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 independant 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.

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.

Consider the real line R and C^1_0 http://latex.mathoverflow.net/png?C%5E1%5F0 , 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 http://latex.mathoverflow.net/png?M%2FM%5E2 has dimension the continuum (because the classes of x^{1+\epsilon} http://latex.mathoverflow.net/png?x%5E%7B1%2B%5Cepsilon%7D are linearly independant in the quotient for 0<\epsilon <1 http://latex.mathoverflow.net/png?0%3C%5Cepsilon%20%3C1 ). Hence the Zariski tangent space of the real line , i.e. the dual of M/M^2 http://latex.mathoverflow.net/png?M%2FM%5E2 , has dimension 2^{continuum} http://latex.mathoverflow.net/png?2%5E%7Bcontinuum%7D . 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.

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 independant 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.