To provide a bit more context, I'm gonna give a seminar talk where I will have to use fundamental classes and since the audience is not necessarily familiar with them, I will introduce them as a black box. The audience is not necessarily familiar with the notion of orientation of a manifold, but they're familiar with the notion of an orientation of a vector bundle. Since we only care about smooth manifolds, I think defining orientability as orientability of the tangent bundle is the most appropriate definition. The reason I'd like to avoid local homology is cause I think it's the least intu…

So say we have $A$ a noetherian domain of dim. 1 and we're trying to show that $A_M$ having the property of unique factorization for all ideals $M \in Max(A)$ implies $A$ has the property.

So for $I$ a nonzero proper ideal of $A$, we can get a finite set of maximal ideals $\{ M_1, ..., M_s \}$ of $A$ that contain $I$. We consider all the maps $\varphi_i : A \to A_{M_i}$ and use our assumption to say that $\varphi_i(I) = (MA_M)^{a_i}$ for a unique $a_i >0$. This being so, we can write $I \subset \varphi_1^{-1}((M_1 A_{M_1})^{a_1}) \cap \dots \cap \varphi_s^{-1}((M_s A_{M_s})^{a_s})$.

Ok, let's see. Pick coordinates $x=(x_1,...,x_n)$ on $M$, then you get coordinates on $T^{\ast}M$ coming from $x_1,...,x_n$ in the base and picking out the coefficients relative to the basis $dx_1,...,dx_n$ in the fibers. In these charts, $df\colon M\rightarrow T^{\ast}M$ looks as follows: take a point $p$ in Euclidean space, map it to $x^{-1}(p)$, take this to $df\vert_{x^{-1}(p)}$, the first $n$ coordinate of this are just $p$ again, but $df\vert_{x^{-1}(p)}=\sum_{i=1}^n\frac{\partial f}{\partial x_i}\Big\vert_{x^{-1}(p)}dx_i\vert_{x^{-1}(p)}$, so the last $n$ coordinates are just $\frac{…