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.

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