Real manifolds and affine schemes

(1) This is a highly productive way of looking at smooth manifolds. It is responsible for synthetic differential geometry and derived smooth manifolds. Both of these subjects heavily rely on this identification.

Synthetic differential geometry looks at spectra of finite-dimensional real algebras, and interprets these as geometric spaces of infinitesimal shape. This allows one to make infinitesimal arguments of the type used by Élie Cartan and Sophus Lie perfectly rigorous. (Text)books have been written about this approach:

• Anders Kock: Synthetic differential geometry;
• Anders Kock: Synthetic geometry of manifolds;
• René Lavendhomme: Basic concepts of synthetic differential geometry;
• Ieke Moerdijk, Gonzalo Reyes: Models for smooth infinitesimal analysis.

Derived smooth manifolds start by enlarging the category of real algebras of smooth functions to a bigger, cocomplete category (such as all real algebras, or, better, C^∞-rings, see below). One then takes simplicial objects in this category of algebras, i.e., simplicial real algebras. This category admits a good theory of homotopy colimits, which are given by derived tensor products and other constructions from homological algebra. The opposite category of simplicial real algebras should be thought of as a category of geometric spaces of some kind. This category has an excellent theory of homotopy limits, in particular, one compute correct (i.e., derived) intersections of nontransversal submanifolds in it, and get expected answers. For example, this can be used to define canonical representative of characteristic classes, one could take the Euler class to be the derived zero locus of the zero section, for example. Other (potential) applications include the Fukaya ∞-category, which in presence of nontransversal intersections can be turned into a category in a more straightforward way.

Dominic Joyce wrote a book about this:

• Dominic Joyce: Algebraic geometry over C^∞-rings.

Other sources:

• David Spivak: Derived smooth manifolds.
• Dennis Borisov, Justin Noel: Simplicial approach to derived differential manifolds.
• David Carchedi, Dmitry Roytenberg: On theories of superalgebras of differentiable functions.
• David Carchedi, Dmitry Roytenberg: Homological algebra for superalgebras of differentiable functions.
• David Carchedi, Pelle Steffens: On the universal property of derived smooth manifolds.

(2) Yes, for example, Kähler differentials are not smooth differential 1-forms, even though derivations are smooth vector fields. This is (one of) the reasons for passing to C^∞-rings instead. In the category of C^∞-rings, C^∞-Kähler differentials are precisely smooth differential 1-forms.

(3) Not really, this question was discussed here before, here is one of the discussions: Algebraic description of compact smooth manifolds?