Fan Is to Monoid as Scheme Is to Ring

This paper generalizes the notion of a toric variety. In particular, these generalized toric varieties include examples of non-normal non-quasiprojective toric varieties. Such an example seems not to have been noted before in the literature. This generalization is achieved by replacing a fan of strictly rational polyhedral cones in a lattice with a monoided space, that is a topological space equipped with a distinguished sheaf of monoids. For a classic toric variety, the underlying topological space of this monoided space is its orbit space under the action of the torus. And, if σ is a cone, then the stalk of the structure sheaf of the monoided space at the point corresponding to σ is the monoid Sσ = σ ∩M that is usually associated to σ. When applied to the affine toric variety associated to some cone σ, the monoided space so obtained is isomorphic to the spectrum of the monoid Sσ, where the spectrum of a monoid is defined in analogy with the definition of the spectrum of a ring. More generally, we will call a monoided space that is locally isomorphic to the spectrum of a monoid a fan and we form schemes from these fans by taking monoid algebras and glueing.