I know that if $(M,\mathcal{A})$ is a smooth manifold, the dual tangent space at $p\in M$ can be defined as $$ T^*_pM=I_p/I_p^2, $$ where $I_p$ is the ideal of the ring $C^\infty(M)$ consisting of smooth functions that vanish at $p$ and $I_p^2$ is the second power of this ideal.
I understand that this definition is useful, because, unlike other definitions, this one can be generalized to situations when you don't have the smooth structure $\mathcal{A}$, however this definition is so unintuitive I have a hard time grasping it.
- According to wikipedia, the product of ideals $A$ and $B$ is defined as $$ AB=\{a_1b_1+...a_nb_n|\ a_i\in A,b_i\in B,n\in\mathbb{N}\}, $$ I assume the point of this definition is that by demanding that the functions vanish, we make sure that the functions' Taylor expansion has no zeroth order terms, and since the elements of the product ideal are second order expressions, I assume the quotient is needed to get rid of second order terms.
Seems logical, since by the usual definition we can infer that the contangent space is generated by differentials of functions, which are, of course, the first order part. Am I correct in this?
If I am correct on the first point, what about higher than second order terms? Shouldn't we need to quotient those out too?
Thinking about it as I write this post, elements of $I_p^2$ are second order algebraic expressions from $C^\infty(M)$, but they are not polynomials, a Taylor expansion, however is polynomial. I think I don't understand it now.
This line of thought seems to depend on the functions having Taylor expansions. However I know this definition is generalizable to algebraic varieties as well as locally ringed spaces in general. Without having a structure that allows Taylor expansions, how do we know that this grasps the concept of "functions having the same first order behaviour at $p$" for those cases as well?
How can we see that this quotient space is finite $n$ dimensional? If the proof is not particularily pleasant, I don't expect anyone to actually post it here, but a reference to a manifold theory textbook that utilizes this definition heavily enough to also contain related proofs would be nice.
I would appreciate any response, even if it doesn't address these points directly, if it can help me see intuitively why this particular quotient space has the same meaning as the usual definition of dual tangent space.
EDIT: Since there have been misunderstandings, I wish to clarify: In my last bullet point I am not asking how to prove that the tangent/cotangent space is $n$ dimensional for an $n$ dimensional manifold. I am asking how to see that the space $I_p/I_p^2$ is $n$ dimensional, or alternatively, how to see that it is isomorphic to the cotangent space (defined by any alternative means).
Not a full answer, but too long for a comment.
Terms of order greater than $2$ will be quotiented out automatically. Recall that an ideal is closed under multiplication by ring elements. So, if $x^2 \in I^2_p$, then $x^{2+k}=x^2x^k\in I^2_p$ also. Intuitively, if $x^2=0$ in the quotient, then $x^{2+k}=x^2x^k=0x^k=0$.
This does not in fact rely on full Taylor expansion; it seems that just a few terms are sufficient, like in Peano form: $f(x)=f(p)+f'(p)(x-p)+o(|x-p|^2)$. The non-trivial part is to show that $o(|x-p|^2)$ equals $I^2_p$.
The fact that for $n$-dimensional manifold the tangent space is $n$-dimensional follows from the chart that provides us with coordinates $x_1, \dots x_n$: the vectors $\frac{\partial}{\partial x_k}$ will form a basis.