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This paper studies the problem of which sequences of non-negative integers arise as the functions $\operatorname{reg} I^{n-1}/I^n$, $\operatorname{reg} R/I^n$, $\operatorname{reg} I^n$ for an ideal $I$ generated by forms of degree $d$ in a standard graded algebra $R$. These functions are asymptotically linear with slope $d$. If $\dim R/I = 0$, we give a complete characterization of all numerical functions which arise as the functions $\operatorname{reg} I^{n-1}/I^n$, $\operatorname{reg} R/I^n$ and show that $\operatorname{reg} I^n$ can be any numerical function $f(n) \ge dn$ that weakly decreases until it becomes a linear function with slope $d$. The latter result gives a negative answer to a question of Eisenbud and Ulrich. If $\dim R/I \ge 1$, we show that $\operatorname{reg} I^{n-1}/I^n$ can be any numerical asymptotically linear function $f(n) \ge dn-1$ with slope $d$ and $\operatorname{reg} R/I^n$ can be any numerical asymptotically linear function $f(n) \ge dn-1$ with slope $d$ that is weakly increasing. Inspired of a recent work of Ein, Ha and Lazarsfeld on non-singular complex projective schemes, we also prove that the function of the saturation degree of $I^n$ is asymptotically linear for an arbitrary graded ideal $I$ and study the behavior of this function., Comment: 24 pages

By a classical result of Brodmann, the function $\operatorname{depth} R/I^t$ is asymptotically a constant, i.e. there is a number $s$ such that $\operatorname{depth} R/I^t = \operatorname{depth} R/I^s$ for $t > s$. One calls the smallest number $s$ with this property the index of depth stability of $I$ and denotes it by $\operatorname{dstab}(I)$. This invariant remains mysterious til now. The main result of this paper gives an explicit formula for $\operatorname{dstab}(I)$ when $I$ is an arbitrary ideal generated by squarefree monomials of degree 2. That is the first general case where one can characterize $\operatorname{dstab}(I)$ explicitly. The formula expresses $\operatorname{dstab}(I)$ in terms of the associated graph. The proof involves new techniques which relate different topics such as simplicial complexes, systems of linear inequalities, graph parallelizations, and ear decompositions. It provides an effective method for the study of powers of edge ideals., Comment: 26 pages, 6 figures; This is the revised manuscript which is accepted for TAMS

Let $I$ be the edge ideal of a connected non-bipartite graph and $R$ the base polynomial ring. Then $\operatorname{depth} R/I \ge 1$ and $\operatorname{depth} R/I^t = 0$ for $t \gg 1$. We give combinatorial conditions for $\operatorname{depth} R/I^t = 1$ for some $t$ in between and show that the depth function is non-increasing thereafter. Especially, the depth function quickly decreases to 0 after reaching 1. We show that if $\operatorname{depth} R/I = 1$ then $\operatorname{depth} R/I^2 = 0$ and if $\operatorname{depth} R/I^2 = 1$ then $\operatorname{depth} R/I^5 = 0$. Other similar results suggest that if $\operatorname{depth} R/I^t = 1$ then $\operatorname{depth} R/I^{t+3} = 0$. This a surprising phenomenon because the depth of a power can determine a smaller depth of another power. Furthermore, we are able to give a simple combinatorial criterion for $\operatorname{depth} R/I^{(t)} = 1$ for $t \gg 1$ and show that the condition $\operatorname{depth} R/I^{(t)} = 1$ is persistent, where $I^{(t)}$ denotes the $t$-th symbolic powers of $I$., Comment: 15 pages, 3 figures

We survey recent studies and results on the following problem: which numerical functions can be the depth function of powers and symbolic powers of homogeneous ideals., Comment: 13 pages

Projective monomial curves correspond to rings generated by monomials of the same degree in two variables. Such rings always have finite Macaulayfication. We show how to characterize the Buchsbaumness and the Castelnuovo-Mumford regularity of these rings by means of their finite Macaulayfication, and we use this method to study the Buchsbaumness and to estimate the Castelnuovo-Mumford regularity of large classes of non-smooth monomial curves in terms of the given monomials., Comment: 22 pages

Let $(R,\mathfrak m)$ be a local ring and $I, J$ two arbitrary ideals of $R$. Let $\operatorname{gr}_J(R/I)$ denote the associated ring of $R/I$ with respect to $J$, which corresponds to the normal cone in geometry. The main result of this paper shows that if $I = (f_1,...,f_r)$, where $f_1,...,f_r$ is a $J$-filter regular sequence, there exists a number $N$ such that if $f_i' \equiv f_i \mod J^N$ and $I' = (f_1',...,f_r')$, then $\operatorname{gr}_J(R/I) \cong \operatorname{gr}_J(R/I')$. If $J$ is an $\mathfrak m$-primary ideal, this result implies a long standing conjecture of Srinivas and Trivedi on the invariance of the Hilbert-Samuel function under small perturbations, which has been solved recently by Ma, Quy and Smirnov. As a byproduct, the Artin-Rees number of $I$ and $I'$ with respect to $J$ are the same. Furthermore, we give explicit upper bounds for the smallest number $N$ with the above property. These results solve two problems raised by Ma, Quy and Smirnov. There are other interesting consequences on the invariance of the Achilles-Manaresi function, the relation type, the Castelnuovo-Mumford regularity, the Cohen-Macaulayness and the Gorensteiness of the Rees algebra of $R/I$ with respect to $J$ under small perturbation of $I$. We also prove a converse of the main result showing that the condition $I$ being generated by a $J$-filter regular sequence is the best possible for its validity. The main result can be also extended to perturbations with respect to filtrations of ideals. As a consequence, if $R$ is a power series ring, $f_1,...,f_r$ is a filter regular sequence, and $f_i'$ is the $n$-jet of $f_i$ for $n \gg 0$, then $I$ and $I'$ have the same initial ideal with respect to any Noetherian monomial order. A special case of this consequence was a conjecture of Adamus and Seyedinejad on approximations of analytic complete intersection singularities., Comment: 23 pages, to be published in Trans. Amer. Math. Soc

We prove that two arbitrary ideals $I \subset J$ in an equidimensional and universally catenary Noetherian local ring have the same integral closure if and only if they have the same multiplicity sequence. We also obtain a Principle of Specialization of Integral Dependence, which gives a condition for integral dependence in terms of the constancy of the multiplicity sequence in families., Comment: 17 pages, to be published in Math. Ann

These notes are an introduction to some basic aspects of the Castelnuovo-Mumford regularity and related topics such as weak regularity, a*-invariant and partial regularities., Comment: 23 pages, Lecture notes at the International Conference on Commutative Algebra and Combinatorics at Harish-Chandra Research Institute, Allahabad, December 2003

This paper addresses the problem of comparing minimal free resolutions of symbolic powers of an ideal. Our investigation is focused on the behavior of the function depth R/I^(t) = dim R - pd I^(t) - 1, where I^(t) denotes the t-th symbolic power of a homogeneous ideal I in a noetherian polynomial ring R and pd denotes the projective dimension. It has been an open question whether the function depth R/I^(t) is non-increasing if I is a squarefree monomial ideal. We show that depth R/I^(t) is almost non-increasing in the sense that depth R/I^(s) \ge depth R/I^(t) for all s \ge 1 and t \in E(s), where E(s) = \cup_{i \ge 1} {t \in N| i(s-1)+1 \le t \le is} (which contains all integers t \ge (s-1)^2+1). The range E(s) is the best possible since we can find squarefree monomial ideals I such that depth R/I^(s) < depth R/I^(t) for t \not\in E(s), which gives a negative answer to the above question. Another open question asks whether the function depth R/I^(t) is always constant for t \gg 0. We are able to construct counter-examples to this question by monomial ideals. On the other hand, we show that if I is a monomial ideal such that I^(t) is integrally closed for t \gg 0 (e.g. if I is a squarefree monomial ideal), then depth R/I^(t) is constant for t \gg 0 with lim_{t \to \infty} depth R/I^(t) = dim R - dim \oplus_{t \ge 0} I^(t)/m I^(t). Our last result (which is the main contribution of this paper) shows that for any positive numerical function \phi(t) which is periodic for t \gg 0, there exist a polynomial ring R and a homogeneous ideal I such that depth R/I^(t) = \phi(t) for all t \ge 1. As a consequence, for any non-negative numerical function \psi(t) which is periodic for t \gg 0, there is a homogeneous ideal I and a number c such that pd I^(t) = \psi(t) + c for all t \ge 1., Comment: 45 pages; this version contains a corrigendum, which gives a corrected proof to Theorem 3.3. The original paper and the corrigendum will appear with different DOI

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., Comment: 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 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

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., Comment: 20 pages, published version, Journal of Algebraic Combinatorics 2018

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., Comment: 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

In this paper we develop a theory of monomial preorders, which differ from the classical notion of monomial orders in that they allow ties between monomials. Since for monomial preorders, the leading ideal is less degenerate than for monomial orders, our results can be used to study problems where monomial orders fail to give a solution. Some of our results are new even in the classical case of monomial orders and in the special case in which the leading ideal defines the tangent cone., Comment: 24 pages; corrected typos, added references and an acknowledgment. The paper is accepted for publication in Mathematics of Computation

We establish strong relationships between the Castelnuovo-Mumford regularity and the Ratliff-Rush closure of an ideal. Our results have several interesting consequences on the computation of the Ratliff-Rush closure, the stability of the Ratliff-Rush filtration, the invariance of the reduction number, and the computation of the Castelnuovo-Mumford regularity of the Rees algebra and the fiber ring. In particular, we prove that the Castelnuovo-Mumford regularity of the Rees algebra and of the fiber ring are equal for large classes of monomial ideals in two variables, thereby verifying a conjecture of Eisenbud and Ulrich for these cases., Comment: 16 pages; to appear in Journal of Algebra

In this paper, we give a complete description of the associated primes of every power of the edge ideal in terms of generalized ear decompositions of the graph. This result establishes a surprising relationship between two seemingly unrelated notions of Commutative Algebra and Combinatorics. It covers all previous major results in this topic and has several interesting consequences., Comment: 28 pages, 7 figures. This revision contains more comments on the methods and the results of the paper. Important terminology and notations now appear in Definition environments for the convenience of the readers. That changes the numeration of the results considerably. This paper will appear in Trans. Amer. Math. Soc

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., Comment: 19 pages; to appear in Math. Z

For the edge ideal I of an arbitrary simple graph G we describe the monomials of the saturation of a power of I in terms of (vertex) weighted graphs associated with the monomials. This description allows us to characterize the embedded associated primes of a power of I as covers of G which contain certain types of subgraphs of G. As an application, we completely classify the associated primes of the second and the third power of I in terms of G., Comment: 16 pages, 2 figures

We give explicit formulas for the Bhattacharya function of m-primary complete monomial ideals in two variables in terms of the vertices of the Newton polyhedra or in terms of the decompositions of the ideals as products of simple ideals., Comment: 11 pages, to appear in Comm. Algebra

Absolutely superficial sequences was introduced by P. Schenzel in order to study generalized Cohen-Macaulay (resp. Buchsbaum) modules. For an arbitrary local ring, they turned out to be d-sequences. This paper established properties of absolutely superficial sequences with respect to a module. It is shown that they are closely related to other sequences in the theory of generalized Cohen-Macaulay (resp. Buchsbaum) modules. In particular, there is a bounding function for the Hilbert-Samuel function of every parameter ideal such that this bounding function is attained if and only if the ideal is generated by an absolutely superficial sequence.

We present combinatorial characterizations for the associated primes of the second power of squarefree monomial ideals and criteria for this power to have positive depth or depth greater than one., Comment: To be published in Journal of Pure and Applied Algebra

We introduce the notion of independent sequences with respect to a monomial order by using the least terms of polynomials vanishing at the sequence. Our main result shows that the Krull dimension of a Noetherian ring is equal to the supremum of the length of independent sequences. The proof has led to other notions of independent sequences, which have interesting applications. For example, we can characterize the maximum number of analytically independent elements in an arbitrary ideal of a local ring and that dim B is not greater than dim A if B is a subalgebra of A and A is a (not necessarily finitely generated) subalgebra of a finitely generated algebra over a Noetherian Jacobson ring., Comment: This is a revised version of the submitted manuscript

We prove that for m > 2, the m-th symbolic power of a Stanley-Reisner ideal is Cohen-Macaulay if and only if the simplicial complex is a matroid. Similarly, the m-th ordinary power is Cohen-Macaulay for some m > 2 if and only if the complex is a complete intersection. These results solve several open questions on the Cohen-Macaulayness of ordinary and symbolic powers of Stanley-Reisner ideals. Moreover, they have interesting consequences on the Cohen-Macaulayness of symbolic powers of facet ideals and cover ideals., Comment: 20 pages

This paper studies properties of simplicial complexes for which the m-th symbolic power of the Stanley-Reisner ideal equals to the m-th ordinary power for a given m > 1. The main results are combinatorial characterizations of such complexes in the two-dimensional case. It turns out that there exist only a finite number of complexes with this property and that these complexes can be described completely. As a consequence we are able to determine all complexes for which the m-th ordinary power of the Stanley-Reisner ideal is Cohen-Macaulay for a given m > 1., Comment: 19 pages

We present criteria for the Cohen-Macaulayness of a monomial ideal in terms of its primary decomposition. These criteria allow us to use tools of graph theory and of linear programming to study the Cohen-Macaulayness of monomial ideals which are intersections of prime ideal powers. We can characterize the Cohen-Macaulayness of the second symbolic power or of all symbolic powers of a Stanley-Reisner ideal in terms of the simplicial complex. These characterizations show that the simplicial complex must be very compact if some symbolic power is Cohen-Macaulay. In particular, all symbolic powers are Cohen-Macaulay if and only if the simplicial complex is a matroid complex. We also prove that the Cohen-Macaulayness can pass from a symbolic power to another symbolic powers in different ways., Comment: The published version of this paper contains a gap in the proofs of Theorem 2.5 and Theorem 3.5. This version corrects the proofs with almost the same arguments. Moreover, we have to modify the definition of tight complexes in Theorem 2.5. These changes don't affect other things in the published version. A corrigendum has been sent to the journal

Two-dimensional squarefree monomial ideals can be seen as the Stanley-Reisner ideals of graphs. The main results of this paper are combinatorial characterizations for the Cohen-Macaulayness of ordinary and symbolic powers of such an ideal in terms of the associated graph., Comment: 9 pages

It is proved that all vertex cover algebras of a hypergraph are standard graded if and only if the hypergraph is unimodular. This has interesting consequences on the symbolic powers of monomial ideals., Comment: 6 pages

We prove that the integral closures of the powers of a squarefree monomial ideal I equal the symbolic powers if and only if I is the edge ideal of a Fulkersonian hypergraph., Comment: 5 pages

We establish a uniform bound for the Castelnuovo-Mumford regularity of associated graded rings of parameter ideals in a generalized Cohen-Macaulay ring. As consequences, we obtain uniform bounds for the relation type and the postulation number. Moreover, we show that generalized Cohen-Macaulay rings can be characterized by the existence of such uniform bounds., Comment: 12 pages

A forest is the clique complex of a strongly chordal graph and a quasi-forest is the clique complex of a chordal graph. Kruskal--Katona type theorems for forests, quasi-forests, pure forests and pure quasi-forests will be presented. In addition, it will be shown that a quasi-forest is shellable if and only if its $h$-vector $(h_0, h_1, h_2, ...)$ satisfies $h_i = 0$ for $i > 1$., Comment: 7pages, simplify the characterization of face vectors, remove appendix, add references

The aim of this paper is to characterize simplicial complexes which have standard graded vertex cover algebras. This property has several nice consequences for the squarefree monomial ideals defining these algebras. It turns out that such simplicial complexes are closely related to a range of hypergraphs which generalize bipartite graphs and trees. These relationships allow us to obtain very general results on standard graded vertex cover algebras which cover previous major results on Rees algebras of squarefree monomial ideals., Comment: 20 pages, 6 figures

We introduce and study vertex cover algebras of weighted simplicial complexes. These algebras are special classes of symbolic Rees algebras. We show that symbolic Rees algebras of monomial ideals are finitely generated and that such an algebra is normal and Cohen-Macaulay if the monomial ideal is squarefree. For a simple graph, the vertex cover algebra is generated by elements of degree 2, and it is standard graded if and only if the graph is bipartite. We also give a general upper bound for the maximal degree of the generators of vertex cover algebras., Comment: 19 pages

The main results of this paper interpret mixed volumes of lattice polytopes as mixed multiplicities of ideals and mixed multiplicities of ideals as Samuel's multiplicities. In particular, we can give a purely algebraic proof of Bernstein's theorem which asserts that the number of common zeros of a system of Laurent polynomial equations in the torus is bounded above by the mixed volume of their Newton polytopes., Comment: 19 pages, to appear in Trans. Amer. Math. Soc

This paper proves that the Castelnuovo-Mumford regularities of the product and sum of two monomial complete intersection ideals are at most the sum of the regularities of the two ideals, and provides examples showing that these inequalities do not hold for general complete intersections., Comment: 10 pages

The notion of regularity has been used by S. Kleiman in the construction of bounded families of ideals or sheaves with given Hilbert polynomial, a crucial point in the construction of Hilbert or Picard scheme. In a related direction, Kleiman proved that if I is an equidimensional reduced ideal in a polynomial ring S over an algebraically closed field, then the coefficients of the Hilbert polynomial of R = S/I can be bounded by the dimension and the multiplicity of R. Srinivas and Trivedi proved that the corresponding result does not hold for a local domain. However, they proved that there exist a finite number of Hilbert functions for a local Cohen-Macaulay ring of given multiplicity and dimension. The proofs of the above results are very difficult and involve deep results from Algebraic Geometry. The aim of this paper is to introduce a unified approach which gives more general results and easier proofs of the above mentioned results. This approach is based on the fact that a class C of standard graded algebras has a finite number of Hilbert functions if and only if there are upper bounds for the regularity and the embedding dimension of the members of C., Comment: 13 pages

This paper addresses problems related to the existence of arithmetic Macaulayfications of projective schemes. Let Y be the blow-up of a projective scheme X = Proj R along the ideal sheaf of I \subset R. It is known that there are embeddings Y \cong Proj k[(I^e)_c] for c \ge d(I)e + 1, where d(I) denotes the maximal generating degree of I, and that there exists a Cohen-Macaulay ring of the form k[(I^e)_c] if and only if H^0(Y,O_Y) = k, H^i(Y,O_Y) = 0 for i = 1,...,dim Y-1, Y is equidimensional and Cohen-Macaulay. Cutkosky and Herzog asked when there is a linear bound on c and e ensuring that k[(I^e)_c] is a Cohen-Macaulay ring. We obtain a surprising compelte answer to this question, namely, that under the above conditions, there are well determined invariants a and b such that k[(I^e)_c] is Cohen-Macaulay for all c > d(I)e + a and e > b. Our approach is based on recent results on the asymptotic linearity of the Castelnuovo-Mumford regularity of ideal powers. We also investigate the existence of a Cohen-Macaulay Rees algebra of the form R[(I^e)_ct] (which provides an arithmetic Macaulayfication for X). If R has negative a*-invariant, we prove that such a Cohen-Macaulay Rees algebra exists if and only if f_*O_Y = O_X, R^i f*O_Y = 0 for i > 0, Y is equidimensional and Cohen-Macaulay. Especially, these conditions imply the Cohen-Macaulayness of R[(I^e)_ct] for all c > d(I)e + a and e > b. The above results can be applied to obtain several new classes of Cohen-Macaulay algebras., Comment: 21 pages, to appear in Trans. Amer. Math. Soc

Our focus in this paper is in effective computation of the core core(I) of an ideal I which is defined to be the intersection of all minimal reductions of I. The first main result is a closed formula for the graded core(m) of the maximal graded ideal m of an arbitrary standard graded algebra A over a field k. This formula allows us to study basic properties of the graded core and to construct counter-examples to some open questions on the core of ideals in a local ring. For instance, we can show that in general, core(m \otimes E) \neq core(m)\otimes E, where E is a field extension of k. From this it follows that the equation core(I R') = core(I)R' does not hold for an arbitrary flat local homomorphism R \to R' of Cohen-Macaulay local rings. The second main result proves the formulae core(I)= (J^r:I^r)I = (J^r:I^r)J = J^{r+1}:I^r for any equimultiple ideal I in a Cohen-Macaulay ring R with with characteristic zero residue field, where J is a minimal reduction of I and r is its reduction number. This result has been obtained independently by Polini-Ulrich and Hyry-Smith in the one-dimensional case or when R is a Gorenstein ring. Moreover, we can prove that core(I) = IK, where K is the conductor of R in the blowing-up ring at I., Comment: 20 pages, to appear in Compositio Math

The aim of this paper is to study the relationship between reduction numbers and Borel-fixed ideals in all characteristics. By definition, Borel-fixed ideals are closed under certain specializations which is similar to the strong stability. We will estimate the number of monomials which can be specialized to a given monomial. As a consequence, we obtain a combinatorial version of the well-known Eakin-Sathaye's theorem which bounds the reduction number in terms of the Hilbert function. Furthermore, we show that the bound of Eakin-Sathaye's theorem is attained by the reduction number of a lex-segment monomial ideal. This result answers a question of Conca in the affirmative. We will also show that the reduction number of the lex-segment ideal is bounded exponentially by the reduction number of the given ideal., Comment: 11 pages