3 Sheaves
References: [ GW1 ] (2.5)–(2.8) or [ Ha ] II.1.
We have attached to every ring \(R\) a topological space \(\operatorname{Spec}(R)\), but this topological space “retains very little information” about the ring \(R\). For example, for every field \(k\) we get the same topological space as its prime spectrum. To achieve our goal of “making \(\operatorname{Spec}(R)\) into a geometric object”, we follow the slogan that the geometry of a “space” is determined by the functions on the space, meaning that for each kind of geometry (topology, differential geometry, complex geometry) there is a natural notion of function (continuous, differentiable, holomorphic), and this is a characteristic feature of the whole theory.
For us, the intuition is that elements of the ring \(R\) should be viewed as functions on \(\operatorname{Spec}(R)\), but since the elements of \(R\) are not “really” functions, it is useful to introduce a more abstract framework that allows us to talk about (and gain intuition from) the previously mentioned cases, but which also applies to the prime spectra of rings.
As it turns out, the following properties are crucial for the “functions” we want to consider. Let \(X\) be a topological space.
A function might be defined on all of \(X\), or on some smaller open subset of \(X\) (the “domain of definition” of the function). We want to allow the functions to have poles at some point of \(X\) and therefore do not ask that the domain of definition is always equal to \(X\).
Functions should be determined by “local information” – since we do not want to talk of the values of a function, we will instead talk about restrictions of a function to open subsets within its “domain of definition”, and require that it is determined by the restrictions to open subsets that cover the domain of definition, and also that functions can be defined by “gluing with respect to an open cover” (see below).
(We are deliberately vague about the “target” of our “functions”. In differential geometry it would be \(\mathbb {R}\), in complex geometry it would be \(\mathbb {C}\), but in algebraic geometry we do not really have functions and therefore do not really have a target of functions at our disposal.)
for each open \(U \subseteq X\), a set \({\mathscr F}(U)\),
for each pair \(U \subseteq V \subseteq X\) of open subsets, a map (“restriction map”) \(\operatorname{res}^V_U \colon {\mathscr F}(V)\to {\mathscr F}(U)\),
![\begin{tikzcd}
\Fscr(V)\ar[d, "\res^V_U"] \ar[r, "\varphi_{V}"] & \Gscr(V)\ar[d, "\res^V_U"] \\
\Fscr(U) \ar[r, "\varphi_{U}"] & \Gscr(U)
\end{tikzcd}](images/img-0007.svg)
Notation: We often write \(s_{|U}\) for \(\operatorname{res}^V_U(s)\) (\(s\in {\mathscr F}(V)\)). The elements of \({\mathscr F}(U)\) are also called sections of the presheaf on the open set \(U\). One also writes \(\Gamma (U, {\mathscr F})\) instead of \({\mathscr F}(U)\).
While the notion of presheaf provides the basic framework to talk about (generalizations of) functions on a topological space, it is much too general and does not capture enough properties that (even generalized) “functions” should have. It turns out that the crucial property is that functions should be determined by their restrictions to an open cover, and that it should be possible to “specify a function locally”, i.e., on an open cover, provided that the obvious compatibility condition on intersections is satisfied. This observation is turned into the definition of sheaf, as follows.
Let \(X\) be a topological space. A presheaf \({\mathscr F}\) (of sets) on \(X\) is called a sheaf (of sets), if the following condition is satisfied. For every open subset \(U \subseteq X\) and every cover \(U = \bigcup _{i\in I} U_i\) by open subsets of \(X\), the diagram
![\begin{tikzcd}
\Fscr(U) \ar[r, "\rho"] & \prod_{i\in I} \Fscr(U_i) \ar[r, shift left, "\sigma"] \ar[r, shift right, swap, "\sigma'"] & \prod_{(i,j)\in I\times I}\Fscr(U_i\cap U_j)\\[-.5cm]
s \ar[r, mapsto] & (s_{|U_i})_i\\[-.5cm]
& (s_i)_i\ar[r, mapsto, "\sigma"] & (s_{i|U_{i}\cap U_{j}})_{i,j}\\[-.5cm]
& (s_i)_i\ar[r, mapsto, swap, "\sigma'"] & (s_{j|U_{i}\cap U_{j}})_{i,j}
\end{tikzcd}](images/img-0008.svg)
is exact, i.e., the map \(\rho \) is injective, and the image of \(\rho \) is the set of elements \((s_i)_{i\in I}\) such that \(\sigma ((s_i)_{i\in I}) = \sigma '((s_i)_{i\in I})\).
For sheaves \({\mathscr F}\), \({\mathscr G}\) on \(X\), a morphism \({\mathscr F}\to {\mathscr G}\) of sheaves is a morphism between the presheaves \({\mathscr F}\) and \({\mathscr G}\).
Pedantic remark: It follows from the definition (applied to \(U=\emptyset \), \(I=\emptyset \)) that for every sheaf \({\mathscr F}\), the set \({\mathscr F}(\emptyset )\) has precisely one element.
Nov. 12, 2025
Let \(X\) be a topological space and \(Y\) a set. Setting, for \(U \subseteq X\) open, \({\mathscr F}(U) = {\rm Map}(X,Y)\) (the set of all maps \(U\to Y\)) defines a sheaf on \(X\).
Let \(X\) and \(Y\) be topological spaces. Setting \({\mathscr F}(U) = {\rm Map}_{\rm cont}(X,Y)\) (the set of all continuous maps \(U\to Y\)) defines a sheaf on \(X\).
Let \(X \subseteq \mathbb {R}^n\) open (or any differentiable manifold). Setting \({\mathscr F}(U) = C^\infty (U)\), the set of infinitely often differentiable functions \(U\to \mathbb {R}\), defines a sheaf on \(X\).
Let \(X \subseteq \mathbb {C}^n\) open (or any complex manifold). Setting \({\mathscr F}(U) = {\rm Hol}(U)\), the set of holomorphic functions \(U\to \mathbb {C}\), defines a sheaf on \(X\).
Let \(X = \mathbb {R}\) (with the analytic topology). Setting \({\mathscr F}(U) = \{ f\colon U\to \mathbb {R}\ \text{bounded}\} \) defines a presheaf on \(\mathbb {R}\) which is not a sheaf. (Similarly: bounded and continuous; or bounded and differentiable.)
Typically, the restriction maps \(\operatorname{res}^V_U\) are neither injective nor surjective. If the map is not surjective, we may think of it as saying that there are “functions” (more precisely, sections of the sheaf) defined on \(U\) but which do not extend (because they may “have poles” at points of \(V\setminus U\)) to the larger set \(V\). On the other hand, a “function” on \(V\) cannot usually be expected to be determined by its values on a smaller open set \(U\); so in the above examples (1), (2), (3) the restriction maps will be injective only in trivial cases. However in complex analysis (Example (4) above) there is the interesting result (the “identity theorem”) that the restriction map \(\operatorname{res}^V_U\) is injective whenever \(\emptyset \ne U \subseteq V\) and \(V\) is connected.
It follows easily from the sheaf axioms that for \(U = U_1 \cap U_2\) with \(U_1\), \(U_2\) open and \(U_1\cap U_2 = \emptyset \), the natural map \({\mathscr F}(U)\to {\mathscr F}(U_1)\times {\mathscr F}(U_2)\) induced by the restriction maps is an isomorphism.
Often the sets of sections carry more structure, for example, in the above examples of sheaves of actual functions (with certain properties such as continuity or differentiability) with target a ring, we can actually add and multiply functions by using the addition and multiplication on the target, so that the sets \({\mathscr F}(U)\) in this case naturally carry a ring structure and the restriction maps are ring homomorphism. This is of course a useful piece of information to remember, and we therefore make the following definition.
Our next goal is to define a sheaf of rings on \(X=\operatorname{Spec}(R)\) for a ring \(R\). This sheaf will be denoted \({\mathscr O}_X\) and called the structure sheaf on \(\operatorname{Spec}(R)\). The underlying idea is the following. We said that we want to view elements of \(R\) as (a kind of) functions on \(X\), so we will start by setting \({\mathscr O}_X(X)=R\). For a general open subset \(U \subset X\) it is however not so clear what to do. However, for principal opens \(D(f) \subseteq X\), there is a natural candidate. Namely, we have seen that there is a natural homeomorphism \(\operatorname{Spec}(R_f)\cong D(f)\), and since we have already made a guess what the ring of functions on the left hand side should be (namely \(R_f\)), we will set \({\mathscr O}_X(D(f)) = R_f\). (We will check later that this is well-defined, i.e., that whenever \(D(f) = D(g)\), we have a canonical identification \(R_f = R_g\).) At this point we can already (and will, see below) check that this definition satisfies the conditions in the definition of sheaves (i.e., the “gluing of sections”); the computation is not so difficult, but this is a crucial point of the theory. Having checked this, philosophically, we can expect that this should be enough information in order to define \({\mathscr O}_X\), because the \(D(f)\) form a basis of the topology, and a sheaf should be determined by local information. This is in fact a general result on sheaves, and we will prove it below.
In this section we fix a topological space \(X\) and a basis \(\mathcal B\) of the topology of \(X\) (recall that this means that \(\mathcal B\) is a set of open subsets of \(X\) such that every open subset of \(\mathcal B\) can be written as a union of elements of \(\mathcal B\)). Things simplify if \(\mathcal B\) satisfies in addition the property that any finite intersections of open subsets lying in \(\mathcal B\) is again an element of \(\mathcal B\). This is satisfied for the basis of principal open subsets of the Zariski topology of the spectrum of a ring, the situation relevant for us, so the reader is advised to make this extra assumption.
A presheaf \({\mathscr F}\) on the basis \(\mathcal B\) of the topology is given by a set \({\mathscr F}(U)\) for every \(U\in \mathcal B\) and a restriction map \(\operatorname{res}^V_U\colon {\mathscr F}(V)\to {\mathscr F}(U)\) for every pair of open subsets \(U, V\in \mathcal B\) with \(U\subseteq V\), such that \(\operatorname{res}^U_U = \operatorname{id}_{{\mathscr F}(U)}\) for every \(U\in \mathcal B\) and \(\operatorname{res}^W_U = \operatorname{res}^V_U\circ \operatorname{res}^W_V\) for all open subsets \(U,V,W\in \mathcal B\), \(U \subseteq V \subseteq W\).
A presheaf \({\mathscr F}\) on \(\mathcal B\) is called a sheaf on \(\mathcal B\), if for every \(U\in \mathcal B\), every cover \(U=\bigcup _i U_i\) with \(U_i\in \mathcal B\) and every open cover \(U_i\cap U_j = \bigcup _k U_{ijk}\) with \(U_{ijk}\in \mathcal B\), the sequence
![\begin{tikzcd}
\Fscr(U) \ar[r, "\rho"] & \prod_{i\in I} \Fscr(U_i) \ar[r, shift left, "\sigma"] \ar[r, shift right, swap, "\sigma'"] & \prod_{(i,j)\in I\times I}\prod_k\Fscr(U_{ijk})\\[-.5cm]
s \ar[r, mapsto] & (s_{|U_i})_i\\[-.5cm]
& (s_i)_i\ar[r, mapsto, "\sigma"] & (s_{i|U_{ijk}})_{i,j}\\[-.5cm]
& (s_i)_i\ar[r, mapsto, swap, "\sigma'"] & (s_{j|U_{ijk}})_{i,j}
\end{tikzcd}](images/img-0009.svg)
is exact.
Equivalently, in (2) it suffices to ask the exactness for every open cover \(U=\bigcup _i U_i\), but to fix one open cover \(U_i\cap U_j = \bigcup _k U_{ijk}\) for each pair \(i,j\), rather than check it for all such covers. In particular, if \(\mathcal B\) is stable under finite intersections, then one can just cover \(U_i\cap U_j\) “by itself”, so that one can use “the same sequence as in the definition of a sheaf”.
Let \({\mathscr F}\), \({\mathscr G}\) be presheaves on \(\mathcal B\). A morphism \(f\colon {\mathscr F}\to {\mathscr G}\) is given by a collection of maps \(f_U\colon {\mathscr F}(U)\to {\mathscr G}(U)\) for all \(U\in \mathcal B\), such that for all \(U \subseteq V\), \(U, V\in \mathcal B\), we have \(\operatorname{res}^V_U\circ f_V = f_U\circ \operatorname{res}^V_U\) (where on the left we use the restriction map for \({\mathscr G}\), on the left hand side that for \({\mathscr F}\)).
For sheaves \({\mathscr F}\), \({\mathscr G}\) a morphism \({\mathscr F}\to {\mathscr G}\) of sheaves on \(\mathcal B\) is a morphism of the underlying presheaves.
Together with the obvious identity morphisms and composition of morphisms we obtain the categories of presheaves on \(\mathcal B\) and of sheaves on \(\mathcal B\).
It is clear that we can restrict sheaves (and morphisms) on \(X\) to \(\mathcal B\).
For every sheaf \({\mathscr F}\) on \(X\) (in the sense of Definition 3.3), the restriction \({\mathscr F}_{|\mathcal B}\) given by \({\mathscr F}_{|\mathcal B}(U)={\mathscr F}(U)\) for all \(U\in \mathcal B\), and similarly for the restriction maps, is a sheaf on \(\mathcal B\).
Similarly any morphism \(f\colon {\mathscr F}\to {\mathscr G}\) of sheaves on \(X\) induces by restriction a morphism \(f_{|\mathcal B}\colon {\mathscr F}_{|\mathcal B}\to {\mathscr G}_{|\mathcal B}\).
For sheaves, it is reasonable to expect that we can also go in the other direction, i.e., recover a sheaf from its values (including the restriction maps) on \(\mathcal B\), or more generally, given any sheaf \({\mathscr F}'\) on \(\mathcal B\) construct a sheaf \({\mathscr F}\) on \(X\) such that \({\mathscr F}_{|\mathcal B}\) is \({\mathscr F}'\). Furthermore this construction should also be compatible with morphisms. (Of course, a similar result cannot hold true for arbitrary presheaves.)
Nov. 18, 2025
The restriction to \(\mathcal B\) and extension to all open subsets of \(X\) are inverse to each other – but only if this is formulated in the right way. It is impossible to achieve that the extension of \({\mathscr F}_{|\mathcal B}\) is equal to \({\mathscr F}\); rather the best one can hope for is that it is isomorphic to \({\mathscr F}\), and that these isomorphisms are compatible with morphisms of sheaves. This situation is best captured by the notion of equivalence of categories, see the next section for a short discussion.The key point is the construction of a sheaf \({\mathscr F}\) on \(X\) given a sheaf \({\mathscr F}'\) on \(\mathcal B\). The idea of defining \({\mathscr F}(U)\) for an arbitrary open \(U \subseteq X\) is easy to explain. Assume that we already had constructed the sheaf \({\mathscr F}\). Then for every cover \(U=\bigcup U_i\) with \(U_i\in \mathcal B\), we could recover \({\mathscr F}(U)\) from the sheaf sequence as a subset of \(\prod _i {\mathscr F}(U_i)=\prod _i {\mathscr F}'(U_i)\). If \(\mathcal B\) is stable under finite intersections, then the intersections \(U_i\cap U_j\) are also in \(\mathcal B\) and we can directly express the compatibility condition cutting our \({\mathscr F}(U)\) inside this products in terms of \({\mathscr F}'\). In general, one can proceed similarly as in the definition of the notion of sheaf on \(\mathcal B\).
In order to check that this construction has the correct properties, it is however slightly inconvenient that it depends on the choice of a cover of \(U\). This problem may be circumvented by simply using the cover of \(U\) given by all elements of \(\mathcal B\) that are contained in \(U\). This has the additional advantage that all intersections arising in the definition of the sheaf axioms are covered by open subsets that themselves occur in the cover, whence it is enough to simply ask for the compatibility with all restrictions, in the following sense: We define
Similarly as in the first paragraph, the sheaf axioms imply that this is the only possible candidate for \({\mathscr F}(U)\). It is then not difficult to define restriction maps, to define the extension of morphisms of sheaves, and to show that this extension functor is a quasi-inverse of the restriction functor.
All of the above (definitions and) results carry over to the settings of (pre-)sheaves of (abelian) groups, rings, modules over a fixed ring, etc.
References: [ GW1 ] Appendix A; [ Alg2 ] Section 3.1.
A category \(\mathcal C\) is given by a collection (“class”) of objects \(\operatorname{Ob}(\mathcal C)\), for any two \(X, Y\in \operatorname{Ob}(\mathcal C)\) a collection \(\operatorname{Hom}_{\mathcal C}(X, Y)\) of morphisms, for any object \(X\) a morphism \(\operatorname{id}_X\in \operatorname{Hom}_{\mathcal C}(X, X)\), and for any three objects \(X, Y, Z\) a map
such that \(f\circ \operatorname{id}= f\), \(g\circ \operatorname{id}= g\), \((f\circ g)\circ h = f\circ (g\circ h)\) whenever these expressions are defined. We write \(f\colon X\to Y\) if \(f\in \operatorname{Hom}_{\mathcal C}(X, Y)\), and accordingly sometimes speak (and think) of morphisms in a category as arrows. We sometimes write \(X\in \mathcal C\) instead of \(X\in \operatorname{Ob}(\mathcal C)\). Set-theoretic remark: We usually implicitly make the assumption that for all \(X\), \(Y\), \(\operatorname{Hom}_{\mathcal C}(X, Y)\) is a set (i.e., that \(\mathcal C\) is what is usually called a locally small category).
Examples. The categories of sets (with maps of sets as morphisms), of finite sets, of groups (with group homomorphisms), of abelian groups, of rings (with ting homomorphisms), of modules over a fixed ring (with module homomorphisms), of finitely generated modules over a fixed ring, of topological spaces (with continuous maps as morphisms).
For objects \(X, Y\) in \(\mathcal C\) we say that \(X\) and \(Y\) are isomorphic and write \(X\cong Y\), if there exists an isomorphism \(X\to Y\) in \(\mathcal C\).
Let \(\mathcal C\), \(\mathcal D\) be categories. A functor \(F\colon \mathcal C\to \mathcal D\) is given by the following data: For each object \(X\) of \(\mathcal C\) an object \(F(X)\) of \(\mathcal D\), and for every morphism \(f\colon X\to Y\) in \(\mathcal C\) a morphism \(F(f)\colon F(X)\to F(Y)\), such that \(F(\operatorname{id}_X) = \operatorname{id}_{F(X)}\) for all \(X\) and such that \(F(f\circ g) = F(f)\circ F(g)\).
It is useful to extend this notion in the following way. A contravariant functor \(F\) from \(\mathcal C\) to \(\mathcal D\) is given by the following data: For each object \(X\) of \(\mathcal C\) an object \(F(X)\) of \(\mathcal D\), and for every morphism \(f\colon X\to Y\) in \(\mathcal C\) a morphism \(F(f)\colon F(Y)\to F(X)\), such that \(F(\operatorname{id}_X) = \operatorname{id}_{F(X)}\) for all \(X\) and such that \(F(f\circ g) = F(g)\circ F(f)\).
In order to distinguish between the two sorts of functors, the first variant is called a covariant functor. A slightly different way to define (and denote) contravariant functor is as follows. Given a category \(\mathcal C\), we define the opposite (or dual) category \(\mathcal C^{\rm opp}\) as follows. It has the same objects as \(\mathcal C\), and for any two objects \(X, Y\), we set
i.e. “all arrows switch direction”. As identity morphisms we use the identity morphisms in \(\mathcal C\). Composition in \(\mathcal C^{\rm opp}\) is defined using the composition in \(\mathcal C\) in the obvious way. Then a contravariant functor from \(\mathcal C\) to \(\mathcal D\) is a (covariant) functor \(\mathcal C^{\rm opp}\to \mathcal D\). In view of this definition, one usually denotes contravariant functors in this way, i.e., as \(\mathcal C^{\rm opp}\to \mathcal D\).
If \(F\) is a functor and \(f\) is an isomorphism, then \(F(f)\) is an isomorphism.
The following properties of functors are often interesting, and we will need them later on.
faithful, if for all objects \(X, Y\in {\mathcal C}\) the map \(\operatorname{Hom}_{{\mathcal C}}(X,Y)\to \operatorname{Hom}_{{\mathcal D}}(F(X), F(Y))\) is injective,
full, if for all objects \(X, Y\in {\mathcal C}\) the map \(\operatorname{Hom}_{{\mathcal C}}(X,Y)\to \operatorname{Hom}_{{\mathcal D}}(F(X), F(Y))\) is surjective,
fully faithful, if it is full and faithful, i.e., if for all objects \(X, Y\in {\mathcal C}\) the map \(\operatorname{Hom}_{{\mathcal C}}(X,Y)\to \operatorname{Hom}_{{\mathcal D}}(F(X), F(Y))\) is bijective,
essentially surjective, if for every object \(Z\in {\mathcal D}\), there exists an object \(X\in {\mathcal C}\) such that \(F(X)\cong Z\) (NB: isomorphism, not necessarily equality!).
Next we define morphisms of functors (also called natural transformations).
Let \(F, G\colon \mathcal C\to \mathcal D\) be functors. A morphism \(\Phi \colon F\to G\) of functors is given by a collection \(\Phi _X\colon F(X)\to G(X)\) of morphisms in \({\mathcal D}\) for every \(X\in {\mathcal C}\), such that for every morphism \(f\colon X\to Y\) in \({\mathcal C}\), the diagram
![\begin{tikzcd}
F(X)\ar[r, "\Phi_X"]\ar[d, "F(f)"] & G(X) \ar[d, "G(f)"]\\
F(Y)\ar[r, "\Phi_Y"] & G(Y)
\end{tikzcd}](images/img-0010.svg)
commutes. (Applying the definition to functors \({\mathcal C}^{\rm opp}\to {\mathcal D}\), one similarly obtinas the notion of morphism between two contravariant functors.)
With this notion of morphism, together with the obvious identity morphisms and composition of morphisms of functors, the collection of all functors between fixed categories \({\mathcal C}\), \({\mathcal D}\) is itself a category, the functor category (however, the collection of all morphisms between two functors might not be a set). In particular, we also obtain the notion of isomorphism between two functors \({\mathcal C}\to {\mathcal D}\).
Nov. 19, 2025
Functors are the natural “morphisms” between categories. In fact, we can define the category of all categories, where functors are the morphisms (again the collections of morphisms in this category are not necessarily sets). Note that we have obvious identity functors and can form the composition of functors. (Since we also defined morphisms between functors, there is, so to say, another level to the story in this case; this is formalized by the notion of 2-category, but we will not have to go into this.) In particular, we obtain the notion of isomorphism between categories. However, it turns out that isomorphisms of categories are rather rare. A much more useful notion is the following weaker one.The functor \(F\) has a quasi-inverse \(G\) (i.e., \(G\) is a functor \({\mathcal D}\to {\mathcal C}\) such that \(G\circ F\cong \operatorname{id}_{{\mathcal C}}\), \(F\circ G\cong \operatorname{id}_{{\mathcal D}}\); it is crucial here that we only ask for isomorphisms, not equality, of these functors!).
The functor \(F\) is fully faithful and essentially surjective.
\(\operatorname{Spec}\colon (\text{Rings})^{\rm opp}\to (\text{Top})\)
forgetful functors
Hom functors
“adjointness tensor-Hom” as example of isomorphism of functors: for every \(X\), have \(\operatorname{Hom}(Y\otimes X, Z) \cong \operatorname{Hom}(Y, \operatorname{Hom}(X, Z))\) functorially in \(Y\) and \(Z\).
dual vector space, morphism to double dual
localization of a ring, base change (of modules or rings)
\(GL_n(-)\), \(\det \colon GL_n(-)\to GL_1(-)\).
Let \(X\) be a topological space, and define the category \({\rm Ouv}(X)\) as follows. The objects of \({\rm Ouv}(X)\) are the open subsets of \(X\). For open subsets \(U, V \subseteq X\), we set
Here \(\{ * \} \) denotes a set with one element. There is then a unique way to define identity morphisms and composition, and one obtains a category.
With this definition, a presheaf of sets on \(X\) is the same as a functor \({\rm Ouv}(X)^{\rm opp}\to (\text{Sets})\). A morphism of presheaves is the same as a morphism of the corresponding functors. With this interpretation we in particular obtain a natural notion of presheaf on \(X\) with values in any category \({\mathcal C}\) (namely a functor \({\rm Ouv}(X)^{\rm opp}\to {\mathcal C}\)) and of morphisms between such presheaves (namely a morphism of the functors).
We can now define the structure sheaf on the spectrum of a ring. So fix a ring \(R\) and let \(X=\operatorname{Spec}(R)\). We want to define a (“natural”) sheaf of rings on \(X\). As we have seen above, it is enough to define a sheaf (of rings) on the basis of the topology given by the principal opens, and we want to set \({\mathscr O}_X(D(f)) = R_f\).
The first step now is to check that this is well-defined (note that we may have \(D(f)=D(g)\) for \(f\ne g\)).
For \(f,g\in R\), we have \(D(f) \subseteq D(g)\) if and only if \(\frac g1 \in R_f\) is a unit, i.e., \(\frac g1\in R_f^\times \). In this case, we obtain a commutative diagram
![\begin{tikzcd}
R_g\ar[rr] & & R_f\\
& R\ar[lu]\ar[ru]
\end{tikzcd}](images/img-0011.svg)
of ring homomorphisms (where \(R\to R_f\) and \(R\to R_g\) are the natural maps into the localizations).
If \(f, g\in R\) satisfy \(D(f) = D(g)\), then there is a unique isomorphism \(R_f\cong R_g\) of \(R\)-algebras.
(1) We have
and this condition is equivalent to \(\frac g1\in R_f^\times \). In fact, if \(f^n = gh\), then \(\frac{g}{1}\cdot \frac{h}{f^n} = 1\) in \(R_f\). Conversely, if \(\frac{g}{1}\cdot \frac{h'}{f^{n'}} = 1\) in \(R_f\), then there is \(m\) such that \(gh'f^m = f^{n'+m}\).
The existence of the \(R\)-algebra homomorphism \(R_g\to R_f\) then follows from properties of the localization of a ring (and in fact is also equivalent to the condition that \(g\) maps to a unit in \(R_f\)). Furthermore, this homomorphism is the unique \(R\)-algebra homomorphism \(R_f\to R_g\).
(2) follows from (1).
Using Part (2) of the lemma, we see that we can attach to each principal open \(D(f)\) the ring \(R_f\) (well-defined up to unique isomorphism, so we can identify the rings \(R_f\), \(R_g\) for \(D(f)=D(g)\) in a specific way). Alternatively, the lemma implies that we have a unique isomorphism
of \(R\)-algebras, and the right hand side \(S^{-1}R\) only depends on \(D(f)\), not on \(f\).
Nov. 25, 2025
The condition \(\bigcup _{i\in I} D(f_i) = \operatorname{Spec}(R)\) is equivalent to saying that the ideal \((f_i;\ i\in I)\) is not contained in any prime ideal, but then the quotient \(R/(f_i;\ i\in I)\) cannot have a maximal ideal, so is the zero ring.
Note that the lemma proves that \(\operatorname{Spec}(R)\) is quasi-compact. It also shows that whenever \(f_1,\dots , f_r\in R\) generate the unit ideal, then for every \(N\ge 1\), also \(f_1^N, \dots , f_r^N\) generate the unit ideal.
We denote the sheaf on \(X\) that we obtain from \({\mathscr O}_X'\) by Proposition 3.9 by \({\mathscr O}_X\) and call it the structure sheaf on \(X\).
We need to show: For all \(f, f_i\in R\) such that \(D(f) = \bigcup _{i\in I} D(f_i)\), the sequence
where the maps are \(\rho (s)= \left( \frac s1 \right)_i\) (with the \(i\)-th entry in \(R_{f_i}\)), and \(\sigma ((s_i)_i) = \left( \frac{s_i}{1}-\frac{s_j}{1} \right)_{i,j}\), is exact.
We first do the following reduction steps:
Replacing \(R\) by \(R_f\), we may assume that \(f=1\), and hence that \(R_f=R\).
Since all principal open subsets are quasi-compact, we may assume that the index set \(I\) of the open cover if finite. (This requires a small “computation”.)
As the previous lemma shows, the assumption that \(\bigcup _i D(f_i) = \operatorname{Spec}(R)\) is equivalent to saying that the elements \(f_i\) generate the unit ideal in \(R\). This implies that for every \(N\), the powers \(f_i^N\) also generate the unit ideal. We will refer to this property by (*).
Injectivity in the above sequence. Let \(s\in R\) such that the image of \(s\) in each localization \(R_{f_i}\) vanishes. Then for each \(i\) there exists \(N_i\) such that \(f^{N_i}s=0\). Since \(I\) is finite, we find \(N\) with the property that \(f_i^Ns=0\) for all \(i\). Now use (*) to write \(1 = \sum _i g_if_i^N\). We then see that
Exactness “in the middle”: \({\rm Im}(\rho ) = \operatorname{Ker}(\sigma )\). The inclusion \(\subseteq \) is clear (in fact, it holds for any presheaf). So let \((s_i)_i\in \operatorname{Ker}(\sigma )\). We write
(again we use that \(I\) is finite, so that we can find an \(N\) that works for all \(s_i\)).
The assumption that \((s_i)_i\in \operatorname{Ker}(\sigma )\) means that all the differences \(\frac{s_i}{1}-\frac{s_j}{1}\in R_{f_if_j}\) vanish, so we find \(M\ge 0\) such that
Now we use (*) to write \(1 = \sum g_i f_i^{M+N}\) (these are other \(g_i\)’s than above).
Define \(a = \sum _j g_j f_j^M a_j\). We will check that \(\rho (a) = (s_i)_i\). For this we need to prove that \(\frac{a}{1} - \frac{a_i}{f_i^N} = 0\in R_{f_i}\) for all \(i\). But from the definition of \(a\) it follows that
and that implies the result.
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A. Knapp, Elliptic Curves, Princeton Univ. Press 1992.
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