I.2.6

This problem has Hartshorne height 1; I.1.10 is suggested to solve this problem.

Results To Utilize

 * Proposition 1.7: $$\dim A(Y) = \dim Y$$ for a variety Y.
 * Thm 1.8A: $$\dim B = $$ tr.deg. of $$Frac(B)$$ over $$k$$ when $$B$$ if finitely generated k-algebra and a domain; $$ ht\ P + \dim B/P = \dim B$$.
 * Proposition 1.10: Y quasi-affine, then $$\dim Y = \dim \overline{Y}$$
 * Proof of Proposition 2.2, basically the part of about the dehomoginization and homogenization maps $$\alpha$$ and $$\beta$$

Outline

 * Use homogenization/dehomoginization to establish that $$A(Y_i) \cong $$ degree 0 part of $$S(Y)_{x_i}$$ and ultimately conclude $$A(Y_i)[x_i,x_i^{-1} \cong S(Y)_{x_i}$$.
 * Argue $$\dim S(Y) = \dim S(Y)_{x_i}$$.
 * Use 1.7, 1.8A, 1.10 to calculate $$\dim S(Y)_{x_i}$$ in terms of $$\dim Y_i$$.
 * Use that that $$Y_i$$ form a covering and use I.1.10 to finish the problem.

Conclusions

 * From the last step of the outline, it can be concluded that $$\dim Y = \dim Y_i$$ whenever $$Y_i$$ is nonempty.