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Changed \\, -> \, ; fixed a few other typos (and HTML -> MathJax) to get over 6 characters

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edit approved Aug 5, 2016 at 12:34

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The philosophy I subscribe to: tangent vector is an object defined by a base point and a direction.

When you want to make a formal definition out of it you do the following things:

for schemes, you define tangent vector as a map from $\text{Spec}\\,\Bbbk[\epsilon]$$\text{Spec}\,\Bbbk[\epsilon]$;
for C^r$\text C^r$-manifolds, you define tangent vector as a map from a neighborhood of 0 up to $\sim$

Is it possible to make a definition based on neighborhoods for schemes? Yes, you just have to do it technically right and consider formal neighborhood (which is the same for any smooth point of dimension n$n$) rather thenthan Zariski neighborhood.

Similarly, is it possible to define tangent vectors for C^r$\text C^r$-manifolds using ideals? Yes, simplesimply take C^r$C^r$-differentiable maps modulo C^r$\text C^r$-maps with 0 differential. Those are proper analogues of $\mathfrak m$ and $\mathfrak m^2$ in this context. The second set, for a curve, can be described as set of C^r-1$\text C^{r-1}$-maps multiplied by x$x$, but this is just one of the possible presentations.

Conclusion: when you look at it this way, the definitions are really similar on a deep level, but may be presented differently because of the technical details peculiar to each of the two cases.

The philosophy I subscribe to: tangent vector is an object defined by a base point and a direction.

When you want to make a formal definition out of it you do the following things:

for schemes, you define tangent vector as a map from $\text{Spec}\\,\Bbbk[\epsilon]$;
for C^r-manifolds, you define tangent vector as a map from a neighborhood of 0 up to $\sim$

Similarly, is it possible to define tangent vectors for C^r-manifolds using ideals? Yes, simple take C^r-differentiable maps modulo C^r-maps with 0 differential. Those are proper analogues of $\mathfrak m$ and $\mathfrak m^2$ in this context. The second set, for a curve, can be described as set of C^r-1-maps multiplied by x, but this is just one of the possible presentations.

Conclusion: when you look at it this way, the definitions are really similar on a deep level, but may be presented differently because of the technical details peculiar to each of the two cases.

The philosophy I subscribe to: tangent vector is an object defined by a base point and a direction.

When you want to make a formal definition out of it you do the following things:

for schemes, you define tangent vector as a map from $\text{Spec}\,\Bbbk[\epsilon]$;
for $\text C^r$-manifolds, you define tangent vector as a map from a neighborhood of 0 up to $\sim$

Similarly, is it possible to define tangent vectors for $\text C^r$-manifolds using ideals? Yes, simply take $C^r$-differentiable maps modulo $\text C^r$-maps with 0 differential. Those are proper analogues of $\mathfrak m$ and $\mathfrak m^2$ in this context. The second set, for a curve, can be described as set of $\text C^{r-1}$-maps multiplied by $x$, but this is just one of the possible presentations.

Conclusion: when you look at it this way, the definitions are really similar on a deep level, but may be presented differently because of the technical details peculiar to each of the two cases.

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created Dec 9, 2009 at 0:25

Ilya Nikokoshev

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The philosophy I subscribe to: tangent vector is an object defined by a base point and a direction.

When you want to make a formal definition out of it you do the following things:

for schemes, you define tangent vector as a map from $\text{Spec}\\,\Bbbk[\epsilon]$;
for C^r-manifolds, you define tangent vector as a map from a neighborhood of 0 up to $\sim$

Conclusion: when you look at it this way, the definitions are really similar on a deep level, but may be presented differently because of the technical details peculiar to each of the two cases.