Bounds for test exponents

Test ideals were first introduced by Mel Hochster and Craig Huneke in their celebrated theory of tight closure, and since their invention have been closely tied to the theory of Frobenius splittings. Subsequently, test ideals have also found application far beyond their original scope to questions arising in complex analytic geometry. In this paper we give a contemporary survey of test ideals and their wide-ranging applications.

Let $S$ be a polynomial ring in $n$ variables over a field. Let $I$ be a homogeneous ideal in $S$ generated by forms of degree at most $d$ with $\text{dim}(S/I)=r$. In the first part of this paper, we show how to derive from a result of Hoa an upper bound for the regularity of $\sqrt{I}$. More specifically we show that $\text{reg}(\sqrt{I})\leq d^{(n-1)2^{r-1}}$. In the second part, we show that the $r$-th arithmetic degree of $I$ is bounded above by $2\cdot d^{2^{n-r-1}}$. T...

Let I denote a homogeneous R_+-primary ideal in a two-dimensional normal standard-graded domain over an algebraically closed field of characteristic zero. We show that a homogeneous element f belongs to the solid closure I^* if and only if e_{HK}(I) = e_{HK}((I,f)), where e_{HK} denotes the (characteristic zero) Hilbert-Kunz multiplicity of an ideal. This provides a version in characteristic zero of the well-known Hilbert-Kunz criterion for tight closure in positive character...

We prove a result relating the Jacobian ideal and the generalized test ideal associated to a principal ideal in $R=k[x_1,...,x_n]$ with $[k:k^p]<\infty$ or in $R=k[[x_1,...,x_n]]$ with an arbitrary field $k$ of characteristic $p>0$. As a consequence of this result, we establish an upper bound on the number of $F$-jumping coefficients of a principal ideal with an isolated singularity.

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^+...

Let $R$ be a polynomial ring over a field $k$ with irrelevant ideal $\frak m$ and dimension $d$. Let $I$ be a homogeneous ideal in $R$. We study the asymptotic behavior of the length of the modules $H^{i}_{\frak m}(R/I^n)$ for $n\gg 0$. We show that for a fixed number $\alpha \in \mathbb Z$, $\limsup_{n\rightarrow \infty}\frac{\lambda(H^{i}_{\frak m}(R/I^n)_{\geq -\alpha n})}{n^d}<\infty.$ Combining this with recent strong vanishing results gives that $\limsup_{n\rightarrow \...

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.

Let $R=\mathbb{K}[X_1, \ldots , X_n ]$ be a polynomial ring over a field $\mathbb{K}$. We introduce an endomorphism $\mathcal{F}^{[m]}: R \rightarrow R $ and denote the image of an ideal $I$ of $R$ via this endomorphism as $I^{[m]}$ and call it to be the $m$ \textit{-th square power} of $I$. In this article, we study some homological invariants of $I^{[m]}$ such as regularity, projective dimension, associated primes and depth for some families of ideals e.g. monomial ideals.

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 ideal...

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 c...