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In this paper, we show the equality of the (local) v -number and Castelnuovo-Mumford regularity of certain classes of Gorenstein algebras, including the class of Gorenstein monomial algebras. Also, for the same classes of algebras with the assumption of level, we show that the (local) v -number serves as an upper bound for the regularity. As an application, we get the equality between the v -number and regularity for Stanley-Reisner rings of matroid complexes. Furthermore, this paper investigates the v -number of Frobenius powers of graded ideals in prime characteristic setup. In this direction, we demonstrate that the v -numbers of Frobenius powers of graded ideals have an asymptotically linear behaviour. In the case of unmixed monomial ideals, we provide a method for computing the v -number without prior knowledge of the associated primes. [ABSTRACT FROM AUTHOR]

Let A be a connected cochain DG algebra such that H (A) is a Noetherian graded algebra. We give some criteria for A to be homologically smooth in terms of the singularity category, the cone length of the canonical module k and the global dimension of A . For any cohomologically finite DG A -module M, we show that it is compact when A is homologically smooth. If A is in addition Gorenstein, we get CMreg M = depth A A + Ext. reg M < ∞ , where CMreg M is the Castelnuovo-Mumford regularity of M, depth A A is the depth of A and Ext. reg M is the Ext-regularity of M. [ABSTRACT FROM AUTHOR]

Assume that G is a graph with edge ideal I (G). For every integer s ≥ 1 , we denote the squarefree part of the s -th symbolic power of I (G) by I (G) { s }. We determine an upper bound for the regularity of I (G) { s } when G is a chordal graph. If G is a Cameron-Walker graph, we compute reg (I (G) { s }) in terms of the induced matching number of G. Moreover, for any graph G , we provide sharp upper bounds for reg (I (G) { 2 }) and reg (I (G) { 3 }). [ABSTRACT FROM AUTHOR]

Let G be a finite simple graph, and let NI(G) denote the closed neighborhood ideal of G in a polynomial ring R. We show that if G is a forest, then the Castelnuovo–Mumford regularity of R/NI(G) is the same as the matching number of G, thus proving a conjecture of Sharifan and Moradi in the affirmative. We also show that the matching number of G provides a lower bound for the Castelnuovo–Mumford regularity of R/NI(G) for any G. Furthermore, we prove that if G contains a simplicial vertex, then NI(G) admits a Betti splitting, and consequently, we show that the projective dimension of R/NI(G) is also bounded below by the matching number of G, if G is a forest or a unicyclic graph. [ABSTRACT FROM AUTHOR]

We study the properties of the rook complex ℛ of a polyomino 풫 seen as independence complex of a graph G, and the associated Stanley–Reisner ideal Iℛ. In particular, we characterize the polyominoes 풫 having a pure rook complex, and the ones whose Stanley–Reisner ideal has linear resolution. Furthermore, we prove that for a class of polyominoes the Castelnuovo–Mumford regularity of Iℛ coincides with the induced matching number of G. [ABSTRACT FROM AUTHOR]

We study the category of GL -equivariant modules over the infinite exterior algebra in positive characteristic. Our main structural result is a shift theorem à la Nagpal. Using this, we obtain a Church–Ellenberg type bound for the Castelnuovo–Mumford regularity. We also prove finiteness results for local cohomology. [ABSTRACT FROM AUTHOR]

Matrix Schubert varieties are affine varieties arising in the Schubert calculus of the complete flag variety. We give a formula for the Castelnuovo–Mumford regularity of matrix Schubert varieties, answering a question of Jenna Rajchgot. We follow her proposed strategy of studying the highest-degree homogeneous parts of Grothendieck polynomials, which we call Castelnuovo–Mumford polynomials. In addition to the regularity formula, we obtain formulas for the degrees of all Castelnuovo–Mumford polynomials and for their leading terms, as well as a complete description of when two Castelnuovo–Mumford polynomials agree up to scalar multiple. The degree of the Grothendieck polynomial is a new permutation statistic which we call the Rajchgot index; we develop the properties of Rajchgot index and relate it to major index and to weak order. [ABSTRACT FROM AUTHOR]

It is well known that the Eisenbud-Goto regularity conjecture is true for arithmetically Cohen-Macaulay varieties, projective curves, smooth surfaces, smooth threefolds in \mathbb {P}^5, and toric varieties of codimension two. After J. McCullough and I. Peeva constructed counterexamples in 2018, it has been an interesting question to find the categories such that the Eisenbud-Goto conjecture holds. So far, surface counterexamples have not been found while counterexamples of any dimension greater or equal to 3 are known. In this paper, we construct counterexamples to the Eisenbud-Goto conjecture for projective surfaces in \mathbb {P}^4 and investigate projective invariants, cohomological properties, and geometric properties. The counterexamples are constructed via binomial rational maps between projective spaces. [ABSTRACT FROM AUTHOR]

In this paper, we study the class of affine semigroups generated by integral vectors, whose components are in generalized arithmetic progression and we observe that the defining ideal is determinantal. We also give a sufficient condition on the defining ideal of the semigroup ring for the equality of the Betti numbers of the defining ideal and those of its initial ideal. We introduce the notion of an affine semigroup generated by join of two affine semigroups and show that it preserves some nice properties, including Cohen-Macaulayness, when the constituent semigroups have those properties. [ABSTRACT FROM AUTHOR]

Consider a Noetherian domain R and a finite group G ⊆ G l n (R). We prove that if the ring of invariants R [ x 1 , ... , x n ] G is a Cohen-Macaulay ring, then it is generated as an R -algebra by elements of degree at most max ⁡ (| G | , n (| G | − 1)). As an intermediate result we also show that if R is a Noetherian local ring with infinite residue field then such a ring of invariants of a finite group G over R contains a homogeneous system of parameters consisting of elements of degree at most | G |. [ABSTRACT FROM AUTHOR]

Let M be a finitely generated module of dimension d over a Noetherian local ring (A, 픪) and I an 픪-primary ideal. Let be a pair of good I-filtrations 픽 and 픽′ of M. We show that the Hilbert coefficients ei(픽) are bounded below and above in terms of i, e0(픽′),…,e i(픽′), and reduction numbers of 픽 and 픽′, for all i ≥ 1. [ABSTRACT FROM AUTHOR]

For a finite dimensional vector space V of dimension n, we consider the incidence correspondence (or partial flag variety) X\subset \mathbb {P}V \times \mathbb {P}V^{\vee }, parametrizing pairs consisting of a point and a hyperplane containing it. We completely characterize the vanishing and non-vanishing behavior of the cohomology groups of line bundles on X in characteristic p>0. If n=3 then X is the full flag variety of V, and the characterization is contained in the thesis of Griffith from the 70s. In characteristic 0, the cohomology groups are described for all V by the Borel–Weil–Bott theorem. Our strategy is to recast the problem in terms of computing cohomology of (twists of) divided powers of the cotangent sheaf on projective space, which we then study using natural truncations induced by Frobenius, along with careful estimates of Castelnuovo–Mumford regularity. When n=3, we recover the recursive description of characters from recent work of Linyuan Liu, while for general n we give character formulas for the cohomology of a restricted collection of line bundles. Our results suggest truncated Schur functions as the natural building blocks for the cohomology characters. [ABSTRACT FROM AUTHOR]

We study the continuous CM-regularity of torsion-free coherent sheaves on polarized irregular smooth projective varieties (X, OX(1)), and its relation with the theory of generic vanishing. This continuous variant of the Castelnuovo–Mumford regularity was introduced by Mustopa, and he raised the question whether a continuously 1-regular such sheaf F is GV. Here we answer the question in the affirmative for many pairs (X, OX(1)) which includes the case of any polarized abelian variety. Moreover, for these pairs, we show that if F is continuously k-regular for some positive integer k ≤ dim X, then F is a GV−(k−1) sheaf. Further, we extend the notion of continuous CM-regularity to a real valued function on the ℚ-twisted bundles on polarized abelian varieties (X, OX(1)), and we show that this function can be extended to a continuous function on N1(X)ℝ. We also provide syzygetic consequences of our results for Oℙ(E)(1) on ℙ(ɛ) associated to a 0-regular bundle ɛ on polarized abelian varieties. In particular, we show that Oℙ(E)(1) satisfies the Np property if the base-point freeness threshold of the class of OX(1) in N1(X) is less than 1/(p + 2). This result is obtained using a theorem in the Appendix A written by Atsushi Ito. [ABSTRACT FROM AUTHOR]

Let G be a simple graph and I 3 (G) be its 3-path ideal in the corresponding polynomial ring R. In this article, we prove that for an arbitrary graph G, reg (R / I 3 (G)) is bounded below by 2 ν 3 (G) , where ν 3 (G) denotes the 3-path induced matching number of G. We give a class of graphs, namely trees for which the lower bound is attained. Also, for a unicyclic graph G, we show that reg (R / I 3 (G)) ≤ 2 ν 3 (G) + 2 and provide an example that shows that the given upper bound is sharp. [ABSTRACT FROM AUTHOR]

We study chains of nonzero edge ideals that are invariant under the action of the monoid Inc of increasing functions on the positive integers. We prove that the sequence of Castelnuovo–Mumford regularity of ideals in such a chain is eventually constant with limit either 2 or 3, and we determine explicitly when the constancy behavior sets in. This provides further evidence to a conjecture on the asymptotic linearity of the regularity of Inc -invariant chains of homogeneous ideals. The proofs reveal unexpected combinatorial properties of Inc -invariant chains of edge ideals. [ABSTRACT FROM AUTHOR]

Let ≥ 2 be an integer. The graph G n ¯ is obtained by letting all the elements of {0, ... , − 1} to be the vertices and defining distinct vertices and to be adjacent if and only if gcd(+ ,) ≠ 1. In this paper, we give some bounds for the Castelnuovo–Mumford regularity of the edge ideals and their powers for G n ¯ . [ABSTRACT FROM AUTHOR]

We prove that level binomial edge ideals with regularity 2 and pseudo-Gorenstein binomial edge ideals with regularity 3 are cones, and we describe them completely. Also, we characterize level and pseudo-Gorenstein binomial edge ideals of bipartite graphs. [ABSTRACT FROM AUTHOR]

Let A be a noetherian connected graded algebra. We introduce and study homological invariants that are weighted sums of the homological and internal degrees of cochain complexes of graded A-modules, providing weighted versions of Castelnuovo–Mumford regularity, Tor-regularity, Artin–Schelter regularity, and concavity. In some cases an invariant (such as Tor-regularity) that is infinite can be replaced with a weighted invariant that is finite, and several homological invariants of complexes can be expressed as weighted homological regularities. We prove a few weighted homological identities some of which unify different classical homological identities and produce interesting new ones. [ABSTRACT FROM AUTHOR]

Let G = K n 1 , n 2 , ... , n r be a complete multipartite graph on [n] with n > r > 1 and JG being its binomial edge ideal. It is proved that the Castelnuovo-Mumford regularity reg(JGt) is 2t +1 for any positive integer t. [ABSTRACT FROM AUTHOR]

Let A = { a 0 , … , a n - 1 } be a finite set of n ≥ 4 non-negative relatively prime integers, such that 0 = a 0 < a 1 < ⋯ < a n - 1 = d . The s-fold sumset of A is the set sA of integers that contains all the sums of s elements in A. On the other hand, given an infinite field k, one can associate with A the projective monomial curve C A parametrized by A, C A = { (v d : u a 1 v d - a 1 : ⋯ : u a n - 2 v d - a n - 2 : u d) ∣ (u : v) ∈ P k 1 } ⊂ P k n - 1. The exponents in the previous parametrization of C A define a homogeneous semigroup S ⊂ N 2 . We provide several results relating the Castelnuovo–Mumford regularity of C A to the behavior of the sumsets of A and to the combinatorics of the semigroup S that reveal a new interplay between commutative algebra and additive number theory. [ABSTRACT FROM AUTHOR]

We generalize the results by Eisenbud and Schreyer about Ulrich bundles over Veronese varieties to Segre-Veronese varieties. We discuss the range where we have natural cohomology and we construct multigraded resolutions and monads for Ulrich bundles of any rank. Moreover we give cohomological characterizations for significant families of bundles. [ABSTRACT FROM AUTHOR]

Let = (V () , E ()) be a weighted oriented graph and I () denote the corresponding edge ideal. In this paper, we give a combinatorial characterization of I () which has a linear resolution. As a consequence, we prove that if I () is the edge ideal of a weighted oriented graph , then I () has a linear resolution if and only if all powers of I () have a linear resolution. Also, we prove that if is a weighted oriented graph and w (x) > 1 for all x ∈ V () , then I () has a linear resolution if and only if all powers of I () have linear quotients. We provide a lower bound for the regularity of powers of edge ideals of weighted oriented graphs in terms of induced matching. Finally, we obtain a general upper bound for the regularity of edge ideals of weighted oriented graphs. [ABSTRACT FROM AUTHOR]

We investigate the Castelnuovo–Mumford regularity and the multiplicity of the toric ring associated with a three-dimensional Ferrers diagram. In particular, in the rectangular case, we provide direct formulas for these two important invariants. Then, we compare these invariants for an accompanying pair of Ferrers diagrams under some mild conditions and bound the Castelnuovo–Mumford regularity for more general cases. [ABSTRACT FROM AUTHOR]

This paper gives explicit formulas for the reduction number and the Castelnuovo–Mumford regularity of projective monomial curves. [ABSTRACT FROM AUTHOR]

Let S = k [ x 1 , ... , x n ] be a polynomial ring in n variables over a field k. When I , J and K are monomial ideals of S generated by powers of the variables x i , it is proved that reg (I J K) ≤ reg (I) + reg (J) + reg (K). If n ≤ 4 , the same result for the product of a finite number of ideals as above is proved. [ABSTRACT FROM AUTHOR]

We give degree formulas for Grothendieck polynomials indexed by vexillary permutations and 1432-avoiding permutations via tableau combinatorics. These formulas generalize a formula for degrees of symmetric Grothendieck polynomials which appeared in previous joint work of the authors with Y. Ren and A. St. Dizier. We apply our formulas to compute Castelnuovo-Mumford regularity of classes of generalized determinantal ideals. In particular, we give combinatorial formulas for the regularities of all one-sided mixed ladder determinantal ideals. We also derive formulas for the regularities of certain Kazhdan-Lusztig ideals, including those coming from open patches of Schubert varieties in Grassmannians. This provides a correction to a conjecture of Kummini-Lakshmibai-Sastry-Seshadri (2015). [ABSTRACT FROM AUTHOR]

Let G be a simple graph and I (X G) = φ − 1 (x i 2 − x j 2 : i , j ∈ V G) , where φ : K [ E G ] → K [ V G ] is the homomorphism that sends an edge to the product of its vertices. The ideal I (X G) is Cohen–Macaulay, one-dimensional and binomial. If G is bipartite, it is known that the Castelnuovo–Mumford regularity of I (X G) is equal to the maximum cardinality of a set of edges having no more than half of the edges of any Eulerian subgraph of G. Here, with respect to the grevlex order associated to an ordering of the edge set of G, we describe a Gröbner basis for I (X G) , and we characterize the standard monomials of the ideal (I (X G) , t e) in terms of even sets of vertices marked with a parity. Using these results, we give a combinatorial interpretation of the degree of I (X G) , via the set of even sets of vertices of G; and we show that the Castelnuovo–Mumford regularity of I (X G) , for any graph, is the maximum cardinality of a set of edges having no more than half of the edges of any even Eulerian subgraph of G or, equivalently, the maximum cardinality of a minimum fixed parity T-join. [ABSTRACT FROM AUTHOR]

In this paper we study and relate several invariants connected to the solving degree of a polynomial system. This provides a rigorous framework for estimating the complexity of solving a system of polynomial equations via Gröbner bases methods. Our main results include a connection between the solving degree and the last fall degree and one between the degree of regularity and the Castelnuovo–Mumford regularity. [ABSTRACT FROM AUTHOR]

Let G be a finite simple graph and I (G) denote the corresponding edge ideal in a polynomial ring over a field . In this paper, we obtain upper bounds for the Castelnuovo–Mumford regularity of symbolic powers of certain classes of edge ideals. We also prove that for several classes of graphs, the regularity of symbolic powers of their edge ideals coincides with that of their ordinary powers. [ABSTRACT FROM AUTHOR]

We study the regularity of symbolic powers of square-free monomial ideals. We prove that if I=IΔ is the Stanley-Reisner ideal of a simplicial complex Δ, then reg(I(n))δ(n−1)+ b for all n1, where δ=limn→∞ reg(I(n))/n, b=max{reg(IΓ)|Γ is a subcomplex of Δ with F(Γ)⊆ F(Δ)}, and F(Γ) and F(Δ) are the set of facets of Γ and Δ, respectively. This bound is sharp for any n. When I=I(G) is the edge ideal of a simple graph G, we obtain a general linear upper bound reg(I(n))2n+ord-match(G)−1, where ord-match(G) is the ordered matching number of G. [ABSTRACT FROM AUTHOR]

There are two motivating questions in [M. Mahmoudi, A. Mousivand, M. Crupi, G. Rinaldo, N. Terai and S. Yassemi, arXiv:1006.1087v1] and [M. Mahmoudi, A. Mousivand, M. Crupi, G. Rinaldo, N. Terai and S. Yassemi, J. Pure Appl. Algebra, 215(10) (2011), 2473-2480] about Castelnuovo-Mumford regularity and vertex decomposability of simple graphs. In this paper, we give negative answers to the questions by providing two counterexamples. [ABSTRACT FROM AUTHOR]

Introduction ‎‎ The study of the minimal free resolution of homogenous ideals and their powers is an interesting and active area of research in commutative algebra. Two invariants which measure the complexity of the minimal free resolutions are the so-called “projective dimension” and “Castelnuovo-Mumford regularity” (or simply, regularity) of the given ideal. Projective dimension determines the length of the minimal free resolution, while regularity is defined in terms of the degree of the entries of the matrices defining the differentials of the resolution. The focus of this paper is on the regularity of powers of ideals. One of the main results in this area is obtained by Cutkosky, Herzog, Trung [7], and independently Kodiyalam [8]. They proved that for a homogenous ideal I in a polynomial ring, the regularity of powers of I is asymptotically linear. In other words, there exist integers a(I) and b(I) such that regIs=aIs+b(I) for every integer s≫0. It is known that a(I) is bounded above by the maximum degree of generators of I. Moreover, if I is generated in a single degree d, then aI=d. But in general, it is not so much known about b(I) even if I is monomial ideal. However, when I is a quadratic squarefree monomial ideal, Alilooee, Banerjee, Beyarslan and Ha [9] conjectured that bI≤regI-2. In fact, they conjectured that the inequality regIs≤2s+regI-2 holds for any integer s≥1, when I is quadratic squarefree monomial ideal. Recently, Benerjee and Nevo [10] proved this conjecture for s=2. In this paper, we provide an alternative proof for their result. While the proof in [10] is based on topological arguments and using the Hochster’s formula, our proof is purely algebraic. Material and methods To every simple graph G one associates a quadratic squarefree monomial ideal, called its edge ideal, whose generators are the quadratic squarefree monomials corresponding to the edges of G. This association is a strong tool in the study of squarefree monomial ideals, as one can use the combinatorial properties of G to obtain information about the algebraic and homological properties of it, s edge ideal. One of the main results for bounding the regularity of powers of edge ideals is obtained by Benerjee [1]. He proved that the regularity of the sth power of an edge ideal I(G) has an upper bound which is defined in terms of the regularity of its (s-1)th power and the regularity of the edge ideal of some graphs which are explicitly determined by the structure of the G. This result has an essential role in our proof. Results and discussion The main result of this paper states that for every graph G, with edge ideal I(G), we have regIG2≤reg(IG)+2. In order to prove this inequality, using the aforementioned result of Benerjee, we must prove that the regularity of certain colon ideals are at most regIG. To achieve this goal, we use a short exact sequence argument which allows us to estimate the regularity of the colon ideas in terms of the regularity of edge ideal of some graphs which are strictly smaller than G. Conclusion The following conclusions were drawn from this research. The conjectured inequality of Alilooee, Banerjee, Beyarslan and Ha [9] is true for the case of s=2. It is known that for every graph G with edge ideal I(G) and induced matching number ν(G), we have 2s+νG-1≤reg(IGs), for every integer s≥1. Thus, our result implies that if regIG=νG+1, then regIG2=νG+3. The short exact sequence argument is a common technique in the study of regularity of monomial ideals. So, it would be interesting if one can prove the above-mentioned conjecture, using this method, even in the case of s=3 .

Let \operatorname {pd}(I(G)) and \operatorname {reg}(I(G)) respectively denote the projective dimension and the regularity of the edge ideal I(G) of a graph G. For any positive integer n, we determine all pairs (\operatorname {pd}(I(G)),\, \operatorname {reg}(I(G))) as G ranges over all connected bipartite graphs on n vertices. [ABSTRACT FROM AUTHOR]

Assume that G is a chordal graph with edge ideal I(G) and induced matching number ν(G). We denote the sth symbolic power of I(G) by I(G)(s). It is shown that reg(I(G)(s))=2s + ν(G) − 1 for every integer s ≥ 1. [ABSTRACT FROM AUTHOR]

We show that the v -number of an arbitrary monomial ideal is bounded below by the v -number of its polarization and also find a criteria for the equality. By showing the additivity of associated primes of monomial ideals, we obtain the additivity of the v-numbers for arbitrary monomial ideals. We prove that the v -number v (I (G)) of the edge ideal I(G), the induced matching number im (G) and the regularity reg (R / I (G)) of a graph G, satisfy v (I (G)) ≤ im (G) ≤ reg (R / I (G)) , where G is either a bipartite graph, or a (C 4 , C 5) -free vertex decomposable graph, or a whisker graph. There is an open problem in Jaramillo and Villarreal (J Combin Theory Ser A 177:105310, 2021), whether v (I) ≤ reg (R / I) + 1 , for any square-free monomial ideal I. We show that v (I (G)) > reg (R / I (G)) + 1 , for a disconnected graph G. We derive some inequalities of v -numbers which may be helpful to answer the above problem for the case of connected graphs. We connect v (I (G)) with an invariant of the line graph L(G) of G. For a simple connected graph G, we show that reg (R / I (G)) can be arbitrarily larger than v (I (G)) . Also, we try to see how the v -number is related to the Cohen–Macaulay property of square-free monomial ideals. [ABSTRACT FROM AUTHOR]

Let I be a homogeneous ideal in a polynomial ring S. In this paper, we extend the study of the asymptotic behavior of the minimum distance function δI of I and give bounds for its stabilization point, rI, when I is an F -pure or a square-free monomial ideal. These bounds are related with the dimension and the Castelnuovo–Mumford regularity of I.

Let P be a finite poset and suppose I is the Stanley–Reisner ideal of the chain complex of P. In this paper, we prove the inequality reg (I (s)) ≤ 2 s + rank (P) for every integer s where I (s) denotes the sth symbolic power of I. As a consequence, we deduce that if I(G) is the edge ideal of a graph G such that its complementary graph Gc is bipartite, then the regularity of I (G) (s) is either 2s or 2 s + 1. [ABSTRACT FROM AUTHOR]

The paper provides a connection between Commutative Algebra and Integer Programming and contains two parts. The first one is devoted to the asymptotic behavior of integer programs with a fixed cost linear functional and the constraint sets consisting of a finite system of linear equations or inequalities with integer coefficients depending linearly on n. An integer N ∗ is determined such that the optima of these integer programs are a quasi-linear function of n for all n ≥ N ∗ . Using results in the first part, one can bound in the second part the indices of stability of the Castelnuovo–Mumford regularities of integral closures of powers of a monomial ideal and that of symbolic powers of a square-free monomial ideal. [ABSTRACT FROM AUTHOR]

We explore the dependence of the Betti numbers of monomial ideals on the characteristic of the field. A first observation is that for a fixed prime p either the i-th Betti number of all high enough powers of a monomial ideal differs in characteristic 0 and in characteristic p or it is the same for all high enough powers. In our main results, we provide constructions and explicit examples of monomial ideals all of whose powers have some characteristic-dependent Betti numbers or whose asymptotic regularity depends on the field. We prove that, adding a monomial on new variables to a monomial ideal allows to spread the characteristic dependence to all powers. For any given prime p, this produces an edge ideal such that all its powers have some Betti numbers that are different over Q and over Z p . Moreover, we show that, for every r ≥ 0 and i ≥ 3 there is a monomial ideal I such that some coefficient in a degree ≥ r of the Kodiyalam polynomials P 3 (I) , ... , P i + r (I) depends on the characteristic. We also provide a summary of related results and speculate about the behavior of other combinatorially defined ideals. [ABSTRACT FROM AUTHOR]