Holger Brenner
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.
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On complete intersection projective closures of monomial curves
研究四维仿射空间中单项式曲线射影闭包的完全交性质，给出其定义理想为完全交的刻画，还探讨其是否算术科恩 - 麦考利。
Our aim in this paper is to compute the Poincaré series of the derivation module of the projective closure of certain affine monomial curves.
Prime Ideals and Projective Closure in the Boolean Polynomial Semiring
本文研究二元布尔多项式半环中的素理想，给出线性生成素理想的完全分类，并通过同余引入射影闭包，证明其与素同余射影空间中的拓扑闭包一致。
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 give an example showing that tight closure does not commute with localization.
We give a survey about some recent work on tight closure and Hilbert-Kunz theory from the viewpoint of vector bundles
We present a geometric interpretation of tight closure in terms of vector bundles and projective bundles.
We study different notions of slope of a vector bundle over a smooth projective curve with respect to ampleness and affineness in order to apply this to tight closure problems. This method gives new degree estimates from above and from below for the tight closure of a homogeneous $R_+$-primary ideal in a two-dimensional normal standard-graded algebra $R$ in terms of the minimal and the maximal slope of the sheaf of relations for some ideal generators. If moreover this sheaf of relations is semistable, then both degree estimates coincide and we get a vanishing type theorem.
Tight closure of ideals on Witt rings
引入 Witt 环上理想的紧闭包和参数系的拟紧闭性，刻画拟 F-理性，并研究闭包算子与整闭包的关系。
We use geometric and cohomological methods to show that given a degree bound for membership in ideals of a fixed degree type in the polynomial ring P=k[x_0,..., x_d], one obtains a good generic degree bound for membership in the tight closure of an ideal of that degree type in any standard-graded k-algebra R of dimension d+1. This indicates that the tight closure of an ideal behaves more uniformly than the ideal itself. Moreover, if R is normal, one obtains a generic bound for membership in the Frobenius closure. If d is at most 2, then the bound for ideal membership in P can be computed from …
We study computational aspects of the tight closure of a homogeneous primary ideal in a two-dimensional normal standard-graded domain. We show how to use slope criteria for the sheaf of syzygies for generators of the ideal to compute the tight closure. Our method gives in particular an algorithm to compute the tight closure of three elements under the condition that we are able to compute the Harder-Narasimhan filtration. We apply this to the computation of (x^a,y^a,z^a)^* in K[x,y,z]/(F), where F is a homogeneous polynomial.
We introduce a new variant of tight closure associated to any fixed ideal $\a$, which we call $\a$-tight closure, and study various properties thereof. In our theory, the annihilator ideal $τ(\a)$ of all $\a$-tight closure relations, which is a generalization of the test ideal in the usual tight closure theory, plays a particularly important role. We prove the correspondence of the ideal $τ(\a)$ and the multiplier ideal associated to $\a$ (or, the adjoint of $\a$ in Lipman's sense) in normal $\Q$-Gorenstein rings reduced from characteristic zero to characteristic $p \gg 0$. Also, in fixed prim…
In this paper our aim is twofold. First, we introduce the notion of star gluing of numerical semigroups and show that arithmetically Cohen-Macaulay and Gorenstein properties of the projective closure are preserved under this gluing operation. We then give a condition on Gröbner basis of the defining ideal of an affine monomial curve which ensures that the Betti sequence of the affine curve is the same as the Betti sequence of its projective closure. We also study the effect of simple gluing on Betti sequences of the projective closure. Finally, we construct some numerical semigroups, using a g…
We provide a negative answer to an old question in tight closure theory by showing that the containment x^3y^3 \in (x^4,y^4,z^4)^* in K[x,y,z]/(x^7+y^7-z^7) holds for infinitely many but not for almost all prime characteristics of the field K. This proves that tight closure exhibits a strong dependence on the arithmetic of the prime characteristic. The ideal (x,y,z) \subset K[x,y,z,u,v,w]/(x^7+y^7-z^7, ux^4+vy^4+wz^4+x^3y^3) has then the property that the cohomological dimension fluctuates arithmetically between 0 and 1.
We study the projective closures of three important families of affine monomial curves in dimension $4$, namely the Backelin curve, the Bresinsky curve and the Arslan curve, in order to explore possible connections between syzygies and the arithmetic Cohen-Macaulay property.
The F-threshold $c^J(\a)$ of an ideal $\a$ with respect to the ideal $J$ is a positive characteristic invariant obtained by comparing the powers of $\a$ with the Frobenius powers of $J$. We show that under mild assumptions, we can detect the containment in the integral closure or the tight closure of a parameter ideal using F-thresholds. We formulate a conjecture bounding $c^J(\a)$ in terms of the multiplicities $e(\a)$ and $e(J)$, when $\a$ and $J$ are zero-dimensional ideals, and $J$ is generated by a system of parameters. We prove the conjecture when $J$ is a monomial ideal in a polynomial …
F-coherent rings with applications to tight closure theory
本文引入正特征诺特环的新类F-凝聚环（完全闭包为凝聚的诺特环），研究其性质，并结合紧闭包理论给出应用，讨论了F-凝聚环与几类F-环的关系及凝聚性对一般完全环紧闭包行为的影响。
Building on work of Brenner and Monsky from 2010 and on a Hilbert-Kunz calculation of Monsky from 1998, we exhibit a novel example of a hypersurface over $\overline{\mathbb{F}_2}$ in which tight closure does not commute with localization. Our methods involve a surprising tiling argument using Sierpiński triangles, as well as an inspection of a certain dynamical system in characteristic two.
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.
Projective Closure of Semigroup Algebras
本文研究$\mathbb{N}^{d}$中单纯仿射半群的射影闭包，用Gröbner基刻画其Cohen-Macaulay性质、给出非Cohen-Macaulay情形下Buchsbaum性质的判据，还提出$k$-提升概念并研究其与原半群的关系。
Let I denote an R_+ -primary homogeneous ideal in a normal standard-graded Cohen-Macaulay domain over a field of positive characteristic p. We give a linear degree bound for the Frobenius powers I^[q] of I, q=p^e, in terms of the minimal slope of the top-dimensional syzygy bundle on the projective variety Proj R. This provides an inclusion bound for tight closure. In the same manner we give a linear bound for the Castelnuovo-Mumford regularity of the Frobenius powers I^[q].
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.
Hara and Smith independently proved that in a normal $\mQ$-Gorenstein ring of characteristic $p \gg 0$, the test ideal coincides with the multiplier ideal associated to the trivial divisor. We extend this result for a pair $(R, Δ)$ of a normal ring $R$ and an effective $\mQ$-Weil divisor $Δ$ on $\Spec R$. As a corollary, we obtain the equivalence of strongly F-regular pairs and klt pairs.
We study the relationship between the tight closure of an ideal and the sum of all ideals in its linkage class.
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 an operation on modules over an $F$-finite ring of characteristic $p$. We call this operation \emph{tight interior}. While it exists more generally, in some cases this operation is equivalent to the Matlis dual of tight closure. Moreover, the interior of the ring itself is simply the big test ideal. We directly prove, without appeal to tight closure, results analogous to persistence, colon capturing, and working modulo minimal primes, and we begin to develop a theory dual to phantom homology. Using our dual notion of persistence, we obtain new and interesting transformation rules …
Closure operations induced via resolutions of singularities in characteristic zero
在零特征中，通过解析奇点引入KH闭包操作，用于检测理性奇点并改进紧闭操作，同时探索相关闭包操作及其与正特征紧闭操作的联系。
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…
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.
This is an expanded version of the author's lecture at the Conference in Commutative Algebra and Algebraic Geometry in Messina Italy in June 1999. The purpose of the talk was to give a brief introduction to the subject of tight closure, aimed at commutative algebraists who have not before studied this topic. The first part focused mainly on the definition and basic properties, with the second part focusing on some applications to algebraic geometry, particularly to global generation of adjoint linear series. These lecture notes follow even more closely a series of two lectures I gave in Kashik…
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 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.
Tight closure of products and F-rational singularities
本文通过极小闭包理论研究F-理性质，并在高维情况下扩展了曲面奇点的特征化结果。
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.
Tight closure, coherence, and localization at single elements
本文提出开放持续性与可粘合性条件，证明满足二者的闭包运算在单元素局部化与保持拟凝聚性之间等价，并将其用于紧闭问题，引入半 F-正则性。
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.
We provide bounds on the Castelnuovo-Mumford regularity in terms of ``defining equations'' by using elements that annihilates some cohomology modules, inspired by works of Miyazaki, Nagel, Schenzel and Vogel. The elements in these annihilators are provided either by liaison or by tight closure theories. Our results hold in any characteristic.
We associate to every equicharacteristic zero Noetherian local ring $R$ a faithfully flat ring extension which is an ultraproduct of rings of various prime characteristics, in a weakly functorial way. Since such ultraproducts carry naturally a non-standard Frobenius, we can define a new tight closure operation on $R$ by mimicking the positive characteristic functional definition of tight closure. This approach avoids the use of generalized Néron Desingularization and only relies on Rotthaus' result on Artin Approximation in characteristic zero. If $R$ is moreover equidimensional and universall…
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.
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…
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…
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…
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…
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…
Let $F$ be a field, let $D$ be a subring of $F$, and let ${\mathfrak{X}}$ be the Zariski-Riemann space of valuation rings containing $D$ and having quotient field $F$. We consider the Zariski, inverse and patch topologies on ${\mathfrak{X}}$ when viewed as a projective limit of projective integral schemes having function field contained in $F$, and we characterize the locally ringed subspaces of ${\mathfrak{X}}$ that are affine schemes.
Let $R$ be a (commutative Noetherian) local ring of prime characteristic that is $F$-pure. This paper studies a certain finite set ${\mathcal I}$ of radical ideals of $R$ that is naturally defined by the injective envelope of the simple $R$-module. This set ${\mathcal I}$ contains $0$ and $R$, and is closed under taking primary components. For a multiplicatively closed subset $S$ of $R$, the concept of tight closure with respect to $S$, or $S$-tight closure, is discussed, together with associated concepts of $S$-test element and $S$-test ideal. It is shown that an ideal of $R$ belongs to ${\ma…
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