SPLINTER-TYPE SINGULARITIES VIA ULTRAPOWER

Abstract

We study the notions of splinter, birational derived splinter, and derived splinter for a ring $A$.They are defined by the splitting of the maps $A\to Rf_*\cO_X$ for certain morphisms $X\to\Spec(A)$. For Noetherian $A$, these notions can be considered as notions of singularities of $A$. We establish numerous fundamental properties, including behaviours with respect to direct limit and \'etale extensions. Then we study the more subtle problem of the ascent along a regular map.We make use the construction of ultrapower. A special case of ultraproduct, this construction is seen more in model theory. We give an alternative definition for local rings. Taking ultrapower does not preserve Noetherianness. We therefore need to involve non-Noetherian rings and schemes. We also give a short discussion on the equicharacteristic zero case. In the appendix, we perform a model-theoretic study of splinters.This is where the idea for this work originates from. For the reader's convenience, we review the basics of model theory of rings. We also discuss a variant of (birational) derived splinter that behaves better in the non-Noetherian case.