I've been studying smooth manifolds in order to learn more about differential geometry. Recently I ran into a construction of the cotangent space that I hadn't seen much before. In many resources that I have used, the tangent space is created first and then the cotangent is defined as its dual. However, I have also seen it defined like the Zariski tangent space using alebraic rather than differential techniques. My question is about the motivations behind this construction.
As I understand it, we first consider the ring of $C^r$ functions $M\to \mathbb{R}$ and look at the ideal $I_p$ of functions vanishing at a point $p$. That is,
$$I_p = \bigg\{f\in C^r(M) : f(p)=0\bigg\}$$
My first question is related to this ideal. Specifically, how would one be lead to consider this ideal? I have worked out that this ideal is the kernel of a ring homomorphism sending each function to its value at $p$, and I have read that all maximal ideas in this ring have this form. I've seen the tangent space motivated through partial derivatives, curves on the manifold, and derivations. Is there a starting point for wanting to consider functions that vanish at $p$?
Next, we can take the square of this ideal $J_p=I_p^2 = \big\{ \sum_{i=1}^nf_ig_i:n\in\mathbb{N}\text{ and all }f_i, g_i\in I_p \big\}$. This is another ideal in the ring. Therefore, there should be a ring homomorphism with kernel $J$. I have not been able to construct this homomorphism, could I have some help getting to it? On top of this, I am wondering why one would want to consider this subset of the larger ideal. That is why is the square of $I$ relevant to the construction of the space?
To complete the construction, the cotangent space is defined as the quotient $I_p/J_p$. Because both $I_p$ and $J_p$ are rngs, can I consider this to be a quotient ring, as $J_p$ must be an ideal of $I_p$ when it is considered a a rng separate from the larger ring? Furthermore,I don't know what the equivalent classes of this ring would be because I do not have a good handle on $J$. For example, is there an example of a function that is in $I_p^2$ that is not in $I_p$? Grasping the difference between $I_p$ and $J_p$ could be helpful.
Many thanks in advance. I hope that someone can help to make this construction a bit clearer.
Best Answer
One way of motivating and building intuition for this process is looking at things concretely using charts.
Let $h\in J_p$. by definition, $h=fg$ for some $f,g\in I_p$. Using a chart $x^1,\dots,x^n$ containing $p$, we note that $h$ has vanishing first derivative at $p$. $$ \partial_i h(p)=\partial_i(fg)(p)=g(p)\partial_if(p)+f(p)\partial_ig(p)=0 $$ It's easy to verify that this property is independent of the choice of chart. This states that every element of $J_p$ vanishes at $p$ and has (in an intrinsic sense) vanishing first derivative at $p$. The converse of this statement is tricky; in $r\ge 2$ it is true, and follows from Taylor's theorem. In any case, we can think of $J_p$ in this way.
The quotient $I_p/J_p$ can then be thought of as generated by the equivalence $f\sim g$ if $f-g\in J_p$, that is, the quotient consists of equivalence classes of functions which "agree up to linear order".
These equivalence classes are canonically isomorphic to $T^*M$, Given an equivalence class $[f]$ in the above construction with representative $f$, and a tangent vector (derivation) $v$ at $p$, we can define $[f](v)=v(f)$. This construction is independent of representative. We can also define the differential at $p$ of a $C^r$ function by $d_pf=[f-f(p)]$, and show it agrees with the more common definition of $d_p$.