Étale cohomology

:$F\; \backslash to\; \backslash Gamma(F)$

:$F\; \backslash to\; \backslash Gamma(F)$

(where the space of sections Γ(''F'') of ''F'' is ''F''(''X'')). The sections of a sheaf can be thought of as Hom('''Z''', ''F'') where '''Z''' is the sheaf that returns the integers as an [[abelian group]]. The idea of ''derived functor'' here is that the functor of sections doesn't respect [[exact sequence]]s as it is not right exact; according to general principles of [[homological algebra]] there will be a sequence of functors ''H'' ⁰, ''H'' ¹, ... that represent the 'compensations' that must be made in order to restore some measure of exactness (long exact sequences arising from short ones). The ''H'' ⁰ functor coincides with the section functor Γ.

(where the space of sections Γ(''F'') of ''F'' is ''F''(''X'')). The sections of a sheaf can be thought of as Hom('''Z''', ''F'') where '''Z''' is the sheaf that returns the integers as an [[abelian group]]. The idea of ''derived functor'' here is that the functor of sections doesn't respect [[exact sequence]]s as it is not right exact; according to general principles of [[homological algebra]] there will be a sequence of functors ''H''⁰, ''H''¹, ... that represent the 'compensations' that must be made in order to restore some measure of exactness (long exact sequences arising from short ones). The ''H''⁰ functor coincides with the section functor Γ.

More generally, a morphism of schemes ''f'' : ''X'' → ''Y'' induces a map ''f''_∗ from étale sheaves over ''X'' to étale sheaves over ''Y'', and its right derived functors are denoted by ''R^qf''_∗, for ''q'' a non-negative integer. In the special case when ''Y'' is the spectrum of an algebraically closed field (a point), ''R''^''q''''f''_∗(''F'' ) is the same as ''H^q''(''F'' ).

More generally, a morphism of schemes ''f'' : ''X'' → ''Y'' induces a map ''f''_∗ from étale sheaves over ''X'' to étale sheaves over ''Y'', and its right derived functors are denoted by ''R^qf''_∗, for ''q'' a non-negative integer. In the special case when ''Y'' is the spectrum of an algebraically closed field (a point), ''R''^''q''''f''_∗(''F''{{hairsp}}) is the same as ''H^q''(''F''{{hairsp}}).

Suppose that ''X'' is a [[Noetherian scheme]]. An abelian étale sheaf ''F'' over ''X'' is called '''finite locally constant''' if it is represented by an étale cover of ''X''. It is called '''[[constructible sheaf|constructible]]''' if ''X'' can be covered by a finite family of subschemes on each of which the restriction of ''F'' is finite locally constant. It is called '''[[torsion sheaf|torsion]]''' if ''F''(''U'') is a torsion group for all étale covers ''U'' of ''X''. Finite locally constant sheaves are constructible, and constructible sheaves are torsion. Every torsion sheaf is a filtered inductive limit of constructible sheaves.

Suppose that ''X'' is a [[Noetherian scheme]]. An abelian étale sheaf ''F'' over ''X'' is called '''finite locally constant''' if it is represented by an étale cover of ''X''. It is called '''[[constructible sheaf|constructible]]''' if ''X'' can be covered by a finite family of subschemes on each of which the restriction of ''F'' is finite locally constant. It is called '''[[torsion sheaf|torsion]]''' if ''F''(''U'') is a torsion group for all étale covers ''U'' of ''X''. Finite locally constant sheaves are constructible, and constructible sheaves are torsion. Every torsion sheaf is a filtered inductive limit of constructible sheaves.

:$H^q(X,\backslash varprojlim\; F\_i)\; \backslash to\; \backslash varprojlim\; H^q(X,\; F\_i),$

:$H^q(X,\backslash varprojlim\; F\_i)\; \backslash to\; \backslash varprojlim\; H^q(X,\; F\_i),$

this is '''not''' usually an isomorphism. An [[l-adic sheaf|'''ℓ-adic sheaf''']] is a special sort of inverse system of étale sheaves ''F_i'', where ''i'' runs through positive integers, and ''F_i'' is a module over '''Z'''/ℓ^''i'' '''Z''' and the map from ''F''_''i''+1 to ''F_i'' is just reduction mod '''Z'''/ℓ^''i'' '''Z'''.

this is '''not''' usually an isomorphism. An [[l-adic sheaf|'''ℓ-adic sheaf''']] is a special sort of inverse system of étale sheaves ''F_i'', where ''i'' runs through positive integers, and ''F_i'' is a module over '''Z'''/ℓ^''i'''''Z''' and the map from ''F''_''i''+1 to ''F_i'' is just reduction mod '''Z'''/ℓ^''i'''''Z'''.

When ''V'' is a [[Algebraic curve#Singularities|non-singular]] [[algebraic curve]] of [[genus (mathematics)|genus]] ''g'', ''H''¹ is a free '''Z'''_ℓ-module of rank 2''g'', dual to the [[Tate module]] of the [[Jacobian variety]] of ''V''. Since the first [[Betti number]] of a [[Riemann surface]] of genus ''g'' is 2''g'', this is isomorphic to the usual singular cohomology with '''Z'''_ℓ coefficients for complex algebraic curves. It also shows one reason why the condition ℓ ≠ ''p'' is required: when ℓ = ''p'' the rank of the Tate module is at most ''g''.

When ''V'' is a [[Algebraic curve#Singularities|non-singular]] [[algebraic curve]] of [[genus (mathematics)|genus]] ''g'', ''H''¹ is a free '''Z'''_ℓ-module of rank 2''g'', dual to the [[Tate module]] of the [[Jacobian variety]] of ''V''. Since the first [[Betti number]] of a [[Riemann surface]] of genus ''g'' is 2''g'', this is isomorphic to the usual singular cohomology with '''Z'''_ℓ coefficients for complex algebraic curves. It also shows one reason why the condition ℓ ≠ ''p'' is required: when ℓ = ''p'' the rank of the Tate module is at most ''g''.

\end{align}

\end{align}

Here ''j'' is the injection of the [[generic point]], ''i_x'' is the injection of a [[closed point]] ''x'', '''G'''_''m'',''K'' is the sheaf '''G'''_''m'' on {{nowrap|Spec ''K''}} (the generic point of ''X''), and '''Z'''_''x'' is a copy of '''Z''' for each closed point of ''X''. The groups ''H ⁱ''(''i_x*'' '''Z''') vanish if ''i'' > 0 (because ''i_x*'' '''Z''' is a [[Stalk (sheaf)#Skyscraper sheaf|skyscraper sheaf]]) and for ''i'' = 0 they are '''Z''' so their sum is just the divisor group of ''X''. Moreover, the first cohomology group ''H'' ¹(''X'', ''j''_∗'''G'''_''m'',''K'') is isomorphic to the Galois cohomology group ''H'' ¹(''K'', ''K''*) which vanishes by [[Hilbert's theorem 90]]. Therefore, the long exact sequence of étale cohomology groups gives an exact sequence

Here ''j'' is the injection of the [[generic point]], ''i_x'' is the injection of a [[closed point]] ''x'', '''G'''_''m'',''K'' is the sheaf '''G'''_''m'' on {{nowrap|Spec ''K''}} (the generic point of ''X''), and '''Z'''_''x'' is a copy of '''Z''' for each closed point of ''X''. The groups ''Hⁱ''(''i_x*'''''Z''') vanish if ''i'' > 0 (because ''i_x*'''''Z''' is a [[Stalk (sheaf)#Skyscraper sheaf|skyscraper sheaf]]) and for ''i'' = 0 they are '''Z''' so their sum is just the divisor group of ''X''. Moreover, the first cohomology group ''H''¹(''X'', ''j''_∗'''G'''_''m'',''K'') is isomorphic to the Galois cohomology group ''H''¹(''K'', ''K''*) which vanishes by [[Hilbert's theorem 90]]. Therefore, the long exact sequence of étale cohomology groups gives an exact sequence

:$K\backslash to\; \backslash operatorname\{Div\}(X)\backslash to\; H^1(\backslash mathbf\{G\}\_m)\backslash to\; 1$

:$K\backslash to\; \backslash operatorname\{Div\}(X)\backslash to\; H^1(\backslash mathbf\{G\}\_m)\backslash to\; 1$

where Div(''X'') is the group of divisors of ''X'' and ''K'' is its function field. In particular ''H'' ¹(''X'', '''G'''_''m'') is the [[Picard group]] Pic(''X'') (and the first cohomology groups of '''G'''_''m'' are the same for the étale and Zariski topologies). This step works for varieties ''X'' of any dimension (with points replaced by codimension 1 subvarieties), not just curves.

where Div(''X'') is the group of divisors of ''X'' and ''K'' is its function field. In particular ''H''¹(''X'', '''G'''_''m'') is the [[Picard group]] Pic(''X'') (and the first cohomology groups of '''G'''_''m'' are the same for the étale and Zariski topologies). This step works for varieties ''X'' of any dimension (with points replaced by codimension 1 subvarieties), not just curves.

===Calculation of ''Hⁱ''(''X'', G_''m'')===

===Calculation of ''Hⁱ''(''X'', G_''m'')===

The same long exact sequence above shows that if ''i'' ≥ 2 then the cohomology group ''H ⁱ''(''X'', '''G'''_''m'') is isomorphic to ''H ⁱ''(''X'', ''j''_*'''G'''_''m'',''K''), which is isomorphic to the Galois cohomology group ''H ⁱ''(''K'', ''K''*). [[Tsen's theorem]] implies that the Brauer group of a function field ''K'' in one variable over an algebraically closed field vanishes. This in turn implies that all the Galois cohomology groups ''H ⁱ''(''K'', ''K''*) vanish for ''i'' ≥ 1, so all the cohomology groups ''H ⁱ''(''X'', '''G'''_''m'') vanish if ''i'' ≥ 2.

The same long exact sequence above shows that if ''i'' ≥ 2 then the cohomology group ''Hⁱ''(''X'', '''G'''_''m'') is isomorphic to ''Hⁱ''(''X'', ''j''_*'''G'''_''m'',''K''), which is isomorphic to the Galois cohomology group ''Hⁱ''(''K'', ''K''*). [[Tsen's theorem]] implies that the Brauer group of a function field ''K'' in one variable over an algebraically closed field vanishes. This in turn implies that all the Galois cohomology groups ''Hⁱ''(''K'', ''K''*) vanish for ''i'' ≥ 1, so all the cohomology groups ''Hⁱ''(''X'', '''G'''_''m'') vanish if ''i'' ≥ 2.

===Calculation of ''Hⁱ''(''X'', ''μ_n'')===

===Calculation of ''Hⁱ''(''X'', ''μ_n'')===

If ''n'' is divisible by ''p'' this argument breaks down because ''p''-th roots of unity behave strangely over fields of characteristic ''p''. In the Zariski topology the Kummer sequence is not exact on the right, as a non-vanishing function does not usually have an ''n''-th root locally for the Zariski topology, so this is one place where the use of the [[étale topology]] rather than the Zariski topology is essential.

If ''n'' is divisible by ''p'' this argument breaks down because ''p''-th roots of unity behave strangely over fields of characteristic ''p''. In the Zariski topology the Kummer sequence is not exact on the right, as a non-vanishing function does not usually have an ''n''-th root locally for the Zariski topology, so this is one place where the use of the [[étale topology]] rather than the Zariski topology is essential.

===Calculation of ''H ⁱ''(''X'', Z/''n''Z)===

===Calculation of ''Hⁱ''(''X'', Z/''n''Z)===

By fixing a primitive ''n''-th root of unity we can identify the group '''Z'''/''n'''''Z''' with the group ''μ_n'' of ''n''-th roots of unity. The étale group ''H ⁱ''(''X'', '''Z'''/''n'''''Z''') is then a free module over the ring '''Z'''/''n'''''Z''' and its rank is given by:

By fixing a primitive ''n''-th root of unity we can identify the group '''Z'''/''n'''''Z''' with the group ''μ_n'' of ''n''-th roots of unity. The étale group ''Hⁱ''(''X'', '''Z'''/''n'''''Z''') is then a free module over the ring '''Z'''/''n'''''Z''' and its rank is given by:

:

:

where ''g'' is the genus of the curve ''X''. This follows from the previous result, using the fact that the Picard group of a curve is the points of its [[Jacobian variety]], an [[abelian variety]] of dimension ''g'', and if ''n'' is coprime to the characteristic then the points of order dividing ''n'' in an abelian variety of dimension ''g'' over an algebraically closed field form a group isomorphic to ('''Z'''/''n'''''Z''')^2''g''. These values for the étale group ''H ⁱ''(''X'', '''Z'''/''n'''''Z''') are the same as the corresponding singular cohomology groups when ''X'' is a complex curve.

where ''g'' is the genus of the curve ''X''. This follows from the previous result, using the fact that the Picard group of a curve is the points of its [[Jacobian variety]], an [[abelian variety]] of dimension ''g'', and if ''n'' is coprime to the characteristic then the points of order dividing ''n'' in an abelian variety of dimension ''g'' over an algebraically closed field form a group isomorphic to ('''Z'''/''n'''''Z''')^2''g''. These values for the étale group ''Hⁱ''(''X'', '''Z'''/''n'''''Z''') are the same as the corresponding singular cohomology groups when ''X'' is a complex curve.

===Calculation of ''H ⁱ''(''X'', Z/''p''Z)===

===Calculation of ''H ⁱ''(''X'', Z/''p''Z)===