Perfectoid pure singularities

Abstract: Fix a prime number $p$. Inspired by the notion of $F$-pure or $F$-split singularities, we study the condition that a Noetherian ring with $p$ in its Jacobson radical is pure inside some perfectoid (classical) ring, a condition we call \emph{perfectoid pure}. We also study a related a priori weaker condition which asks that $R$ is pure in its absolute perfectoidization, a condition we call \emph{lim-perfectoid pure}. We show that both these notions coincide when $R$ is LCI. Mixed characteristic analogs of $F$-injective and Du Bois singularities are also explored. We study these notions of singularity, proving that they are weakly normal and that they are Du Bois after inverting $p$. We also explore the behavior of perfectoid singularities under finite covers and their relation to log canonical singularities. Finally, we prove an inversion of adjunction result in the LCI setting, and use it to prove that many common examples are perfectoid pure.

• The paper introduces perfectoid pure singularities by defining when a Noetherian ring is pure inside a perfectoid ring, extending classical F-purity concepts.
• It establishes that, for locally complete intersection rings, perfectoid pure and lim-perfectoid pure conditions coincide, while also exploring mixed characteristic analogs.
• The study demonstrates that these singularities are weakly normal and Du Bois after inverting p, with inversion of adjunction results linking them to log canonical models.

The paper "Perfectoid pure singularities" investigates new concepts in the study of singularities in algebraic geometry, particularly through the lens of perfectoid spaces. The authors focus on the following key aspects:

The notion of perfectoid pure singularities is central. These singularities are defined by examining when a Noetherian ring, with a prime $p$ in its Jacobson radical, is pure inside some perfectoid ring. This extends the ideas from $F$-pure or $F$-split singularities, which are well-known in characteristic $p$ settings.

Another important concept introduced in the paper is lim-perfectoid pure. This is considered a potentially weaker condition, where a ring $R$ is pure in its absolute perfectoidization. The study reveals that, for locally complete intersection (LCI) rings, both conditions coincide.

The paper also explores mixed characteristic analogs of familiar singularities. Specifically, the authors investigate the conditions under which singularities are $F$-injective and possess Du Bois properties after inverting $p$. This highlights the relevance of these notions in broader contexts.

Key results include demonstrating that perfectoid pure singularities are weakly normal and Du Bois after inverting $p$. Additionally, the paper examines how these singularities behave under finite covers and their connection to log canonical singularities, expanding upon traditional singularity theory.

An inversion of adjunction result is proved within the LCI setting, enabling the authors to apply the theory to various common examples, showing they exhibit perfectoid pure characteristics.

Overall, this paper enhances understanding of singularities through innovative approaches, particularly by integrating concepts from perfectoid spaces, and provides useful frameworks for future exploration in algebraic geometry.