So I'm looking at $X=V(xv-yu)\subseteq\Bbb A^2_k\times\Bbb P^1_k$, where $x,y$ are coordinates on $\Bbb A^2_k$ and $u,v$ on $\Bbb P^1_k$ (the blowup of the affine plane at the origin) and at the projection $X\to\Bbb A^2_k$
I want to compute $\Omega_{X/\Bbb A^2_k}$ on an affine cover, so I want to use the usual two patches on $\Bbb P^1_k$ given by $U_1=\{u\neq 0\}$ with coordinate $a=v/u$ and $U_2$ defined similarly with coordinate $b=u/v$
Let $X_1=\Bbb A^2_k\times U_1\simeq \Bbb A^3_k$, here $X$ is cut out by $xa-y$, so $X\cap X_1$ is $\mathrm{Spec}A$ where $A= k[x,y,a]/(xa-y)$
And $\Omega_{X_1/\Bbb A^2_k}$ should be the $A$-module generated by $da$, with relation $xda$. Makes sense so far?
So now I want to find the support of $\Omega_{X/\Bbb A^2}$. I think that $\Omega_{X_1/\Bbb A^2}$ is supported at the single point $(x)$ and the other one at $(y)$, but this feels suspicious
hi , does anyone here know where I could find a script with useful properties for union and interesection of indexed families of sets ( such like distributivity and stuff ) ?
Well if it's supported at the single point $(x)$ (which is not closure), it's also supported along the closure of this point - which is exactly the exceptional divisor in your chart
here's maybe how you can think about it: the blow up is an isomorphism away from the origin, and you should expect $\Omega_{X/\mathbb{A}^2}$ to be supported away from the locus where your map is an isomorphism
Hmm so the module $(Ada)/(xda)$ should be isomorphic to $A/(x)$ and its support should be $\mathrm{Spec}(A/(x))$ which is isomorphic to $\mathrm{Spec}(k[a])$ as you pointed out so it's supported on a line
So the other half is supported on $\mathrm{Spec}(k[b])$ by symmetry and now it's a question of deciding how those two supports fit together and that should be as a copy of $\Bbb P^1$ (which makes sense with the origin heuristic you were talking about!)
That situation reminds me of how you cannot trust theories talking about their own consistency. A theory can swear to be consistent while it is in fact inconsistent, but even worse a theory can prove its own inconsistency while being consistent
@AlessandroCodenotti might be interesting to note that in your charts, your \P^1 is cut out by a single element. This is in fact a general phenomenon -- it will turn out to define the universal property of blow up -- i.e. the pre-image of the locus you're blowing up will be something which is locally cut out by a single equation
Problem: Give an example of a bounded positive function on [0,1] that is not Lebesgue integrable....Since the measure space is finite and the function of interest is bounded, isn't this equivalent to finding a non-measurable function? If so, by Vitali's theorem there is a nonmeasurable subset $A$ of $[0,1]$, so take $f = 1_{A} + \frac{1}{2}$. Would this work?
