Grothendieck topology

In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category C which makes the objects of C act like the open sets of a topological space. A category together with a choice of Grothendieck topology is called a site.

Grothendieck topologies axiomatize the notion of an open cover. Using the notion of covering provided by a Grothendieck topology, it becomes possible to define sheaves on a category and their cohomology. This was first done in algebraic geometry and algebraic number theory by Alexander Grothendieck to define the étale cohomology of a scheme. It has been used to define other cohomology theories since then, such as l-adic cohomology, flat cohomology, and crystalline cohomology. While Grothendieck topologies are most often used to define cohomology theories, they have found other applications as well, such as to John Tate's theory of rigid analytic geometry.

There is a natural way to associate a site to an ordinary topological space, and Grothendieck's theory is loosely regarded as a generalization of classical topology. Under meager point-set hypotheses, namely sobriety, this is completely accurate—it is possible to recover a sober space from its associated site. However simple examples such as the indiscrete topological space show that not all topological spaces can be expressed using Grothendieck topologies. Conversely, there are Grothendieck topologies which do not come from topological spaces.

Introduction

André Weil's famous Weil conjectures proposed that certain properties of equations with integral coefficients should be understood as geometric properties of the algebraic variety that they defined. His conjectures postulated that there should be a cohomology theory of algebraic varieties which gave number-theoretic information about their defining equations. This cohomology theory was known as the "Weil cohomology", but using the tools he had available, Weil was unable to construct it.

In the early 1960s, Alexander Grothendieck introduced étale maps into algebraic geometry as algebraic analogues of local analytic isomorphisms in analytic geometry. He used étale coverings to define an algebraic analogue of the fundamental group of a topological space. Soon Jean-Pierre Serre noticed that some properties of étale coverings mimicked those of open immersions, and that consequently it was possible to make constructions which imitated the cohomology functor H¹. Grothendieck saw that it would be possible to use Serre's idea to define a cohomology theory which he suspected would be the Weil cohomology. To define this cohomology theory, Grothendieck needed to replace the usual, topological notion of an open covering with one that would use étale coverings instead. Grothendieck also saw how to phrase the definition of covering abstractly; this is where the definition of a Grothendieck topology comes from.

Definition

Motivation

The classical definition of a sheaf begins with a topological space X. A sheaf associates information to the open sets of X. This information can be phrased abstractly by letting O(X) be the category whose objects are the open subsets U of X and whose morphisms are the inclusion maps U → V of open sets U and V of X. We will call such maps open immersions, just as in the context of schemes. Then a presheaf on X is a contravariant functor from O(X) to the category of sets, and a sheaf is a presheaf which satisfies the gluing axiom. The gluing axiom is phrased in terms of pointwise covering, i.e., {U_i} covers U if and only if $cup$_i U_i = U. A Grothendieck topology encodes the information about collections of open subsets of U and which collections cover U without any reference to the space itself.

Complete collections of opens in a space

If X is a topological space, and U is an open subset, then a collection, s, of open subsets of U can be called "complete" if it is closed under inclusion, i.e., if V $in$ s and W $subseteq$ V then W $in$ s. (The term "complete" is not standard and is simply used to help the explanation in this article.) Notice that given an arbitrary collection of open subsets of U, "completing" it by adding in subsets of its elements has no effect on the union of the collection, and specifically, whether it covers U.

Sieves

In a Grothendieck topology, the notion of a complete collection of open subsets of U is replaced by the notion of a sieve. If c is any given object in C, a sieve on c is a subfunctor of the functor Hom(−, c); (this is the Yoneda embedding applied to c). In the case of O(X), a sieve S on an open set U corresponds to a complete collection of open subsets of U "selected" by S. More precisely, consider that for any open subset V of U, S(V) will be a subset of Hom(V, U), which has only one element, the open immersion V → U. Then V will be considered "selected" by S if and only if S(V) is nonempty.

If S is a sieve on X, and f: Y → X is a morphism, then left composition by f gives a sieve on Y called the pullback of S along f, denoted by f^{$^ast$}S. It is defined as the fibered product S ×_{Hom(−, X)} Hom(−, Y) together with its natural embedding in Hom(−, Y). More concretely, for each object Z of C, f^{$^ast$}S(Z) = { g: Z → Y | fg $in$S(Z) }, and f^{$^ast$}S inherits its action on morphisms by being a subfunctor of Hom(−, Y). In the classical example, the pullback of a collection {V_i} of subsets of of U along an inclusion W → U is the collection {V_i∩W}.

Covering Sieves

A classical topology on a set X is a collection of distinguished subsets, called open sets. This selection is subject to certain conditions: the axioms of a topological space. By comparison, a Grothendieck topology J on a category C is a collection, for each object c of C, of distinguished sieves on c, called covering sieves of c and denoted by J(c). This selection will be subject to certain axioms, stated below. Continuing the previous example, a sieve S on an open set U in O(X) will be a covering sieve if and only if the union of all the open sets V for which S(V) is nonempty equals U; in other words, if and only if S gives us a collection of open sets which cover U in the classical sense.

Axioms

The conditions we impose on a Grothendieck topology are:

• (T 1) (Base change) If S is a covering sieve on X, and f: Y → X is a morphism, then the pullback f^{$^ast$}S is a covering sieve on Y.
• (T 2) (Local character) Let S be a covering sieve on X, and let T be any sieve on X. Suppose that for each object Y of C and each arrow f: Y → X in S(Y), the pullback sieve f^{$^ast$}T is a covering sieve on Y. Then T is a covering sieve on X.
• (T 3) (Identity) Hom(−, X) is a covering sieve on X for any object X in C.

The base change axiom corresponds to the idea that if {$U\_i$} covers U, then {U_i ∩ V} should cover U ∩ V. The local character axiom corresponds to the idea that if {U_i} covers U and {V_ij}_{j $in$Ji} covers U_i for each i, then the collection {V_ij} for all i and j should cover U. Lastly, the identity axiom corresponds to the idea that any set is covered by all its possible subsets.

Alternative Axioms

In fact, it is possible to put these axioms in another form where their geometric character is more apparent, assuming that the underlying category C contains certain fibered products. In this case, instead of specifying sieves, we can specify that certain collections of maps with a common codomain should cover their codomain. These collections are called covering families. If the collection of all covering families satisfies certain axioms, then we say that they form a Grothendieck pretopology. These axioms are:

• (PT 0) (Existence of fibered products) For all objects X of C, and for all morphisms X₀ → X which appear in some covering family of X, and for all morphisms Y → X, the fibered product X₀ ×_X Y exists.
• (PT 1) (Stability under base change) For all objects X of C, all morphisms Y → X, and all covering families {X_α → X}, the family {X_α ×_X Y → Y} is a covering family.
• (PT 2) (Local character) If {X_α → X} is a covering family, and if for all α, {X_βα → X_α} is a covering family, then the family of composites {X_βα → X_α → X} is a covering family.
• (PT 3) (Isomorphisms) If f: Y → X is an isomorphism, then {f} is a covering family.

For any pretopology, the collection of all sieves that contain a covering family from the pretopology is always a Grothendieck topology.

For categories with fibered products, there is a converse. Given a collection of arrows {X_α → X}, we construct a sieve S by letting S(Y) be the set of all morphisms Y → X that factor through some arrow X_α → X. This is called the sieve generated by {X_α → X}. Now choose a topology. Say that {X_α → X} is a covering family if and only if the sieve that it generates is a covering sieve for the given topology. It is easy to check that this defines a pretopology.

(PT 3) is sometimes replaced by a weaker axiom:

• (PT 3') (Identity) If 1_X : X → X is the identity arrow, then {1_X} is a covering family.

(PT 3) implies (PT 3'), but not conversely. However, suppose that we have a collection of covering families that satisfies (PT 0) through (PT 2) and (PT 3'), but not (PT 3). These families generate a pretopology. The topology generated by the original collection of covering families is then the same as the topology generated by the pretopology, because the sieve generated by an isomorphism Y → X is Hom(−, X). Consequently, if we restrict our attention to topologies, (PT 3) and (PT 3') are equivalent.

Sites and sheaves

Let C be a category and let J be a Grothendieck topology on C. The pair (C, J) is called a site.

A presheaf on a category is a contravariant functor from C to the category of all sets. Note that for this definition C is not required to have a topology. A sheaf on a site, however, should allow gluing, just like sheaves in classical topology. Consequently, we define a sheaf on a site to be a presheaf F such that for all objects X and all covering sieves S on X, the natural map Hom(Hom(−, X), F) → Hom(S, F) induced by the inclusion of S into Hom(−, X) is a bijection. Halfway in between a presheaf and a sheaf is the notion of a separated presheaf, where the natural map above is required to be only an injection, not a bijection, for all sieves S.

Using the Yoneda lemma, it is possible to show that a presheaf on the category O(X) is a sheaf on the topology defined above if and only if it is a sheaf in the classical sense.

Sheaves on a pretopology have a particularly simple description: For each covering family {X_α → X}, the diagram

$F(X)\; rightarrow\; prod\_\{alphain\; A\}\; F(X\_alpha)\; prod\_\{alpha,beta\; in\; A\}\; F(X\_alphatimes\_X\; X\_beta)$

must be an equalizer. For a separated presheaf, the first arrow need only be injective.

Similarly, one can define presheaves and sheaves of abelian groups, rings, modules, and so on. One can require either that a presheaf F is a contravariant functor to the category of abelian groups (or rings, or modules, etc.), or that F be an abelian group (ring, module, etc.) object in the category of all contravariant functors from C to the category of sets. These two definitions are equivalent.

Examples

The discrete and indiscrete topologies

Let C be any category. To define the discrete topology, we declare all sieves to be covering sieves. If C has all fibered products, this is equivalent to declaring all families to be covering families. To define the indiscrete topology, we declare only the sieves of the form Hom(−, X) to be covering sieves. The indiscrete topology is also known as the biggest or chaotic topology, and it is generated by the pretopology which has only isomorphisms for covering families. A sheaf on the indiscrete site is the same thing as a presheaf.

The canonical topology

Let C be any category. The Yoneda embedding gives a functor Hom(−, X) for each object X of C. The canonical topology is the biggest topology such that every representable presheaf Hom(−, X) is a sheaf. A covering sieve or covering family for this site is said to be strictly universally epimorphic. A topology which is less fine than the canonical topology, that is, for which every covering sieve is strictly universally epimorphic, is called subcanonical. Subcanonical sites are exactly the sites for which every presheaf of the form Hom(−, X) is a sheaf. Most sites encountered in practice are subcanonical.

Small site associated to a topological space

We repeat the example which we began with above. Let X be a topological space. We defined O(X) to be the category whose objects are the open sets of X and whose morphisms are inclusions of open sets. The covering sieves on an object U of O(X) were those sieves S satisfying the following condition:

• If W is the union of all the sets V such that S(V) is non-empty, then W = U.

This topology can also naturally be expressed as a pretopology. We say that a family of inclusions {V_α $sube$ U} is a covering family if and only if the union $cup$V_α equals U. This site is called the 'small site associated to a topological space X''.

Big site associated to a topological space

Let Spc be the category of all topological spaces. Given any family of functions {u_α : V_α → X}, we say that it is a surjective family or that the morphisms u_α are jointly surjective if $cup$ u_α(V_α) equals X. We define a pretopology on Spc by taking the covering families to be surjective families all of whose members are open immersions. Let S be a sieve on Spc. S is a covering sieve for this topology if and only if:

• For all Y and every morphism f : Y → X in S(Y), there exists a V and a g : V → X such that g is an open immersion, g is in S(V), and f factors through g.
• If W is the union of all the sets f(Y), where f : Y → X is in S(Y), then W = X.

Fix a topological space X. Consider the comma category Spc/X of topological spaces with a fixed continuous map to X. The topology on Spc induces a topology on Spc/X. The covering sieves and covering families are almost exactly the same; the only difference is that now all the maps involved commute with the fixed maps to X. This is the big site associated to a topological space X . Notice that Spc is the big site associated to the one point space. This site was first considered by Jean Giraud.

The big and small sites of a manifold

Let M be a manifold. M has a category of open sets O(M) because it is a topological space, and it gets a topology as in the above example. For two open sets U and V of M, the fiber product U ×_M V is the open set U ∩ V, which is still in O(M). This means that the topology on O(M) is defined by a pretopology, the same pretopology as before.

Let Mfd be the category of all manifolds and continuous maps. (Or smooth manifolds and smooth maps, or real analytic manifolds and analytic maps, etc.) Mfd is a subcategory of Spc, and open immersions are continuous (or smooth, or analytic, etc.), so Mfd inherits a topology from Spc. This lets us construct the big site of the manifold M as the site Mfd/M. We can also define this topology using the same pretopology we used above. Notice that to satisfy (PT 0), we need to check that for any continuous map of manifolds X → Y and any open subset U of Y, the fibered product U ×_Y X is in Mfd/M. This is just the statement that the preimage of an open set is open. Notice, however, that not all fibered products exist in Mfd because the preimage of a smooth map at a critical value need not be a manifold.

Topologies and schemes

Fix a scheme X. There is more than one natural site associated to X. All of the following sites are subcanonical, and they are ordered from coarsest to finest.

The big and small Zariski sites

All schemes are topological spaces. We get the small Zariski site of X by considering X as a topological space and looking at the site O(X). To define the big Zariski site, let Zar be the category whose objects are schemes and whose morphisms are morphisms of schemes. We define a pretopology on Zar by taking the covering families to be surjective families of scheme-theoretic open immersions. This defines a topology whose covering sieves S are characterized by the following two properties:

• For all Y and every morphism f : Y → X in S(Y), there exists a V and a g : V → X such that g is an open immersion, g is in S(V), and f factors through g.
• If W is the union of all the sets f(Y), where f : Y → X is in S(Y), then W = X.

Despite their outward similarities, the topology on Zar is not the restriction of the topology on Spc! This is because there are morphisms of schemes which are topologically open immersions but which are not scheme-theoretic open immersions. For example, let A be a non-reduced ring and let N be its ideal of nilpotents. The quotient map A → A/N induces a map Spec A/N → Spec A which is the identity on underlying topological spaces. To be a scheme-theoretic open immersion it must also induce an isomorphism on structure sheaves, which this map does not do. In fact, this map is a closed immersion.

We call Zar/X the big Zariski site of X .

The big and small Nisnevich sites

A family of morphisms {u_α : X_α → X} is a Nisnevich cover if the family is jointly surjective and each u_α is an étale morphism with the following additional property: For every point x ∈ X, there exists an α and a point u ∈ X_α such that the induced map of residue fields k(x) → k(u) is an isomorphism. (We call such an étale morphism with this property a Nisnevich morphism.) We define a pretopology on the category of schemes and morphisms of schemes by declaring covering families to be exactly the Nisnevich covers. This generates a topology called the Nisnevich topology. We write Nis for the category of schemes with the Nisnevich topology.

The small Nisnevich site of X is the category O(X_Nis) whose objects are schemes U with a fixed Nisnevich morphism U → X. The morphisms are morphisms of schemes compatible with the fixed maps to X. The big Nisnevich site of X is the category Nis/X, that is, the category of schemes with a fixed map to X, considered with the Nisnevich topology.

Nisnevich called this topology the completely decomposed topology. He introduced it in order to provide a cohomological interpretation of the class set of an affine group scheme (originally defined in adelic terms) and used it to partially prove the Grothendieck-Serre conjecture on rationally trivial torsors. This topology has also found important applications in algebraic K-theory, A¹ homotopy theory and the theory of motives.

The big and small étale sites

We say that a family of morphisms {u_α : X_α → X} is an étale cover if the family is jointly surjective and each u_α is an étale morphism. We define a pretopology on the category of schemes and morphisms of schemes by declaring covering families to be exactly the étale covers. This generates a topology called the étale topology. We write Ét for the category of schemes with the étale topology.

The small étale site of X is the category O(X_ét) whose objects are schemes U with a fixed étale morphism U → X. The morphisms are morphisms of schemes compatible with the fixed maps to X. The big étale site of X is the category Ét/X, that is, the category of schemes with a fixed map to X, considered with the étale topology.

We can define the étale topology using less data. First, we notice that the étale topology is finer than the Zariski topology. Consequently, to define an étale cover of a scheme X, it suffices to first cover X by open affine subschemes, that is, to take a Zariski cover, and then to define an étale cover of an affine scheme. We define an étale cover of an affine scheme X to be a surjective family {u_α : X_α → X} such that the set of all α is finite, each X_α is affine, and each u_α is étale. Then an étale cover of X is a family {u_α : X_α → X} which becomes an étale cover after base changing to any open affine subscheme of X.

Flat sites

Bibliography

• Demazure, Michel; Alexandre Grothendieck, eds. (1970). Séminaire de Géométrie Algébrique du Bois Marie - 1962-64 - Schémas en groupes - (SGA 3) - vol. 1 (Lecture notes in mathematics 151). Berlin; New York: Springer-Verlag.
• Artin, Michael; Alexandre Grothendieck, Jean-Louis Verdier, eds. (1972). Séminaire de Géométrie Algébrique du Bois Marie - 1963-64 - Théorie des topos et cohomologie étale des schémas - (SGA 4) - vol. 1 (Lecture notes in mathematics 269). Berlin; New York: Springer-Verlag.
• Nisnevich, Yevsey A. (1989). "The completely decomposed topology on schemes and associated descent spectral sequences in algebraic K-theory". Algebraic K-theory: connections with geometry and topology. Proceedings of the NATO Advanced Study Institute held in Lake Louise, Alberta, December 7--11, 1987. NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, 279, 241-342. .