This chapter summarizes D. Joyce, ‘Algebraic Geometry over $C^\infty$-rings’, Memoirs of the AMS, 2019. If X is a manifold then the set $C^\infty$(X) of smooth functions on X is a ‘ $C^\infty$-ring’, a rich algebraic structure with many operations. A module over a $C^\infty$-ring is a module over it as an real algebra. A $C^\infty$-ring has a cotangent module. For $C^\infty$(X) this is the sections of the cotangent bundle T*X.

‘ $C^\infty$-Schemes’ are schemes over $C^\infty$-rings, a way of using Algebro-Geometric techniques in Differential Geometry, and of allowing Differential Geometers to study spaces far more general than manifolds. They include smooth manifolds, but also many singular and infinite-dimensional spaces. They have applications to Synthetic Differential Geometry, and to ‘derived manifolds’ in Derived Differential Geometry. The category of $C^\infty$-schemes has good properties: it is Cartesian closed and has all finite limits and directed colimits.

We also study sheaves of modules over $C^\infty$-schemes, as for (quasi-)coherent sheaves in Algebraic Geometry. A $C^\infty$-scheme has a cotangent sheaf, generalizing the cotangent bundle of a manifold.