## 26.5 Affine schemes

Let $R$ be a ring. Consider the topological space $\mathop{\mathrm{Spec}}(R)$ associated to $R$, see Algebra, Section 10.17. We will endow this space with a sheaf of rings $\mathcal{O}_{\mathop{\mathrm{Spec}}(R)}$ and the resulting pair $(\mathop{\mathrm{Spec}}(R), \mathcal{O}_{\mathop{\mathrm{Spec}}(R)})$ will be an affine scheme.

Recall that $\mathop{\mathrm{Spec}}(R)$ has a basis of open sets $D(f)$, $f \in R$ which we call standard opens, see Algebra, Definition 10.17.3. In addition, the intersection of two standard opens is another: $D(f) \cap D(g) = D(fg)$, $f, g\in R$.

Lemma 26.5.1. Let $R$ be a ring. Let $f \in R$.

1. If $g\in R$ and $D(g) \subset D(f)$, then

1. $f$ is invertible in $R_ g$,

2. $g^ e = af$ for some $e \geq 1$ and $a \in R$,

3. there is a canonical ring map $R_ f \to R_ g$, and

4. there is a canonical $R_ f$-module map $M_ f \to M_ g$ for any $R$-module $M$.

2. Any open covering of $D(f)$ can be refined to a finite open covering of the form $D(f) = \bigcup _{i = 1}^ n D(g_ i)$.

3. If $g_1, \ldots , g_ n \in R$, then $D(f) \subset \bigcup D(g_ i)$ if and only if $g_1, \ldots , g_ n$ generate the unit ideal in $R_ f$.

Proof. Recall that $D(g) = \mathop{\mathrm{Spec}}(R_ g)$ (see Algebra, Lemma 10.17.6). Thus (a) holds because $f$ maps to an element of $R_ g$ which is not contained in any prime ideal, and hence invertible, see Algebra, Lemma 10.17.2. Write the inverse of $f$ in $R_ g$ as $a/g^ d$. This means $g^ d - af$ is annihilated by a power of $g$, whence (b). For (c), the map $R_ f \to R_ g$ exists by (a) from the universal property of localization, or we can define it by mapping $b/f^ n$ to $a^ nb/g^{ne}$. The equality $M_ f = M \otimes _ R R_ f$ can be used to obtain the map on modules, or we can define $M_ f \to M_ g$ by mapping $x/f^ n$ to $a^ nx/g^{ne}$.

Recall that $D(f)$ is quasi-compact, see Algebra, Lemma 10.29.1. Hence the second statement follows directly from the fact that the standard opens form a basis for the topology.

The third statement follows directly from Algebra, Lemma 10.17.2. $\square$

In Sheaves, Section 6.30 we defined the notion of a sheaf on a basis, and we showed that it is essentially equivalent to the notion of a sheaf on the space, see Sheaves, Lemmas 6.30.6 and 6.30.9. Moreover, we showed in Sheaves, Lemma 6.30.4 that it is sufficient to check the sheaf condition on a cofinal system of open coverings for each standard open. By the lemma above it suffices to check on the finite coverings by standard opens.

Definition 26.5.2. Let $R$ be a ring.

1. A standard open covering of $\mathop{\mathrm{Spec}}(R)$ is a covering $\mathop{\mathrm{Spec}}(R) = \bigcup _{i = 1}^ n D(f_ i)$, where $f_1, \ldots , f_ n \in R$.

2. Suppose that $D(f) \subset \mathop{\mathrm{Spec}}(R)$ is a standard open. A standard open covering of $D(f)$ is a covering $D(f) = \bigcup _{i = 1}^ n D(g_ i)$, where $g_1, \ldots , g_ n \in R$.

Let $R$ be a ring. Let $M$ be an $R$-module. We will define a presheaf $\widetilde M$ on the basis of standard opens. Suppose that $U \subset \mathop{\mathrm{Spec}}(R)$ is a standard open. If $f, g \in R$ are such that $D(f) = D(g)$, then by Lemma 26.5.1 above there are canonical maps $M_ f \to M_ g$ and $M_ g \to M_ f$ which are mutually inverse. Hence we may choose any $f$ such that $U = D(f)$ and define

$\widetilde M(U) = M_ f.$

Note that if $D(g) \subset D(f)$, then by Lemma 26.5.1 above we have a canonical map

$\widetilde M(D(f)) = M_ f \longrightarrow M_ g = \widetilde M(D(g)).$

Clearly, this defines a presheaf of abelian groups on the basis of standard opens. If $M = R$, then $\widetilde R$ is a presheaf of rings on the basis of standard opens.

Let us compute the stalk of $\widetilde M$ at a point $x \in \mathop{\mathrm{Spec}}(R)$. Suppose that $x$ corresponds to the prime $\mathfrak p \subset R$. By definition of the stalk we see that

$\widetilde M_ x = \mathop{\mathrm{colim}}\nolimits _{f\in R, f\not\in \mathfrak p} M_ f$

Here the set $\{ f \in R, f \not\in \mathfrak p\}$ is preordered by the rule $f \geq f' \Leftrightarrow D(f) \subset D(f')$. If $f_1, f_2 \in R \setminus \mathfrak p$, then we have $f_1f_2 \geq f_1$ in this ordering. Hence by Algebra, Lemma 10.9.9 we conclude that

$\widetilde M_ x = M_{\mathfrak p}.$

Next, we check the sheaf condition for the standard open coverings. If $D(f) = \bigcup _{i = 1}^ n D(g_ i)$, then the sheaf condition for this covering is equivalent with the exactness of the sequence

$0 \to M_ f \to \bigoplus M_{g_ i} \to \bigoplus M_{g_ ig_ j}.$

Note that $D(g_ i) = D(fg_ i)$, and hence we can rewrite this sequence as the sequence

$0 \to M_ f \to \bigoplus M_{fg_ i} \to \bigoplus M_{fg_ ig_ j}.$

In addition, by Lemma 26.5.1 above we see that $g_1, \ldots , g_ n$ generate the unit ideal in $R_ f$. Thus we may apply Algebra, Lemma 10.24.1 to the module $M_ f$ over $R_ f$ and the elements $g_1, \ldots , g_ n$. We conclude that the sequence is exact. By the remarks made above, we see that $\widetilde M$ is a sheaf on the basis of standard opens.

Thus we conclude from the material in Sheaves, Section 6.30 that there exists a unique sheaf of rings $\mathcal{O}_{\mathop{\mathrm{Spec}}(R)}$ which agrees with $\widetilde R$ on the standard opens. Note that by our computation of stalks above, the stalks of this sheaf of rings are all local rings.

Similarly, for any $R$-module $M$ there exists a unique sheaf of $\mathcal{O}_{\mathop{\mathrm{Spec}}(R)}$-modules $\mathcal{F}$ which agrees with $\widetilde M$ on the standard opens, see Sheaves, Lemma 6.30.12.

Definition 26.5.3. Let $R$ be a ring.

1. The structure sheaf $\mathcal{O}_{\mathop{\mathrm{Spec}}(R)}$ of the spectrum of $R$ is the unique sheaf of rings $\mathcal{O}_{\mathop{\mathrm{Spec}}(R)}$ which agrees with $\widetilde R$ on the basis of standard opens.

2. The locally ringed space $(\mathop{\mathrm{Spec}}(R), \mathcal{O}_{\mathop{\mathrm{Spec}}(R)})$ is called the spectrum of $R$ and denoted $\mathop{\mathrm{Spec}}(R)$.

3. The sheaf of $\mathcal{O}_{\mathop{\mathrm{Spec}}(R)}$-modules extending $\widetilde M$ to all opens of $\mathop{\mathrm{Spec}}(R)$ is called the sheaf of $\mathcal{O}_{\mathop{\mathrm{Spec}}(R)}$-modules associated to $M$. This sheaf is denoted $\widetilde M$ as well.

We summarize the results obtained so far.

Lemma 26.5.4. Let $R$ be a ring. Let $M$ be an $R$-module. Let $\widetilde M$ be the sheaf of $\mathcal{O}_{\mathop{\mathrm{Spec}}(R)}$-modules associated to $M$.

1. We have $\Gamma (\mathop{\mathrm{Spec}}(R), \mathcal{O}_{\mathop{\mathrm{Spec}}(R)}) = R$.

2. We have $\Gamma (\mathop{\mathrm{Spec}}(R), \widetilde M) = M$ as an $R$-module.

3. For every $f \in R$ we have $\Gamma (D(f), \mathcal{O}_{\mathop{\mathrm{Spec}}(R)}) = R_ f$.

4. For every $f\in R$ we have $\Gamma (D(f), \widetilde M) = M_ f$ as an $R_ f$-module.

5. Whenever $D(g) \subset D(f)$ the restriction mappings on $\mathcal{O}_{\mathop{\mathrm{Spec}}(R)}$ and $\widetilde M$ are the maps $R_ f \to R_ g$ and $M_ f \to M_ g$ from Lemma 26.5.1.

6. Let $\mathfrak p$ be a prime of $R$, and let $x \in \mathop{\mathrm{Spec}}(R)$ be the corresponding point. We have $\mathcal{O}_{\mathop{\mathrm{Spec}}(R), x} = R_{\mathfrak p}$.

7. Let $\mathfrak p$ be a prime of $R$, and let $x \in \mathop{\mathrm{Spec}}(R)$ be the corresponding point. We have $\widetilde M_ x = M_{\mathfrak p}$ as an $R_{\mathfrak p}$-module.

Moreover, all these identifications are functorial in the $R$ module $M$. In particular, the functor $M \mapsto \widetilde M$ is an exact functor from the category of $R$-modules to the category of $\mathcal{O}_{\mathop{\mathrm{Spec}}(R)}$-modules.

Proof. Assertions (1) - (7) are clear from the discussion above. The exactness of the functor $M \mapsto \widetilde M$ follows from the fact that the functor $M \mapsto M_{\mathfrak p}$ is exact and the fact that exactness of short exact sequences may be checked on stalks, see Modules, Lemma 17.3.1. $\square$

Definition 26.5.5. An affine scheme is a locally ringed space isomorphic as a locally ringed space to $\mathop{\mathrm{Spec}}(R)$ for some ring $R$. A morphism of affine schemes is a morphism in the category of locally ringed spaces.

It turns out that affine schemes play a special role among all locally ringed spaces, which is what the next section is about.

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