$C^\infty$-algebraic geometry with corners

TL;DR

This paper extends $C^$-algebraic geometry to include corners, defining new categories of $C^$-rings with corners and schemes with corners, to support future theories of derived manifolds and orbifolds.

Contribution

It introduces and studies $C^$-rings with corners and $C^$-schemes with corners, generalizing manifolds with corners within the algebraic geometric framework.

Findings

Defined categories of $C^$-rings with corners.

Established properties of $C^$-schemes with corners.

Laid foundations for derived manifolds with corners.

Abstract

If $X$ is a manifold then the set $C^\infty(X)$ of smooth functions $f:X\to\mathbb R$ 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 this book, a sequel to the second author's monograph on $C^\infty$-algebraic geometry arXiv:1001.0023, 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 as the foundations of theories of derived manifolds and derived orbifolds with corners. This book is based on the PhD thesis of the first author, supervised by the second author.