We study new categories of ’ $C^\infty$-schemes with corners’, following the definition of schemes in Algebraic Geometry. A ‘(local) $C^\infty$-ringed space with corners’ is a topological space with a sheaf of $C^\infty$-rings with corners. We define a spectrum functor from $C^\infty$-rings with corners to local $C^\infty$-ringed spaces with corners, adjoint to the global sections functor. Then a $C^\infty$-scheme with corners is a local $C^\infty$-ringed space with corners which is locally isomorphic to the spectrum of a $C^\infty$-ring with corners.

Manifolds with (g-)corners embed into $C^\infty$-schemes with corners as full subcategories. Thus, $C^\infty$-schemes with corners are a vast generalization of manifolds with corners and manifolds with g-corners.

In ordinary Algebraic Geometry, the spectrum functor is full and faithful, but for $C^\infty$-rings with corners neither holds. ‘Semi-complete’ $C^\infty$-rings with corners are a special class on which the spectrum functor is faithful and surjective, but not full.

We study special classes of $C^\infty$-schemes with corners, and existence of fibre products of $C^\infty$-schemes with corners.

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. ‘ $C^\infty$-schemes’ are schemes over $C^\infty$-rings, a way of using Algebro-Geometric techniques in Differential Geometry. 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.

Manifolds with corners, such as a triangle or a cube, are generalizations of manifolds, with boundaries and corners. They occur in many places in Differential Geometry. In this book we define and study new categories of ‘ $C^\infty$-rings with corners’ and ‘ $C^\infty$-schemes with corners’, which generalize manifolds with corners in the same way that $C^\infty$-rings and $C^\infty$-schemes generalize manifolds. These will be used in future work by the second author as the foundations of theories of derived manifolds and derived orbifolds with corners. These have important applications in Symplectic Geometry, as moduli spaces of pseudo-holomorphic curves should be derived orbifolds with corners.

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.