Projectivization, Homogenization, and Dehomogenezation

Simple Dehomogenization in a Polynomial Ring

I’ve endeavored to be as general with this example as possible so that it may generalize to the proof of Proposition (Proj is a Scheme). I also wrote this before I wrote those notes, and so the notes might be a little more well-organized. It might be good pedagogy to struggle through this example first because I included some more motivational remarks here, then read those notes, and then come back to this example with a better idea of the structure of the thing.

Consider $k[x, y]$ again. Note that $y$ an element of degree at least one, as in Proposition (Proj is a Scheme). We can see that $k[x, y] / (y-1)$ is $k[x]$, and the zero-graded peice of $k[x, y][y^{-1}]$ is the set of ratios of homogeneous elements of total degree zero. Consider the map $\psi: k[x, y][y^{-1}]_ {0}\to k[x, y]/(y-1)$ induced by the quotient map; this map is clearly surjective and we wish to show it is injective as well. If $\psi\left({f}/{g}\right) = 0$, with $\deg f= \deg g$ and $g = y^n$, then $\psi(f/g) = \psi(f)\psi(y^{-n}) = \psi(f)$, because $\psi(y^{-n}) = (\psi(y^{-1})^n = (\psi(y))]^{-n}) = 1^{-n} = 1$. Then $f/g\neq 0 \implies f\neq 0\implies \psi(y)\neq 0$ and we are done.

We further wish to show that $U(y)\cong \spec k[x]$. Note a point $\mf p\in U(y)$ is a relevent homogeneous prime ideal which does not contain $y$; we wish to define a map $\phi:U(y)\to \spec k[x]$ given by the rule $\mf p\mapsto \mf p / ((y-1)\cap \mf p)$. Since $(y)\not\subset \mf p$, this does not give the unit ideal; furthermore $k[x, y]/ ((y-1)\cap \mf p) / (\mf p)/((y-1)\cap \mf p) \cong k[x, y]/(\mf p)$ (by an appropriate isomorphism theorem) is an integral domain, and thus $\mf p/((y-1)\cap \mf p)$ is a prime ideal of $k[x, y]/(y-1)\cong k[x]$, and so the map is well defined. We wish to show this map is a homeomorphism. First, we demonstrate that it has an inverse $\phi^{-1}$ given by taking a prime ideal $\mf q\in \spec k[x]$ to the ideal generated by the “homogenezation” of elements in $\mf q$, where the homogenezation of an element $f\in \mf q$ is given by writing $f$ as a sum of homogeneous terms and multiplying each by $y$ times the difference between the maximum of the degrees of the terms of $f$ and the degree of that term. That is:

This is a well defined map; moreover, following it by the quotient map recovers the original polynomial, and given a homogeneous polynomial, quotienting by $y -1$ and then homogenizing yields the same polynomial. We now show that $\phi$ is a homeomorphism; i.e. $\phi$ and $\phi^{-1}$ are continuous. First we show for $\phi$; it suffices to show that for all ideals $\mf a$ of $k[x]$ there is a homogeneous ideal $\mf b$ such that $\phi^{-1}(V(\mf a)) = V(\mf b)\cap U(y)$. But then $\mf b$ is just the ideal generated by the homogenezation of every element of $\mf a$, as $\mf p$ contains $\mf a$ if and only if the ideal generated by the homogenezation of $\mf p$ contains the ideal generated by the homogenezation of $\mf a$ and does not contain $y$ (the if is a little difficult to see; suppose that $\mf p$ is an ideal who’s homogenezation contains the ideal generated by the homogenezation of every element of $\mf a$, but does not contain $y$. Then applying $\phi$ gives the intended result, because $\phi$ and $\phi^{-1}$ are inverses not just on the set of prime ideals but also between $k[x]$ and the homogeneous prime ideals of $k[x, y]$ which do not contain $y$.) A similar argument shows $\phi^{-1}$ is continuous.

Finally, we extend $\psi$ above to an isomorphism on structure sheaves. This is easy; the map $\psi$ defined above extends to the degree zero peice of any localization of $k[x, y][y^{-1}]$, and $\Gamma(\mc O_{\proj(k[x, y])}, U(y)\cap V)$ is the zero degree peice of a localization of $k[x, y][y^{-1}]$ for all open $V$. This holds because $\Gamma(\mc O_{\proj(k[x, y])}, U(y))$ is the set of all locally constant functions of total degree zero which have denominator not contained in $\mf p$ for all $\mf p\in U(y)\cap V$, which is the degree zero peice of the localization of $k[x, y][y^{-1}]$ by a multiplicatively closed set $S$ of functions of degree $\geq 1$ such that $V = X-V(S)$.