Both schemes and manifolds are local ringed spaces which are locally isomorphic to spaces in some full subcategory of local ringed spaces (local models). Now, there is the inherent notion of the Zariski tangent space in a point (dual of maximal ideal modulo its square) which is the "right" definition for schemes and for $C^\infty$-manifolds (over $\mathbb{R}$ and $\mathbb{C}$). But for $C^r$-manifolds over $\mathbb{R}$ with $r<\infty$ this is not the correct definition. Here one has to take equivalence classes of $C^r$-curves through the point. Isn't there some general definition of tangent spaces which is always the right one?
I am also not completely sure what "right" means. So far, I think that one wants the dimension of the tangent space to be equal to the dimension of the point. This is for example the problem with the Zariski tangent spaces for $C^r$-manifolds. Can this failure be explained geometrically?
Best Answer
Consider the real line $\mathbb R$ and $C^1_0$ , the ring of germs of continuously differentiable functions at zero. Now take the ideal $M$ of germs vanishing at zero. The Zariski cotangent space $M/M^2$ has dimension the continuum (because the classes of $x^{1+\epsilon}$ are linearly independent in the quotient for $0<\epsilon <1$ ). Hence the Zariski tangent space of the real line, i.e. the dual of $M/M^2$, has dimension $2^{continuum}$. Some geometers might think this is a bit large for the real line.
This result is essentially exercise 13 of Chapter 3 of Spivak's Differential Geometry, Volume I.