# Projective Nullstellensatz (Tuesday, November 02)

## Quotients of Graded Rings

:::{.proposition title="Quotients of graded rings by homogeneous ideals are again graded"}
If $R$ is a graded ring and $I\normal R$ is a homogeneous ideal, then $R/I$ is a graded ring.
:::

## Cones and Projectivization

:::{.definition title="Cones"}
An affine variety $X \subseteq \AA^{n+1}$ is a **cone** iff

- $\vector 0 \in X$
- $kX \subseteq X$

:::

:::{.remark}
This says that $X$ is the origin and a union of lines through the origin.
For the following definitions, we define a map
\[  
\pi: \AA^{n+1}\smz &\to \PP^n \\
\tv{x_0, \cdots, x_n} &\mapsto \tv{x_0 : \cdots :x_n}
.\]
:::

:::{.definition title="Projectivization of a Cone"}
For a cone $X \subseteq \AA^{n+1}$, the **projectivization** of $X$ is defined as
\[  
\PP(X) \da \pi(X\smz) = \ts{ \tv{x_0: \cdots : x_n } \in \PP^n \st \tv{x_0, \cdots, x_n} \in X } \subseteq \PP^n
.\]
:::

:::{.definition title="Cone Over a Projective Variety"}
For a projective variety $X \subseteq \PP^n$, the **cone over $X$** is the cone defined by
\[  
C(X) \da \ts{0} \union \pi^{-1}(X) = \ts{0} \union \ts{ \tv{x_0, \cdots, x_n} \st \tv{x_0: \cdots : x_n} \in X } \subseteq \AA^{n+1}
.\]
:::

:::{.remark}
We have
\[  
\PP V_a(S) = V_p(S) &\text{and}& C(V_p(S)) = V_a(S)
.\]
:::

## Projective Nullstellensatz

:::{.proposition title="Projective Nullstellensatz Construction"}
Define
\[  
V_p(J) &\da \ts{\vector x \in \PP^n \st f(\vector x) = 0 \text{ for all homogeneous } f\in J} \subseteq \PP^n \\
I_p(X) &\da \gens{ f \in k[x_0, \cdots, x_n] \text{ homogeneous } \st f(\vector x) = 0\,\, \forall x\in X} \normal k[x_0, \cdots, x_n]
.\]
:::

\todo[inline]{Missing some info, fill in.}