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].
The study of Frobenius actions on local cohomology modules over a local ring of prime characteristic has interesting connections with the theory of tight closure. This paper establishes new connections by developing the notion of relative Frobenius actions on local cohomology. As an application, we show that a ring has $F$-nilpotent singularities if and only if the tight closure of every parameter ideal is equal to its Frobenius closure.
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 …
The main aim of this article is to study the relation between $F$-injective singularity and the Frobenius closure of parameter ideals in Noetherian rings of positive characteristic. The paper consists of the following themes, including many other topics. We prove that if every parameter ideal of a Noetherian local ring of prime characteristic $p>0$ is Frobenius closed, then it is $F$-injective. We prove a necessary and sufficient condition for the injectivity of the Frobenius action on $H^i_{\fm}(R)$ for all $i \le f_{\fm}(R)$, where $f_{\fm}(R)$ is the finiteness dimension of $R$. As applicat…
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 an inclusion result for graded dagger closure for primary ideals in symmetric section rings of abelian varieties over an algebraically closed field of arbitrary characteristic.
This paper is concerned with ideals in a commutative Noetherian ring $R$ of prime characteristic. The main purpose is to show that the Frobenius closures of certain ideals of $R$ generated by regular sequences exhibit a desirable type of `uniform' behaviour. The principal technical tool used is a result, proved by R. Hartshorne and R. Speiser in the case where $R$ is local and contains its residue field which is perfect, and subsequently extended to all local rings of prime characteristic by G. Lyubeznik, about a left module over the skew polynomial ring $R[x,f]$ (associated to $R$ and the Fro…
Suppose that R is a two-dimensional normal standard-graded domain over a finite field. We prove that there exists a uniform Frobenius test exponent b for the class of homogeneous ideals in R generated by at most n elements. This means that for every ideal I in this class we have that f^(p^b) belongs to I^([p^b]) if and only if f belongs to the Frobenius closure I^F. This gives in particular a finite test for the Frobenius closure. On the other hand we show that there is no uniform bound for Frobenius test exponent for all homogeneous ideals independent of the number of generators. Under simila…
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 $(R,\frak m)$ be an excellent generalized Cohen-Macaulay local ring of dimension $d$ that is $F$-injective on the punctured spectrum. Let $\frak q$ be a standard parameter ideal of $R$. The aim of the paper is to prove that $$\ell_R({\frak q}^F/{\frak q})\leq \sum\limits_{i=0}^{d}\binom{d}{i}\ell_R(0^F_{H^i_{\frak m}(R)}).$$ Moreover, if $\frak q$ is contained in a large enough power of $\frak m$, we have $${\frak q}^F/{\frak q} \cong \bigoplus_{i=0}^d (0^F_{H^i_{\frak m}(R)})^{\binom{d}{i}}.$$
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…
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…
The Frobenius test exponent $\operatorname{Fte}(R)$ of a local ring $(R,\mathfrak{m})$ of prime characteristic $p > 0$ is the smallest $e_0 \in \mathbb{N}$ such that for every ideal $\mathfrak{q}$ generated by a (full) system of parameters, the Frobenius closure $\mathfrak{q}^F$ has $(\mathfrak{q}^F)^{[p^{e_0}]} = \mathfrak{q}^{[p^{e_0}]}$. We establish a suffcient condition for $\operatorname{Fte}(R)<\infty$ and use it to show that if $R$ is such that the Frobenius closure of the zero submodule in the lower local cohomology modules has finite colength, i.e. $H^j_{\mathfrak{m}}(R) / 0^F_{H^j_{…
This paper proves some special cases in which localization of tight closure holds. In particular it studies the condition LC relating to bounding the Loewy lengths of local cohomology of Frobenius iterates of quotient rings.
Let $X$ be an equivariant embedding of a connected reductive group $G$ over an algebraically closed field $k$ of positive characteristic. Let $B$ denote a Borel subgroup of $G$. A $G$-Schubert variety in $X$ is a subvariety of the form $\diag(G) \cdot V$, where $V$ is a $B \times B$-orbit closure in $X$. In the case where $X$ is the wonderful compactification of a group of adjoint type, the $G$-Schubert varieties are the closures of Lusztig's $G$-stable pieces. We prove that $X$ admits a Frobenius splitting which is compatible with all $G$-Schubert varieties. Moreover, when $X$ is smooth, proj…
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 …
We construct normal hypersurfaces whose local cohomology modules have infinitely many associated primes. These include unique factorization domains of characteristic zero with rational singularities, as well as F-regular unique factorization domains of positive characteristic. As a consequence, we answer a question on the associated primes of Frobenius powers of ideals, which arose from the localization problem in tight closure theory.
It is an open question whether tight closure commutes with localization in quotients of a polynomial ring in finitely many variables over a field. Katzman showed that tight closure of ideals in these rings commutes with localization at one element if for all ideals I and J in a polynomial ring there is a linear upper bound in q on the degree in the least variable of reduced Groebner bases in reverse lexicographic ordering of the ideals of the form J + I^{[q]}. Katzman conjectured that this property would always be satisfied. In this paper we prove several cases of Katzman's conjecture. We also…
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 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).
Let $R$ be a commutative Noetherian local ring of prime characteristic $p$. The main purposes of this paper are to show that if the injective envelope $E$ of the simple $R$-module has a structure as a torsion-free left module over the Frobenius skew polynomial ring over $R$, then $R$ has a tight closure test element (for modules) and is $F$-pure, and to relate the test ideal of $R$ to the smallest '$E$-special' ideal of $R$ of positive height. A byproduct is an analogue of a result of Janet Cowden Vassilev: she showed, in the case where $R$ is an $F$-pure homomorphic image of an $F$-finite reg…
We introduce a new variant of tight closure and give an interpretation of adjoint ideals via this tight closure. As a corollary, we prove that a log pair $(X,Δ)$ is plt if and only if the modulo $p$ reduction of $(X,Δ)$ is divisorially F-regular for all large $p \gg 0$. Here, divisorially F-regular pairs are a class of singularities in positive characteristic introduced by Hara and Watanabe in terms of Frobenius splitting.
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.
It was previously known, by work of Smith-Swanson and of Sharp-Nossem, that the linear growth property of primary decompositions of Frobenius powers of ideals in rings of prime characteristic has strong connections to the localization problem in tight closure theory. The localization problem has recently been settled in the negative by Brenner and Monsky, but the linear growth question is still open. We study growth of primary decompositions of Frobenius powers of dimension one homogeneous ideals in graded rings over fields. If the ring is positively graded we prove that the linear growth prop…
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 …
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…
This paper studies Frobenius powers of parameter ideals in a commutative Noetherian local ring $R$ of prime characteristic $p$. For a given ideal $\fa$ of $R$, there is a power $Q$ of $p$, depending on $\fa$, such that the $Q$-th Frobenius power of the Frobenius closure of $\fa$ is equal to the $Q$-th Frobenius power of $\fa$. The paper addresses the question as to whether there exists a {\em uniform} $Q_0$ which `works' in this context for all parameter ideals of $R$ simultaneously. In a recent paper, Katzman and Sharp proved that there does exists such a uniform $Q_0$ when $R$ is Cohen--Maca…
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 …
Let R be a regular, local and F-finite ring defined over a field of finite characteristic. Let I be an ideal of height c with normal quotient $A=R/I$. It is shown that the local cohomology module H^c_I(R) contains a unique simple D_R--submodule L(A,R). This should be viewed as a finite characteristic analog of the Kashiwara--Brylinski D_R--module in characteristic zero which corresponds to the intersection cohomology complex via the Riemann--Hilbert correspondence. Besides the existence of L(A,R), more importantly, we give its construction as a certain dual of the tight closure of zero in $H^d…
Let $B$ be an affine Cohen-Macaulay algebra over a field of characteristic $p$. For every prime ideal $\mathfrak{p}\subset B$, let $\text{H}_\mathfrak{p}$ denote $H^{\dim B_\mathfrak{p}}_{\mathfrak{p} B_\mathfrak{p}}\left( \widehat{B_\mathfrak{p}} \right)$. Each such $\text{H}_\mathfrak{p}$ is an Artinian module endowed with a natural Frobenius map $Θ$ and if $\text{Nil}(\text{H}_\mathfrak{p})$ denotes the set of all elements in $\text{H}_\mathfrak{p}$ killed by some power of $Θ$ then a theorem by Hartshorne-Speiser and Lyubeznik shows that there exists an $e\geq 0$ such that $Θ^e \text{Nil}(\…
