Mathematics

One of the fundamental invariants connecting algebra and geometry is the degree of an ideal. In this paper we derive the probabilistic behavior of degree with respect to the versatile Erdős-Rényi-type model for random monomial ideals defined in \cite{rmi}. We study the staircase structure associated to a monomial ideal, and show that in the random case the shape of the staircase diagram is approximately hyperbolic, and this behavior is robust across several random models. Since the discrete volume under this staircase is related to the summatory higher-order divisor function studied in number theory, we use this connection and our control over the shape of the staircase diagram to derive the asymptotic degree of a random monomial ideal. Another way to compute the degree of a monomial ideal is with a standard pair decomposition. This paper derives bounds on the number of standard pairs of a random monomial ideal indexed by any subset of the ring variables. The standard pairs indexed by maximal subsets give a count of degree, as well as being a more nuanced invariant of the random monomial ideal.

Let M and N be finitely generated graded modules over a graded complete intersection R such that Ext i R (M,N) has finite length for all i?? . We show that the even and odd Hilbert polynomials, which give the lengths of Ext i R (M,N) for all large even i and all large odd i , have the same degree and leading coefficient whenever the highest degree of these polynomials is at least the dimension of M or N . Refinements of this result are given when R is regular in small codimensions.

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.

Recently, chains of increasing symmetric ideals have attracted considerable attention. In this note, we summarize some results and open problems concerning the asymptotic behavior of several algebraic and homological invariants along such chains, including codimension, projective dimension, Castelnuovo-Mumford regularity, and Betti tables.

We study the set of star operations on local Noetherian domains D of dimension 1 such that the conductor (D:T) (where T is the integral closure of D ) is equal to the maximal ideal of D . We reduce this problem to the study of a class of closure operations (more precisely, multiplicative operations) in a finite extension k⊆B , where k is a field, and then we study how the cardinality of this set of closures vary as the size of k varies while the structure of B remains fixed.

A Puiseux monoid is a submonoid of (Q,+) consisting of nonnegative rational numbers. Although the operation of addition is continuous with respect to the standard topology, the set of irreducibles of a Puiseux monoid is, in general, difficult to describe. In this paper, we use topological density to understand how much a Puiseux monoid, as well as its set of irreducibles, spread through R ≥0 . First, we separate Puiseux monoids according to their density in R ≥0 , and we characterize monoids in each of these classes in terms of generating sets and sets of irreducibles. Then we study the density of the difference group, the root closure, and the conductor semigroup of a Puiseux monoid. Finally, we prove that every Puiseux monoid generated by a strictly increasing sequence of rationals is nowhere dense in R ≥0 and has empty conductor.

A Puiseux monoid is an additive submonoid of the nonnegative cone of rational numbers. Although Puiseux monoids are torsion-free rank-one monoids, their atomic structure is rich and highly complex. For this reason, they have been important objects to construct crucial examples in commutative algebra and factorization theory. In 1974 Anne Grams used a Puiseux monoid to construct the first example of an atomic domain not satisfying the ACCP, disproving Cohn's conjecture that every atomic domain satisfies the ACCP. Even recently, Jim Coykendall and Felix Gotti have used Puiseux monoids to construct the first atomic monoids with monoid algebras (over a field) that are not atomic, answering a question posed by Robert Gilmer back in the 1980s. This dissertation is focused on the investigation of the atomic structure and factorization theory of Puiseux monoids. Here we established various sufficient conditions for a Puiseux monoid to be atomic (or satisfy the ACCP). We do the same for two of the most important atomic properties: the finite-factorization property and the bounded-factorization property. Then we compare these four atomic properties in the context of Puiseux monoids. This leads us to construct and study several classes of Puiseux monoids with distinct atomic structure. Our investigation provides sufficient evidence to believe that the class of Puiseux monoids is the simplest class with enough complexity to find monoids satisfying almost every fundamental atomic behavior.

Let (K,ν) be a valued field, the notions of \emph{augmented valuation}, of \emph{limit augmented valuation} and of \emph{admissible family} of valuations enable to give a description of any valuation μ of K[x] extending ν . In the case where the field K is algebraically closed, this description is particularly simple and we can reduce it to the notions of \emph{minimal pair} and \emph{pseudo-convergent family}. Let (K,ν) be a henselian valued field and ν ¯ the unique extension of ν to the algebraic closure K ¯ of K and let μ be a valuation of K[x] extending ν , we study the extensions μ ¯ from μ to K ¯ [x] and we give a description of the valuations μ ¯ i of K ¯ [x] which are the extensions of the valuations μ i belonging to the admissible family associated with μ .

One of the most stunning results in the representation theory of Cohen-Macaulay rings is Auslander's well known theorem which states a CM local ring of finite CM type can have at most an isolated singularity. There have been some generalizations of this in the direction of countable CM type by Huneke and Leuschke. In this paper, we focus on a different generalization by restricting the class of modules. Here we consider modules which are high syzygies of MCM modules over non-commutative rings, exploiting the fact that non-commutative rings allow for finer homological behavior. We then generalize Auslander's Theorem in the setting of complete Gorenstein local domains by examining path algebras, which preserve finiteness of global dimension.

We study the two dual notions of prime avoidance and prime absorbance. We generalize the classical prime avoidance lemma to radical ideals. A number of new criteria are provided for an abstract ring to be C.P. (every set of primes satisfies avoidance) or P.Z. (every set of primes satisfies absorbance). Special consideration is given to the interaction with chain conditions and Noetherian-like properties. It is shown that a ring is both C.P. and P.Z. iff it has finite spectrum.