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.
Often the following construction is useful.
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 obtains 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 3.10 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.
References: For foundational material on the notion of colimit, see [ GW1 ] Appendix A (and the problem sheets).
Let \({\mathscr F}\) be a presheaf on a topological space \(X\), and let \(x\). The “stalk” of the sheaf is the collection of all sections defined on some (possibly very small) open neighborhood of \(x\), in the following precise sense.
For presheaves with values in some category \({\mathcal C}\) (rather than sets), we would take the colimit above in the category \({\mathcal C}\) (assuming that it exists). However, the following argument shows that for the categories that we will be concerned with in this class, this distinction is not important.
The index set of the colimit is filtered (for \(U\), \(V\) open neighborhoods of \(x\), we have \(U\cap V \subseteq U, V\), so \(U, V \le U\cap V\), and \(U\cap V\) is again an open neighborhood of \(x\)). Therefore, if \({\mathscr F}\) is a presheaf of groups / abelian groups / rings, then the stalk of \({\mathscr F}\) at a point \(x\) in the category of sets (i.e., as constructed above) has a natural structure of group / abelian group / ring, etc., and this gives the colimit in the category of groups / etc. For every open neighborhood \(U\) of \(x\), we have a natural map \({\mathscr F}(U)\to {\mathscr F}_x\); if \({\mathscr F}\) is a presheaf of groups (etc.), then this is a group homomorphism (etc.).
If \({\mathscr F}\to {\mathscr G}\) is a morphism of sheaves, for all open neighborhoods \(U \subseteq V\) of \(x\) we have a commutative diagram
![\begin{tikzcd}
\Fscr(V) \ar[d]\ar[r]\ar[dd, bend right=50] & \Gscr(V)\ar[d]\ar[dd, bend left=50]\\
\Fscr(U) \ar[d]\ar[r] & \Gscr(U)\ar[d]\\
\Fscr_x & \Gscr_x.
\end{tikzcd}](images/img-0012.svg)
Here the vertical maps are the restriction map and the natural maps to the stalk.
These diagrams induce a morphism \({\mathscr F}_x\to {\mathscr G}_x\) between the stalks. This shows that the construction of stalks is a functor from the category of presheaves (of sets) to the category of sets. Likewise, we obtain functors from the category of presheaves of abelian groups to the category of abelian groups, and similarly for presheaves of groups, rings, etc.
In the theory of holomorphic functions (complex analysis in one or several variables), the stalk of the structure sheaf can be interpreted in terms of convergent power series as follows. Let \(X = \mathbb {C}\) (or more generally an open subset of \(\mathbb {C}^n\), \(n\ge 1\), or even more generally any complex manifold), and define the “structure sheaf” \({\mathscr O}_X\) by setting
Then the stalk \({\mathscr O}_{X,x}\) can be identified with the ring of convergent power series at \(x\) (i.e., power series in \(n\) variables that converge in some open neighborhood of \(x\)). The identity theorem can be phrased as saying that for every connected open neighborhood \(U\) of \(x\), the natural map \({\mathscr O}_X(U)\to {\mathscr O}_{X,x}\) is injective.
Nov. 26, 2025
Recall that for the computation of the limit of a convergent sequence of real numbers, we may pass to a subsequence, i.e., we may replace the index set \(\mathbb {N}\) of the limit by any infinite subset of \(\mathbb {N}\). Something similar holds for colimits:
Let \((I, \le )\) be a partially ordered set. We call a subset \(I' \subseteq I\) cofinal, if for every \(i\in I\) there exists \(i'\in I'\) with \(i \le i'\). We equip \(I'\) with the induced partial order. Then for every inductive system \((F_i)_{i\in I}\), we have a natural isomorphism
Let \(R\) be a ring, \(X = \operatorname{Spec}(R)\). Let \({\mathfrak p}\in \operatorname{Spec}(R)\). Let us compute the stalk of the structure sheaf \({\mathscr O}_X\) at the point \({\mathfrak p}\). By the previous remark, we may compute the stalk as
The latter colimit is isomorphic to the localization \(R_{{\mathfrak p}}\) of \(R\) with respect to \({\mathfrak p}\) (i.e., the localization with respect to the open subset \(R\setminus {\mathfrak p}\)). In fact, the universal property of the colimit gives us a ring homomorphism \(\mathop{\rm colim}\limits _{f\in R,\ {\mathfrak p}\in D(f)} R_f\to R_{{\mathfrak p}}\) (here we use that \({\mathfrak p}\in D(f)\) by definition is equivalent to \(f\not \in {\mathfrak p}\), and in this case \(\frac f1\) is a unit in \(R_{{\mathfrak p}}\)).
It is easy to see that this map is surjective. For the injectivity, we need to show that for all \(s, f\in R\), \({\mathfrak p}\in D(f)\) and \(i\ge 0\), \(\frac{s}{f^i} = 0\) in \(R_{{\mathfrak p}}\) implies that \(\frac{s}{f^i} = 0\) in some localization \(R_{fg}\) with \({\mathfrak p}\in D(g)\). This also follows immediately from properties of the localization.
The following propositions illustrate in which sense the stalks capture “local information about a sheaf”.
Let \(s\in {\mathscr F}(U)\) be an element which maps to \(0\) in every stalk \({\mathscr F}_x\), \(x\in U\). By definition of the stalk, thus for every \(x\), there exists an open neighborhood \(V_x \subseteq U\) of \(x\) such that \(s_{V_x} = 0\) (in \({\mathscr F}(V_x)\)). But then clearly all the \(V_x\) cover \(U\), and the sheaf axioms imply that \(s=0\).
The following are equivalent:
For every open \(U \subseteq X\), the map \({\mathscr F}(U)\to {\mathscr G}(U)\) is injective.
For every \(x\in X\), the map \({\mathscr F}_x\to {\mathscr G}_x\) is injective.
The following are equivalent:
For every open \(U \subseteq X\), the map \({\mathscr F}(U)\to {\mathscr G}(U)\) is bijective.
The morphism \({\mathscr F}\to {\mathscr G}\) is an isomorphism.
For every \(x\in X\), the map \({\mathscr F}_x\to {\mathscr G}_x\) is bijective.
(1) The implication (ii) \(\Rightarrow \) (i) follows from the previous lemma. For the implication (i) \(\Rightarrow \) (ii), let \(x\in X\), and \(s\in {\mathscr F}_x\) mapping to \(0\) in \({\mathscr G}_x\). Let \(\dot{s}\in {\mathscr F}(V)\) be a representative of \(s\), i.e., an element with image \(s\), where \(V\) is a suitable open neighborhood of \(x\). The image of \(\dot{s}\) in \({\mathscr G}(V)\) maps to \(0\) in the stalk at \(x\), hence its restriction to a suitable open neighborhood \(U\) of \(x\) is \(0\) (in \({\mathscr G}(U)\)). But then the injectivity in (i) implies that \(\dot{s}_{|U} = 0\), and a fortiori \(s = 0\).
(2) See Problem sheet 6.
Note that the analogous statement to (1) for surjective maps is not true! (Cf. Problem sheet 7.) It turns out that the stalks provide the correct perspective on these properties, and we make the following definition.
The morphisms \(\varphi \), \(\psi \) are equal.
For every \(x\in X\), the induced morphisms \(\varphi _x, \psi _x\colon {\mathscr F}_x\to {\mathscr G}_x\) between the stalks are equal.
It is clear that (i) implies (ii). The converse follows from Lemma 3.25
Dec. 2, 2025
We now study a “natural way” of attaching to an arbitrary presheaf a sheaf that is “as close as possible” to the given presheaf (in particular, both have the same stalk at each point of the underlying space). This sheaf will be called the sheafification (German: Garbifizierung) of the given presheaf. We will define it by a universal property, as follows:A sheaf \(\widetilde{{\mathscr F}}\) together with a morphism \(\iota _{{\mathscr F}}\colon {\mathscr F}\to \widetilde{{\mathscr F}}\) of presheaves is called a sheafification of \({\mathscr F}\), if for every morphism \({\mathscr F}\to {\mathscr G}\) from \({\mathscr F}\) to a sheaf \({\mathscr G}\), there exists a unique morphism \(\widetilde{{\mathscr F}}\to {\mathscr G}\) such that the diagram
![\begin{tikzcd}
\Fscr\ar[rr]\ar[dr] & & \Gscr\\
& \widetilde{\Fscr} \ar[ur]
\end{tikzcd}](images/img-0013.svg)
commutes.
A sheafification of the presheaf \({\mathscr F}\) exists. If \({\mathscr F}\) is a presheaf of groups, abelian groups, rings, …, then so is the sheafification.
The morphism \({\mathscr F}\to \widetilde{{\mathscr F}}\) induces an isomorphism on the stalks for each \(x\in X\).
For every morphism \({\mathscr F}\to {\mathscr G}\) we obtain a morphism \(\widetilde{{\mathscr F}}\to \widetilde{{\mathscr G}}\), so that the diagram
![\begin{tikzcd}
\Fscr\ar[r]\ar[d] & \Gscr\ar[d]\\
\widetilde{\Fscr}\ar[r] & \widetilde{\Gscr}
\end{tikzcd}](images/img-0014.svg)
commutes. Thus sheafification defines a functor from the category of presheaves on \(X\) to the category of sheaves on \(X\).
It follows immediately from the definition that the sheafification is uniquely determined up to isomorphism, and that the sheafification of a sheaf \({\mathscr F}\) is simply the identity morphism.
We need to prove the existence of the sheafification. We may construct it explicitly as follows. For \(U \subseteq X\) open, we set
(This is a natural candidate in view of Lemma 3.25.) The restriction maps are defined as the obvious projection maps. One checks that this defines a sheaf. From this points, the rest of the proof is not difficult, but doing things in the right order saves some work. We have a morphism \({\mathscr F}\to \widetilde{{\mathscr F}}\) of presheaves by defining
where \(s_x\) denotes the image of \(s\) in the stalk \({\mathscr F}_x\). At this point one checks that this induces an isomorphism on all stalks (thus proving Part (3)).
For \(\varphi \colon {\mathscr F}\to {\mathscr G}\) a morphism of presheaves we obtain a morphism \(\widetilde{{\mathscr F}}\to \widetilde{{\mathscr G}}\) by setting
By Part (3) and Proposition 3.28, this is the unique morphism which makes the diagram in (4) commutative. Applying this in the special case where the target \({\mathscr G}\) of the morphism \({\mathscr F}\to {\mathscr G}\) is a sheaf, shows the universal property, because in that case \({\mathscr G}= \widetilde{{\mathscr G}}\) (more precisely, the morphism \({\mathscr G}\to \widetilde{{\mathscr G}}\) is an isomorphism, by Proposition 3.26.
A typical use of the sheafification is to do certain constructions of sheaves that can be naturally carried out for presheaves, but may not themselves yield sheaves (even if one starts with sheaves, so to say), as in the following example.
Let \(f\colon {\mathscr F}\to {\mathscr G}\) be a morphism of sheaves. The image sheaf \(\mathop{\rm Im}(f)\) is defined as the sheafification of the presheaf
(Note that this presheaf usually is not a sheaf!)
The image sheaf comes with a natural injective sheaf morphism \(\mathop{\rm Im}(f)\to {\mathscr G}\). This is an isomorphism if and only of \(f\) is surjective.
We can express the universal property of the sheafification by saying that, for every presheaf \({\mathscr F}\) and sheaf \({\mathscr G}\), there are bijections
(given by composition with the natural map \({\mathscr F}\to \widetilde{{\mathscr F}}\); these maps are functorial in \({\mathscr F}\) and in \({\mathscr G}\)). On the right, we “view \({\mathscr G}\) as a presheaf” (forgetting that it is a sheaf), and have \(\operatorname{Hom}\) in the category of presheaves. On the left, we have \(\operatorname{Hom}\) in the category of sheaves, because source and target are sheaves. (Since morphisms of sheaves by definition are just morphisms of presheaves, this may sound overly pedantic, but at this point it is useful to distinguish between the two categories.)
We can thus express this by saying that the sheafification functor is left adjoint to the inclusion functor from the category of sheaves into the category of presheaves.
Let \(f\colon X\to Y\) be a continuous map between topological spaces. We want to think about how we could “transport” sheaves from \(X\) to \(Y\) and vice versa.
Given a presheaf \({\mathscr F}\) on \(X\), it is fairly straighforward to define its “pushforward” or “direct image” in \(Y\).
It is easy to check that for a sheaf \({\mathscr F}\), the direct image \(f_*{\mathscr F}\) is again a sheaf. While this construction is fairly simple, note that it is usually not easy to express the stalks of \(f_*{\mathscr F}\) in terms of the stalks of \({\mathscr F}\).
If \({\mathscr F}\to {\mathscr F}'\) is a morphism of sheaves on \(X\), then we obtain a morphism \(f_*{\mathscr F}\to f_*{\mathscr F}'\) in the obvious way. It is easy to check that \(f_*\) defines a functor from the category of sheaves on \(X\) to the category of sheaves on \(Y\). Similarly for sheaves of groups, abelian groups, rings, etc.
If \(f\colon X\to Y\), \(g\colon Y\to Z\) are continuous maps, and \({\mathscr F}\) is a presheaf on \(X\), then \((g\circ f)_*{\mathscr F}= g_*f_*{\mathscr F}\).
Going in the other direction is a little more cumbersome, but with the sheafification we have all the necessary tools at our disposal. We will proceed in two steps and first define a presheaf.
Even if \({\mathscr G}\) is a sheaf, the presheaf \(f^+{\mathscr G}\) usually is not a sheaf, and we define
The construction \({\mathscr G}\mapsto f^+{\mathscr G}\) is functorial in \({\mathscr G}\). Since sheafification is also functorial in \({\mathscr G}\), we see that the inverse image \(f^{-1}\) is a functor from the category of presheaves on \(Y\) to the category of sheaves on \(X\). If \({\mathscr G}\) is a presheaf of (abelian) groups, rings, etc., then so are \(f^+{\mathscr G}\) and \(f^{-1}{\mathscr G}\).
It is not difficult to compute the stalks of the inverse image presheaf:
We have
For the final equality we use that \(f\) is continuous and that therefore every open neighborhood \(V \subseteq Y\) of \(f(x)\) contains \(f(U)\) for some \(U\) (e.g., \(U=f^{-1}(V)\)).
From the computation of the stalks, we obtain the compatibility with composition of morphisms in the following sense.
One first checks that \((g\circ f)^{+}{\mathscr H}\cong f^{+}(g^{+}{\mathscr H})\). Passing to the sheafification, this implies that \((g\circ f)^{-1}{\mathscr H}\cong f^{-1}(g^{+}{\mathscr H})\). On the other hand, (the proof of) Proposition 3.37 implies that \(f^{+}(g^{+}{\mathscr H})\cong f^{-1}(g^{+}{\mathscr H})\), because the natural morphism between these sheaves induces isomorphisms on all stalks.
Finally we note the following useful relation between the functors \(f_*\) and \(f^{-1}\).
In the situation of the proposition, of \({\mathscr F}\) is a sheaf of abelian groups and \({\mathscr G}\) is a presheaf of abelian groups, then the \(\operatorname{Hom}\) sets above are abelian groups and the adjunction isomorphisms are group isomorphisms.
First note that since \({\mathscr F}\) is a sheaf, we may identify \(\operatorname{Hom}_{{\rm Sh}(X)}(f^{-1}{\mathscr G}, {\mathscr F})\) with \(\operatorname{Hom}_{{\rm PreSh}(X)}(f^{+}{\mathscr G}, {\mathscr F})\). It is then not difficult to explicitly construct natural maps in both directions and to check that they are inverse to each other and functorial.
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