global regular functions projective variety

In A let D be the homogeneous ideal generated by all homogeneous denominators h for which $fh\in A_{\mathrm{deg}h}$. Global regularity says that at each point of X at least one such h is nonzero. This uses the set of all admissible denominators, not a simultaneous choice of local representatives. Let $\tilde{D}$ be the inverse image of D in $k[x_0,\dots ,x_n]$. Its affine zero set is contained in the cone over X, and has no nonzero point: any such point represents a point of X where all admissible denominators would vanish.

Each coordinate x_i therefore vanishes on $V(\tilde{D})$. F2 implies $x_i^{N_i}\in \tilde{D}$ for some $N_i\ge 1$, so its class lies in D. This is the precise inherited AC use in the argument. Expressing that homogeneous class as a finite sum of homogeneous generators of D and taking the component of degree N_i shows $f{x}_i^{N_i}\in A_{N_i}$: multiply each generator by f in its prescribed degree. This also covers coordinates zero in A.

Choose $N\ge 1$ at least every N_i. Then $f{x}_i^N\in A_N$ for all i. Choose an integer $d>(n+1)(N-1)$. Every monomial of degree d contains some x_i to exponent at least N, so multiplying by f carries its class into A_d. The finitely many monomials of degree d span A_d, hence multiplication by f defines a k-linear endomorphism of this finite-dimensional space. It is nonzero as a space: at any point of X some coordinate x_i is nonzero, and then $x_i^d$ is nonzero in A_d.

F3 supplies a monic polynomial $P\in k[T]$ with $P(f)a=0$ for every $a\in A_d$. Choose nonzero $a\in A_d$; cancellation in $\mathrm{Frac}(A)$ gives $P(f)=0$. As k is algebraically closed, P splits into linear factors, and the field has no zero divisors. Thus $f=c$ in the fraction field for some $c\in k$. Its local evaluations are consequently that same constant at every point of X.