Direct Images and Inverse Images of Sheaves

In this post we will be working with the etale topology once again, so we start by formalizing some concepts. We want to first define the category (see Category Theory) called \text{Et}(X) or \text{Et}/X in the literature. The objects of this category are etale morphisms (see Cohomology in Algebraic Geometry) \varphi:U\rightarrow X of schemes to  X, while the morphisms are etale morphisms \psi:U\rightarrow U' such that if \varphi ':U'\rightarrow X is another etale morphism to X, then \varphi' \circ \psi=\varphi.

Presheaves of sets, abelian groups, etc. on the category \text{Et}(X) are defined as contravariant functors from the category \text{Et}(X) to sets, abelian groups, etc. They are sheaves if they satisfy the sheaf conditions commonly referred to as local identity and gluing (see Even More Category Theory: The Elementary Topos). We will refer to presheaves (resp. sheaves) on \text{Et}(X) as etale presheaves (resp. etale sheaves) or simply as presheaves (resp. sheaves) on X.

Let f:X\rightarrow Y be a morphism of schemes, and let \mathcal{F} be a sheaf on X. There is a sheaf on Y determined by the f and \mathcal{F}, called the direct image sheaf, written f_{*}\mathcal{F} and defined by

\displaystyle f_{*}\mathcal{F}(U)=\mathcal{F}(X\times_{Y}U)

for an etale morphism U\rightarrow Y.

The direct image functor f_{*} is the functor that assigns to a sheaf \mathcal{F} the direct image sheaf f_{*}\mathcal{F}. The derived functor (see More on Chain Complexes and The Hom and Tensor Functors) of f_{*} is called the higher direct image functor and written R^{n}f_{*}.

On the other hand, a morphism f:X\rightarrow Y of schemes and a sheaf \mathcal{G} on Y also determine a sheaf on X called the inverse image sheaf, written f^{*}\mathcal{F} and obtained via the following construction:

Let U\rightarrow X be an etale morphism of schemes. The presheaf \mathcal{G}' is given by

\mathcal{G}'(U)=\varinjlim \mathcal{G}(V)

where the direct limit (see Etale Cohomology of Fields and Galois Cohomology) is taken over all V\rightarrow Y such that the morphisms commute (i.e. the composition of morphisms U\rightarrow V and V\rightarrow Y is equal to the composition of morphisms U\rightarrow X and X\rightarrow Y).

We then define the sheaf f^{*}\mathcal{G} as the sheaf associated to the presheaf \mathcal{G}' (the process of associating a sheaf to a presheaf, also known as sheafification, is left to the references for now).

We now introduce the notions of open subschemes and closed subschemes, and open immersions and closed immersions. We quote directly from the book Algebraic Geometry by Robin Hartshorne:

An open subscheme of a scheme X is a scheme U whose topological space is an open subset of X, and whose structure sheaf \mathcal{O}_{U} is isomorphic to the restriction \mathcal{O}_{X|U} of the structure sheaf of X. An open immersion is a morphism f:X\rightarrow Y which induces an isomorphism of X with an open subscheme of Y.

Note that every open subset of a scheme carries a unique structure of open subscheme.

A closed immersion is a morphism f: Y\rightarrow X of schemes such that induces a homeomorphism of \text{sp}(Y) onto a closed subset of \text{sp}(X) and furthermore the induced map f^{\#}:\mathcal{O}_{X}\rightarrow f_{*}\mathcal{O}_{Y} of sheaves on X is surjective. A closed subscheme of a scheme X is an equivalence class of closed immersions, where we say f:Y\rightarrow X and f':Y'\rightarrow X are equivalent if there is an isomorphism i: Y'\rightarrow Y such that f'=f\circ i.

Now it may happen that given an open immersion j:U\rightarrow X and a sheaf \mathcal{F} on U, the stalks of j_{*}\mathcal{F} may not be zero for points outside U. Therefore we define another sheaf j_{!} on X, given by the following construction:

Given a sheaf \mathcal{F} on U, and an etale morphism \varphi:V\rightarrow X, let

\displaystyle \mathcal{F}_{!}(V)=\mathcal{F}(V) if \varphi(V)\subseteq U

\displaystyle \mathcal{F}_{!}(V)=0 if \varphi(V)\nsubseteq U

Once again, \mathcal{F}_{!} is a presheaf on X, but it need not be a sheaf, therefore we define instead j_{!}\mathcal{F} to be the sheaf associated to the presheaf \mathcal{F}_{!}.

One concept related to this “extension by zero” functor j_{!} is cohomology with compact support, written H_{c}^{n}(U,\mathcal{F})=H^{r}(X,j_{!}\mathcal{F}).

The functors f_{*}, f^{*}, and the generalization of j_{!}, called the direct image with compact support and denoted f_{!}, are part of the so-called “six operations” which play an important role in modern algebraic geometry.

References:

Image Functors for Sheaves on Wikipedia

Direct Image Functor on Wikipedia

Inverse Image Functor on Wikipedia

Direct Image with Compact Support on Wikipedia

Six Operations on Wikipedia

Six Operations on the nLab

Lectures on Etale Cohomology by J.S. Milne

Algebraic Geometry by Robin Hartshorne

Etale Cohomology and the Weil Conjecture by Eberhard Freitag and Reinhardt Kiehl

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Etale Cohomology of Fields and Galois Cohomology

In Cohomology in Algebraic Geometry we have introduced sheaf cohomology and Cech cohomology as well as the concept of etale morphisms, and the Grothendieck topology (see More Category Theory: The Grothendieck Topos) that it defines. In this post, we give one important application of these ideas, related to the ideas discussed in Galois Groups.

Let K be a field (see Rings, Fields, and Ideals). A field has only two ideals: (0) and (1), the latter of which is the unit ideal and is therefore the entire field itself as well. Its only prime ideal (which is also a maximal ideal) is (0); recall that in algebraic geometry (see Basics of Algebraic Geometry), the “points” of the mathematical object we call a scheme correspond (locally, at least) to the prime ideals of a ring R, and we refer to this set of “points” as \text{Spec }R. Therefore, for the field K, \text{Spec }K=(0), in other words, \text{Spec }K is made up of a single point.

Now we need to define sheaves on \text{Spec }K. Using ordinary concepts of topology will not be very productive, since our topological space consists only of a single point; therefore, we will not be able to obtain any interesting open covers out of this topological space. However, using the ideas in More Category Theory: The Grothendieck Topos, we can “expand” our idea of open covers. Instead of inclusions of open sets, we will instead make use of etale morphisms, as we have discussed in Cohomology in Algebraic Geometry.

Let K\rightarrow L be an etale morphism. This also means that L is an etale K-algebra (see also The Hom and Tensor Functors for the definition of algebra in our context). It is a theorem that an etale K-algebra is a direct product of finitely many separable field extensions of K (see Algebraic Numbers).

The definition of presheaf and sheaf remains the same, however the sheaf conditions can be restated in our case as the following (perhaps easier to understand) statement, which we copy verbatim from the book Etale Cohomology and the Weil Conjecture by Eberhard Freitag and Reinhardt Kiehl:

The elements s\in \mathcal{F}(B) correspond one-to-one to families of elements

s_{i}\in\mathcal{F}(B_{i}), i\in I

having the property

\text{Image }(s_{i})=\text{Image }(s_{j}), in (B_{i}\otimes_{B}B_{j})

This condition must also hold for i=j!

A separable closure \bar{K} of K is a separable field extension of K (see Cohomology in Algebraic Geometry) that is a subfield of the algebraic closure of K. The algebraic closure of K is an algebraic extension (see Algebraic Numbers) of K which is algebraically closed, i.e., it contains all the roots of polynomials with coefficients in this algebraic extension. Both the algebraic closure and the separable closure of K are unique up to isomorphism. In the case of the field of rational numbers \mathbb{Q}, the separable closure and the algebraic closure coincide and they are both equal to the field of algebraic numbers.

Given the separable closure \bar{K} of K, we define \mathcal{F}(\bar{K}) as the stalk (see Localization) of the sheaf at \bar{K}. It is also written using the language of direct limits (also called an inductive limit):

\displaystyle \mathcal{F}(\bar{K})=\varinjlim\mathcal{F}(L)

We digress slightly in order to explain what this means. The language of direct limits and inverse limits (the latter are also called projective limits) are ubiquitous in abstract algebra, algebraic geometry, and algebraic number theory, and are special cases of the notion of limits we have discussed in Even More Category Theory: The Elementary Topos.

A directed set I is an ordered set in which for every pair i,j there exists k such that i\leq k,j\leq k. A direct, resp. inverse system over I is a family \{A_{i},f_{ij}|i,j\in I,i\leq j\} of objects A_{i} and morphisms f_{ij}: A_{i}\rightarrow A_{j}, resp. f_{ij}: A_{j}\rightarrow A_{i} such that

\displaystyle f_{ii} is the identity map of A_{i}, and

\displaystyle f_{ik}=f_{jk}\circ f_{ij} resp. f_{ik}=f_{ij}\circ f_{jk}

for all i\leq j\leq k .

The direct limit of a direct system is then defined as the quotient

\displaystyle \varinjlim_{i\in I} A_{i}=\coprod_{i\in I} A_{i}/\sim

where two elements x_{i}\in A_{i} and x_{j}\in A_{j} are considered equivalent, x_{i}\sim x_{j} if there exists k such that f_{ik}(x_{i})=f_{jk}(x_{j}).

Meanwhile, the inverse limit of an inverse system is the subset

\displaystyle \varprojlim_{i\in I} A_{i}=\{(x_{i})_{i\in I}\in \prod_{i\in I}A_{i}|f_{ij}(x_{j})=x_{i}\text{ for }i\leq j \}

of the product \displaystyle \prod_{i\in I}A_{i}.

The classical definition of stalk, for a sheaf \mathcal{F} can then also be expressed as the direct limit of the direct system given by the sets (or abelian groups, or modules, etc.) \mathcal{F}(U) and the restriction maps \rho_{UV}: \mathcal{F(U)}\rightarrow \mathcal{F}(V) for open sets V\subseteq U. In our case, of course, instead of inclusion maps V\subseteq U we instead have more general maps induced by etale morphisms.

An example of an etale sheaf over \text{Spec }K is given by the following: Let

\displaystyle \mathcal{G}_{m}(B)=B^{*} where B^{*} is the multiplicative group of the etale K-algebra B.

In this case we have \mathcal{F}(\bar{K})=\bar{K}^{*}, the multiplicative group of the separable closure \bar{K} of K. We note that the multiplicative group of a field F is just the group F-\{0\}, with the law of composition given by multiplication.

In order to make contact with the theory of Galois groups, we now define the concept of G-modules, where G is a group. A left G-module is given by an abelian group M and a map \rho: G\times M\rightarrow M such that

\displaystyle \rho(e,a)=x,

\displaystyle \rho (gh,a)=\rho(g,\rho(h,a)),

and

\displaystyle \rho(g,(ab))=\rho(g,a)\rho(g,b).

Instead of \rho(g,a) we usually just write g\cdot a. A right G-module may be similarly defined, and may be obtained from a left G-module by defining a\cdot g=g^{-1}\cdot a.

The abelian group \mathcal{F}(\bar{K}) has the structure of a G-module, where G is the Galois group \text{Gal}(\bar{K}/K) (also written as G(\bar{K}/K)), the group of field automorphisms of \bar{K} that keep K fixed.

We see now that there is a connection between Galois theory and etale sheaves over a field. More generally, there is a connection between the Etale cohomology of a field and “Galois cohomology“, an important part of algebraic number theory that we now define. Galois cohomology is the derived functor (see More on Chain Complexes and The Hom and Tensor Functors) of the fixed module functor.

First we construct the standard resolution of the the Galois module (a G-module where G is the Galois group of some field extension) A. It is given by X^{n}(G,A), the abelian group of all functions from the direct product G^{n+1} to A, and the coboundary map

\displaystyle \partial^{n}: X^{n-1}\rightarrow X^{n}

given by

\displaystyle \partial^{n}x(\sigma_{0},...,\sigma{n})=\sum_{i=0}^{n}(-1)^{i}x(\sigma_{0},...,\hat{\sigma_{i}},...,\sigma_{n})

where \hat{\sigma_{i}} signifies that \sigma_{i} is to be omitted.

We now apply the fixed module functor to obtain the cochain complex

\displaystyle C^{n}(G,A)=X^{n}(G,A)^{G}.

The elements of C^{n}(G,A) are the functions x: G^{n+1}\rightarrow A such that

\displaystyle x(\sigma\sigma_{0},...,\sigma\sigma_{n})=\sigma x(\sigma_{0},...,\sigma_{n})

for all \sigma\in G.

The Galois cohomology groups H^{n}(G,A) are then obtained by taking the cohomology of this cochain complex, i.e.

\displaystyle H^{n}(G,A)=\text{Ker }\partial^{n+1}/\text{Im }\partial^{n}

Note: We have adopted here the notation of the book Cohomology of Number Fields by Jurgen Neukirch, Alexander Schmidt, and Kay Wingberg. Some references use a different notation; for instance X_{n} may be defined as the abelian group of functions from G^{n} to A instead of from G^{n+1} to A. This results in different notation for the cochain complexes and their boundary operators; however, the Galois cohomology groups themselves will remain the same.

It is a basic result of Galois cohomology that H^{0}(G,A) gives A^{G}, the subset of A such that \sigma\cdot a=a for all \sigma\in G. In other words, A^{G} is the subset of A that is fixed by G.

We have the following connection between Etale cohomology for fields and Galois cohomology:

\displaystyle H^{n}(K,\mathcal{F})=H^{n}(G,\mathcal{F}(\bar{K}))

We now mention some other basic results of the theory. In analogy with sheaf cohomology, the group H^{0}(K,\mathcal{F}) is just the set of “global sections” \Gamma(K,\mathcal{F})=\mathcal{F}(K) of \mathcal{F}. Letting \mathcal{F}=\mathcal{G}_{m} which we have defined earlier, we have

\displaystyle H^{0}(K,\mathcal{G}_{m})=\mathcal{G}_{m}(K)=K^{*}

In the language of Galois cohomology,

\displaystyle H^{0}(G,\mathcal{G}_{m}(\bar{K}))=(\bar{K}^{*})^{G}=K^{*}

Meanwhile, for H^{1}, we have the following result, called Hilbert’s Theorem 90:

\displaystyle H^{1}(K,\mathcal{G}_{m})=H^{1}(G,\bar{K}^{*})=\{1\}.

The group H^{2}(K,\mathcal{G}_{m})=H^{2}(G,\bar{K}^{*}) is called the Brauer group and also plays an important part in algebraic number theory. The etale cohomology of fields, or equivalently, Galois cohomology, are the topic of famous problems in modern mathematics such as the Milnor conjecture and its generalization, the Bloch-Kato conjecture, which was solved by Vladimir Voevodsky in 2009. They also play an important part in the etale cohomology of more general rings.

References:

Etale Cohomology on Wikipedia

Stalk on Wikipedia

Direct Limit on Wikipedia

Inverse Limit on Wikipedia

Hilbert’s Theorem 90 on Wikipedia

Group Cohomology on Wikipedia

Galois Cohomology on Wikipedia

Milnor Conjecture on Wikipedia

Norm Residue Isomorphism Theorem

Etale Cohomology and the Weil Conjecture by Eberhard Freitag and Reinhardt Kiehl

Cohomology of Number Fields by Jurgen Neukirch, Alexander Schmidt, and Kay Wingberg

Cohomology in Algebraic Geometry

In Homology and Cohomology we discussed cohomology as used to study topological spaces. In this post we study cohomology in the context of algebraic geometry. We will need the concepts we have discussed in PresheavesSheavesMore on Chain Complexes, and The Hom and Tensor Functors.

Sheaf cohomology is simply the derived functor (see More on Chain Complexes and The Hom and Tensor Functors) of the global section functor \Gamma (X,-), which assigns to a sheaf \mathcal{F} its set of global sections \Gamma (X,\mathcal{F})=\mathcal{F}(X).

One thing to note here, in constructing our resolutions, is that we are dealing with sheaves of modules, instead of just modules. The morphisms of sheaves of modules on a topological space X are defined as homomorphisms of modules \mathcal{F}(U)\rightarrow \mathcal{G}(U) for every open set U of X.

Since our definition is quite abstract, we also discuss here Cech cohomology, which is more concrete. Let X be a topological space, and let \mathfrak{U}=(U_{i})_{i\in I} be an open covering of X. We let

\displaystyle C^{p}(\mathfrak{U},\mathcal{F})=\prod_{i_{0}<...<i_{p}}\mathcal{F}(U_{i_{0},...,i_{p}})

where

\displaystyle \mathcal{F}(U_{i_{0},...,i_{p}})=U_{i_{0}}\cap ...\cap U_{i_{p}}

The coboundary maps d^{p}:C^{p}\rightarrow C^{p+1} are given by

\displaystyle (d^{p}\alpha)_{i_{0},...,i_{p+1}}=\sum_{k=0}^{p+1}(-1)^{k}\alpha_{i_{0},...,\hat{i_{k}},...,i_{p+1}}|_{U_{i_{0},...,i_{p}}}

where \hat{i_{k}} means that the index i_{k} is to be omitted. The Cech cohomology is then given by the cohomology of this complex.

The Cech cohomology is equivalent to the sheaf cohomology. The injective resolution of the sheaf \mathcal{F} is given by a “sheafified” version of the chain complex we constructed earlier. For an open subset V of X, let f:V\rightarrow X be the inclusion map. We define

\displaystyle \mathcal{C}^{p}(\mathfrak{U},\mathcal{F})=\prod_{i_{0}<...<i_{p}}f_{*}(\mathcal{F}|_{U_{i_{0},...,i_{p}}})

with the same definition for the coboundary map as earlier. Then

\Gamma(X,\mathcal{C}^{p}(\mathfrak{U},\mathcal{F}))=C^{p}(\mathfrak{U},\mathcal{F})

from which it can be seen that the Cech cohomology is indeed  the derived functor of the global section functor \Gamma (X,-).

Up to now, for our topological spaces we have always used the Zariski topology. We now introduce another kind of topology called the Etale topology. The Etale topology is not a topology in the sense of Basics of Topology and Continuous Functions, but a Grothendieck topology, which we have discussed in More Category Theory: The Grothendieck Topos. Our underlying category will be written \text{Et}/X, and its objects will be etale morphisms (to be explained later) h:U\rightarrow X, and its morphisms will be etale morphisms f:U\rightarrow U' such that if g:U'\rightarrow X then f\circ g=h.

An etale morphism is a morphism of schemes that is both flat (see The Hom and Tensor Functors) and unramified. A morphism f:Y\rightarrow X is said to be unramified if for all points y of Y the morphism \mathcal{O}_{X,f(y)}\rightarrow\mathcal{O}_{Y,y} of local rings (see Localization) has the property that \mathfrak{m}_{Y,y}=\mathfrak{m}_{X,f(y)}\cdot \mathcal{O}_{Y,y} and the residue field \mathcal{O}_{Y,y}/\mathfrak{m}_{Y,y} is a finite separable field extension of \mathcal{O}_{X,f(y)}/\mathfrak{m}_{X,f(y)}.

The concept of field extensions was discussed in Algebraic Numbers. We have explored in that same post how field extensions F\subset K may be “generated” by the roots of polynomials with coefficients in F. The field extension is called separable if the aforementioned polynomial (called the minimal polynomial) has distinct roots.

An unramified morphism f:Y\rightarrow X is also required to be locally of finite type, which means that for every open subset \text{Spec }A (we recall that in Localization we have updated our definition of schemes to mean something that “locally” looks like our “old” definition of schemes – the mathematical objects referred to by this “old” definition will henceforth be referred to as affine schemes) of X and every open subset \text{Spec }B of f^{-1}(\text{Spec }A) the induced morphism A\rightarrow B makes B into a “finitely generated” A-algebra.

Using the etale topology to define the sheaves to be used in cohomology results in etale cohomology, the original driving force for the development of the concept of the Grothendieck topos. Hopefully we will be able to flesh out more of this interesting theory in future posts.

References:

Sheaf Cohomology on Wikipedia

Cech Cohomology on Wikipedia

Etale Morphism on Wikipedia

Etale Cohomology on Wikipedia

Algebraic Geometry by Andreas Gathmann

The Rising Sea: Foundations of Algebraic Geometry by Ravi Vakil

Lectures on Etale Cohomology by J.S. Milne

Algebraic Geometry by Robin Hartshorne