Order reduction of $\mathrm{\Lambda }$-marked monomial ideals and weak resolutions

Borger’s theory of $\mathrm{\Lambda }$-spaces imbues algebraic spaces, which include schemes, with an additional structure defined by an extension of the Witt vector functor. Motivated by ${\mathbb{F}}_1$-geometry, we prove the existence of a weak resolution of singularities in the category of $\mathrm{\Lambda }$-schemes. Our arguments are based on standard arguments in characteristic $0$ using the order reduction of an ideal marked with $\mathrm{\Lambda }$-equivariant data. This paper is based on work from the author’s PhD thesis.