Exercise 111.6.1. Compute $\mathop{\mathrm{Spec}}(\mathbf{Z})$ as a set and describe its topology.
111.6 The Spectrum of a ring
Exercise 111.6.2. Let $A$ be any ring. For $f\in A$ we define $D(f):= \{ \mathfrak p \subset A \mid f \not\in \mathfrak p\} $. Prove that the open subsets $D(f)$ form a basis of the topology of $\mathop{\mathrm{Spec}}(A)$.
Exercise 111.6.3. Prove that the map $I\mapsto V(I)$ defines a natural bijection
\[ \{ I\subset A\text{ with }I = \sqrt{I}\} \longrightarrow \{ T\subset \mathop{\mathrm{Spec}}(A)\text{ closed}\} \]
Definition 111.6.4. A topological space $X$ is called quasi-compact if for any open covering $X = \bigcup _{i\in I} U_ i$ there is a finite subset $\{ i_1, \ldots , i_ n\} \subset I$ such that $X = U_{i_1}\cup \ldots U_{i_ n}$.
Exercise 111.6.5. Prove that $\mathop{\mathrm{Spec}}(A)$ is quasi-compact for any ring $A$.
Definition 111.6.6. A topological space $X$ is said to verify the separation axiom $T_0$ if for any pair of points $x, y\in X$, $x\not= y$ there is an open subset of $X$ containing one but not the other. We say that $X$ is Hausdorff if for any pair $x, y\in X$, $x\not= y$ there are disjoint open subsets $U, V$ such that $x\in U$ and $y\in V$.
Exercise 111.6.7. Show that $\mathop{\mathrm{Spec}}(A)$ is not Hausdorff in general. Prove that $\mathop{\mathrm{Spec}}(A)$ is $T_0$. Give an example of a topological space $X$ that is not $T_0$.
Remark 111.6.8. Usually the word compact is reserved for quasi-compact and Hausdorff spaces.
Definition 111.6.9. A topological space $X$ is called irreducible if $X$ is not empty and if $X = Z_1\cup Z_2$ with $Z_1, Z_2\subset X$ closed, then either $Z_1 = X$ or $Z_2 = X$. A subset $T\subset X$ of a topological space is called irreducible if it is an irreducible topological space with the topology induced from $X$. This definition implies $T$ is irreducible if and only if the closure $\bar T$ of $T$ in $X$ is irreducible.
Exercise 111.6.10. Prove that $\mathop{\mathrm{Spec}}(A)$ is irreducible if and only if $Nil(A)$ is a prime ideal and that in this case it is the unique minimal prime ideal of $A$.
Exercise 111.6.11. Prove that a closed subset $T\subset \mathop{\mathrm{Spec}}(A)$ is irreducible if and only if it is of the form $T = V({\mathfrak p})$ for some prime ideal ${\mathfrak p}\subset A$.
Definition 111.6.12. A point $x$ of an irreducible topological space $X$ is called a generic point of $X$ if $X$ is equal to the closure of the subset $\{ x\} $.
Exercise 111.6.13. Show that in a $T_0$ space $X$ every irreducible closed subset has at most one generic point.
Exercise 111.6.14. Prove that in $\mathop{\mathrm{Spec}}(A)$ every irreducible closed subset does have a generic point. In fact show that the map ${\mathfrak p} \mapsto \overline{\{ {\mathfrak p}\} }$ is a bijection of $\mathop{\mathrm{Spec}}(A)$ with the set of irreducible closed subsets of $X$.
Exercise 111.6.15. Give an example to show that an irreducible subset of $\mathop{\mathrm{Spec}}(\mathbf{Z})$ does not necessarily have a generic point.
Definition 111.6.16. A topological space $X$ is called Noetherian if any decreasing sequence $Z_1\supset Z_2 \supset Z_3\supset \ldots $ of closed subsets of $X$ stabilizes. (It is called Artinian if any increasing sequence of closed subsets stabilizes.)
Exercise 111.6.17. Show that if the ring $A$ is Noetherian then the topological space $\mathop{\mathrm{Spec}}(A)$ is Noetherian. Give an example to show that the converse is false. (The same for Artinian if you like.)
Definition 111.6.18. A maximal irreducible subset $T\subset X$ is called an irreducible component of the space $X$. Such an irreducible component of $X$ is automatically a closed subset of $X$.
Exercise 111.6.19. Prove that any irreducible subset of $X$ is contained in an irreducible component of $X$.
Exercise 111.6.20. Prove that a Noetherian topological space $X$ has only finitely many irreducible components, say $X_1, \ldots , X_ n$, and that $X = X_1\cup X_2\cup \ldots \cup X_ n$. (Note that any $X$ is always the union of its irreducible components, but that if $X = {\mathbf R}$ with its usual topology for instance then the irreducible components of $X$ are the one point subsets. This is not terribly interesting.)
Exercise 111.6.21. Show that irreducible components of $\mathop{\mathrm{Spec}}(A)$ correspond to minimal primes of $A$.
Definition 111.6.22. A point $x\in X$ is called closed if $\overline{\{ x\} } = \{ x\} $. Let $x, y$ be points of $X$. We say that $x$ is a specialization of $y$, or that $y$ is a generalization of $x$ if $x\in \overline{\{ y\} }$.
Exercise 111.6.23. Show that closed points of $\mathop{\mathrm{Spec}}(A)$ correspond to maximal ideals of $A$.
Exercise 111.6.24. Show that ${\mathfrak p}$ is a generalization of ${\mathfrak q}$ in $\mathop{\mathrm{Spec}}(A)$ if and only if ${\mathfrak p}\subset {\mathfrak q}$. Characterize closed points, maximal ideals, generic points and minimal prime ideals in terms of generalization and specialization. (Here we use the terminology that a point of a possibly reducible topological space $X$ is called a generic point if it is a generic points of one of the irreducible components of $X$.)
Exercise 111.6.25. Let $I$ and $J$ be ideals of $A$. What is the condition for $V(I)$ and $V(J)$ to be disjoint?
Definition 111.6.26. A topological space $X$ is called connected if it is nonempty and not the union of two nonempty disjoint open subsets. A connected component of $X$ is a maximal connected subset. Any point of $X$ is contained in a connected component of $X$ and any connected component of $X$ is closed in $X$. (But in general a connected component need not be open in $X$.)
Exercise 111.6.27. Let $A$ be a nonzero ring. Show that $\mathop{\mathrm{Spec}}(A)$ is disconnected iff $A\cong B \times C$ for certain nonzero rings $B, C$.
Exercise 111.6.28. Let $T$ be a connected component of $\mathop{\mathrm{Spec}}(A)$. Prove that $T$ is stable under generalization. Prove that $T$ is an open subset of $\mathop{\mathrm{Spec}}(A)$ if $A$ is Noetherian. (Remark: This is wrong when $A$ is an infinite product of copies of ${\mathbf F}_2$ for example. The spectrum of this ring consists of infinitely many closed points.)
Exercise 111.6.29. Compute $\mathop{\mathrm{Spec}}(k[x])$, i.e., describe the prime ideals in this ring, describe the possible specializations, and describe the topology. (Work this out when $k$ is algebraically closed but also when $k$ is not.)
Exercise 111.6.30. Compute $\mathop{\mathrm{Spec}}(k[x, y])$, where $k$ is algebraically closed. [Hint: use the morphism $\varphi : \mathop{\mathrm{Spec}}(k[x, y]) \to \mathop{\mathrm{Spec}}(k[x])$; if $\varphi ({\mathfrak p}) = (0)$ then localize with respect to $S = \{ f\in k[x] \mid f \not= 0\} $ and use result of lecture on localization and $\mathop{\mathrm{Spec}}$.] (Why do you think algebraic geometers call this affine 2-space?)
Exercise 111.6.31. Compute $\mathop{\mathrm{Spec}}(\mathbf{Z}[y])$. [Hint: as above.] (Affine 1-space over $\mathbf{Z}$.)
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