\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 $V(P + Q)$.)

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0280. Beware of the difference between the letter 'O' and the digit '0'.