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 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…
In this note, we derive a formula for the F-pure threshold of diagonal hypersurfaces over a perfect field of prime characteristic. We also calculate the associated test ideal at the F-pure threshold, and give formulas for higher jumping numbers of Fermat hypersurfaces.
Let $R=k[x_1, ..., x_n]/(x_1^d + ... + x_n^d)$, where $k$ is a field of characteristic $p$, $p$ does not divide $d$ and $n \geq 3$. We describe a method for computing the test ideal for these diagonal hypersurface rings. This method involves using a characterization of test ideals in Gorenstein rings as well as developing a way to compute tight closures of certain ideals despite the lack of a general algorithm. In addition, we compute examples of test ideals in diagonal hypersurface rings of small characteristic (relative to $d$) including several that are not integrally closed. These examples…
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.
The third named author and Pérez proved that under certain conditions the test ideal of a module closure agrees with the trace ideal of the module closure. We use this fact to compute the test ideals of various rings with respect to the closures coming from their indecomposable maximal Cohen-Macaulay modules. We also give an easier way to compute the test ideal of a hypersurface ring in 3 variables coming from a module with a particular type of matrix factorization.
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 characterize the ideals $I$ of $\mathcal O_n$ of finite colength whose integral closure is equal to the integral closure of an ideal generated by pure monomials. This characterization, which is motivated by an inequality proven by Demailly and Pham, is given in terms of the log canonical threshold of $I$ and the sequence of mixed multiplicities of $I$.
In 1936, Krull asked if the integral closure of a primary ideal is still primary. Fifty years later, Huneke partially answered this question by giving a primary polynomial ideal whose integral closure is not primary in a regular local ring of characteristic $p=2$. We provide counterexamples to Krull's question regarding polynomial rings with any characteristics. We also find that the Jacobian ideal $J$ of the polynomial $f = x^6 + y^6 + x^4 z t + z^3$ given by Briançon and Speder in 1975 is a counterexample to Krull's question. Let $V_1$ be the hypersurface defined by $f = 0$ and $V_2$ be its …
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…
Diagonal F-thresholds for determinants and Pfaffians
Otto-von-Guericke-University Magdeburg（马格德堡奥托·冯·古里克大学）；University of Notre Dame（圣母大学）；Institute of Mathematics “Simion Stoilow” of the Romanian Academy（罗马尼亚科学院西米翁·斯托伊洛数学研究所）；University of Cassino and Southern Lazio（卡西诺与南拉齐奥大学）；Visvesvaraya National Institute of Technology, Nagpur（那格浦尔维斯瓦纳特国家技术学院）
Let $k$ be a field. For each pair of positive integers $(n,N)$, we resolve $Q=R/(x^N,y^N,z^N)$ as a module over the ring $R=k[x,y,z]/(x^n+y^n+z^n)$. Write $N$ in the form $N=a n+r$ for integers $a$ and $r$, with $r$ between $0$ and $n-1$. If $n$ does not divide $N$ and the characteristic of $k$ is fixed, then the value of $a$ determines whether $Q$ has finite or infinite projective dimension. If $Q$ has infinite projective dimension, then value of $r$, together with the parity of $a$, determines the periodic part of the infinite resolution. When $Q$ has infinite projective dimension we give an…
Let $I$ be an ideal generated by a system of parameters in an excellent Cohen-Macaulay local domain. We show that the associated graded ring $G^*(I)$ of the filtration $\{(I^n)^*: n\in \mathbb{N}\}$ is Cohen-Macaulay. We prove that if $R$ is an excellent Buchsbaum local domain then $G^*(I)$ is a Buchsbaum module over the Rees ring $\mathcal R^*(I)=\oplus_{n\in \mathbb{N}}(I^n)^*.$ We provide quick proofs of well-known results of I. Aberbach, Huneke-Itoh and Huneke-Hochster about the filtration $\{(I^n)^*: n\in \mathbb{N}\}$ in excellent local domains. An important tool used in the proofs is a …
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.
We study the properties of F-rationality and F-regularity in multigraded rings and their diagonal subalgebras. The main focus is on diagonal subalgebras of bigraded rings: these constitute an interesting class of rings since they arise naturally as homogeneous coordinate rings of blow-ups of projective varieties. As a consequence of some of the results obtained here, it is shown that there exist standard bigraded hypersurfaces whose rings of invariants under torus actions have rational singularities, but are not of F-regular type. Another application is the construction of families of rings wi…
Analogues of Eakin-Sathaye theorem for reductions of ideals are proved for ${\mathbb N}^s$-graded good filtrations. These analogues yield bounds on joint reduction vectors for a family of ideals and reduction numbers for $\mathbb N$-graded filtrations. Several examples related to lex-segment ideals, contracted ideals in $2$-dimensional regular local rings and the filtration of integral and tight closures of powers of ideals in hypersurface rings are constructed to show effectiveness of these bounds.
We provide a formula for $F$-thresholds of a Thom-Sebastiani type polynomial over a perfect field of prime characteristic. This result extends the formula for the $F$-pure threshold of a diagonal hypersurface. We also compute the first test ideal of Thom-Sebastiani type polynomials. Finally, we apply our result to find hypersurfaces where the log canonical thresholds equals the $F$-pure thresholds for infinitely many prime numbers.
We give a formula for the smallest degree of a non-Koszul relation on $x_1^{d_1}, \ldots, x_n^{d_n}, (x_1+\ldots +x_n)^{d_{n+1}}\in k[x_1, \ldots, x_n]$ (under certain assumptions on $d_1, \ldots, d_{n+1}$) where $k$ is a field of positive characteristic $p$. As an application of our result, we give a formula for the diagonal F-threshold of a diagonal hypersurface. Another application is a characterization, depending on the characteristic $p$ of $k$, of the values of $d_1, \ldots, d_{n+1}$ (satisfying certain assumptions) such that the ring $k[x_1, \ldots, x_{n+1}]/(x_1^{d_1}, \ldots, x_{n+1}^…
