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The so-called Dao numbers are a sort of measure of the asymptotic behaviour of full properties of certain product ideals in a Noetherian local ring $R$ with infinite residue field and positive depth. In this paper, we answer a question of H. Dao on how to bound such numbers. The auxiliary tools range from Castelnuovo-Mumford regularity of appropriate graded structures to reduction numbers of the maximal ideal. In particular, we substantially improve previous results (and answer questions) by the authors. As an application, we provide new characterizations of when $R$ is regular; for instance, we show that this holds if and only if the maximal ideal of $R$ can be generated by a $d$-sequence (in the sense of Huneke) if and only if the third Dao number of any (minimal) reduction of the maximal ideal vanishes.

Let $G$ be a simple graph on the vertex set $\{v_{1},\ldots,v_{n}\}$. An algebraic object attached to $G$ is the toric ideal $I_G$. We say that $I_G$ is splittable if there exist subgraphs $G_1$ and $G_2$ of $G$ such that $I_G=I_{G_1}+I_{G_2}$, where both $I_{G_1}$ and $I_{G_2}$ are not equal to $I_G$. We show that $I_G$ is splittable if and only if it is edge splittable. We also prove that the toric ideal of a complete bipartite graph is not splittable. In contrast, we show that the toric ideal of a complete graph $K_n$ is always splittable when $n \geq 4$. Additionally, we show that the toric ideal of $K_n$ has a minimal splitting if and only if $4 \leq n \leq 5$. Finally, we prove that any minimal splitting of $I_G$ is also a reduced splitting.

Inspired by the ideas and techniques used in the study of cluster algebras we construct a new class of algebras, called bistellar cluster algebras, from closed oriented triangulated even-dimensional manifolds by performing middle-dimensional bistellar moves. This class of algebras exhibit the algebraic behaviour of middle-dimensional bistellar moves but do not satisfy the classical cluster algebra axiom: "every cluster variable in every cluster is exchangeable". Thus the construction of bistellar cluster algebras is quite different from that of a classical cluster algebra. Secondly, using bistellar cluster algebras and the techniques of combinatorial topology, we construct a direct system associated with a set of PL homeomorphic PL manifolds of dimension 2 or 4, and show that the limit of this direct system is a PL invariant., Comment: 26 pages

We give a direct construction of the ring spectrum of spherical Witt vectors of a perfect $\mathbb{F}_p$-algebra R as the completion of the spherical monoid algebra $\mathbb{S}[R]$ of the multiplicative monoid $(R,\cdot)$ at the ideal $I = \mathrm{fib}(\mathbb{S}[R] \to R)$. This generalizes a construction of Cuntz and Deninger. We also use this to give a description of the category of p-complete modules over the spherical Witt vectors and a universal property for spherical Witt vectors as an $\mathbb{E}_1$-ring., Comment: 7 pages

The ideal of the arc scheme of a double point or, equivalently, the differential ideal generated by the ideal of a double point is a primary ideal in an infinite-dimensional polynomial ring supported at the origin. This ideal has a rich combinatorial structure connecting it to singularity theory, partition identities, representation theory, and differential algebra. Macaulay inverse system is a powerful tool for studying the structure of primary ideals which describes an ideal in terms of certain linear differential operators. In the present paper, we show that the inverse system of the ideal of the arc scheme of a double point is precisely a vector space spanned by all the Wronskians of the variables and their formal derivatives. We then apply this characterization to extend our recent result on Poincar\'e-type series for such ideals.

We prove that cycles, wheels and block graphs have sequentially Cohen-Macaulay binomial edge ideals. Moreover, we provide a construction of new families of sequentially Cohen-Macaulay graphs by cones.

This article considers cuspidal curves whose coordinate rings are numerical semigroup algebras. Their rings of differential operators are shown to be cocommutative and conilpotent left Hopf algebroids. If the semigroups are symmetric so that the curves are Gorenstein, they are full Hopf algebroids (admit an antipode).

We give an elementary proof and generalize some Hochsters's type formulae on local cohomology and Ext's of squarefree modules

In this paper, we present OliVier a new Public Key Exchange cryptosystem that is based on a multivariate quadratic polynomial system: Oil & Vinegar polynomials together with fully quadratic ones. We describe its designing process, usage, complexity

We describe a general procedure of constructing non-trivial cup-products from non-trivial compositions in the derived category of a commutative ring. Using this, we show that there exist faithfully flat commutative boolean ring maps that are not descendable., Comment: Preprint, all comments are welcome

In this work, we present a mathematical formulation for machine learning of (1) functions on symmetric matrices that are invariant with respect to the action of permutations by conjugation, and (2) functions on point clouds that are invariant with respect to rotations, reflections, and permutations of the points. To achieve this, we construct $O(n^2)$ invariant features derived from generators for the field of rational functions on $n\times n$ symmetric matrices that are invariant under joint permutations of rows and columns. We show that these invariant features can separate all distinct orbits of symmetric matrices except for a measure zero set; such features can be used to universally approximate invariant functions on almost all weighted graphs. For point clouds in a fixed dimension, we prove that the number of invariant features can be reduced, generically without losing expressivity, to $O(n)$, where $n$ is the number of points. We combine these invariant features with DeepSets to learn functions on symmetric matrices and point clouds with varying sizes. We empirically demonstrate the feasibility of our approach on molecule property regression and point cloud distance prediction., Comment: 28 pages, 2 figures, 2 tables

We describe a method for parallelizing the lexicographic enumeration algorithm for the factorization set of an element in a numerical semigroup via bounds. This enables the use of GPU and distributed computing methods. We provide a CUDA implementation with measured runtimes., Comment: 9 pages, 1 figure

We construct proper moduli algebraic spaces of K-polystable $\mathbb{Q}$-Fano cones (a.k.a. Calabi-Yau cones) or equivalently their links i.e., Sasaki-Einstein manifolds with singularities. As a byproduct, it gives alternative algebraic construction of proper K-moduli of $\mathbb{Q}$-Fano varieties. In contrast to the previous algebraic proof of its properness ([BHLLX, LXZ]), we do not use the $\delta$-invariants ([FO, BJ]) nor the $L^2$-normalized Donaldson-Futaki invariants. We use the local normalized volume of [Li] and the higher $\Theta$-stable reduction instead., Comment: 56 pages

We prove a new kind of homological stability theorem for automorphism groups of finitely-generated projective modules over Dedekind domains, which takes into account all possible stabilisation maps between these, rather than only stabilisation by the free module of rank 1. We show the same kind of stability holds for Clausen and Jansen's reductive Borel--Serre spaces., Comment: 18 pages

Let $G$ be a simple graph with binomial edge ideal $J_G$. We prove how to calculate the multidegree of $J_G$ based on combinatorial properties of $G$. In particular, we study the set $S_{\min}(G)$ defined as the collection of subsets of vertices whose prime ideals have minimum codimension. We provide results which assist in determining $S_{\min}(G)$, then calculate $S_{\min}(G)$ for star, horned complete, barbell, cycle, wheel, and friendship graphs, and use the main result of the paper to obtain the multidegrees of their binomial edge ideals., Comment: 12 pages, 3 figures, submitted to the Proceedings of the AMS

In this note, we show that a ring $R$ is $S$-coherent if and only if every finitely presented $R$-module is $S$-coherent, providing a positive answer to a question proposed in [D. Bennis, M. El Hajoui, {\it On $S$-coherence}, J. Korean Math. Soc. \textbf{55} (2018), no.6, 1499-1512].

Linear hypersurfaces over a field $k$ have been playing a central role in the study of some of the challenging problems on affine spaces. Breakthroughs on such problems have occurred by examining two questions on linear polynomials of the form\\ $H:=\alpha(X_1,\dots,X_m)Y - F(X_1,\dots, X_m,Z,T)\in D:=k[X_1,\ldots,X_m, Y,Z,T]$: (i) Whether the affine variety $\mathbb{V}\in \mathbb{A}^{m+3}_k$ defined by $H$ is isomorphic to $\mathbb{A}^{m+2}_k$. (ii) If $\mathbb{V}$ is isomorphic to an affine space, then whether $H$ is a coordinate in $D$. In \cite{adv2}, the first two authors had addressed these questions when $\alpha$ is a monomial of the form $\alpha(X_1,\ldots,X_m) = X_1^{r_1}\dots X_m^{r_m}$; $r_i>1,\, 1 \leqslant i \leqslant m$ and $F$ is of a certain type. In this paper, using $K$-theory and $\mathbb{G}_a$-actions, we address these questions for a wider family of linear varieties. In particular, we show that when the characteristic of $k$ is zero, $F \in k[Z,T]$ and $H$ defines a hyperplane (i.e., the affine variety $\mathbb{V}$ defined by $H$ is an affine space), then $H$ is a coordinate in $D$ along with $X_1, X_2, \dots, X_m$. As a consequence we obtain a certain families of higher dimensional linear hyperplanes satisfying the Abhyankar-Sathaye conjecture on the Epimorphism Problem. Our results in arbitrary characteristic yield counter examples to the Zariski Cancellation Problem in positive characteristic.

For a commutative finite $\mathbb{Z}$-algebra, i.e., for a commutative ring $R$ whose additive group is finitely generated, it is known that the group of units of $R$ is finitely generated, as well. Our main results are algorithms to compute generators and the structure of this group. This is achieved by reducing the task first to the case of reduced rings, then to torsion-free reduced rings, and finally to an order in a reduced ring. The simplified cases are treated via a calculation of exponent lattices and various algorithms to compute the minimal primes, primitive idempotents, and other basic objects. All algorithms have been implemented and are available as a SageMath package. Whenever possible, the time complexity of the described methods is tracked carefully., Comment: 16 pages; submitted to jGCC

We study the behavior of multidegrees in families and the existence of numerical criteria to detect integral dependence. We show that mixed multiplicities of modules are upper semicontinuous functions when taking fibers and that projective degrees of rational maps are lower semicontinuous under specialization. We investigate various aspects of the polar multiplicities and Segre numbers of an ideal and introduce a new invariant that we call polar-Segre multiplicities. In terms of polar multiplicities and our new invariants, we provide a new integral dependence criterion for certain families of ideals. By giving specific examples, we show that the Segre numbers are the only invariants among the ones we consider that can detect integral dependence. Finally, we generalize the result of Gaffney and Gassler regarding the lexicographic upper semicontinuity of Segre numbers.

A conjecture raised in 1990 by C. Huneke predicts that, for a $d$-dimensional Noetherian local ring $R$, local cohomology modules of finitely generated $R$-modules have finitely many associated primes. Although counterexamples do exist, the conjecture has been confirmed in several cases, for instance if $d\leq 3$, and witnessed some progress in special cases for higher $d$. In this paper, assuming that $R$ is a factorial domain, we establish the case $d=4$, and under different additional conditions (in a couple of results) also the case $d=5$. Finally, when $R$ is regular and contains a field, we apply the Hartshorne-Lichtenbaum vanishing theorem as a tool to deal with the case $d=6$., Comment: 14 pages

Let G be a finite simple graph and let indm(G) and ordm(G) denote the induced matching number and the ordered matching number of G, respectively. We characterize all bipartite graphs G with indm(G) = ordm(G). We establish the Castelnuovo-Mumford regularity of powers of edge ideals and depth of powers of cover ideals for such graphs. We also give formulas for the count of connected non-isomorphic spanning subgraphs of the complete bipartite graph K_{m,n} for which indm(G) = ordm(G) = 2, with an explicit expression for the count when m = 2,3,4 and m <= n.

Let $(R,\mathfrak{m},k)$ be a Noetherian local ring and let $M$ be a finitely generated $R$-module. The main focus of this paper is to give positive answers for some long-standing homological conjectures over the idealization ring $R\ltimes M$. First, if $N$ is a $R\ltimes k$-module, we show that the vanishing of $\operatorname{Ext}_{R\ltimes k}^{i}(N,N\oplus (R\ltimes k))$ for $i=1,2,3$ gives that $N$ is free, and this provides a sharpened version of the Auslander-Reiten conjecture over $R\ltimes k$. Also, we give a characterization of the Betti numbers of an $R$-module over the idealization ring $R\ltimes M$ and, as a biproduct, we derive that the Jorgensen-Leuschke conjecture holds true for $R\ltimes M$. Further, we show that the true of Buchsbaum-Eisenbud-Horrocks and Total Rank conjectures over $R$ implies the true over $R\ltimes M$. This establishes particular answers for both conjectures for modules with infinite projective dimension, especially when $R$ is regular or a complete intersection ring. As applications of the idealization ring theory, we show that the Zariski-Lipman conjecture holds for any ring $R$ provided the Betti numbers of the $R$-derivation module $\operatorname{Der}_k(R)$, seen as $R\ltimes k$-module, satisfy the inequality $\beta_{n}^{R\ltimes k}(\operatorname{Der}_k(R))\leq\beta_{n-1}^{R\ltimes k}(\operatorname{Der}_k(R))$ for some $n>0$. Some implications regarding the Herzog-Vasconcelos conjecture are also provided., Comment: 18 pages, 0 figures

Given a right adjoint functor between triangulated categories and an object in the target category, we show that the unit map of adjunction on that object is a split monomorphism if and only if the object belongs to the additive closure of (all possible) shifts of an object in the image of the functor. Applications to geometric context related to (derived) splinters and rational singularities are given.

We prove that for finitely generated abelian groups $A$ and $B$, the space of $\mathbb{E}_\infty$-ring maps between the spherical groups rings $\mathbb{S}[A] \to \mathbb{S}[B]$ is equivalent to the discrete set of group homomorphisms $A \to B$. We also prove generalizations where the sphere is replaced by other ring spectra, e.g. we give a formula for the strict units in group rings of the form $R[A]$ for $A$ a finite $p$-group and $R$ $p$-completely chromatically complete., Comment: 41 pages, comments welcome

Almost commuting operators were introduced in 1985 by George Wilson to present generalizations the Korteweg-de Vries hierarchy, nowadays known as Gelfand-Dikey (GD) hierarchies. In this paper, we review the formal construction of the vector space of almost commuting operators with a given ordinary differential operator (ODO), with the ultimate goal of obtaining a basis by computational routines, using the language of differential polynomials. We use Wilson's results on weigheted ODOs to guarantee the solvability of the triangular system that allows to compute the homogeneous almost commuting operator of a given order in the ring of ODOs. As a consequence the computation of the equations of the GD hierarchies is obtained without using pseudo-differential operators. The algorithms to calculate the almost commuting basis and the GD hierarchies in the ring of ODOs are implemented in SageMath, and explicit examples are provided.

The paper systematically classifies rings based on the dominant metric dimensions (Ddim) of their associated CZDG, establishing consequential bounds for the Ddim of these compressed zero-divisor graphs. The authors investigate the interplay between the ring-theoretic properties of a ring ( R ) and associated CZDG. An undirected graph consisting of vertex set ( Z(R_E)\{[0]}\ =\ R_E\{[0],[1]}), where ( R_E=\{[x]:\ x\in R\} ) and ([x]=\{y\in R:\ \text{ann}(x)=\text{ann}(y)\} ) is called a compressed zero-divisor graph, denoted by ( \Gamma_E(R) ). An edge is formed between two vertices ([x]) and ([y]) of ( Z(R_E) ) if and only if ([x][y]=[xy]=[0]), that is, iff ( xy=0 ). For a ring ( R ), graph ( G ) is said to be realizable as ( \Gamma_E(R) ) if ( G ) is isomorphic to ( \Gamma_E(R) ). Moreover, an exploration into the Ddim of realizable graphs for rings is conducted, complemented by illustrative examples reinforcing the presented results. A recent discussion within the paper elucidates the nuanced relationship between Ddim, diameter, and girth within the domain of compressed zero-divisor graphs. This research offers a comprehensive and insightful analysis at the intersection of algebraic structures and graph theory, providing valuable contributions to the current mathematical discourse., Comment: 13

Let $I_{n,m} = (x_1\cdots x_{m},x_2 \cdots x_{m+1},\ldots,x_{n+1}x_{n+2}\cdots x_{n+m})$ be the $m$-path ideal of a path of length $n + m-1$ over a polynomial ring $S = \mathrm{k}[x_1,\ldots,x_{n+m}]$. We compute all the graded Betti numbers of all powers of $I_{n,m}$.

We provide an effective method to compute multiplier ideals of meromorphic functions in dimension two. We also prove that meromorphic functions only have integer jumping numbers after reaching some threshold., Comment: 12 pages, 0 figures

Quillen's fundamental spectral sequences relate Andr\'{e}-Quillen homology and cohomology to Tor and Ext functors. The five-term exact sequences arising from these spectral sequences are leveraged to characterize regular and complete intersection local rings. Despite their immense importance in the theory of Andr\'{e}-Quillen homology and cohomology, these spectral sequences are not treated in the literature as they merit. A sketchy argument with gaps for a special case of the first spectral sequence appears in an unpublished manuscript of Quillen, while the second spectral sequence is only stated without proof in another paper of Quillen. A rigorous proof of the spectral sequences requires a delicate investigation of the subtle structures involved. The goal of this expository article is to present a comprehensive and detailed proof of Quillen's fundamental spectral sequences in a readable and well-structured manner.

Given a valuation $v$ with quotient field $K$ and a sequence $\mathcal{K} :K_0\subseteq K_1\subseteq\cdots$ of finite extensions of $K$, we construct a weighted tree $\mathcal{T}(v,\mathcal{K})$ encoding information about the ramification of $v$ in the extensions $K_i$; conversely, we show that a weighted tree $\mathcal{T}$ can be expressed as $\mathcal{T}(v,\mathcal{K})$ under some mild hypothesis on $v$ or on $\mathcal{T}$. We use this construction to construct, for every countable successor ordinal number $\alpha$, an almost Dedekind domain $D$, integral over $V$ (the valuation domain of $v$) whose SP-rank is $\alpha$. Subsequently, we extend this result to countable limit ordinal numbers by considering integral extensions of Dedekind domains with countably many maximal ideals.

An integral domain $D$ is called an SP-domain if every ideal is a product of radical ideals. Such domains are always almost Dedekind domains, but not every almost Dedekind domain is an SP-domain. The SP-rank of $D$ provides a natural measure of the deviation of $D$ from being an SP-domain. In the present paper we show that every ordinal number $\alpha$ can be realized as the SP-rank of an almost Dedekind domain.

We study the distribution of the Hodge spectral exponents of an irreducible plane curve by comparing it with a continuous distribution. We provide a closed formula for this difference in terms of numerical invariants of the curve. We characterize those families of irreducible plane curves whose limit distribution of Hodge spectral exponents is the continuous distribution and we provide intervals of dominating values., Comment: 25 pages, 3 figures

This work presents a range of triangulated characterizations for important classes of singularities such as derived splinters, rational singularities, and Du Bois singularities. An invariant called 'level' in a triangulated category can be used to measure the failure of a variety to have a prescribed singularity type. We provide explicit computations of this invariant for reduced Nagata schemes of Krull dimension one and for affine cones over smooth projective hypersurfaces. Furthermore, these computations are utilized to produce upper bounds for Rouquier dimension on the respective bounded derived categories., Comment: Current: Removed properness assumptions, removed Section 4, improved exposition. Previous: Initial version

Let $R$ be a commutative ring and let $n \geq 1.$ We study $\Gamma(s)$, the generating function and Ann$(s)$, the ideal of characteristic polynomials of $s$, an $n$--dimensional sequence over $R$. We express $f(X_1,\ldots,X_n) \cdot \Gamma(s)(X_1^{-1},\ldots ,X_n^{-1})$ as a partitioned sum. That is, we give (i) a $2^n$--fold ``border'' partition (ii) an explicit expression for the product as a $2^n$--fold sum; the support of each summand is contained in precisely one member of the partition. A key summand is $\beta_0(f,s)$, the ``border polynomial'' of $f$ and $s$, which is divisible by $X_1\cdots X_n$. We say that $s$ is {\em eventually rectilinear} if the elimination ideals Ann$(s)\cap R[X_i]$ contain an $f_i(X_i)$ for $1 \leq i \leq n$. In this case, we show that $\mbox{Ann}(s)$ is the ideal quotient $(\sum_{i=1}^n(f_i)\ :\ \beta_0(f,s)/(X_1\cdots X_n)).$ When $R$ and $R[[X_1,X_2, \ldots ,X_n]]$ are factorial domains (e.g. $R$ a principal ideal domain or ${\Bbb F}[X_1,\ldots,X_n]$), we compute {\em the monic generator} $\gamma _i$ of $\mbox{Ann}(s) \cap R[X_i]$ from known $f_i \in \mbox{Ann}(s) \cap R[X_i]$ or from a finite number of $1$--dimensional linear recurring sequences over $R$. Over a field ${\Bbb F}$ this gives an $O(\prod_{i=1}^n \delta \gamma _i^3)$ algorithm to compute an ${\Bbb F}$--basis for $\mbox{Ann}(s)$., Comment: This is my original latex document submitted to Journal of Symbolic Computation without the typographical errors which were introduced: 'The Journal apologizes for the typographical errors in Norton (1995) introduced in the subediting process'; see this journal, (1995)20, 769-770

We prove that if $G$ is a totally bounded abelian group \st\ its dual group $\widehat{G}_p$ equipped with the finite-open topology is a Baire group, then every compact subset of $G$ must be finite. This solves an open question by Chasco, Dom\'inguez and Tkachenko. {Among other consequences, we obtain an example of a group that is $g$-dense in its completion but is not $g$-barrelled. This solves a question proposed by Au${\beta}$enhofer and Dikranjan.}

In this paper we adress the question of I. Smirnov and K. Tucker on the dual $F$-signature of the Veronese subrings of polynomial rings in $n$ variables using methods of commutative algebra.

This paper is concerned with local cohomology sheaves on generalized flag varieties supported in closed Schubert varieties, which carry natural structures as (mixed Hodge) D-modules. We employ Kazhdan--Lusztig theory and Saito's theory of mixed Hodge modules to describe a general strategy to calculate the simple composition factors, Hodge filtration, and weight filtration on these modules. Our main tool is the Grothendieck--Cousin complex, introduced by Kempf, which allows us to relate the local cohomology modules in question to parabolic Verma modules over the corresponding Lie algebra. We show that this complex underlies a complex of mixed Hodge modules, and is thus endowed with Hodge and weight filtrations. As a consequence, strictness implies that computing cohomology commutes with taking associated graded with respect to both of these filtrations. We execute this strategy to calculate the composition factors and weight filtration for Schubert varieties in the Grassmannian, in particular showing that the weight filtration is controlled by the augmented Dyck patterns of Raicu--Weyman. As an application, upon restriction to the opposite big cell, we recover the simple composition factors and weight filtration on local cohomology with support in generic determinantal varieties., Comment: 27 pages, comments welcome

Gotzmann's persistence theorem enables us to confirm the Hilbert polynomial of a subscheme of projective space by checking the Hilbert function in just two points, regardless of the dimension of the ambient space. We generalise this result to products of projective spaces, and then extend our result to any smooth projective toric variety. In the case of products of projective spaces, the number of points depends solely on the Picard rank of the ambient space, rather than on the dimension. For a more general smooth projective toric variety this number depends on the number of Hilbert basis elements of the nef cone.

In this paper, we compute the dual $F$-signatures of certain toric rings by using combinatorial techniques. Specifically, we calculate the dual $F$-signatures of Veronese subrings of polynomial rings. Moreover, we give an upper bound for the dual $F$-signatures of Segre products of polynomial rings and show that this upper bound is attained in some cases., Comment: 14 pages

Each numerical semigroup $S$ with smallest positive element $m$ corresponds to an integer point in a polyhedral cone $C_m$, known as the Kunz cone. The faces of $C_m$ form a stratification of numerical semigroups that has been shown to respect a number of algebraic properties of $S$, including the combinatorial structure of the minimal free resolution of the defining toric ideal $I_S$. In this work, we prove that the structure of the infinite free resolution of the ground field $\Bbbk$ over the semigroup algebra $\Bbbk[S]$ also respects this stratification, yielding a new combinatorial approach to classifying homological properties like Golodness and rationality of the poincare series in this setting. Additionally, we give a complete classification of such resolutions in the special case $m = 4$, and demonstrate that the associated graded algebras do not generally respect the same stratification.

Over a commutative Noetherian ring, we show that Auslander-Reiten conjecture holds true for the class of (finitely generated) modules whose dual has finite complete intersection dimension. Our another result validates the conjecture for the class of modules whose self dual has finite complete intersection dimension and either the module or its dual has finite Gorenstein dimension., Comment: Theorem 3.6 in the first version is modified to Theorem 3.7 in this version. Comments and suggestions are welcome!

The notions of N-hyperideals and J-hyperideals as two classes of hyperideals were recently defined in the context of Krasner (m,n)-hyperrings. These concepts are created on the basis of the intersection of all n-ary prime hyperideals and the intersection of all maximal hyperideals, respectively. Despite being vastly different in many aspects, they shar numerous similar properties. The aim of this research work is to merge them under one frame called n-ary delta(0)-hyperideals where the function delta assigns to each hyperideals of a Krasner (m,n)-hyperring a hyperideal of the same hyperring. We give various properties of n-ary delta(0)-hyperideals and use them to characerize certain classes of hyperring such as hyperintegral domains and local hyperrings. Moreover, we introduce the notions of (s,n)-absorbing delta(0)-hyperideals and weakly (s,n)-absorbing delta(0)-hyperideals.

We reobtain and often refine prior criteria due to Kaplansky, McGovern, Roitman, Shchedryk, Wiegand, and Zabavsky--Bilavska and obtain new criteria for a Hermite ring to be an \textsl{EDR}. We mention three criteria: (1) a Hermite ring $R$ is an \textsl{EDR} iff for all pairs $(a,c)\in R^2$, the product homomorphism $U(R/Rac)\times U\bigl(R/Rc(1-a)\bigr)\to U(R/Rc)$ between groups of units is surjective; (2) a reduced Hermite ring is an \textsl{EDR} iff it is a pre-Schreier ring and for each $a\in R$, every zero determinant unimodular $2\times 2$ matrix with entries in $R/Ra$ lifts to a zero determinant matrix with entries in $R$; (3) a B\'{e}zout domain $R$ is an \textsl{EDD} iff for all triples $(a,b,c)\in R^3$ there exists a unimodular pair $(e,f)\in R^2$ such that $(a,e)$ and $(be+af,1-a-bc)$ are unimodular pairs. We use these criteria to show that each B\'{e}zout ring $R$ that is an $(SU)_2$ ring (as introduced by Lorenzini) such that for each nonzero $a\in R$ there exists no nontrivial self-dual projective $R/Ra$-module of rank $1$ generated by $2$ elements (e.g., all its elements are squares), is an \textsl{EDR}., Comment: 19 pages. arXiv admin note: substantial text overlap with arXiv:2303.08413

In this paper we adapt the notion of the nucleus defined by Benson, Carlson, and Robinson to compact Lie groups in non-modular characteristic. We show that it describes the singularities of the projective scheme of the cohomology of its classifying space. A notion of support for singularity categories of ring spectra (in the sense of Greenlees and Stevenson) is established, and is shown to be precisely the nucleus in this case, consistent with a conjecture of Benson and Greenlees for finite groups., Comment: 15 Pages

In this paper, we explore a class of numerical semigroups initiated by Kunz and Waldi containing two coprime numbers $p < q$, which we call KW semigroups. We characterize KW numerical semigroups by their principal matrices. We present a necessary and sufficient criterion for a matrix to be the principal matrix of a KW semigroup. An explicit description of the minimal resolutions of numerical semigroups in the same class with small embedding dimensions 3 and 4 is given. We give a generalization of this notion to three dimensions using lattice paths under a plane and present some preliminary results and questions., Comment: 19 pages, 6 figures

Our purpose in this work is multifold. First, we provide general criteria for the finiteness of the projective and injective dimensions of a finite module $M$ over a (commutative) Noetherian ring $R$. Second, in the other direction, we investigate the impact of the finiteness of certain homological dimensions of $M$ if $R$ is local, mainly when $R$ is Cohen-Macaulay and with a partial focus on duals. Along the way, we produce various freeness criteria for modules. Finally, we give applications, including characterizations of when $R$ is Gorenstein (and other ring-theoretic properties as well, sometimes in the prime characteristic setting), particularly by means of its anticanonical module, and in addition we address special cases of some long-standing conjectures; for instance, we confirm the 1985 conjecture of Vasconcelos on normal modules in case the module of differentials is almost Cohen-Macaulay., Comment: Comments are welcome!

Let $G$ be a finite group. A contravariant functor from the category of finite free $G$-sets to vector spaces has an associated Hilbert series, which records the underlying sequence of $G^n$ representations, $n \in \mathbb N$. We prove that this Hilbert series is rational with denominator given by linear polynomials with coefficients in the field generated by the character table of $G$., Comment: 8 pages

Let A be a commutative noetherian local DG-ring with bounded cohomology. The Intersection Theorem for DG-modules is examined and some of its applications are provided. The first is to prove the DG-setting of the amplitude inequality, New Intersection Theorem and Krull's principle ideal theorem. The second is to solve completely the Minamoto's conjecture in [Israel J. Math. 242 (2021) 1-36]. The third is to show the DG-version of the Bass conjecture about Cohen-Macaulay rings and the Vasconcelos conjecture about Gorenstein rings., Comment: 18 pages, comments welcome!

Let $R$ be a commutative ring If $\mathcal{C}_1$ and $\mathcal{C}_2$ are $R$-linear triangulated categories then we can give an obvious triangulated structure on $\mathcal{C} = \mathcal{C}_1 \oplus \mathcal{C}_2$ where $Hom_\mathcal{C}(U, V) = 0$ if $U \in \mathcal{C}_i$ and $V \in \mathcal{C}_j$ with $i \neq j$. We say a $R$-linear triangulated category $\mathcal{C}$ is disconnected if $\mathcal{C} = \mathcal{C}_1 \oplus \mathcal{C}_2$ where $\mathcal{C}_i$ are non-zero triangulated subcategories of $\mathcal{C}$. Let $\mathcal{C}_i$ and $\mathcal{D}_j$ be connected triangulated $R$ categories with $i \in \Gamma$ and $j \in \Lambda$. Suppose there is an equivalence of triangulated $R$-categories \[ \Phi \colon \bigoplus_{i \in \Gamma}\mathcal{C}_i \xrightarrow{\cong} \bigoplus_{j \in \Lambda}\mathcal{D}_j \] Then we show that there is a bijective function $\pi \colon \Gamma \rightarrow \Lambda$ such that we have an equivalence $\mathcal{C}_i \cong \mathcal{D}_{\pi(i)} $ for all $i \in \Gamma$. We give several examples of connected triangulated categories and also of triangulated subcategories which decompose into utmost finitely many components.

It was recently conjectured that every component of a discrete rational dynamical system is a solution to an algebraic difference equation that is linear in its highest-shift term (a quasi-linear equation). Holonomic sequences are trivially seen as solutions to such dynamical systems. We prove that the conjecture holds for holonomic sequences and propose two algorithms for converting holonomic recurrence equations into such quasi-linear equations. The two algorithms differ in their efficiency and the minimality of orders in their outputs., Comment: 9 pages