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\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*}

10.16 The spectrum of a ring

We arbitrarily decide that the spectrum of a ring as a topological space is part of the algebra chapter, whereas an affine scheme is part of the chapter on schemes.

Definition 10.16.1. Let $R$ be a ring.

  1. The spectrum of $R$ is the set of prime ideals of $R$. It is usually denoted $\mathop{\mathrm{Spec}}(R)$.

  2. Given a subset $T \subset R$ we let $V(T) \subset \mathop{\mathrm{Spec}}(R)$ be the set of primes containing $T$, i.e., $V(T) = \{ \mathfrak p \in \mathop{\mathrm{Spec}}(R) \mid \forall f\in T, f\in \mathfrak p\} $.

  3. Given an element $f \in R$ we let $D(f) \subset \mathop{\mathrm{Spec}}(R)$ be the set of primes not containing $f$.

Lemma 10.16.2. Let $R$ be a ring.

  1. The spectrum of a ring $R$ is empty if and only if $R$ is the zero ring.

  2. Every nonzero ring has a maximal ideal.

  3. Every nonzero ring has a minimal prime ideal.

  4. Given an ideal $I \subset R$ and a prime ideal $I \subset \mathfrak p$ there exists a prime $I \subset \mathfrak q \subset \mathfrak p$ such that $\mathfrak q$ is minimal over $I$.

  5. If $T \subset R$, and if $(T)$ is the ideal generated by $T$ in $R$, then $V((T)) = V(T)$.

  6. If $I$ is an ideal and $\sqrt{I}$ is its radical, see basic notion (27), then $V(I) = V(\sqrt{I})$.

  7. Given an ideal $I$ of $R$ we have $\sqrt{I} = \bigcap _{I \subset \mathfrak p} \mathfrak p$.

  8. If $I$ is an ideal then $V(I) = \emptyset $ if and only if $I$ is the unit ideal.

  9. If $I$, $J$ are ideals of $R$ then $V(I) \cup V(J) = V(I \cap J)$.

  10. If $(I_ a)_{a\in A}$ is a set of ideals of $R$ then $\cap _{a\in A} V(I_ a) = V(\cup _{a\in A} I_ a)$.

  11. If $f \in R$, then $D(f) \amalg V(f) = \mathop{\mathrm{Spec}}(R)$.

  12. If $f \in R$ then $D(f) = \emptyset $ if and only if $f$ is nilpotent.

  13. If $f = u f'$ for some unit $u \in R$, then $D(f) = D(f')$.

  14. If $I \subset R$ is an ideal, and $\mathfrak p$ is a prime of $R$ with $\mathfrak p \not\in V(I)$, then there exists an $f \in R$ such that $\mathfrak p \in D(f)$, and $D(f) \cap V(I) = \emptyset $.

  15. If $f, g \in R$, then $D(fg) = D(f) \cap D(g)$.

  16. If $f_ i \in R$ for $i \in I$, then $\bigcup _{i\in I} D(f_ i)$ is the complement of $V(\{ f_ i \} _{i\in I})$ in $\mathop{\mathrm{Spec}}(R)$.

  17. If $f \in R$ and $D(f) = \mathop{\mathrm{Spec}}(R)$, then $f$ is a unit.

Proof. We address each part in the corresponding item below.

  1. This is a direct consequence of (2) or (3).

  2. Let $\mathfrak {A}$ be the set of all proper ideals of $R$. This set is ordered by inclusion and is non-empty, since $(0) \in \mathfrak {A}$ is a proper ideal. Let $A$ be a totally ordered subset of $\mathfrak A$. Then $\bigcup _{I \in A} I$ is in fact an ideal. Since 1 $\notin I$ for all $I \in A$, the union does not contain 1 and thus is proper. Hence $\bigcup _{I \in A} I$ is in $\mathfrak {A}$ and is an upper bound for the set $A$. Thus by Zorn's lemma $\mathfrak {A}$ has a maximal element, which is the sought-after maximal ideal.

  3. Since $R$ is nonzero, it contains a maximal ideal which is a prime ideal. Thus the set $\mathfrak {A}$ of all prime ideals of $R$ is nonempty. $\mathfrak {A}$ is ordered by reverse-inclusion. Let $A$ be a totally ordered subset of $\mathfrak {A}$. It's pretty clear that $J = \bigcap _{I \in A} I$ is in fact an ideal. Not so clear, however, is that it is prime. Let $xy \in J$. Then $xy \in I$ for all $I \in A$. Now let $B = \{ I \in A | y \in I\} $. Let $K = \bigcap _{I \in B} I$. Since $A$ is totally ordered, either $K = J$ (and we're done, since then $y \in J$) or $K \supset J$ and for all $I \in A$ such that $I$ is properly contained in $K$, we have $y \notin I$. But that means that for all those $I, x \in I$, since they are prime. Hence $x \in J$. In either case, $J$ is prime as desired. Hence by Zorn's lemma we get a maximal element which in this case is a minimal prime ideal.

  4. This is the same exact argument as (3) except you only consider prime ideals contained in $\mathfrak {p}$ and containing $I$.

  5. $(T)$ is the smallest ideal containing $T$. Hence if $T \subset I$, some ideal, then $(T) \subset I$ as well. Hence if $I \in V(T)$, then $I \in V((T))$ as well. The other inclusion is obvious.

  6. Since $I \subset \sqrt{I}, V(\sqrt{I}) \subset V(I)$. Now let $\mathfrak {p} \in V(I)$. Let $x \in \sqrt{I}$. Then $x^ n \in I$ for some $n$. Hence $x^ n \in \mathfrak {p}$. But since $\mathfrak {p}$ is prime, a boring induction argument gets you that $x \in \mathfrak {p}$. Hence $\sqrt{I} \subset \mathfrak {p}$ and $\mathfrak {p} \in V(\sqrt{I})$.

  7. Let $f \in R \setminus \sqrt{I}$. Then $f^ n \notin I$ for all $n$. Hence $S = \{ 1, f, f^2, \ldots \} $ is a multiplicative subset, not containing $0$. Take a prime ideal $\bar{\mathfrak {p}} \subset S^{-1}R$ containing $S^{-1}I$. Then the pull-back $\mathfrak {p}$ in $R$ of $\bar{\mathfrak {p}}$ is a prime ideal containing $I$ that does not intersect $S$. This shows that $\bigcap _{I \subset \mathfrak p} \mathfrak p \subset \sqrt{I}$. Now if $a \in \sqrt{I}$, then $a^ n \in I$ for some $n$. Hence if $I \subset \mathfrak {p}$, then $a^ n \in \mathfrak {p}$. But since $\mathfrak {p}$ is prime, we have $a \in \mathfrak {p}$. Thus the equality is shown.

  8. $I$ is not the unit ideal if and only if $I$ is contained in some maximal ideal (to see this, apply (2) to the ring $R/I$) which is therefore prime.

  9. If $\mathfrak {p} \in V(I) \cup V(J)$, then $I \subset \mathfrak {p}$ or $J \subset \mathfrak {p}$ which means that $I \cap J \subset \mathfrak {p}$. Now if $I \cap J \subset \mathfrak {p}$, then $IJ \subset \mathfrak {p}$ and hence either $I$ of $J$ is in $\mathfrak {p}$, since $\mathfrak {p}$ is prime.

  10. $\mathfrak {p} \in \bigcap _{a \in A} V(I_ a) \Leftrightarrow I_ a \subset \mathfrak {p}, \forall a \in A \Leftrightarrow \mathfrak {p} \in V(\cup _{a\in A} I_ a)$

  11. If $\mathfrak {p}$ is a prime ideal and $f \in R$, then either $f \in \mathfrak {p}$ or $f \notin \mathfrak {p}$ (strictly) which is what the disjoint union says.

  12. If $a \in R$ is nilpotent, then $a^ n = 0$ for some $n$. Hence $a^ n \in \mathfrak {p}$ for any prime ideal. Thus $a \in \mathfrak {p}$ as can be shown by induction and $D(f) = \emptyset $. Now, as shown in (7), if $a \in R$ is not nilpotent, then there is a prime ideal that does not contain it.

  13. $f \in \mathfrak {p} \Leftrightarrow uf \in \mathfrak {p}$, since $u$ is invertible.

  14. If $\mathfrak {p} \notin V(I)$, then $\exists f \in I \setminus \mathfrak {p}$. Then $f \notin \mathfrak {p}$ so $\mathfrak {p} \in D(f)$. Also if $\mathfrak {q} \in D(f)$, then $f \notin \mathfrak {q}$ and thus $I$ is not contained in $\mathfrak {q}$. Thus $D(f) \cap V(I) = \emptyset $.

  15. If $fg \in \mathfrak {p}$, then $f \in \mathfrak {p}$ or $g \in \mathfrak {p}$. Hence if $f \notin \mathfrak {p}$ and $g \notin \mathfrak {p}$, then $fg \notin \mathfrak {p}$. Since $\mathfrak {p}$ is an ideal, if $fg \notin \mathfrak {p}$, then $f \notin \mathfrak {p}$ and $g \notin \mathfrak {p}$.

  16. $\mathfrak {p} \in \bigcup _{i \in I} D(f_ i) \Leftrightarrow \exists i \in I, f_ i \notin \mathfrak {p} \Leftrightarrow \mathfrak {p} \in \mathop{\mathrm{Spec}}(R) \setminus V(\{ f_ i\} _{i \in I})$

  17. If $D(f) = \mathop{\mathrm{Spec}}(R)$, then $V(f) = \emptyset $ and hence $fR = R$, so $f$ is a unit.

$\square$

The lemma implies that the subsets $V(T)$ from Definition 10.16.1 form the closed subsets of a topology on $\mathop{\mathrm{Spec}}(R)$. And it also shows that the sets $D(f)$ are open and form a basis for this topology.

Definition 10.16.3. Let $R$ be a ring. The topology on $\mathop{\mathrm{Spec}}(R)$ whose closed sets are the sets $V(T)$ is called the Zariski topology. The open subsets $D(f)$ are called the standard opens of $\mathop{\mathrm{Spec}}(R)$.

It should be clear from context whether we consider $\mathop{\mathrm{Spec}}(R)$ just as a set or as a topological space.

Lemma 10.16.4. Suppose that $\varphi : R \to R'$ is a ring homomorphism. The induced map

\[ \mathop{\mathrm{Spec}}(\varphi ) : \mathop{\mathrm{Spec}}(R') \longrightarrow \mathop{\mathrm{Spec}}(R), \quad \mathfrak p' \longmapsto \varphi ^{-1}(\mathfrak p') \]

is continuous for the Zariski topologies. In fact, for any element $f \in R$ we have $\mathop{\mathrm{Spec}}(\varphi )^{-1}(D(f)) = D(\varphi (f))$.

Proof. It is basic notion (41) that $\mathfrak p := \varphi ^{-1}(\mathfrak p')$ is indeed a prime ideal of $R$. The last assertion of the lemma follows directly from the definitions, and implies the first. $\square$

If $\varphi ' : R' \to R''$ is a second ring homomorphism then the composition

\[ \mathop{\mathrm{Spec}}(R'') \longrightarrow \mathop{\mathrm{Spec}}(R') \longrightarrow \mathop{\mathrm{Spec}}(R) \]

equals $\mathop{\mathrm{Spec}}(\varphi ' \circ \varphi )$. In other words, $\mathop{\mathrm{Spec}}$ is a contravariant functor from the category of rings to the category of topological spaces.

Lemma 10.16.5. Let $R$ be a ring. Let $S \subset R$ be a multiplicative subset. The map $R \to S^{-1}R$ induces via the functoriality of $\mathop{\mathrm{Spec}}$ a homeomorphism

\[ \mathop{\mathrm{Spec}}(S^{-1}R) \longrightarrow \{ \mathfrak p \in \mathop{\mathrm{Spec}}(R) \mid S \cap \mathfrak p = \emptyset \} \]

where the topology on the right hand side is that induced from the Zariski topology on $\mathop{\mathrm{Spec}}(R)$. The inverse map is given by $\mathfrak p \mapsto S^{-1}\mathfrak p$.

Proof. Denote the right hand side of the arrow of the lemma by $D$. Choose a prime $\mathfrak p' \subset S^{-1}R$ and let $\mathfrak p$ the inverse image of $\mathfrak p'$ in $R$. Since $\mathfrak p'$ does not contain $1$ we see that $\mathfrak p$ does not contain any element of $S$. Hence $\mathfrak p \in D$ and we see that the image is contained in $D$. Let $\mathfrak p \in D$. By assumption the image $\overline{S}$ does not contain $0$. By basic notion (54) $\overline{S}^{-1}(R/\mathfrak p)$ is not the zero ring. By basic notion (62) we see $S^{-1}R / S^{-1}\mathfrak p = \overline{S}^{-1}(R/\mathfrak p)$ is a domain, and hence $S^{-1}\mathfrak p$ is a prime. The equality of rings also shows that the inverse image of $S^{-1}\mathfrak p$ in $R$ is equal to $\mathfrak p$, because $R/\mathfrak p \to \overline{S}^{-1}(R/\mathfrak p)$ is injective by basic notion (55). This proves that the map $\mathop{\mathrm{Spec}}(S^{-1}R) \to \mathop{\mathrm{Spec}}(R)$ is bijective onto $D$ with inverse as given. It is continuous by Lemma 10.16.4. Finally, let $D(g) \subset \mathop{\mathrm{Spec}}(S^{-1}R)$ be a standard open. Write $g = h/s$ for some $h\in R$ and $s\in S$. Since $g$ and $h/1$ differ by a unit we have $D(g) = D(h/1)$ in $\mathop{\mathrm{Spec}}(S^{-1}R)$. Hence by Lemma 10.16.4 and the bijectivity above the image of $D(g) = D(h/1)$ is $D \cap D(h)$. This proves the map is open as well. $\square$

Lemma 10.16.6. Let $R$ be a ring. Let $f \in R$. The map $R \to R_ f$ induces via the functoriality of $\mathop{\mathrm{Spec}}$ a homeomorphism

\[ \mathop{\mathrm{Spec}}(R_ f) \longrightarrow D(f) \subset \mathop{\mathrm{Spec}}(R). \]

The inverse is given by $\mathfrak p \mapsto \mathfrak p \cdot R_ f$.

Proof. This is a special case of Lemma 10.16.5. $\square$

It is not the case that every “affine open” of a spectrum is a standard open. See Example 10.26.4.

Lemma 10.16.7. Let $R$ be a ring. Let $I \subset R$ be an ideal. The map $R \to R/I$ induces via the functoriality of $\mathop{\mathrm{Spec}}$ a homeomorphism

\[ \mathop{\mathrm{Spec}}(R/I) \longrightarrow V(I) \subset \mathop{\mathrm{Spec}}(R). \]

The inverse is given by $\mathfrak p \mapsto \mathfrak p / I$.

Proof. It is immediate that the image is contained in $V(I)$. On the other hand, if $\mathfrak p \in V(I)$ then $\mathfrak p \supset I$ and we may consider the ideal $\mathfrak p /I \subset R/I$. Using basic notion (51) we see that $(R/I)/(\mathfrak p/I) = R/\mathfrak p$ is a domain and hence $\mathfrak p/I$ is a prime ideal. From this it is immediately clear that the image of $D(f + I)$ is $D(f) \cap V(I)$, and hence the map is a homeomorphism. $\square$

Remark 10.16.8. A fundamental commutative diagram associated to a ring map $\varphi : R \to S$, a prime $\mathfrak q \subset S$ and the corresponding prime $\mathfrak p = \varphi ^{-1}(\mathfrak q)$ of $R$ is the following

\[ \xymatrix{ \kappa (\mathfrak q) = S_{\mathfrak q}/{\mathfrak q}S_{\mathfrak q} & S_{\mathfrak q} \ar[l] & S \ar[r] \ar[l] & S/\mathfrak q \ar[r] & \kappa (\mathfrak q) \\ \kappa (\mathfrak p) \otimes _ R S = S_{\mathfrak p}/{\mathfrak p}S_{\mathfrak p} \ar[u] & S_{\mathfrak p} \ar[u] \ar[l] & S \ar[u] \ar[r] \ar[l] & S/\mathfrak pS \ar[u] \ar[r] & (R \setminus \mathfrak p)^{-1}S/\mathfrak pS \ar[u] \\ \kappa (\mathfrak p) = R_{\mathfrak p}/{\mathfrak p}R_{\mathfrak p} \ar[u] & R_{\mathfrak p} \ar[u] \ar[l] & R \ar[u] \ar[r] \ar[l] & R/\mathfrak p \ar[u] \ar[r] & \kappa (\mathfrak p) \ar[u] } \]

In this diagram the arrows in the outer left and outer right columns are identical. The horizontal maps induce on the associated spectra always a homeomorphism onto the image. The lower two rows of the diagram make sense without assuming $\mathfrak q$ exists. The lower squares induce fibre squares of topological spaces. This diagram shows that $\mathfrak p$ is in the image of the map on Spec if and only if $S \otimes _ R \kappa (\mathfrak p)$ is not the zero ring.

Lemma 10.16.9. Let $\varphi : R \to S$ be a ring map. Let $\mathfrak p$ be a prime of $R$. The following are equivalent

  1. $\mathfrak p$ is in the image of $\mathop{\mathrm{Spec}}(S) \to \mathop{\mathrm{Spec}}(R)$,

  2. $S \otimes _ R \kappa (\mathfrak p) \not= 0$,

  3. $S_{\mathfrak p}/\mathfrak p S_{\mathfrak p} \not= 0$,

  4. $(S/\mathfrak pS)_{\mathfrak p} \not= 0$, and

  5. $\mathfrak p = \varphi ^{-1}(\mathfrak pS)$.

Proof. We have already seen the equivalence of the first two in Remark 10.16.8. The others are just reformulations of this. $\square$

Lemma 10.16.10. Let $R$ be a ring. The space $\mathop{\mathrm{Spec}}(R)$ is quasi-compact.

Proof. It suffices to prove that any covering of $\mathop{\mathrm{Spec}}(R)$ by standard opens can be refined by a finite covering. Thus suppose that $\mathop{\mathrm{Spec}}(R) = \cup D(f_ i)$ for a set of elements $\{ f_ i\} _{i\in I}$ of $R$. This means that $\cap V(f_ i) = \emptyset $. According to Lemma 10.16.2 this means that $V(\{ f_ i \} ) = \emptyset $. According to the same lemma this means that the ideal generated by the $f_ i$ is the unit ideal of $R$. This means that we can write $1$ as a finite sum: $1 = \sum _{i \in J} r_ i f_ i$ with $J \subset I$ finite. And then it follows that $\mathop{\mathrm{Spec}}(R) = \cup _{i \in J} D(f_ i)$. $\square$

Lemma 10.16.11. Let $R$ be a ring. The topology on $X = \mathop{\mathrm{Spec}}(R)$ has the following properties:

  1. $X$ is quasi-compact,

  2. $X$ has a basis for the topology consisting of quasi-compact opens, and

  3. the intersection of any two quasi-compact opens is quasi-compact.

Proof. The spectrum of a ring is quasi-compact, see Lemma 10.16.10. It has a basis for the topology consisting of the standard opens $D(f) = \mathop{\mathrm{Spec}}(R_ f)$ (Lemma 10.16.6) which are quasi-compact by the first remark. The intersection of two standard opens is quasi-compact as $D(f) \cap D(g) = D(fg)$. Given any two quasi-compact opens $U, V \subset X$ we may write $U = D(f_1) \cup \ldots \cup D(f_ n)$ and $V = D(g_1) \cup \ldots \cup D(g_ m)$. Then $U \cap V = \bigcup D(f_ ig_ j)$ which is quasi-compact. $\square$


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