Galois cohomology ($H^i(G,M)$)

Galois cohomology is the group cohomology of a Galois group acting on a module. For a discrete $G$-module $M$, the first group is the quotient of crossed homomorphisms $c(\sigma \tau )=c(\sigma )+\sigma c(\tau )$ by maps $c(\sigma )=\sigma m-m$.

For an abelian variety $A$ over a field $K$, its Weil–Châtelet group is the Galois cohomology group $H^1(K,A(K^{\mathrm{sep}}))$, classifying torsors under $A$. Such an algebraic-group torsor becomes $A$ after passing to a separable closure but need not have a $K$-point. A chosen geometric point identifies the torsor with $A$ and its Galois descent with a cocycle; changing that point adds a coboundary. The zero class is precisely the torsor with a rational point. For an elliptic curve, these torsors are genus-one curves with a specified action and Jacobian. Over a number field the kernel of restriction to every completion is the Tate–Shafarevich group.

For a Finite Galois extension $L/K$ and a finite-dimensional $L$-space $V$ with a semilinear action of its Galois group, the map $L{\otimes }_K{V}^G\to V$ is an isomorphism. The vectors ${\sum }_{\sigma }\sigma (a)\sigma (v)$ are invariant and span $V$ by the Artin independence theorem. Choosing an invariant $L$-basis then identifies the invariant space with its $K$-span. The proof does not divide by the extension degree.

A splitting isomorphism of a central simple algebra with $M_n(L)$ transports its Galois action to $T_{\sigma }=c_{\sigma }{\sigma }_0$, giving a nonabelian first cohomology class. Conversely a cocycle gives a semilinear algebra action whose invariants descend to a central simple algebra. Equivalent cocycles give isomorphic fixed algebras. This gives a pointed-set bijection ${\mathrm{CSA}}_n(L/K)\cong H^1(\mathrm{Gal}(L/K),{\mathrm{PGL}}_n(L))$.

For an extension $L/K$ with Galois group $G$, a semilinear action on an $L$-space is an additive group action satisfying $\sigma (av)=\sigma (a)\sigma (v)$. Scalar coefficients are transformed along with vectors. Invariants form a $K$-space, and Galois descent of vector spaces reconstructs the original space from it.

For a finite Galois module $M$ unramified outside a finite set $S$, define $H_S^1(K,M)$ as the classes in Galois cohomology whose restriction to inertia is zero at every finite prime outside $S$. This means zero as a cohomology class, rather than zero for every possible representative. Passing to a finite Galois extension on which $M$ is constant reduces the finiteness problem to characters with bounded ramification and a finite restriction kernel.

A discrete Galois module over $K$ is an abelian group with a continuous action of the absolute Galois group $G_K$. Continuity means that each element has an open stabilizer. Examples are the algebraic points of an elliptic curve, its finite torsion module $E[m]$, and the group of roots of unity ${\mu }_m$. Galois cohomology takes continuous cocycles with values in such a module.

The module ${\mu }_m$ consists of the $m$th roots of unity in a separable closure. If the field characteristic does not divide $m$, it has $m$ elements and fits into the Kummer sequence $1\to {\mu }_m\to K_s^{\times }\stackrel{m}{\to }{K}_s^{\times }\to 1$. Its Galois action is the cyclotomic action; a generator gives a noncanonical identification with $\mathbb{Z}/m\mathbb{Z}$.

For a finite Galois extension $L/K$, multiplicative Hilbert theorem 90 says $H^1(\mathrm{Gal}(L/K),L^{\ast })=0$. A multiplicative cocycle has the form $c_{\sigma }=\sigma (b)/b$. One proof forms a nonzero sum $A={\sum }_{\sigma }{c}_{\sigma }\sigma (a)$, using linear independence of field automorphisms. Its transformation law is $\tau A=c_{\tau }^{-1}A$, so $b=A^{-1}$ gives $c_{\tau }=\tau (b)/b$. Passing to the algebraic closure gives $H^1(K,{\overline{K}}^{\ast })=0$, so the multiplicative Kummer sequence identifies $H^1(K,{\mu }_m)$ with $K^{\ast }/K^{\ast m}$.

For an unramified extension of non-Archimedean local fields, let $\varphi$ generate its cyclic Galois group. The ordinary Hilbert theorem 90 supplies $b\in L^{\times }$ with $x=\varphi (b)/b$. A common uniformizer $\pi \in K$ gives $y={\pi }^{-v_L(b)}b\in {\mathcal{O}}_L^{\times }$, retaining the same quotient because $\varphi$ fixes $\pi$. The reverse implication follows by telescoping the field norm.