Knots & Global sections of projective varieties

I’m trying to post something everyday.  In any case, I spent a good chunk of time today editing a knot paper.  Its not on the arxiv yet, but maybe I’ll put a link when it is.  Anyways some math.  The goal is to show if $\ Y$ is projective over an algebraically closed field k then $\ \Gamma(Y, \mathcal{O}_Y )= k$.  The idea in Hartshorne is simple.  Take $\ f \in \mathcal{O}_Y(Y)$ and show it is integral over $\ k$, hence constant as $\ k = \overline{k}$.

By identifying $\ f$ with some degree 0 element in some localization of $\ S(Y)$ it will be enough to show $\ f$ is integral over $\ S(Y)$ (once we have this we just take the degree 0 part).  In particular we look at the standard affine patches $\ Y_i \subset Y$ and note $\ f \in A(Y_i)$.  Also there is an isomorphism $\ A(Y_i) \cong S(Y)_{(x_i)}$ which descends from the isomorphism $\ k[y_1,\dotsc,y_n] \cong k[x_0,\dotsc,x_n]_{(x_i)}$ sending $\ g(y_1,\dotsc,y_n) \mapsto g(x_0/x_i, \dotsc, \widehat{x_i/x_i},\dotsc, x_n/x_i)$.

So now you identify $\ f$  with some fraction $\ g_i/x_i^{n_i}$ and verify for $N \ge \sum_i n_i$ that $S(Y)_N \cdot f \subset S(Y)_N$.  Recall $f$ is a degree zero, so considering $S(Y)_N$ guarantees than any monomial of degree N will cancel the denominator of $f$.  Consequently, $f\cdot f \cdot S(Y)_N \subset f \cdot S(Y)_N \subset S(Y)_N$.  So in fact any power of $f$ times something of degree N is in $S(Y)_N$.  Use this to show  $\ S(Y)[f] \subset x_i^{-N}S(Y)$.  Then $\ x_i^{-N}S(Y)$ is finite over $\ S(Y)$ which shows $\ f$ is integral over $\ S(Y)$ (basic comm. alg. result).  And that’s about it.