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.
If g\in R and D(g) \subset D(f), then
f is invertible in R_ g,
g^ e = af for some e \geq 1 and a \in R,
there is a canonical ring map R_ f \to R_ g, and
there is a canonical R_ f-module map M_ f \to M_ g for any R-module M.
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).
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.
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.
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.
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.
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).
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.
We have \Gamma (\mathop{\mathrm{Spec}}(R), \mathcal{O}_{\mathop{\mathrm{Spec}}(R)}) = R.
We have \Gamma (\mathop{\mathrm{Spec}}(R), \widetilde M) = M as an R-module.
For every f \in R we have \Gamma (D(f), \mathcal{O}_{\mathop{\mathrm{Spec}}(R)}) = R_ f.
For every f\in R we have \Gamma (D(f), \widetilde M) = M_ f as an R_ f-module.
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.
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}.
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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