Test ideals are an important concept in tight closure theory and their behavior via flat base change can be very difficult to understand. Our paper presents results regarding this behavior under flat maps with reasonably nice (but far from smooth) fibers. This involves analyzing, in depth, a special type of ideal of test elements, called the CS test ideal. Besides providing new results, the paper also contains extensions of a theorem by G. Lyubeznik and K. E. Smith on the completely stable test ideal and of theorems by F. Enescu and, independently, M. Hashimoto on the behavior of F-rationality under flat base change.
We show that for ideals primary to a maximal ideal in a normal domain of finite type over the complex numbers, its tight closure is contained inside the continuous closure.
We prove that in normal rings the tight closure of an ideal can be computed as the sum of the ideal and a piece of the tight closure, called the special tight closure.
An analogue of the theory of integral closure and reductions is developed for a more general class of closures, called Nakayama closures. It is shown that tight closure is a Nakayama closure by proving a ``Nakayama lemma for tight closure''. Then, after strengthening A. Vraciu's theory of $*$-independence and the special part of tight closure, it is shown that all minimal $*$-reductions of an ideal in an analytically irreducible excellent local ring of positive characteristic have the same minimal number of generators. This number is called the $*$-spread of the ideal, by analogy with the noti…
We give a treatment of the Brenner-Monsky example based on polynomial algebra and linear algebra. No prior knowledge of tight closure theory, Hilbert-Kunz theory, algebraic geometry or local cohomology is assumed.
We introduce two closure operations on ideals in commutative rings related to the ring operation of root closure. One closure is the result of iterating a root-like operation on ideals infinitely many times, and the other closure arises as a graded component of the root closure of the Rees algebra applied to an ideal. We study the properties of these closures and prove they are distinct operations.
It is shown that tight closure commutes with localization in any two dimensional ring $R$ of prime characteristic if either $R$ is a Nagata ring or $R$ possesses a weak test element. Moreover, it is proved that tight closure commutes with localization at height one prime ideals in any ring of prime characteristic.
In this paper we find the tight closure of powers of parameter ideals of certain diagonal hypersurface rings. In many cases the associated graded ring with respect to tight closure filtration turns out to be Cohen-Macaulay. This helps us find the tight Hilbert polynomial in these diagonal hypersurfaces. We determine the tight Hilbert polynomial in the following cases: (1) F-pure diagonal hypersurfaces where number of variables is equal to the degree of defining equation, (2) diagonal hypersurface rings where characteristic of the ring is one less than the degree of defining equation and (3) qu…
It is proved that tight closure commutes with localization in any domain which has a module finite extension in which tight closure is known to commute with localization. It follows that tight closure commutes with localization in binomial rings, in particular in semigroup or toric rings.
An equidimensional local ring is F-rational if and only if one ideal generated by a system of parameters is tightly closed. The question of whether a non-equidimensional local ring can have a tightly closed ideal generated by a system of parameters has been a long-standing open problem, and for certain classes of non-equidimensional rings we prove that this is not possible. A key point is that tight closure has a colon capturing property in equidimensional rings that it does not have in non-equidimensional rings. We define a new closure operation, one that rectifies the absence of the colon ca…
This paper studies the Ratliff-Rush closure of ideals in integral domains. By definition, the Ratliff-Rush closure of an ideal $I$ of a domain $R$ is the ideal given by $\tilde{I}:=\bigcup(I^{n+1}:_{R}I^{n})$ and an ideal $I$ is said to be a Ratliff-Rush ideal if $\tilde{I}=I$. We completely characterize integrally closed domains in which every ideal is a Ratliff-Rush ideal and we give a complete description of the Ratliff-Rush closure of an ideal in a valuation domain.
We establish an inequality involving colengths of the tight closure of ideals of systems of parameters in local rings with some mild conditions. As an application, we prove and refine a result by Goto and Nakamura, conjectured by Watanabe and Yoshida, which states that the Hilbert-Samuel multiplicity of a parameter ideal is greater than or equal to the colength of the tight closure of the ideal.
In the ring R=K[X,Y,Z]/(X^3+Y^3+Z^3), where K is a field of prime characteristic p other than 3, determining the tight closure of the ideal (X^2, Y^2, Z^2)R had existed as a classic example of the difficulty involved in tight closure computations. We settle this question, compute the Frobenius closure of this ideal, and generalize these results to the diagonal hypersurfaces K[X_1,...,X_n]/(X_1^n + ... + X_n^n).
In this paper we study various equivalent conditions for tight closure to commute with localization. If N is a submodule of a finitely generated module M over a Noetherian commutative ring of characteristic p, then a test exponent for c,N,M is defined to be a power q' of p such that u is in the tight closure of N in M whenever cu^q is in the qth Frobenius power of N for some q \ge q'. We prove that that a test exponent for a locally stable test element c and for N,M as above exists if and only if the tight closure of N in M commutes with localization. Other equivalent conditions are given for …
This paper is concerned with tight closure in a commutative Noetherian ring $R$ of prime characteristic $p$, and is motivated by an argument of K. E. Smith and I. Swanson that shows that, if the sequence of Frobenius powers of a proper ideal $I$ of $R$ has linear growth of primary decompositions, then tight closure (of $I$) commutes with localization at the powers of a single element. It is shown in this paper that, provided $R$ has a weak test element, linear growth of primary decompositions for other sequences of ideals of $R$ that approximate, in a certain sense, the sequence of Frobenius p…
Hara and Yoshida introduced a notion of $\aaa$-tight closure in 2003, and they proved that the test ideals given by this operation correspond to multiplier ideals. However, their operation is not a true closure. The alternative operation introduced here is a true closure. Moreover, we define a joint Hilbert-Kunz multiplicity that can be used to test for membership in this closure. We study the connections between the Hara-Yoshida operation and the one introduced here, primarily from the point of view of test ideals. We also consider variants with positive real exponents.
We determine when graded rings have F-rational or F-regular Veronese subrings, and develop techniques of constructing F-rational rings which are not F-regular.
This paper studies Frobenius maps on injective hulls of residue fields of complete local rings with a view toward providing constructive descriptions of objects originating from the theory of tight closure. Specifically, the paper describes algorithms for computing parameter test ideals, and tight closure of certain submodules of the injective hull of residue fields of a class of well-behaved rings which includes all quasi-Gorenstein complete local rings.
We prove that the initial ideal of the defining ideal of a monomial curve that corresponds to an almost arithmetic sequence of positive integers is Ratliff-Rush closed.
In this paper, we prove a result similar to results of Itoh and Hong-Ulrich, proving that integral closure of an ideal is compatible with specialization by a general element of that ideal for ideals of height at least two in a large class of rings. Moreover, we show integral closure of sufficiently large powers of the ideal is compatible with specialization by a general element of the original ideal. In a polynomial ring over an infinite field, we give a class of squarefree monomial ideals for which the integral closure is compatible with specialization by a general linear form.
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 given in [Epstein-Schwede 2014] and allows us to extend commonly used results on tight closure test ideals to operations such as those above.
We characterize the rings in which the equality $(τI:τ)= I^*$ holds for every ideal $I \subset R$. Under certain assumptions, these rings must be either weakly F-regular or one-dimensional.
This paper is concerned with the tight closure of an ideal $I$ in a commutative Noetherian ring $R$ of prime characteristic $p$. The formal definition requires, on the face of things, an infinite number of checks to determine whether or not an element of $R$ belongs to the tight closure of $I$. The situation in this respect is much improved by Hochster's and Huneke's test elements for tight closure, which exist when $R$ is a reduced algebra of finite type over an excellent local ring of characteristic $p$. More recently, Hochster and Huneke have introduced the concept of test exponent for tigh…
Let $(R,\mathfrak m)$ be an analytically unramified local ring of positive prime characteristic $p.$ For an ideal $I$, let $I^*$ denote its tight closure. We introduce the tight Hilbert function $H^*_I(n)=\ell(R/(I^n)^*)$ and the corresponding tight Hilbert polynomial $P_I^*(n)$ where $I$ is an $\mathfrak m$-primary ideal. It is proved that $F$-rationality can be detected by the vanishing of the first coefficient of $P_I^*(n).$ We find the tight Hilbert polynomial of certain parameter ideals in hypersurface rings and Stanley-Reisner rings of simplicial complexes.
In a formally unmixed Noetherian local ring, if the colength and multiplicity of an integrally closed ideal agree, then $R$ is regular. We deduce this using the relationship between multiplicity and various ideal closure operations.
We look at how the equivalence of tight closure and plus closure (or Frobenius closure) in the homogeneous m-coprimary case implies the same closure equivalence in the non-homogeneous m-coprimary case in standard graded rings. Although our result does not depend upon dimension, the primary application is based on results known in dimension 2 due to the recent results of H. Brenner. We also show that unlike the Noetherian case, the injective hull of the residue field over $R^+$ or $R^\infty$ contains elements that are not killed by any power of the maximal ideal of R.
This paper establishes the fundamental properties of the $s$-closures, a recently introduced family of closure operations on ideals of rings of positive characteristic. The behavior of the $s$-closure of homogeneous ideals in graded rings is studied, and criteria are given for when the $s$-closure of an ideal can be described exactly in terms of its tight closure and rational powers. Sufficient conditions are established for the weak $s$-closure to equal to the $s$-closure. A generalization of the Briancon-Skoda theorem is given which compares any two different $s$-closures applied to powers o…
It is well known that a domain without proper strongly divisorial ideals is completely integrally closed. In this paper we show that a domain without {\em prime} strongly divisorial ideals is not necessarily completely integrally closed, although this property holds under some additional assumptions.
This article outgrew from an effort to understand our basic question: Are the annihilators of the non-zero Koszul homology modules $H_i$ of an unmixed ideal $I$ contained in the integral closure $\bar{I}$ of $I$? We also obtain some variations on a result of Burch, which continue the theme of the paper in that they deal with annihilators of homology and integrally closed ideals.
Let R be an excellent local ring, m its maximal ideal and I an ideal. Then there exists a positive integer c such that for all integers n, the integral closure of (I + m^n) is contained in m^(n/c) + the integral closure of I. In the proof, a version of the linear Artin approximation theorem is proved.
Using a result of M. Hochster and C. Huneke on $F$-rational rings a criterion for complete intersection rings of characteristic $p>0$ is presented. As an application, we give a completely different proof for an algebraic result of G. Faltings that was used by Taylor and Wiles in \cite{TW} for a simplification of the proof of the minimal deformation problem.
Among the several types of closures of an ideal $I$ that have been defined and studied in the past decades, the integral closure $\bar{I}$ has a central place being one of the earliest and most relevant. Despite this role, it is often a difficult challenge to describe it concretely once the generators of $I$ are known. Our aim in this note is to show that in a broad class of ideals their radicals play a fundamental role in testing for integral closedness, and in case $I\neq \bar{I}$, $\surd{I}$ is still helpful in finding some fresh new elements in $\bar{I}\setminus I$. Among the classes of id…
Let R be an integral domain and I a nonzero ideal of R. A sub-ideal J of I is a t-reduction of I if (JI^{n})_{t}=(I^{n+1})_{t} for some positive integer n. An element x in R is t-integral over I if there is an equation x^{n} + a_{1}x^{n-1} +...+ a_{n-1}x + a_{n} = 0 with a_{i} in (I^{i})_{t} for I = 1,...,n. The set of all elements that are t-integral over I is called the t-integral closure of I. This paper investigates the t-reductions and t-integral closure of ideals. Our objective is to establish satisfactory t-analogues of well-known results, in the literature, on the integral closure of i…
We characterize the tight closure of graded primary ideals in a homogeneous coordinate ring over an elliptic curve by numerical conditions and we show that it is in positive characteristic the same as the plus closure.
We prove that the tight closure and the graded plus closure of a homogeneous ideal coincide for a two-dimensional N-graded domain of finite type over the algebraic closure of a finite field. This answers in this case a ``tantalizing question'' of Hochster.
We prove that the integral closedness of any ideal of height at least two is compatible with specialization by a generic element. This opens the possibility for proofs using induction on the height of an ideal. Also, with additional assumptions, we show that an element is integral over a module if it is integral modulo a generic element of the module. This turns questions about integral closures of modules into problems about integral closures of ideals, by means of a construction known as Bourbaki ideal.
Let $(R, \mathfrak{m}, k)$ be an excellent equidimensional local ring of characteristic $p>0$. The aim of this paper is to show that $\ell_R(\mathfrak{q}^*/\mathfrak{q})$ does not depend on the choice of parameter ideal $\mathfrak{q}$ provided $R$ is an $F$-injective local ring that is $F$-rational on the punctured spectrum.
This paper is concerned with the tight closure of an ideal in a commutative Noetherian local ring $R$ of prime characteristic $p$. Several authors, including R. Fedder, K.-i. Watanabe, K. E. Smith, N. Hara and F. Enescu, have used the natural Frobenius action on the top local cohomology module of such an $R$ to good effect in the study of tight closure, and this paper uses that device. The main part of the paper develops a theory of what are here called 'special annihilator submodules' of a left module over the Frobenius skew polynomial ring associated to $R$; this theory is then applied in th…
Let I\subset K[x,y] be a <x,y>-primary monomial ideal where K is a field. This paper produces an algorithm for computing the Ratliff-Rush closure I for the ideal I=<m_0,...,m_{n}> whenever m_{i} is contained in the integral closure of the ideal <x^{a_{n}},y^{b_0}>. This generalizes of the work of Crispin \cite{Cri}. Also, it provides generalizations and answers for some questions given in \cite{HJLS}, and enables us to construct infinite families of Ratliff-Rush ideals.
This paper is concerned with existence of big tight closure test elements for a commutative Noetherian ring $R$ of prime characteristic $p$. Let $R^{\circ}$ denote the complement in $R$ of the union of the minimal prime ideals of $R$. A big test element for $R$ is an element of $R^{\circ}$ which can be used in every tight closure membership test for every $R$-module, and not just the finitely generated ones. The main results of the paper are that, if $R$ is excellent and satisfies condition $(R_0)$, and $c \in R^{\circ}$ is such that $R_c$ is Gorenstein and weakly $F$-regular, then some power …
A Noetherian local ring $(R,\mathfrak{m})$ is called Buchsbaum if the difference $e(\mathfrak{q}, R)-\ell(R/\mathfrak{q})$, where $\mathfrak{q}$ is an ideal generated by a system of parameters, is a constant independent of $\mathfrak{q}$. In this article, we study the tight closure analog of this condition. We prove that in an unmixed excellent local ring $(R,\mathfrak{m})$ of prime characteristic $p>0$ and dimension at least one, the difference $e(\mathfrak{q}, R)-\ell(R/\mathfrak{q}^*)$ is independent of $\mathfrak{q}$ if and only if the parameter test ideal $τ_{\text{par}}(R)$ contains $\ma…
Let $R$ be a Cohen-Macaulay local ring with maximal ideal $\max$. In this paper we present a procedure for computing the Ratllif-Rush closure of a $\max-$primary ideal $I \subset R$.
We formulate problems of tight closure theory in terms of projective bundles and subbundles. This provides a geometric interpretation of such problems and allows us to apply intersection theory to them. This yields new results concerning the tight closure of a primary ideal in a two-dimensional graded domain.
The paper shows that if the set of associated primes of Frobenius powers of ideals or a closely related set of primes is finite then if tight closure does not commute with localisation one can find a counter-example where $R$ is complete local and we are localizing at a prime ideal $P \subset R$ with $\dim (R/P)=1$. If one assumes further that for any local ring $(R,m)$ of prime characteristic $p$ and every finitely generated $R$-module $\bar M$ the set $ \bigcup_e \Ass G^e (\bar M) $ has finitely many maximal elements and, in addition, for every $R$-module $\bar M$ there exists a positive int…
Let $I$ be a monomial ideal $I$ in a polynomial ring $R = k[x_1,...,x_r]$. In this paper we give an upper bound on $\overline{\dstab} (I)$ in terms of $r$ and the maximal generating degree $d(I)$ of $I$ such that $\depth R/\overline{I^n}$ is constant for all $n\geqslant \overline{\dstab}(I)$. As an application, we classify the class of monomial ideals $I$ such that $\overline{I^n}$ is Cohen-Macaulay for some integer $n\gg 0$.
In two recent papers, the author has developed a theory of graded annihilators of left modules over the Frobenius skew polynomial ring over a commutative Noetherian ring $R$ of prime characteristic $p$, and has shown that this theory is relevant to the theory of test elements in tight closure theory. One result of that work was that, if $R$ is local and the $R$-module structure on the injective envelope $E$ of the simple $R$-module can be extended to a structure as a torsion-free left module over the Frobenius skew polynomial ring, then $R$ is $F$-pure and has a tight closure test element. One…
