General construction of non-flat toric schemes in spectral algebraic geometry

Construct a general method to define non-flat toric schemes associated to fans within spectral algebraic geometry, extending beyond the flat toric schemes X_Σ built from monoid algebras over the sphere spectrum. Specify how to glue local affine pieces and identify appropriate algebra objects and morphisms so that the resulting global objects model non-flat toric schemes analogous to classical toric varieties.

Background

Throughout the paper, toric schemes over the sphere spectrum are constructed in the flat case by gluing affine pieces Spet(𝕊[S_σ]) arising from monoid algebras associated to cones in a fan Σ. This framework yields flat toric schemes, aligning with classical constructions that are flat over ℤ.

The authors note that, while certain non-flat projective lines can be modeled in spectral algebraic geometry (as referenced in SAG), there is no known general procedure to construct non-flat toric schemes. Developing such a method would extend the scope of toric geometry in the spectral setting and clarify the limits of the flat-only approach adopted in this work.