7 - Modules, and sheaves of modules

Summary

A module over a $C^\infty$-ring, or $C^\infty$-ring with corners, is a just a module over the underlying R-algebra. Similarly, a sheaf of modules over a $C^\infty$-scheme, or $C^\infty$-scheme with corners, is a sheaf of modules over the underlying sheaf of R-algebras. So the general theory of modules, sheaves of modules, spectrum functors for modules, and (quasi-)coherent sheaves, extends from ordinary Algebraic Geometry with little extra work.

A manifold with corners has two notions of (co)tangent bundle: the (co)tangent bundles TX, T*X, independent of the corners, and the b-(co)tangent bundles bTX, bT*X, depending on the corners. Similarly, a $C^\infty$-ring with corners has two cotangent modules: the cotangent module, which depends only on the underlying $C^\infty$-ring, and the b-cotangent module, which also depends on the corner structure. Thus a $C^\infty$-scheme with corners has a cotangent sheaf and b-cotangent sheaf.

If X is a manifold with corners, regarded as a $C^\infty$-scheme with corners, the (b-)cotangent sheaf is the sheaf of sections of the (b-)cotangent bundle.

(b-)Cotangent sheaves are covariantly functorial, and have exact sequences for fibre products.