A note on higher almost ring theory

TL;DR

This paper explores a derived approach to localizations in almost ring theory, providing an example where Frobenius endomorphism fails to induce an isomorphism on the smashing spectrum in positive characteristic.

Contribution

It introduces a derived construction of localizations in almost ring theory and demonstrates a counterexample related to Frobenius endomorphism behavior.

Findings

Derived localization construction in almost ring theory

Counterexample involving Frobenius endomorphism

Failure of Frobenius to induce spectrum isomorphism

Abstract

We explain a derived version of the basic construction of localisations of module categories by means of idempotent ideals, which lie at the heart of Faltings' almost ring theory. We use it to provide an example of a commutative algebra in positive characteristic whose Frobenius endomorphism does not induce an isomorphism on its smashing spectrum.