Fan is to monoid as scheme is to ring: a generalization of the notion of a fan

TL;DR

This paper generalizes the concept of fans by defining a topology and sheaf of monoids, creating monoided spaces that can be associated with schemes, including toric varieties, extending the analogy between fans and monoids to a broader context.

Contribution

It introduces a new framework for monoided spaces that generalize fans, allowing their association with schemes via monoid algebras, broadening the scope of toric geometry.

Findings

Monoided spaces are locally isomorphic to spectra of monoids.

The new monoided space structure differs from Kato's approach.

Associated schemes can be normal varieties or toric varieties depending on the monoid algebra.

Abstract

Following DeMeyer, Ford & Miranda [DFM93], we define a topology on a fan by declaring open sets to be its subfans. Then, like Kato [Kat94], we make our fans into monoided spaces by associating a sheaf of monoids to each fan. (Our sheaf of monoids differs from Kato's.) Observing that this new monoided space is locally isomorphic to the spectrum of some monoid in the same sort of way a scheme is locally isomorphic to the spectrum of some ring, we define any monoided space that is locally isomorphic to the spectra of monoids to be a (generalized) fan. The monoid algebra functor can then be used to associate a scheme to such a fan. If we use the monoid algebra functor over some base field k, the resultant scheme is a normal variety if and only if it is a toric variety.