I have a question regarding the sheaf of differentiable functions. As far as i can tell it gives rise to a locally ringed space but i am wondering if it is also a scheme then. In the literature when defining vector bundles as locally free sheaves, one only defines them over schemes but when motivating this definition one defines them over the sheaf of differentiable functions on M. For example take Ravi Vakils notes:
Maybe someone here can clear up my confusion. Thanks in advance!
Thinking of vector bundles of some sort as locally free sheaves over an appropriate locally ringed space is something that works in many mathematical contexts. It works in the context of smooth manifolds, which is a guiding example, and it works in the context of schemes. Vakil only defines them in the latter case, because his text is only about schemes, not because it doesn't make sense otherwise. And, no, the locally ringed space $(M,\mathcal{C}_M^{\infty})$ for $M$ a smooth manifold and $\mathcal{C}_M^{\infty}$ its sheaf of smooth functions is not a scheme except for when $M$ is $0$-dimensional, i.e. a discrete set of points. This is, for example, because the underlying topological space of a scheme always has a basis consisting of compact subsets, so it can only be Hausdorff if it is discrete.