Test ideals vs. multiplier ideals

Mircea Mustata; Ken-ichi Yoshida

Home/Archive/arXiv

arXiv · 0706.1124

Test ideals vs. multiplier ideals

Mircea Mustata·Ken-ichi Yoshida

View original source ↗Read source document ↗Export metadata ↓

Abstract

The generalized test ideals introduced in [HY] are related to multiplier ideals via reduction to characteristic p. In addition, they satisfy many of the subtle properties of the multiplier ideals, which in characteristic zero follow via vanishing theorems. In this note we give several examples to emphasize the different behavior of test ideals and multiplier ideals. Our main result is that every ideal in an F-finite regular local ring can be written as a generalized test ideal. We also prove the semicontinuity of $F$-pure thresholds (though the analogue of the Generic Restriction Theorem for multiplier ideals does not hold).

Explore related subjects

math.AC math.AG

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Mircea Mustata, Ken-ichi Yoshida. 2008-06-03. Test ideals vs. multiplier ideals. https://arxiv.org/abs/0706.1124

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The structure of almost symmetric almost complete intersection numerical semigroups

We prove a structure theorem for numerical semigroups H that are almost symmetric and almost complete intersections. We show that a row-factorization (RF) matrix of H must possess a highly regular structure, which we call a cascade matrix. Consequently, the defining ideal I_H of the semigroup ring k[H] also exhibits a highly regular structure, derived from this cascade matrix. Both the RF-matrix and the binomial minimal generating set of I_H are unique. Conversely, this structure completely characterizes almost symmetric almost complete intersection numerical semigroups: to every cascade matrix M we associate a monoid H and, whenever this is a numerical semigroup, we prove that H is pseudo-symmetric, almost complete intersection, and has M as RF-matrix. As a consequence of our study, we obtain several key results. 1) Rigidty: if an almost complete intersection semigroup is almost symmetric, then it is forced to have odd embedding dimension and to be pseudo-symmetric. This result is a ``next step'' after Kunz's theorem, which states that an almost complete intersection semigroup is never symmetric. 2) Huneke-Ulrich structure: the toric ideal can be realized as the ideal of entries of an alternating matrix multiplied by a vector of variables, and this yields the licci property. 3) Cascade polynomials: for each odd positive integer e, we construct a multivariate squarefree polynomial P_e with integer coefficients, arising from a cascade matrix of variables. We provide an enumerative interpretation of its coefficients, thereby proving their non-negativity. 4) Herzog--Watanabe question: we prove that every minimal relation of an arbitrary numerical semigroup H can be obtained by subtracting two rows in some RF-matrix of H, affirmatively answering a 2019 question by Herzog and Watanabe.

math.AC↗

arXiv2026

Characteristic-free Knörrer periodicity

We show that the Knörrer functor induces an equivalence ${\underline{\operatorname{MCM}}}(R) \simeq {\underline{\operatorname{MCM}}}(A)$ of stable categories of maximal Cohen-Macaulay modules, where $R = S/(f)$ is a complete hypersurface ring and $A = S[\![u,v]\!]/(f+uv)$ is the hyperbolic extension of $R$. No hypothesis is placed on the residue field or on its characteristic, $f$ need not define an isolated singularity, and $S$ need not contain a field. This is in contrast to the iterated double branched cover $R^{\sharp\sharp} = S[\![z,w]\!]/(f+z^2+w^2)$, which is stably equivalent to $R$ only in characteristic not equal to $2$. We also give a direct construction of a free resolution identifying $\operatorname{syz}_2^A(M)$ with the image of $M \oplus \operatorname{syz}_1^R M$ under the functor, and an example in characteristic two in which the corresponding statement for the double branched cover $S[\![z]\!]/(f+z^2)$ fails.

math.AC↗

arXiv2026

Asymptotic Lech Bounds and Bertini Flags

For a $d$-dimensional Noetherian local ring $(R,\m)$, with $d>0$, let $\cinf(R)$ be the limit, as $N\to\infty$, of the suprema of $e_R(I)/(d!\length_R(R/I))$ over ideals $I$ with $\sqrt I=\m$ and $\length_R(R/I)>N$. We prove that if $R$ is complete, reduced, equidimensional, and Cohen--Macaulay on the punctured spectrum, with $d\geq2$ and infinite residue field, then every nonzerodivisor $x\in\m$ satisfies \[ \cinf(R)\leq \max\left\{\cinf(R/xR),\sup_{\fp\ne\m}e(R_{\fp})\right\}. \] As an application, we prove the asymptotic Lech conjecture of Huneke, Ma, Quy, and Smirnov in arbitrary characteristic: $\cinf(R)=1$ whenever $\widehat R$ has isolated singularities. This also settles the conjecture of Ma and Smirnov that the Lech--Mumford constant $\cLM(R)$ is attained when $\widehat R$ has isolated singularities and $\cLM(R)>1$. For reduced equidimensional Cohen--Macaulay standard graded algebras of dimension $\geq2$ over infinite perfect fields, we prove attainment and rationality of $\cLM$ for the completed vertex local rings of sufficiently high Veronese subrings. In characteristic zero, a refinement of our method produces nonisolated klt affine cones in every dimension $d\geq3$, with $d-2$-dimensional singular loci, whose completed vertex local rings have $\cinf=1$ and arbitrarily large $\cLM$.

math.AC↗