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For an arbitrary ideal $I$ in a polynomial ring $R$ we define the notion of initially regular sequences on $R/I$. These sequences share properties with regular sequences. In particular, the length of an initially regular sequence provides a lower bound for the depth of $R/I$. Using combinatorial information from the initial ideal of $I$ we construct sequences of linear polynomials that form initially regular sequences on $R/I$. We identify situations where initially regular sequences are also regular sequences, and we show that our results can be combined with polarization to improve known depth bounds for general monomial ideals., Comment: Major revision of Section 4. Section 5 moved to a new paper and replaced by applications

We introduce an invariant, associated to a coherent sheaf over a projective morphism of schemes, which controls when sheaf cohomology can be passed through the given morphism. We then use this invariant to estimate the stability indexes of the regularity and a*-invariant of powers of homogeneous ideals., Comment: 16 pages

There are two different notions for symbolic powers of ideals existing in the literature, one defined in terms of associated primes, the other in terms of minimal primes. Elaborating on an idea known to Eisenbud, Herzog, Hibi, and Trung, we interpret both notions of symbolic powers as suitable saturations of the ordinary powers. We prove a binomial expansion formula for saturated powers of sums of ideals. This gives a uniform treatment to an array of existing and new results on both notions of symbolic powers of such sums: binomial expansion formulas, computations of depth and regularity, and criteria for the equality of ordinary and symbolic powers., 15 pages, Comments are welcome

Let $\mathcal{D}$ be a weighted oriented graph and let $I(\mathcal{D})$ be its edge ideal. Under a natural condition that the underlying (undirected) graph of $\mathcal{D}$ contains a perfect matching consisting of leaves, we provide several equivalent conditions for the Cohen-Macaulayness of $I(\mathcal{D})$. We also completely characterize the Cohen-Macaulayness of $I(\mathcal{D})$ when the underlying graph of $\mathcal{D}$ is a bipartite graph. When $I(\mathcal{D})$ fails to be Cohen-Macaulay, we give an instance where $I(\mathcal{D})$ is shown to be sequentially Cohen-Macaulay., Comment: 22 pages, 2 figures

Let $G = (V,E)$ be a simple graph. We investigate the Cohen-Macaulayness and algebraic invariants, such as the Castelnuovo-Mumford regularity and the projective dimension, of the toric ring $k[G]$ via those of toric rings associated to induced subgraphs of $G$., 18 pages; changed title and re-organized sections to better exhibit results; correct the last main theorem

We prove bounds on the saturation degrees of homogeneous ideals (and their powers) defining smooth complex projective varieties. For example, we show that a classical statement due to Macualay for zero-dimensional complete intersection ideals holds for any smooth variety. For curves, we bound the saturation degree of powers in terms of the regularity.

We investigate Demailly's Conjecture for a general set of sufficiently many points. Demailly's Conjecture generalizes Chudnovsky's Conjecture in providing a lower bound for the Waldschmidt constant of a set of points in projective space . We also study a containment between symbolic and ordinary powers conjectured by Harbourne and Huneke that in particular implies Demailly's bound, and prove that a general version of this containment holds for generic determinantal ideals and defining ideals of star configurations.

Castelnuovo-Mumford regularity (or simply regularity) is an important invariant in commutative algebra and algebraic geometry. Computing or finding bounds for the regularity is a difficult problem. In the next three chapters, we shall address the regularity of ordinary and symbolic powers of squarefree monomial ideals.

The most effective tool to deal with the Waring problem for forms is the so-called Apolarity Lemma (see Iarrobino and Kanev and the lecture notes of Carlini, Grieve, and Oeding). To introduce the Apolarity Lemma we need to briefly review some notion from apolarity theory, following Geramita (Inverse systems of fat points: Waring’s problem, secant varieties of veronese varieties and parameter spaces for Gorenstein ideals. In The curves seminar at Queen’s, vol 10, pp 2–114, 1996).

In the previous chapter, we looked at a result of Brodmann (Theorem 1.4) concerning the associated primes of powers of ideals. This theorem inspires a number of natural questions. To state these questions, we introduce some suitable terminology.

In this chapter, we present detailed proofs of a few stated results to illustrate how the inductive techniques introduced in the last chapter can be applied to the study of the regularity of powers of edge ideals.

We derive two general bounds for the depths of powers of squarefree monomial ideals corresponding to hyperforests. These bounds generalize known bounds for the depths of squarefree monomial ideals, which were given in terms of the edgewise domination number of the corresponding hypergraphs and the lengths of initially regular sequences with respect to the ideals.

As is standard at PRAGMATIC, the participants were divided into small groups to work on open research problems, based upon their ranked preferences of the problems. In this iteration of PRAGMATIC, we, as instructors, presented a number of open research problems (see the previous chapter) and some suggested approaches. After the initial assignment of projects, we shifted our focus from lecturing to a focus on mentoring the groups. Not only did we suggest how to make progress on their specific projects, but we also gave more general advice on how to do research and how to present the results.

Let $G$ be a finite simple graph on $n$ vertices, that contains no isolated vertices, and let $I(G) \subseteq S = K[x_1, \dots, x_n]$ be its edge ideal. In this paper, we study the pair of integers that measure the projective dimension and the regularity of $S/I(G)$. We show that if the projective dimension of $S/I(G)$ attains its minimum value $2\sqrt{n}-2$ then, with only one exception, the its regularity must be 1. We also provide a full description for the spectrum of the projective dimension of $S/I(G)$ when the regularity attains its minimum value 1., Comment: 14 pages; restructure and rewrite the introduction to highlight our main algebraic results

Given a fat point scheme \(Z=m_1p_1+\cdots +m_sp_s\subset {\mathbb P}^N\), the containment problem for Z is to determine for which r and m the containment (I(Z))(m) ⊆ (I(Z))r holds. In this section we present some initial results for the containment problem, and we define an asymptotic quantity, the resurgence, that measure to what extent the containment hold for a given Z.

The primary decomposition of ideals in Noetherian rings is a fundamental result in commutative algebra and algebraic geometry. It is a far reaching generalization of the fact that every positive integer has a unique factorization into primes. We recall one version of this result.

The study of ideals underlies both algebra and geometry. For example, the study of homogeneous ideals in polynomial rings is an aspect of both commutative algebra and of algebraic geometry. In both cases, given an ideal, one wants to understand how the ideal behaves. One way in which algebra and geometry differ is in what it means to be “given an ideal”. For an algebraist it typically means being given generators of the ideal. For a geometer it often means being given a locus of points (or a scheme) in projective space, the ideal then being all elements of the polynomial ring which vanish on the given locus or scheme. Determining generators for the ideal defining a scheme sometimes requires significant effort, and if given generators a geometer will usually want to know what vanishing locus they cut out. Thus while both algebraists and geometers study ideals, their starting points are different.

An ubiquitous theme in mathematics is the rewriting of mathematical objects. This is usually done to reveal underlying properties, to classify, to solve problems or just for aesthetic reasons!

In this chapter we collect together the projects that were initially presented to the students of PRAGMATIC. Each project was related to the theme of the workshop, i.e., “Powers of ideals and ideals of powers”. Many of these questions are open-ended (and perhaps not well-defined). The intention, however, was to give each group of students some initial suggestions to guide their own research.

Banerjee’s inductive method has also been successfully applied by various authors, such as Alilooee, Beyarslan, and Selvaraja, Jayanthan, Narayanan, and Selvaraja, and Moghimian, Norouzi Seyed Fakhari, and Yassemi, pushing Theorems 5.1 and 5.2 further to the classes of unicyclic graphs (see Theorem 5.3) and very well-covered graphs (see Theorem 5.4). The core of given arguments in these works is an understanding of ideals of the form Iq+1 : 〈M〉, where I = I(G) is the edge ideal of a simple graph G and M is a minimal generator of Iq.

In this chapter, we present a number of open problems and questions for edge ideals of graphs. These problems and questions fall under the umbrella of Problem 4.8. We shall also discuss inductive techniques that have been applied in the literature.

We settle a conjecture of Herzog and Hibi, which states that the function depth $S/Q^n$, $n \ge 1$, where $Q$ is a homogeneous ideal in a polynomial ring $S$, can be any convergent numerical function. We also give a positive answer to a long-standing open question of Ratliff on the associated primes of powers of ideals., 9 pages. This paper is split from the first version of the paper "Symbolic powers of sums of ideals", arXiv:1702.01766, due to a recommendation of its referee

We investigate the resurgence and asymptotic resurgence numbers of fiber products of projective schemes. Particularly, we show that while the asymptotic resurgence number of the k-fold fiber product of a projective scheme remains unchanged, its resurgence number could strictly increase., 10 pages; remove the part about Chudnovsky's conjecture due to a gap in the proof

We study the relationship between depth and regularity of a homogeneous ideal I and those of (I,f) and I:f, where f is a linear form or a monomial. Our results has several interesting consequences on depth and regularity of edge ideals of hypegraphs and of powers of ideals., 20 pages, published version, Journal of Algebraic Combinatorics 2018

We investigate symbolic and regular powers of monomial ideals. For a square-free monomial ideal $I$ in $k[x_0, \ldots, x_n]$ we show $I^{t(m+e-1)-e+r)}$ is a subset of $M^{(t-1)(e-1)+r-1}(I^{(m)})^t$ for all positive integers $m$, $t$ and $r$, where $e$ is the big-height of $I$ and $M = (x_0, \ldots, x_n)$. This captures two conjectures ($r=1$ and $r=e$): one of Harbourne-Huneke and one of Bocci-Cooper-Harbourne. We also introduce the symbolic polyhedron of a monomial ideal and use this to explore symbolic powers of non-square-free monomial ideals., 15 pages. Fixed typo

Let $(R,\mathfrak{m})$ be a Noetherian local ring of dimension $d > 0$. Let $I_\bullet = \{I_n\}_{n \in \mathbb{N}}$ be a graded family of $\mathfrak{m}$-primary ideals in $R$. We examine how far off from a polynomial can the length function $\ell_R(R/I_n)$ be asymptotically. More specifically, we show that there exists a constant $\gamma > 0$ such that for all $n \ge 0$, $$\ell_R(R/I_{n+1}) - \ell_R(R/I_n) < \gamma n^{d-1}.$$, Comment: 11 pages

Let $G$ be a graph and let $I = I(G)$ be its edge ideal. When $G$ is unicyclic, we give a decomposition of symbolic powers of $I$ in terms of its ordinary powers. This allows us to explicitly compute the Waldschmidt constant and the resurgence number of $I$. When $G$ is an odd cycle, we explicitly compute the regularity of $I^{(s)}$ for all $s \in \mathbb{N}$. In doing so, we also give a natural lower bound for the regularity function $\text{reg } I^{(s)}$, for $s \in \mathbb{N}$, for an arbitrary graph $G$., Comment: 19 pages; remove condition on characteristic of the field, remove an incorrect corollary

We present a close relationship between matching number, covering numbers and their fractional versions in combinatorial optimization and ordinary powers, integral closures of powers, and symbolic powers of monomial ideals. This relationship leads to several new results and problems on the containments between these powers., Comment: 25 pages, 2 figures

Let $I = I(G)$ be the edge ideal of a graph $G$. We give various general upper bounds for the regularity function $\text{reg} I^s$, for $s \ge 1$, addressing a conjecture made by the authors and Alilooee. When $G$ is a gap-free graph and locally of regularity 2, we show that $\text{reg} I^s = 2s$ for all $s \ge 2$. This is a slightly weaker version of a conjecture of Nevo and Peeva. Our method is to investigate the regularity function $\text{reg}I^s$, for $s \ge 1$, via local information of $I$., Comment: 17 pages, 6 figures

Let G be a graph and let I = I(G) be its edge ideal. In this paper, when G is a forest or a cycle, we explicitly compute the regularity of I^s for all s > 0. In particular, for these classes of graphs, we provide the asymptotic linear function reg(I^s) as s > 0, and the initial value of s starting from which reg(I^s) attains its linear form. We also give new bounds on the regularity of I when G contains a Hamiltonian path and when G is a Hamiltonian graph., Changed title, 16 pages, 3 figures

Let $I$ and $J$ be nonzero ideals in two Noetherian algebras $A$ and $B$ over a field $k$. Let $I+J$ denote the ideal generated by $I$ and $J$ in $A\otimes_k B$. We prove the following expansion for the symbolic powers: $$(I+J)^{(n)} = \sum_{i+j = n} I^{(i)} J^{(j)}.$$ If $A$ and $B$ are polynomial rings and if chara$(k) = 0$ or if $I$ and $J$ are monomial ideals, we give exact formulas for the depth and the Castelnuovo-Mumford regularity of $(I+J)^{(n)}$, which depend on the interplay between the symbolic powers of $I$ and $J$. The proof involves a result of independent interest which states that under the above assumption, the induced map Tor$_i^A(k,I^{(n)}) \to$ Tor$_i^A(k,I^{(n-1)})$ is zero for all $i \ge 0$, $n \ge 0$. We also investigate other properties and invariants of $(I+J)^{(n)}$ such as the equality between ordinary and symbolic powers, the Waldschmidt constant and the Cohen-Macaulayness., 22 pages, to appear in Math. Z.; This version does not contain the result on the depth function of powers of a homogeneous ideal, due to a recommendation of the referee

We study Stanley’s long-standing conjecture that the h-vectors of matroid simplicial complexes are pure O-sequences. Our method consists of a new and more abstract approach, which shifts the focus from working on constructing suitable artinian level monomial ideals, as often done in the past, to the study of properties of pure O-sequences. We propose a conjecture on pure O-sequences and settle it in small socle degrees. This allows us to prove Stanley’s conjecture for all matroids of rank 3. At the end of the paper, using our method, we discuss a first possible approach to Stanley’s conjecture in full generality. Our technical work on pure O-sequences also uses very recent results of the third author and collaborators.

Let G be a finite simple graph with edge ideal I(G). Let I(G)∨ denote the Alexander dual of I(G). We show that a description of all induced cycles of odd length in G is encoded in the associated primes of (I(G)∨)2. This result forms the basis for a method to detect odd induced cycles of a graph via ideal operations, e.g., intersections, products and colon operations. Moreover, we get a simple algebraic criterion for determining whether a graph is perfect. We also show how to determine the existence of odd holes in a graph from the value of the arithmetic degree of (I(G)∨)2.

We investigate how to modify a simple graph G combinatorially to obtain a sequentially Cohen-Macaulay graph. We focus on adding configurations of whiskers to G, where to add a whisker one adds a new vertex and an edge connecting this vertex to an existing vertex of G. We give various sufficient conditions and necessary conditions on a subset S of the vertices of G so that the graph [email protected]?W(S), obtained from G by adding a whisker to each vertex in S, is a sequentially Cohen-Macaulay graph. For instance, we show that if S is a vertex cover of G, then [email protected]?W(S) is a sequentially Cohen-Macaulay graph. On the other hand, we show that if [email protected]?S is not sequentially Cohen-Macaulay, then [email protected]?W(S) is not a sequentially Cohen-Macaulay graph. Our work is inspired by and generalizes a result of Villarreal on the use of whiskers to get Cohen-Macaulay graphs.

Given arbitrary homogeneous ideals $I$ and $J$ in polynomial rings $A$ and $B$ over a field $k$, we investigate the depth and the Castelnuovo-Mumford regularity of powers of the sum $I+J$ in $A \otimes_k B$ in terms of those of $I$ and $J$. Our results can be used to study the behavior of the depth and regularity functions of powers of an ideal. For instance, we show that such a depth function can take as its values any infinite non-increasing sequence of non-negative integers., 19 pages; to appear in Math. Z

Let k be a field of characteristic 0, R = k[x1, … , xd] be a polynomial ring, and m its maximal homogeneous ideal. Let I ⊂ R be a homogeneous ideal in R. Let λ(M) denote the length of an Rmodule M. In this paper, we show thatalways exists. This limit has been shown to be e(I)/d! form-primary ideals I in a local Cohen–Macaulay ring, where e(I) denotes the multiplicity of I. But we find that this limit may not be rational in general. We give an example for which the limit is an irrational number thereby showing that the lengths of these extension modules may not have polynomial growth.

In a recent work, Kaiser, Stehl\'ik and \v{S}krekovski provide a family of critically 3-chromatic graphs whose expansions do not result in critically 4-chromatic graphs, and thus give counterexamples to a conjecture of Francisco, Ha and Van Tuyl. The cover ideal of the smallest member of this family also gives a counterexample to the persistence and non-increasing depth properties. In this paper, we show that the cover ideals of all members of their family of graphs indeed fail to have the persistence and non-increasing depth properties., Comment: 14 pp; minor revision; to appear in Acta Mathematica Vietnamica

We survey a number of recent studies of the Castelnuovo–Mumford regularity of squarefree monomial ideals. Our focus is on bounds and exact values for the regularity in terms of combinatorial data from associated simplicial complexes and/or hypergraphs.

Let S = k[x 1,…,x n ], d a positive integer, and suppose that S D is the vector space of all polynomials of degree d in S. Define α n (d) ≔ max { dim k V| V monomial subspace of S d , dim k S 1 V = n dim k V} and ρ n (d +1) ≔ min {dim k V | V monomial subspace of S d , S 1 V = S d+1}. The numbers α n (d) and ρ n (d+ 1) are called the spreading numbers and covering numbers, respectively. We describe an approach to calculate these numbers that uses simplicial complexes.

We study the question of when 0-1 polytopes are normal or, equivalently, having the integer decomposition property. In particular, we shall associate to each 0-1 polytope a labeled hypergraph, and examine the equality between its Ehrhart and polytopal rings via the combinatorial structures of the labeled hypergraph., Comment: 16 pages; major revision with many changes; change title and update combinatorial terminology