# I.2.6

219,431pages on
this wiki

This problem has Hartshorne height 2 and Hartshorne volume 3 (I.1.10 > I.1.6).

### 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.