On some properties of birational derived splinters

TL;DR

This paper investigates birational derived splinters, a class of rings generalizing rational singularities, exploring their fundamental properties and behavior under various algebraic operations across different characteristics.

Contribution

It establishes key properties of birational derived splinters, including their stability under localization, subrings, limits, and extensions, and relates them to rational singularities and splinters.

Findings

Direct limits of rational singularities have rational singularities.

Birational derived splinters behave well under localization and étale extensions.

Residue extensions and regularity properties are studied in positive characteristic.

Abstract

A Noetherian reduced ring $A$ is called a birational derived splinter if for all proper birational maps $X\to\operatorname{Spec}(A)$, the canonical map $A\to Rf_*\mathcal{O}_X$ splits. In equal characteristic zero this property characterizes rational singularities, but much less can be said in positive or mixed characteristics. In this paper, we prove some fundamental properties of this notion, including the behavior under localization, taking a pure subring, taking direct limit, and along an \'etale extension. In particular, direct limit of rational singularities in characteristic zero has rational singularities. Then, we study residue extensions (in arbitrary characteristic), and openness and regular extensions in positive characteristic, parallel to Datta-Tucker and the author's previous works on splinters.