Integral Closure Research Articles

Rees algebras of filtrations of covering polyhedra and integral closure of powers of monomial ideals

The aims of this work are to study Rees algebras of filtrations of monomial ideals associated with covering polyhedra of rational matrices with nonnegative entries and nonzero columns using combinatorial optimization and integer programming and to study powers of monomial ideals and their integral closures using irreducible decompositions and polyhedral geometry. We study the Waldschmidt constant and the ic-resurgence of the filtration associated with a covering polyhedron and show how to compute these constants using linear programming. Then, we show a lower bound for the ic-resurgence of the ideal of covers of a graph and prove that the lower bound is attained when the graph is perfect. We also show lower bounds for the ic-resurgence of the edge ideal of a graph and give an algorithm to compute the asymptotic resurgence of squarefree monomial ideals. A classification of when Newton’s polyhedron is the irreducible polyhedron is presented using integral closure.

Maximal generating degrees of integral closures of powers of monomial ideals

Maximal generating degrees of integral closures of powers of monomial ideals

R+ˆ is surprisingly an integral domain

It is shown that if ( R , m , k ) is a complete local domain with char k = p > 0 and R + is its integral closure in an algebraic closure of the quotient field, then both the m -adic and p -adic completions of R + are integral domains. More generally, this theorem remains true if the completeness assumption is relaxed to allow R to be an analytically irreducible Henselian local ring. It is also shown that these rings, which are Cohen-Macaulay R -modules (even balanced in the m -adic case), will have dimension larger than the dimension of R unless dim ⁡ R ≤ 1 .

Limit behavior of the rational powers of monomial ideals

We investigate the rational powers of ideals. We find that in the case of monomial ideals, the canonical indexing leads to a characterization of the rational powers yielding that symbolic powers of squarefree monomial ideals are indeed rational powers themselves. Using the connection with symbolic powers techniques, we use splittings to show the convergence of depths and normalized Castelnuovo–Mumford regularities. We show the convergence of Stanley depths for rational powers, and as a consequence of this, we show the before-now unknown convergence of Stanley depths of integral closure powers. Additionally, we show the finiteness of asymptotic associated primes, and we find that the normalized lengths of local cohomology modules converge for rational powers, and hence for symbolic powers of squarefree monomial ideals.

Lattices, spectral spaces, and closure operations on idempotent semirings

Spectral spaces, introduced by Hochster, are topological spaces homeomorphic to the prime spectra of commutative rings. In this paper we study spectral spaces in perspective of idempotent semirings which are algebraic structures receiving a lot of attention due to its several applications to tropical geometry. We first prove that a space is spectral if and only if it is the prime k-spectrum of an idempotent semiring. In fact, we enrich Hochster's theorem by constructing a subcategory of idempotent semirings which is antiequivalent to the category of spectral spaces. We further provide examples of spectral spaces arising from sets of congruence relations of semirings. In particular, we prove that the space of valuations and the space of prime congruences on an idempotent semiring are spectral, and there is a natural bijection of sets between the two; this shows a stark difference between rings and idempotent semirings. We then develop several aspects of commutative algebra of semirings. We mainly focus on the notion of closure operations for semirings, and provide several examples. In particular, we introduce an integral closure operation and a Frobenius closure operation for idempotent semirings.

Reprint of: Endomorphism rings of reductions of Drinfeld modules

Let A=Fq[T] be the polynomial ring over Fq, and F be the field of fractions of A. Let ϕ be a Drinfeld A-module of rank r≥2 over F. For all but finitely many primes p◁A, one can reduce ϕ modulo p to obtain a Drinfeld A-module ϕ⊗Fp of rank r over Fp=A/p. The endomorphism ring Ep=EndFp(ϕ⊗Fp) is an order in an imaginary field extension K of F of degree r. Let Op be the integral closure of A in K, and let πp∈Ep be the Frobenius endomorphism of ϕ⊗Fp. Then we have the inclusion of orders A[πp]⊂Ep⊂Op in K. We prove that if EndFalg(ϕ)=A, then for arbitrary non-zero ideals n,m of A there are infinitely many p such that n divides the index χ(Ep/A[πp]) and m divides the index χ(Op/Ep). We show that the index χ(Ep/A[πp]) is related to a reciprocity law for the extensions of F arising from the division points of ϕ. In the rank r=2 case we describe an algorithm for computing the orders A[πp]⊂Ep⊂Op, and give some computational data.

When is (D,K) anS-accr pair?

PurposeThe purpose of this article is to determine necessary and sufficient conditions in order that (D,K) to be anS-accr pair, whereDis an integral domain andKis a field which containsDas a subring andSis a multiplicatively closed subset ofD.Design/methodology/approachThe methods used are from the topic multiplicative ideal theory from commutative ring theory.FindingsLetSbe a strongly multiplicatively closed subset of an integral domainDsuch that the ring of fractions ofDwith respect toSis not a field. Then it is shown that (D,K) is anS-accr pair if and only ifKis algebraic overDand the integral closure of the ring of fractions ofDwith respect toSinKis a one-dimensional Prüfer domain. LetD,S,Kbe as above. If each intermediate domain betweenDandKsatisfiesS-strong accr*, then it is shown thatKis algebraic overDand the integral closure of the ring of fractions ofDwith respect toSis a Dedekind domain; the separable degree ofKoverFis finite andKhas finite exponent overF, whereFis the quotient field ofD.Originality/valueMotivated by the work of some researchers onS-accr, the concept ofS-strong accr* is introduced and we determine some necessary conditions in order that (D,K) to be anS-strong accr* pair. This study helps us to understand the behaviour of the rings betweenDandK.

Asymptotic for the number of star operations on one-dimensional Noetherian domains

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 $k\subseteq B$, 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.

On semigroup algebras with rational exponents

In this paper, a semigroup algebra consisting of polynomial expressions with coefficients in a field F and exponents in an additive submonoid M of is called a Puiseux algebra and denoted by Here we study the atomic structure of Puiseux algebras. To begin with, we answer the isomorphism problem for the class of Puiseux algebras, that is, we show that for a field F if two Puiseux algebras and are isomorphic, then the monoids M 1 and M 2 must be isomorphic. Then we construct three classes of Puiseux algebras satisfying the following well-known atomic properties: the ACCP property, the bounded factorization property, and the finite factorization property. We show that there are bounded factorization Puiseux algebras with extremal systems of sets of lengths, which allows us to prove that Puiseux algebras cannot be determined (up to isomorphism) by their arithmetic of lengths. Finally, we give a full description of the seminormal closure, root closure, and complete integral closure of a Puiseux algebra, and we use this description to provide a class of antimatter Puiseux algebras (i.e., Puiseux algebras containing no irreducibles).

On the transfer of some $t-$locally properties

In this paper, we study the transfer of some $t$-locally properties which are stable under localization to $t$-flat overrings of an integral domain $D$. We show that $D,$ $D[X],$ $D\langle X\rangle,$ $D(X)$ and $D[X]_{N_v}$ are simultaneously $t$-locally P$v$MDs (resp., $t$-locally Krull, $t$-locally G-GCD, $t$-locally Noetherian, $t$-locally Strong Mori). A complete characterization of when a pullback is a $t$-locally P$v$MD (resp., $t$-locally GCD, $t$-locally G-GCD, $t$-locally Noetherian, $t$-locally Strong Mori, $t$-locally Mori) is given. As corollaries, we investigate the transfer of some $t$-locally properties among domains of the form $D+XK[X]$, $D+XK[[X]]$ and amalgamated algebras. A particular attention is devoted to the transfer of almost Krull and locally P$v$MD properties to integral closure of a domain having the same property.

Closure-interior duality over complete local rings

We define a duality operation connecting closure operations, interior operations, and test ideals, and describe how the duality acts on common constructions such as trace, torsion, tight and integral closures, and divisible submodules. This generalizes the relationship between tight closure and tight interior demonstrated by Epstein and Schwede (2014) and allows us to extend commonly used results on tight closure test ideals to operations such as those above.

On the existence of birational maximal Cohen-Macaulay modules over biradical extensions in mixed characteristic

Let $S$ be an unramified regular local ring of mixed characteristic $p\geq 3$ and $S^p$ the subring of $S$ obtained by lifting to $S$ the image of the Frobenius map on $S/pS$. Let $R$ be the integral closure of $S$ in a biradical extension of degree $p^2$ of its quotient field obtained by adjoining $p$-th roots of sufficiently general square free elements $f,g\in S^p$. We show that $R$ admits a birational maximal Cohen-Macaulay module. It is noted that $R$ is not automatically Cohen-Macaulay.

Local cohomology of module of differentials of integral extensions

The main focus of this paper is on determining the highest non-vanishing local cohomology modules of ΩB/R,ΩB/V(ΩB/k) where R is either a complete regular local ring or a complete local normal domain with coefficient ring V (field k) and B is its integral closure in an algebraic extension of Q(R). Similar problem is also studied over a normal domain R containing a field k of characteristic 0. In this connection new observations on the direct summand property for integral extensions are also presented.

Existence of birational small Cohen-Macaulay modules over biquadratic extensions in mixed characteristic

Let S be an unramified regular local ring of mixed characteristic two and R the integral closure of S in a biquadratic extension of its quotient field obtained by adjoining roots of sufficiently general square free elements f,g∈S. Let S2 denote the subring of S obtained by lifting to S the image of the Frobenius map on S/2S. When at least one of f,g∈S2, we characterize the Cohen-Macaulayness of R and show that R admits a birational small Cohen-Macaulay module. It is noted that R is not automatically Cohen-Macaulay in case f,g∈S2 or if f,g∉S2.

∗-Almost super-homogeneous ideals in ∗-h-local domains

In this paper, we introduce ∗-almost independent rings of Krull type (∗-almost IRKTs) and ∗-almost generalized Krull domains (∗-almost GKDs) in the general theory of almost factoriality, neither of which need be integrally closed. This fills a gap left in [D. D. Anderson and M. Zafrullah, On∗-Semi-Homogeneous Integral Domains, Advances in Commutative Algebra (Springer, Singapore, 2019)]. We characterize them by ∗-almost super-SH domains, where a domain [Formula: see text] is called a ∗-almost super-SH domain if every nonzero proper principal ideal of [Formula: see text] is a ∗-product of ∗-almost super-homogeneous ideals. We prove that (1) a domain [Formula: see text] is a ∗-almost IRKT if and only if [Formula: see text] is a ∗-almost super-SH domain, (2) a domain is a ∗-almost GKD if and only if [Formula: see text] is a type 1 ∗-almost super-SH domain and (3) a domain [Formula: see text] is a ∗-almost IRKT and an AGCD-domain if and only if [Formula: see text] is a ∗-afg-SH domain. Further, we characterize them by their integral closures. For example, we prove that a domain [Formula: see text] is an almost IRKT if and only if [Formula: see text] is a root extension with [Formula: see text] [Formula: see text]-linked under [Formula: see text] and [Formula: see text] is an IRKT. Examples are given to illustrate the new concepts.

Algebraic Symmetry and Self–Duality of an Open ASEP

We study the regularity of small symbolic powers and integral closure of small powers of edge ideals. We also prove that the regularity of integral closure of powers of edge ideals of graphs with at most two odd cycles is the same as the regularity of their powers.

Valuation rings are derived splinters

We give three proofs that valuation rings are derived splinters: a geometric proof using absolute integral closure, a homological proof which reduces the problem to checking that valuation rings are splinters (which is done in the second author’s PhD thesis and which we reprise here), and a proof by approximation which reduces the problem to Bhatt’s proof of the derived direct summand conjecture. The approximation property also shows that smooth algebras over valuation rings are splinters.

Bass and Betti numbers of A/In

Let (A,m,k) be a Gorenstein local ring of dimension d≥1. Let I be an ideal of A with ht(I)≥d−1. We prove that the numerical functionn↦ℓ(ExtAi(k,A/In+1)) is given by a polynomial of degree d−1 in the case when i≥d+1 and curv(In)>1 for all n≥1. We prove a similar result for the numerical functionn↦ℓ(ToriA(k,A/In+1)) under the assumption that A is a Cohen-Macaulay local ring. We note that there are many examples of ideals satisfying the condition curv(In)>1, for all n≥1. We also consider more general functions n↦ℓ(ToriA(M,A/In) for a filtration {In} of ideals in A. We prove similar results in the case when M is a maximal Cohen-Macaulay A-module and {In=In‾} is the integral closure filtration, I an m-primary ideal in A.

When is a fixed ring comparable to all overrings?

An overring [Formula: see text] of an integral domain [Formula: see text] is said to be comparable if [Formula: see text], [Formula: see text], and each overring of [Formula: see text] is comparable to [Formula: see text] under inclusion. We do provide necessary and sufficient conditions for which [Formula: see text] has a comparable overring. Several consequences are derived, specially for minimal overrings, or in the case where the integral closure [Formula: see text] of [Formula: see text] is a comparable overring, or also when each chain of distinct overrings of [Formula: see text] is finite.

Real Space Triplets in Quantum Condensed Matter: Numerical Experiments Using Path Integrals, Closures, and Hard Spheres.

Path integral Monte Carlo and closure computations are utilized to study real space triplet correlations in the quantum hard-sphere system. The conditions cover from the normal fluid phase to the solid phases face-centered cubic (FCC) and cI16 (de Broglie wavelengths , densities ). The focus is on the equilateral and isosceles features of the path-integral centroid and instantaneous structures. Complementary calculations of the associated pair structures are also carried out to strengthen structural identifications and facilitate closure evaluations. The three closures employed are Kirkwood superposition, Jackson–Feenberg convolution, and their average (AV3). A large quantity of new data are reported, and conclusions are drawn regarding (i) the remarkable performance of AV3 for the centroid and instantaneous correlations, (ii) the correspondences between the fluid and FCC salient features on the coexistence line, and (iii) the most conspicuous differences between FCC and cI16 at the pair and the triplet levels at moderately high densities (. This research is expected to provide low-temperature insights useful for the future related studies of properties of real systems (e.g., helium, alkali metals, and general colloidal systems).