Some Galois Cohomology
I will give an overview of Galois cohomology in this post. Again you are bewarned.
Before we can define galois cohomology groups, we need to understand what a profinite group is. A profinite group is, by definition, an inverse limit of finite groups. An equivalent formulation of a profinite group is that it is a compact, Hausdorff and totally disconnected topological group.
It can be shown that such a group has a basis of neighbourhoods (consisting of open normal subgroups) around the identity element. By compactness, it follows that these open normal subgroup are of finite index. (In fact, generally every open subgroup of a compact group has finite index) Examples of profinite groups are Galois groups of (not necessarily finite) Galois extensions.
Now let G be a profinite group. We say that an abelian group M is a discrete G-module if M has a continous G-action (here we endow M with the discrete topology). For each n>0, we denote the abelian group of continous maps from G^{n} (with the usual product topology) to M by C^{n} (notations here are not standard!). In the case of n=0, we take the group to be M. There is then the differential maps (which I will not define) that go from C^{n} to C^{n+1}. Hence we have constructed a cochain complex and taking cohomology, we obtain our so-called galois cohomology groups. (There is also the non-abelian Galois cohomology where M is not necessarily an abelian group, but I shall not go into that. Also if G is a finite group, then the Galois cohomology groups will coincide with the usual cohomology groups.)
We will give a sketch of how Galois cohomology can be used in local class field theory. Let F be a nonarchimedean local field. Equivalently to say, F is a finite extension of either the field of p-adic rationals or the field of quotient of the power series ring in one variable over a finite field of order p. (Here p is a prime number in the usual sense.) Fix a separable algebraic closure, say K.
Let G be its associated Galois group. Then one has an isomorphism (denoted by inv) between the second cohomology group of G with coefficients in the multiplicative group of K and the additive group Q/Z. For a finite separable extension L over F of degree n, the inv map induces an isomorphism between the second cohomology group of Gal(L/F) with coefficients in the multiplicative group of L and (1/n Z)/Z. Under this isomorphism, the element of the cohomology group corresponding to 1/n mod Z is called the fundamental class of the extension L/F. The establishment of the inv map is the first step in the proof of the existence of the local Artin map. (We remark that the development of class field theory was classically done by cohomological methods. Today, there are other ways to develop the theory of class field theory which are also important in their own rights.)
Before we can define galois cohomology groups, we need to understand what a profinite group is. A profinite group is, by definition, an inverse limit of finite groups. An equivalent formulation of a profinite group is that it is a compact, Hausdorff and totally disconnected topological group.
It can be shown that such a group has a basis of neighbourhoods (consisting of open normal subgroups) around the identity element. By compactness, it follows that these open normal subgroup are of finite index. (In fact, generally every open subgroup of a compact group has finite index) Examples of profinite groups are Galois groups of (not necessarily finite) Galois extensions.
Now let G be a profinite group. We say that an abelian group M is a discrete G-module if M has a continous G-action (here we endow M with the discrete topology). For each n>0, we denote the abelian group of continous maps from G^{n} (with the usual product topology) to M by C^{n} (notations here are not standard!). In the case of n=0, we take the group to be M. There is then the differential maps (which I will not define) that go from C^{n} to C^{n+1}. Hence we have constructed a cochain complex and taking cohomology, we obtain our so-called galois cohomology groups. (There is also the non-abelian Galois cohomology where M is not necessarily an abelian group, but I shall not go into that. Also if G is a finite group, then the Galois cohomology groups will coincide with the usual cohomology groups.)
We will give a sketch of how Galois cohomology can be used in local class field theory. Let F be a nonarchimedean local field. Equivalently to say, F is a finite extension of either the field of p-adic rationals or the field of quotient of the power series ring in one variable over a finite field of order p. (Here p is a prime number in the usual sense.) Fix a separable algebraic closure, say K.
Let G be its associated Galois group. Then one has an isomorphism (denoted by inv) between the second cohomology group of G with coefficients in the multiplicative group of K and the additive group Q/Z. For a finite separable extension L over F of degree n, the inv map induces an isomorphism between the second cohomology group of Gal(L/F) with coefficients in the multiplicative group of L and (1/n Z)/Z. Under this isomorphism, the element of the cohomology group corresponding to 1/n mod Z is called the fundamental class of the extension L/F. The establishment of the inv map is the first step in the proof of the existence of the local Artin map. (We remark that the development of class field theory was classically done by cohomological methods. Today, there are other ways to develop the theory of class field theory which are also important in their own rights.)
posted by MF, Lim at 10:05 PM
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