# The Stacks Project

## Tag 0280

### 102.26. Proj of a ring

Definition 102.26.1. Let $R$ be a graded ring. A homogeneous ideal is simply an ideal $I \subset R$ which is also a graded submodule of $R$. Equivalently, it is an ideal generated by homogeneous elements. Equivalently, if $f \in I$ and $$f = f_0 + f_1 + \ldots + f_n$$ is the decomposition of $f$ into homogeneous pieces in $R$ then $f_i \in I$ for each $i$.

Definition 102.26.2. We define the homogeneous spectrum $\text{Proj}(R)$ of the graded ring $R$ to be the set of homogeneous, prime ideals ${\mathfrak p}$ of $R$ such that $R_{+} \not \subset {\mathfrak p}$. Note that $\text{Proj}(R)$ is a subset of $\mathop{\mathrm{Spec}}(R)$ and hence has a natural induced topology.

Definition 102.26.3. Let $R = \oplus_{d \geq 0} R_d$ be a graded ring, let $f\in R_d$ and assume that $d \geq 1$. We define $R_{(f)}$ to be the subring of $R_f$ consisting of elements of the form $r/f^n$ with $r$ homogeneous and $\deg(r) = nd$. Furthermore, we define $$D_{+}(f) = \{ {\mathfrak p} \in \text{Proj}(R) | f \not\in {\mathfrak p} \}.$$ Finally, for a homogeneous ideal $I \subset R$ we define $V_{+}(I) = V(I) \cap \text{Proj}(R)$.

Exercise 102.26.4. On the topology on $\text{Proj}(R)$. With definitions and notation as above prove the following statements.

1. Show that $D_{+}(f)$ is open in $\text{Proj}(R)$.
2. Show that $D_{+}(ff') = D_{+}(f) \cap D_{+}(f')$.
3. Let $g = g_0 + \ldots + g_m$ be an element of $R$ with $g_i \in R_i$. Express $D(g) \cap \text{Proj}(R)$ in terms of $D_{+}(g_i)$, $i \geq 1$ and $D(g_0) \cap \text{Proj}(R)$. No proof necessary.
4. Let $g\in R_0$ be a homogeneous element of degree $0$. Express $D(g) \cap \text{Proj}(R)$ in terms of $D_{+}(f_\alpha)$ for a suitable family $f_\alpha \in R$ of homogeneous elements of positive degree.
5. Show that the collection $\{D_{+}(f)\}$ of opens forms a basis for the topology of $\text{Proj}(R)$.
6. Show that there is a canonical bijection $D_{+}(f) \to \mathop{\mathrm{Spec}}(R_{(f)})$. (Hint: Imitate the proof for $\mathop{\mathrm{Spec}}$ but at some point thrown in the radical of an ideal.)
7. Show that the map from (6) is a homeomorphism.
8. Give an example of an $R$ such that $\text{Proj}(R)$ is not quasi-compact. No proof necessary.
9. Show that any closed subset $T \subset \text{Proj}(R)$ is of the form $V_{+}(I)$ for some homogeneous ideal $I \subset R$.

Remark 102.26.5. There is a continuous map $\text{Proj}(R) \longrightarrow \mathop{\mathrm{Spec}}(R_0)$.

Exercise 102.26.6. If $R = A[X]$ with $\deg(X) = 1$, show that the natural map $\text{Proj}(R) \to \mathop{\mathrm{Spec}}(A)$ is a bijection and in fact a homeomorphism.

Exercise 102.26.7. Blowing up: part I. In this exercise $R = Bl_I(A) = A \oplus I \oplus I^2 \oplus \ldots$. Consider the natural map $b : \text{Proj}(R) \to \mathop{\mathrm{Spec}}(A)$. Set $U = \mathop{\mathrm{Spec}}(A) - V(I)$. Show that $$b : b^{-1}(U) \longrightarrow U$$ is a homeomorphism. Thus we may think of $U$ as an open subset of $\text{Proj}(R)$. Let $Z \subset \mathop{\mathrm{Spec}}(A)$ be an irreducible closed subscheme with generic point $\xi \in Z$. Assume that $\xi \not\in V(I)$, in other words $Z \not\subset V(I)$, in other words $\xi \in U$, in other words $Z\cap U \not = \emptyset$. We define the strict transform $Z'$ of $Z$ to be the closure of the unique point $\xi'$ lying above $\xi$. Another way to say this is that $Z'$ is the closure in $\text{Proj}(R)$ of the locally closed subset $Z\cap U \subset U \subset \text{Proj}(R)$.

Exercise 102.26.8. Blowing up: Part II. Let $A = k[x, y]$ where $k$ is a field, and let $I = (x, y)$. Let $R$ be the blowup algebra for $A$ and $I$.

1. Show that the strict transforms of $Z_1 = V(\{x\})$ and $Z_2 = V(\{y\})$ are disjoint.
2. Show that the strict transforms of $Z_1 = V(\{x\})$ and $Z_2 = V(\{x-y^2\})$ are not disjoint.
3. Find an ideal $J \subset A$ such that $V(J) = V(I)$ and such that the strict transforms of $Z_1 = V(\{x\})$ and $Z_2 = V(\{x-y^2\})$ are disjoint.

Exercise 102.26.9. Let $R$ be a graded ring.

1. Show that $\text{Proj}(R)$ is empty if $R_n = (0)$ for all $n >> 0$.
2. Show that $\text{Proj}(R)$ is an irreducible topological space if $R$ is a domain and $R_{+}$ is not zero. (Recall that the empty topological space is not irreducible.)

Exercise 102.26.10. Blowing up: Part III. Consider $A$, $I$ and $U$, $Z$ as in the definition of strict transform. Let $Z = V({\mathfrak p})$ for some prime ideal ${\mathfrak p}$. Let $\bar A = A/{\mathfrak p}$ and let $\bar I$ be the image of $I$ in $\bar A$.

1. Show that there exists a surjective ring map $R: = Bl_I(A) \to \bar R: = Bl_{\bar I}(\bar A)$.
2. Show that the ring map above induces a bijective map from $\text{Proj}(\bar R)$ onto the strict transform $Z'$ of $Z$. (This is not so easy. Hint: Use 5(b) above.)
3. Conclude that the strict transform $Z' = V_{+}(P)$ where $P \subset R$ is the homogeneous ideal defined by $P_d = I^d \cap {\mathfrak p}$.
4. Suppose that $Z_1 = V({\mathfrak p})$ and $Z_2 = V({\mathfrak q})$ are irreducible closed subsets defined by prime ideals such that $Z_1 \not \subset Z_2$, and $Z_2 \not \subset Z_1$. Show that blowing up the ideal $I = {\mathfrak p} + {\mathfrak q}$ separates the strict transforms of $Z_1$ and $Z_2$, i.e., $Z_1' \cap Z_2' = \emptyset$. (Hint: Consider the homogeneous ideal $P$ and $Q$ from part (c) and consider $V(P + Q)$.)

The code snippet corresponding to this tag is a part of the file exercises.tex and is located in lines 1835–1990 (see updates for more information).

\section{Proj of a ring}
\label{section-proj-ring}

\begin{definition}
\label{definition-homogeneous-ideal}
Let $R$ be a graded ring. A {\it homogeneous} ideal is simply an ideal
$I \subset R$ which is also a graded submodule of $R$. Equivalently,
it is an ideal generated by homogeneous elements. Equivalently, if
$f \in I$ and
$$f = f_0 + f_1 + \ldots + f_n$$
is the decomposition of $f$ into homogeneous pieces in $R$ then $f_i \in I$
for each $i$.
\end{definition}

\begin{definition}
\label{definition-Proj-R}
We define the {\it homogeneous spectrum $\text{Proj}(R)$}
of the graded ring $R$ to be the set of homogeneous, prime ideals
${\mathfrak p}$ of $R$ such that $R_{+} \not \subset {\mathfrak p}$.
Note that $\text{Proj}(R)$ is a subset of $\Spec(R)$ and hence has a
natural induced topology.
\end{definition}

\begin{definition}
\label{definition-Dplus-Vplus}
Let $R = \oplus_{d \geq 0} R_d$ be a graded ring, let $f\in R_d$ and
assume that $d \geq 1$. We define {\it $R_{(f)}$} to be the subring of
$R_f$ consisting of elements of the form $r/f^n$ with $r$ homogeneous and
$\deg(r) = nd$. Furthermore, we define
$$D_{+}(f) = \{ {\mathfrak p} \in \text{Proj}(R) | f \not\in {\mathfrak p} \}.$$
Finally, for a homogeneous ideal $I \subset R$ we define
$V_{+}(I) = V(I) \cap \text{Proj}(R)$.
\end{definition}

\begin{exercise}
\label{exercise-topology-proj}
On the topology on $\text{Proj}(R)$. With definitions and notation as
above prove the following statements.
\begin{enumerate}
\item Show that $D_{+}(f)$ is open in $\text{Proj}(R)$.
\item Show that $D_{+}(ff') = D_{+}(f) \cap D_{+}(f')$.
\item Let $g = g_0 + \ldots + g_m$ be an element
of $R$ with $g_i \in R_i$. Express $D(g) \cap \text{Proj}(R)$
in terms of $D_{+}(g_i)$, $i \geq 1$ and $D(g_0) \cap \text{Proj}(R)$.
No proof necessary.
\item Let $g\in R_0$ be a homogeneous element of degree $0$.
Express $D(g) \cap \text{Proj}(R)$ in terms of $D_{+}(f_\alpha)$
for a suitable family $f_\alpha \in R$ of homogeneous elements of
positive degree.
\item Show that the collection $\{D_{+}(f)\}$ of opens forms a
basis for the topology of $\text{Proj}(R)$.
\item
\label{item-bijection}
Show that there is a canonical bijection $D_{+}(f) \to \Spec(R_{(f)})$.
(Hint: Imitate the proof for $\Spec$ but at some point thrown in the
radical of an ideal.)
\item Show that the map from (\ref{item-bijection}) is a homeomorphism.
\item Give an example of an $R$ such that $\text{Proj}(R)$ is not
quasi-compact. No proof necessary.
\item Show that any closed subset $T \subset \text{Proj}(R)$ is of
the form $V_{+}(I)$ for some homogeneous ideal $I \subset R$.
\end{enumerate}
\end{exercise}

\begin{remark}
\label{remark-continuous-proj-spec}
There is a continuous map $\text{Proj}(R) \longrightarrow \Spec(R_0)$.
\end{remark}

\begin{exercise}
\label{exercise-iso-polynomial-ring-one-variable}
If $R = A[X]$ with $\deg(X) = 1$, show that the natural map
$\text{Proj}(R) \to \Spec(A)$ is a bijection and in fact
a homeomorphism.
\end{exercise}

\begin{exercise}
\label{exercise-blowing-up-I}
Blowing up: part I.
In this exercise $R = Bl_I(A) = A \oplus I \oplus I^2 \oplus \ldots$.
Consider the natural map $b : \text{Proj}(R) \to \Spec(A)$.
Set $U = \Spec(A) - V(I)$. Show that
$$b : b^{-1}(U) \longrightarrow U$$
is a homeomorphism.
Thus we may think of $U$ as an open subset of $\text{Proj}(R)$.
Let $Z \subset \Spec(A)$ be an irreducible closed subscheme
with generic point $\xi \in Z$. Assume that $\xi \not\in V(I)$,
in other words $Z \not\subset V(I)$, in other words
$\xi \in U$, in other words $Z\cap U \not = \emptyset$. We define
the {\it strict transform} $Z'$ of $Z$ to be the closure of the unique
point $\xi'$ lying above $\xi$. Another way to say this is that
$Z'$ is the closure in $\text{Proj}(R)$ of the locally closed subset
$Z\cap U \subset U \subset \text{Proj}(R)$.
\end{exercise}

\begin{exercise}
\label{exercise-blowing-up-II}
Blowing up: Part II.
Let $A = k[x, y]$ where $k$ is a field, and let $I = (x, y)$.
Let $R$ be the blowup algebra for $A$ and $I$.
\begin{enumerate}
\item Show that the strict transforms of $Z_1 = V(\{x\})$ and
$Z_2 = V(\{y\})$ are disjoint.
\item Show that the strict transforms of $Z_1 = V(\{x\})$ and
$Z_2 = V(\{x-y^2\})$ are not disjoint.
\item Find an ideal $J \subset A$ such that $V(J) = V(I)$
and such that the strict transforms of $Z_1 = V(\{x\})$ and
$Z_2 = V(\{x-y^2\})$ are disjoint.
\end{enumerate}
\end{exercise}

\begin{exercise}
\label{exercise-proj-when-empty}
Let $R$ be a graded ring.
\begin{enumerate}
\item Show that $\text{Proj}(R)$ is empty if $R_n = (0)$ for all $n >> 0$.
\item Show that $\text{Proj}(R)$ is an irreducible topological space
if $R$ is a domain and $R_{+}$ is not zero. (Recall that the empty
topological space is not irreducible.)
\end{enumerate}
\end{exercise}

\begin{exercise}
\label{exercise-blowing-up-III}
Blowing up: Part III.
Consider $A$, $I$ and $U$, $Z$ as in the definition of strict transform.
Let $Z = V({\mathfrak p})$ for some prime ideal ${\mathfrak p}$. Let $\bar A = A/{\mathfrak p}$ and let
$\bar I$ be the image of $I$ in $\bar A$.
\begin{enumerate}
\item Show that there exists a surjective ring map
$R: = Bl_I(A) \to \bar R: = Bl_{\bar I}(\bar A)$.
\item Show that the ring map above induces a bijective map
from $\text{Proj}(\bar R)$ onto the strict transform $Z'$ of $Z$. (This
is not so easy. Hint: Use 5(b) above.)
\item Conclude that the strict transform $Z' = V_{+}(P)$ where
$P \subset R$ is the homogeneous ideal defined by
$P_d = I^d \cap {\mathfrak p}$.
\item Suppose that $Z_1 = V({\mathfrak p})$ and
$Z_2 = V({\mathfrak q})$ are irreducible
closed subsets defined by prime ideals such that  $Z_1 \not \subset Z_2$,
and $Z_2 \not \subset Z_1$. Show that blowing up the ideal
$I = {\mathfrak p} + {\mathfrak q}$ separates the
strict transforms of $Z_1$ and $Z_2$,
i.e., $Z_1' \cap Z_2' = \emptyset$. (Hint: Consider the homogeneous
ideal $P$ and $Q$ from part (c) and consider $V(P + Q)$.)
\end{enumerate}
\end{exercise}

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