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Ultra-test ideals in rings with finitely generated anti-canonical algebras

Tatsuki Yamaguchi

TL;DR

The paper addresses adjoint ideals in characteristic zero without requiring $\mathbb{Q}$-Gorenstein hypotheses by exploiting ultra-Frobenius methods. It develops divisorial ultra-test ideals and proves that, when the anti-canonical algebra $\bigoplus_{i\ge 0} R(-i(K_R+D))$ is finitely generated, these ultra-test ideals coincide with adjoint ideals along $D$, providing a new route to multiplier-like behavior in non-$\mathbb{Q}$-Gorenstein settings. A key application is that adjoint ideals descend under pure morphisms, yielding inclusions that generalize Zhuang’s result and extend TY24 to broader contexts, including lc/klt-type phenomena. The work thus connects nonstandard analysis techniques with tight closure tools to unify adjoint-ideal theory across characteristic $p$ reductions and characteristic zero, broadening the scope of singularity transfer phenomena under pure morphisms.

Abstract

When anti-canonical rings are finitely generated, we give a characterization of adjoint ideals using ultra-Frobenii, a characteristic zero analogue of Frobenius morphisms. This characterization enables us to give an alternative proof of a result of Zhuang, which states that if a ring is of klt type, then so is any of its pure subrings.

Ultra-test ideals in rings with finitely generated anti-canonical algebras

TL;DR

The paper addresses adjoint ideals in characteristic zero without requiring -Gorenstein hypotheses by exploiting ultra-Frobenius methods. It develops divisorial ultra-test ideals and proves that, when the anti-canonical algebra is finitely generated, these ultra-test ideals coincide with adjoint ideals along , providing a new route to multiplier-like behavior in non--Gorenstein settings. A key application is that adjoint ideals descend under pure morphisms, yielding inclusions that generalize Zhuang’s result and extend TY24 to broader contexts, including lc/klt-type phenomena. The work thus connects nonstandard analysis techniques with tight closure tools to unify adjoint-ideal theory across characteristic reductions and characteristic zero, broadening the scope of singularity transfer phenomena under pure morphisms.

Abstract

When anti-canonical rings are finitely generated, we give a characterization of adjoint ideals using ultra-Frobenii, a characteristic zero analogue of Frobenius morphisms. This characterization enables us to give an alternative proof of a result of Zhuang, which states that if a ring is of klt type, then so is any of its pure subrings.

Paper Structure

This paper contains 6 sections, 20 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Adjoint ideals
  4. Divisorial test ideals
  5. Ultraproducts
  6. Main theorem

Key Result

Theorem 1.1

Let $(R,{\mathfrak m})$ be a local normal domain essentially of finite type over $\mathbb{C}$, $D$ be a prime divisor or $D=0$, $f$ be not in any minimal prime of $R(-D)$ and $t$ be a positive rational number. Suppose that $\bigoplus_{i\geqslant 0}R(-i(K_R+D))$ is a finitely generated $R$-algebra. T where $\tau_D^{\mathrm{u}}(R,D)$ denotes the divisorial ultra-test ideal of the triple $(R,D,f^t)$

Theorems & Definitions (55)

  • Theorem 1.1: Theorem \ref{['divisorial ultra-test ideals = adjoint ideals']}
  • Theorem 1.2: Theorem \ref{['adjoint ideals under pure morphisms']}
  • Theorem 1.3: Theorem \ref{['log canonical under pure morphisms']}
  • Definition 2.1
  • Remark 2.2
  • Proposition 2.3: dFH09,TY24
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Proposition 2.7: cf. HH90
  • ...and 45 more