Exercise 111.27.4. 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 $I \subset R$.

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