Selmer Groups and Étale Cohomology

I recently gave a talk in an étale cohomology seminar where I discussed two topics: étale cohomology of fields = Galois cohomology, and higher direct images and the Leray spectral sequence. Today I read a beautiful blending of these two topics which I want to record before I forget it! This is from (Mazur and Weston 1998) Lecture 7.

Selmer Groups

First of all, we think purely about the Galois cohomology world. In particular, the important construction of Selmer groups. Suppose that we have a global field \(F\), whose Galois group \(G _{ F }\) acts on a finite module \(M\) and is unramified outside a finite set of places of \(F\). Let \(S\) be a set of places containing the primes of bad reduction, the places at infinity and the places dividing the order of \(M\). Then, for places \(\nu \not \in S\), we have the inflation-restriction exact sequence on Galois cohomology which gives us \[0 \rightarrow \mathrm{H}_{ }^{ 1 }( G _{ k _{ \nu } } , M ) \rightarrow \mathrm{H}_{ }^{ 1 }( G _{ F _{ \nu } } , M ) \rightarrow \mathrm{Hom}_{ }( I _{ \nu } , M ) ^{ G _{ k _{ \nu } } } \rightarrow 0\] where the fact that the exact sequence terminates follows from the cohomological dimension of the finite field \(k _{ \nu }\). It is common to define the relaxed Selmer structure to be \[\mathrm{H}_{ f }^{ 1 }( G _{ \nu } , M ) = \begin{cases} \mathrm{H}_{ }^{ 1 }( G _{ k _{ \nu } } , M ) , &\text{ if } \nu \in S \\ \mathrm{H}_{ }^{ 1 }( G _{ F _{ \nu } } , M ) , &\text{ if } \mathrm{ o/w } \end{cases}\] which is a subgroup of \(\mathrm{H}_{ }^{ 1 }( G _{ F _{ \nu } } , M )\) for all places \(\nu\). We then define the relaxed Selmer group to be \[\mathrm{H}_{ f }^{ 1 }( G _{ F } , M ) := \left\{ c \in \mathrm{H}_{ }^{ 1 }( G _{ F } , M ) : \mathrm{ res } _{ \nu } c \in \mathrm{H}_{ f }^{ 1 }( G _{ F _{ \nu } } , M ) \forall \nu \right\} .\]

The idea of this definition is to cut down the cohomology classes we are considering to consider only those that have certain restricted local properties. Restriction like this occur very often in mathematics, see for example the proof of the Mordell-Weil theorem for elliptic curves (or abelian varieties!).

Étale Cohomology

When talking about étale cohomology, a natural place to start is the étale cohomology of a scheme of the form \(\mathrm{Spec}\left( F \right)\) for some field \(F\). In this case, we prove the following important result.

There is an equivalence of categories between sheaves on the étale site \(X _{ et }\) for \(X = \mathrm{Spec}\left( F \right)\), and the category of discrete \(\mathrm{Gal}\left( F ^{ \mathrm{ ^{ \mathrm{sep} } } } /FF \right)\)-modules. Furthermore, if \(\mathcal{ F } _{ M }\) is the étale sheaf corresponding to a discrete module \(M\), then \[\mathrm{H}_{ et }^{ i }( X _{ et } , \mathcal{ F } _{ M } ) \cong \mathrm{H}_{ gal }^{ i }( \mathrm{Gal}\left( F ^{\mathrm{sep}}/ F \right) , M ) .\]

Therefore the theory of étale cohomology over a field becomes the theory of Galois cohomology. This means we can ask the following natural question: what is the significance of a Selmer group from the point of view of étale cohomology?

Higher Direct Images

To ask that an element of the cohomology group \(\mathrm{H}_{ }^{ 1 }( G _{ F } , M )\) restricts to \(\mathrm{H}_{ f }^{ 1 }( G _{ F _{ \nu } } , M )\) for some \(\nu \not \in S\) is equivalent to asking that it vanishes upon restriction to the group \[\mathrm{H}_{ }^{ 1 }( I _{ \nu } , M ) = \mathrm{H}_{ et }^{ 1 }( \mathrm{Spec}\left( F _{ \nu } ^{ ur } \right) , \mathcal{ F } _{ M } ),\] where the equality here follows since \(I _{ \nu } = \mathrm{Gal}\left( (F _{ \nu } ^{ ur }) ^{\mathrm{sep}}/ (F _{ \nu } ^{ ur }) \right)\). We now have a Cartesian diagram

and we note that \(\mathcal{ O } _{ F, \nu } ^{ ur }\) is the strict Henselization of \(\mathcal{ O } _{ F }\) at a geometric point above \(\nu\) (we need to choose a separable closure to say the geometric point above \(\nu\)). And therefore using (Milne 1980) Theorem 1.15 on higher direct images, we see that \[\mathrm{H}_{ }^{ 1 }( I _{ \nu } , M ) = \left( R ^{ 1 } j _{ * } \mathcal{ F } _{ M } \right) _{ \nu }\] where \(j: \mathrm{Spec}\left( F \right) \rightarrow \mathrm{Spec}\left( \mathcal{ O } _{ F } [1/S] \right)\). So, the Selmer group is precisely the kernel of the map \[\mathrm{H}_{ et }^{ 1 }( \mathrm{Spec}\left( F \right) , \mathcal{ F } _{ M } ) \rightarrow R ^{ 1 } (j _{ * } \mathcal{ F } _{ M } ) ( \mathrm{Spec}\left( O _{ F } [1/S] \right) .\] The Leray spectral sequence allows us to determine this kernel! In fact, the Leray spectral sequence \[\mathrm{H}_{ et }^{ p }( \mathrm{Spec}\left( \mathcal{ O } _{ F } [1/S] \right) ,R ^{ q } j _{ * } \mathcal{ F } _{ M } )\implies \mathrm{H}_{ et }^{ p+q }( \mathrm{Spec}\left( F \right) , \mathcal{ F } _{ M } )\] tells us precisely (via the five term exact sequence), that the Selmer group is \[\mathrm{H}_{ f }^{ 1 }( F , M ) \cong \mathrm{H}_{ et }^{ 1 }( \mathrm{Spec}\left( \mathcal{ O } _{ F } [1/S] \right) , j _{ * } \mathcal{ F } _{ M } ).\] That is, the Selmer group measure cohomology classes that can somehow be spread out to cohomology classes on a wider base.