Toric varieties, monoid schemes and $cdh$ descent

TL;DR

This paper establishes conditions under which algebraic K-theory satisfies Mayer-Vietoris for blow-up squares of toric varieties, connecting it to topological cyclic homology via monoid schemes.

Contribution

It introduces new conditions for Mayer-Vietoris in algebraic K-theory of toric varieties using monoid schemes, extending classical geometric notions to this setting.

Findings

Mayer-Vietoris property holds under specified conditions for toric varieties.

Links algebraic K-theory with topological cyclic homology in characteristic p.

Develops geometric notions for monoid schemes analogous to classical algebraic geometry.

Abstract

We give conditions for the Mayer-Vietoris property to hold for the algebraic K-theory of blow-up squares of toric varieties in any characteristic, using the theory of monoid schemes. These conditions are used to relate algebraic K-theory to topological cyclic homology in characteristic p. To achieve our goals, we develop for monoid schemes many notions from classical algebraic geometry, such as separated and proper maps.