\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

The Stacks project

103.26 Proj of a ring

Definition 103.26.1. 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 103.26.2. 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 103.26.3. 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 103.26.4. 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 103.26.5. There is a continuous map $ \text{Proj}(R) \longrightarrow \mathop{\mathrm{Spec}}(R_0) $.

Exercise 103.26.6. 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 103.26.7. 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 103.26.8. 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 103.26.9. 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 103.26.10. 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 $V(P + Q)$.)


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