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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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