai:ghirati.mojtaba"Nazari, Sajad"https://zbmath.org/authors/?q=ai:nazari.sajad"Taherifar, Ali"https://zbmath.org/authors/?q=ai:taherifar.aThe authors characterize rings with identity in which every left ideal generated by an idempotent different from \(0\) and \(1\) is either a maximal left ideal or a minimal left ideal. These rings are called IMm-rings. Several special classes of IMm-rings are considered. In particular, if \(R\) is a semiprimitive commutative ring that has infinitely many maximal ideals, \(R\) being an IMm-ring is characterized by means of the Zariski topology of the maximal spectrum \(\text{Max}(R):=\{M\subseteq R: M \text{ is a maximal ideal of } R\}\). Finally the authors study rings with a weaker form of the ``IMm-property''.Classification of left octonionic moduleshttps://zbmath.org/1472.170012021-11-25T18:46:10.358925Z"Huo, Qinghai"https://zbmath.org/authors/?q=ai:huo.qinghai"Li, Yong"https://zbmath.org/authors/?q=ai:li.yong.5|li.yong.6|li.yong.3|li.yong.8|li.yong.9|li.yong.4|li.yong.7"Ren, Guangbin"https://zbmath.org/authors/?q=ai:ren.guangbinThe article is devoted to left \(\mathbb{O}\)-modules, where \(\mathbb{O}\) denotes the classical octonion (Cayley) algebra over the real field \(\mathbb{R}\). This is a particular case of modules over alternative algebras. A review of some previous publications is given. The octonion algebra contains the classical quaternion skew field \(\mathbb{H}\) of Hamilton. The octonion algebra considered as the vector space over \(\mathbb{R}\) has the basis \(i_0\), \(i_1,\ldots,i_7\) such that \(i_0=1\), \(i_k^2=-1\) for each \(k=1,\ldots,7\), \(i_ki_l=-i_li_k\) for each \(k\ne l\) such that \(k\ge 1\) and \(l\ge 1\). The subsequent doubling procedures are:
\(i_1\) is the doubling generator of the complex field \(\mathbb{C}\) over the real field \(\mathbb{R}\), \(i_2\) is the doubling generator of \(\mathbb{H}\) generated from \(\mathbb{C}\) and \(\mathbb{C}i_2\) such that \(i_3=i_1i_2\), then \(i_4\) denotes the doubling generator of \(\mathbf{O}\) generated from \(\mathbb{H}\) and \(\mathbb{H}i_4\) by the smashed product, where \(i_5\), \(i_6\), \(i_7\) are obtained by multiplication of \(i_1\), \(i_2\), \(i_3\) respectively on \(i_4\) with the corresponding order up to a notation choice and an automorphism of \(\mathbb{O}\). It is nonassociative, for example, \((i_1i_2)i_4=-i_1(i_2i_4)\).
The commutator \((i_k,i_j)\) and the associator \((i_k,i_j,i_l)\) belong to \(\mathbb{Z}_2\) for each \(k\), \(j\), \(l\), where \(ab=(ba)(a,b)\) and \((ab)c=(a(bc))(a,b,c)\) for each \(a\), \(b\), \(c\) in \(\mathbf{O}\setminus \{ 0 \} \), \(\mathbb{Z}_2= \{ -1, 1 \} \). There is an involution \(\mathbb{O}\ni z\mapsto \bar{z}\in \mathbb{O}\) such that \(\overline{ab}=\bar{b} \bar{a}\) for each \(a\), \(b\) in \(\mathbb{O}\), \(|b|^2=b\bar{b}\). The octonion division algebra is nonassociative alternative with center \(\mathbb{ R}\) and the multiplicative norm. It is shown in the article that left \(\mathbb{O}\)-modules are of the type \(M=\mathbb{O}^n\bigoplus \bar{\mathbb{O}}^m\). This induces the algebra structure on \(\mathbb{O}^n\). This matter is also described in:
\([1]\) [\textit{N. Bourbaki}, Éléments de mathématique. Algèbre. Chapitres 1 à 3. Reprint of the 1970 original. Berlin: Springer (2007; Zbl 1111.00001)].
\([2]\) [\textit{R. D. Schafer}, An introduction to nonassociative algebras. New York and London: Academic Press (1966; Zbl 0145.25601)].
\([3]\) [\textit{R. H. Bruck}, A survey of binary systems. Berlin: Springer-Verlag (1958; Zbl 0081.01704)].
Using the opposite algebra \(\mathbb{O}_o\), or \(\overline{\mathbb{O}}\) obtained by the involution from \(\mathbb{O}\), one gets the standard correspondence between left and right modules, the left module over the enveloping algebra \(\mathbb{O}_e\) also corresponds to the two-sided \(\mathbb{O}\)-module as in [\textit{N. Bourbaki}, (loc. cit.); \textit{R. D. Schafer}, (loc. cit.)]. The algebra \(L(\mathbb{O})\) generated by left multipliers \(L_b\) on \(\mathbb{O}\), \(b\in \mathbb{O}\), with the associative composition obtained by the set-theoretic composition of maps, is isomorphic to the proper subalgebra of the matrix algebra \(Mat_{8\times 8}(\mathbb{R})\) (see [\textit{R. D. Schafer}, (loc. cit.)]) satisfying relations \((5.14)\)-\((5.20)\) in [\textit{R. H. Bruck}, (loc. cit.)] implying particularly that \(i_lL_{i_j}L_{i_k}\mathbb{Z}_2=i_lL_{i_ki_j}\mathbb{Z}_2\) for each \(l\), \(j\), \(k\) in \(\{ 0,...,7 \} \). On the other hand, \(\mathbb{Z}_2\) is the normal subgroup in the Moufang multiplicative loop \(G= \{ \pm i_k: k=0,...,7 \} \) such that its quotient by \(\mathbb{Z}_2\) is the commutative group \(G/\mathbb{Z}_2\) by Theorem IV.1.1 in [loc. cit.].
It is proposed in the article to use the Clifford algebra \(Cl_7=Cl(0,7,\mathbb{R})\) over \(\mathbb{R}\) for studying the octonion modules by using \(\hat{e}_1, ...,\hat{e_7}\) as generators of \(Cl_7\) with Clifford multiplication \(\hat{e}_{k_1}\cdot ... \cdot \hat{e}_{k_m}\) instead of \(L_{i_1},...,L_{i_7}\). The Clifford algebra is associative semisimple and isomorphic to \(Mat_{8\times 8}(\mathbb{R})\bigoplus Mat_{8\times 8}(\mathbb{R})\). The octonion algebra is simple. There is no any nontrivial homomorphism from \(Cl_7\) to \(L(\mathbb{O})\), or from \(Cl_7\) to \(\mathbf{O}\). For comparison \(L(\mathbb{H})=\mathbb{H}\), since \(\mathbb{H}\) is associative. Then Theorem 4.1 of Huo, Li, Ren contradicts to [\textit{N. Bourbaki}, (loc. cit.)] and the Cartan-Jacobson Theorem 3.28 and Corollary 3.29 in [\textit{R. D. Schafer}, (loc. cit.)].
In the article under review there is wrongly cited reference \([12]\) in Russian. It is available also in English translation: [\textit{S. V. Ludkovsky}, J. Math. Sci., New York 144, No. 4, 4301--4366 (2007; Zbl 1178.47057); translation from Sovrem. Mat. Prilozh. 35 (2005)]. In the latter paper were considered vector spaces \(X\) over \(\mathbb{O}\), which also have the structure of the two-sided octonion modules, such that \(X=X_0i_0\oplus X_1i_1\oplus ... \oplus X_7i_7\), where \(X_0\), ...,\(X_7\) are real vector spaces such that \(X_l\) is isomorphic to \(X_k\) for each \(l\), \(k\), \((ab)x_k=a(bx_k)\), \(bx_k=x_kb\), \(x_k(ab)=(x_ka)b\) for each \(a\),\( b\) in \(\mathbb{O}\), \(x_k\in X_k\), \(k\), \(l\) in \(\{ 0,...,7 \} \). It has properties: \((bb)x=b(bx)\), \(x(bb)=(xb)b\), \([a,b,x_ki_k]=[a,b,i_k]x_k\) for each \(x_k\in X_k\), \(k\in \{ 0,...,7 \} \), \(a\), \(b\) in \(\mathbb{O}\), implying by the \(\mathbb{R}\)-linearity in \(X\) and the corresponding identities in \(\mathbb{O}\) that \([a,b,x]=[b,x,a]=[x,a,b]\) for each \(x\in X\), \(a\), \(b\) in \(\mathbb{O}\), where \([a,b,x]=(ab)x-a(bx)\).Orthogonal abelian Cartan subalgebra decomposition of $\mathfrak{sl}_n$ over a finite commutative ringhttps://zbmath.org/1472.170212021-11-25T18:46:10.358925Z"Sriwongsa, Songpon"https://zbmath.org/authors/?q=ai:sriwongsa.songpon"Zou, Yi Ming"https://zbmath.org/authors/?q=ai:zou.yimingSummary: Orthogonal decomposition of the special linear Lie algebra over the complex numbers was studied in the early 1980s and attracted further attentions in the past decade due to its application in quantum information theory. In this paper, we study this decomposition problem of the special linear Lie algebra over a finite commutative ring with identity.Lie algebras of derivations with large abelian idealshttps://zbmath.org/1472.170852021-11-25T18:46:10.358925Z"Klymenko, I. S."https://zbmath.org/authors/?q=ai:klymenko.i-s"Lysenko, S. V."https://zbmath.org/authors/?q=ai:lysenko.s-v"Petravchuk, Anatoliy"https://zbmath.org/authors/?q=ai:petravchuk.anatolii-pSummary: Let \(\mathbb{K}\) be a field of characteristic zero, \(A=\mathbb{K}[x_1,\ldots,x_n]\) the polynomial ring and \(R=\mathbb{K}(x_1,\dots,x_n)\) the field of rational functions. The Lie algebra \(\widetilde{W}_n(\mathbb{K}):=\operatorname{Der}_{\mathbb{K}}R\) of all \(\mathbb{K} \)-derivation on \(R\) is a vector space (of dimension n) over \(R\) and every its subalgebra \(L\) has rank \(\operatorname{rk}_RL=\dim_RRL\). We study subalgebras \(L\) of rank \(m\) over \(R\) of the Lie algebra \(\widetilde{W}_n(\mathbb{K})\) with an abelian ideal \(I\subset L\) of the same rank \(m\) over \(R\). Let \(F\) be the field of constants of \(L\) in \(R\). It is proved that there exist a basis \(D_1, \ldots, D_m\) of \(FI\) over \(F\), elements \(a_1, \ldots, a_k\in R\) such that \(D_i(a_j)=\delta_{ij}\), \(i=1, \ldots, m\), \(j=1,\ldots, k\), and every element \(D\in FL\) is of the form \(D=\sum_{i=1}^mf_i(a_1, \ldots, a_k)D_i\) for some \(f_i\in F[t_1, \ldots t_k], \deg f_i\leqslant 1\). As a consequence it is proved that \(L\) is isomorphic to a subalgebra (of a very special type) of the general affine Lie algebra \(\mathrm{aff}_m(F)\).On the structures of hive algebras and tensor product algebras for general linear groups of low rankhttps://zbmath.org/1472.200992021-11-25T18:46:10.358925Z"Kim, Donggyun"https://zbmath.org/authors/?q=ai:kim.donggyun"Kim, Sangjib"https://zbmath.org/authors/?q=ai:kim.sangjib"Park, Euisung"https://zbmath.org/authors/?q=ai:park.euisungMinimal solutions of the rational interpolation problemhttps://zbmath.org/1472.410062021-11-25T18:46:10.358925Z"Benítez, Teresa Cortadellas"https://zbmath.org/authors/?q=ai:cortadellas-benitez.teresa"D'Andrea, Carlos"https://zbmath.org/authors/?q=ai:dandrea.carlos"Montoro, Eulàlia"https://zbmath.org/authors/?q=ai:montoro.eulaliaAuthors' abstract: We explore connections between the approach of solving the rational interpolation problem via resolutions of ideals and Syzygies, and the standard method provided by the Extended Euclidean Algorithm (EEA). As a consequence, we obtain explicit descriptions for solutions of minimal degrees in terms of the degrees of elements appearing in the EEA. This result allows us to describe the minimal degree in a \(\mu\)-basis of a polynomial planar parametrization in terms of a critical degree arising in the EEA.On the \(\cap\)-structure spaceshttps://zbmath.org/1472.540192021-11-25T18:46:10.358925Z"Hashemi, Jamal"https://zbmath.org/authors/?q=ai:hashemi.jamalSummary: The family \(\mathcal{M}_{X}\subseteq\mathcal{P}(X)\) is called an \(\cap\)-structure, when it is closed under the arbitrary intersection. This concept has been studied and considered in algebra, specially in lattices. Using this concept, we define a quasi topological structure which is called \(\cap\)-structure space. By studying this space, we attempt to explain some algebraic concepts through this structure.On polycyclic codes over a finite chain ringhttps://zbmath.org/1472.940772021-11-25T18:46:10.358925Z"Fotue-Tabue, Alexandre"https://zbmath.org/authors/?q=ai:tabue.alexandre-fotue"Martínez-Moro, Edgar"https://zbmath.org/authors/?q=ai:martinez-moro.edgar"Blackford, J. Thomas"https://zbmath.org/authors/?q=ai:blackford.jason-thomasThis paper explore polycyclic codes over finite chain rings. For a finite chain ring \(S\), these codes are identified with ideals of the quotient ring \(S[X]/\left\langle X^n- a(X) \right\rangle,\) where \(a(X)\) is a polynomial in \(S[X]\) such that the constant term be a unit. The intersection and sum of free polycyclic codes over \(S\) are characterized and the structure of non-free polycyclic codes is described by using strong Gröbner bases. It is shown that an \(S\)-linear code is polycyclic if and only if its dual is a sequential code. It is proved that the annihilator code and the euclidean orthogonal have the same type. The fact that the annihilator dual of polycyclic code (respectively free-polycyclic code) is also polycyclic code (respectively free-polycyclic code) is established. It is shown that the Galois image of non-free non-zero polycyclic codes is polycyclic and a Gröbner basis is provided. Necessary and sufficient condition for a free polycyclic code to be Galois-disjoint (respectively complete Galois disjoint) is given. The trace codes and restrictions of free polycyclic codes are characterized giving an analogue of Delsarte's theorem relating the trace code and the annihilator dual code.Skew constacyclic codes over the local Frobenius non-chain rings of order 16https://zbmath.org/1472.940902021-11-25T18:46:10.358925Z"Aydin, Nuh"https://zbmath.org/authors/?q=ai:aydin.nuh"Cengellenmis, Yasemin"https://zbmath.org/authors/?q=ai:cengellenmis.yasemin"Dertli, Abdullah"https://zbmath.org/authors/?q=ai:dertli.abdullah"Dougherty, Steven T."https://zbmath.org/authors/?q=ai:dougherty.steven-t"Saltürk, Esengül"https://zbmath.org/authors/?q=ai:salturk.esengulSummary: We introduce skew constacyclic codes over the local Frobenius non-chain rings of order 16 by defining non-trivial automorphisms on these rings. We study the Gray images of these codes, obtaining a number of binary and quaternary codes with good parameters as images of skew cyclic codes over some of these rings.