/sci/ - Science & Math

FUCK

Can someone explain what a regular function on a (quasiprojective) variety is? I feel like I'm being pushed around by Shafarevich with all these different definitions. First, he defines a regular function on an affine algebraic set to be a polynomial map, basically a tuple of elements of the coordinate ring [math]k[X]:=k[x_1,...,x_n]/I(X)[/math]... ok, easy. Then a rational function on an affine algebraic set as a partially defined map of rational functions, a tuple of elements of the field of fractions of the coordinate ring, [math]k(X)[/math]. Again, understandable.

Now come in quasiprojective varieties. A regular function is now locally a rational function of same-degree homogeneous polynomials that has non-vanishing denominator in some open neighbourhood of every point. A map between quasiprojective varieties is regular if there is some affine piece of the codomain, and a neighbourhood of each point mapped there that restricts to a regular function. I can only assume that an isomorphism of varieties is a biregular bijective map.
There is no other indication yet of what (explicitly) [math]k[X][/math] might be in the quasiprojective case. Now there is this exercise:

Show that the variety [math]\mathbb A^2 \ (0,0)[/math] is not isomorphic to an affine variety. I've looked around and it seems everybody keeps saying that its coordinate ring is [math]k[x,y][/math]. Why?