Author: William A. Stein
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100\begin{document}
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109\part*{Galois Cohomology}
110\pauth{John Tate}
111
112\setcounter{tocdepth}{2}   % all the way to subsections
113\tableofcontents
114
115\mainmatter
116\setcounter{page}{1}
117
118\LogoOn
119
120\lectureseries[Galois Cohomology]{Galois Cohomology}
121
122% The ^{1} is a fake'' footnote indicator.  It's there because
123% the PCMI style file bizarly puts a footnote to nothing at the
124% bottom of the page!
125\auth[J. Tate]{John Tate$\mbox{}^{1}$}
126
128\email{tate@math.utexas.edu}
129
130%The following items will become first page footnotes; they are optional.
131
132\subjclass{11}
133\keywords{Galois cohomology, Elliptic curves}
134%\date{August 22, 1999}
136
137%\lecture{Galois cohomology}
138
139        I thank Helena Verrill and William Stein for their help in
140    getting this account of my talks at Park City into print. After
141    Helena typed up her original notes of the talks, William was a
142    great help with the editing, and put them in the canonical format
143    for this volume.
144
145        The somewhat inefficient organization of this account is
146    mainly a result of the fact that, after the first talk had been
147    given with the idea that it was to be the only one, a second was
148    later scheduled, and these are the notes of the material in the
149    two talks in the order it was presented.
150
151
152The bible for this subject is Serre~\cite{serre:gc}, in
153conjunction with~\cite{serre:local} or \cite{cassels-frohlich}.
154Haberland~\cite{haberland}  is also an excellent reference.
155
156\section{Group modules}
157Consider a group~$G$ and an abelian group~$A$
158equipped with a map
159$$G\times A\rightarrow A,$$
160$$(\sigma,a)\mapsto \sigma a.$$
161We use notation $\sigma,\,\tau,\,\rho,\dots$
162for elements of~$G$, and $a,\,b,\,a',\,b',\dots$ for elements of~$A$.
163To say that~$A$ is a \emp{$G$-set} means that
164$$\tau(\sigma a) = (\tau\sigma)a \quad\text{and}\quad 1a=a,$$
165for all $\sigma, \tau\in G$ and $a\in A$,
166where~$1$ is the identity in~$G$.
167To say that~$A$ is a \emp{$G$-module} means that, in addition, we
168have
169$$\sigma(a+b)=\sigma a + \sigma b,$$
170for all $\sigma\in G$ and $a,b\in A$.
171This is all equivalent to giving~$A$ the structure
172of ${\Z}[G]$-module.
173
174Given a $G$-module~$A$ as above, the subgroup of fixed elements of~$A$ is
175  $$A^G:=\left\{ a\in A \mid \sigma a = a \text{ for all } \sigma \in G\right\}.$$
176We say~$G$ \emp{acts trivially} on~$A$ if $\sigma a=a$
177for all $a\in A$; thus
178$A^G=A$ if and only if the action is trivial.
179When ${\Z}$, ${\Q}$, ${\Q}/{\Z}$ are
180considered as $G$-modules, this is with the trivial action,
181unless stated otherwise.
182
183If we take $G=\Gal(K/k)$, with~$K$ a Galois
184extension of~$k$ of possibly infinite degree,
185then we have the following examples of fixed subgroups of
186$G$-modules:
187$$\begin{array}{l|l} 188A & A^G\\ 189\hline 190K^+ \text{ as an additive group}&k^+\\ 191K^{*} \text{ as a multiplicative group}&k^*\\ 192E(K), \text{ where } E/k \text{ is an elliptic curve}&E(k).\\ 193\end{array} 194$$
195The action on $E(K)$ above is given by
196$\sigma(x,y)=(\sigma x,\sigma y)$ for a point $P=(x,y)$,
197if~$E$ is given as a plane cubic.
198In general, if~$C$ is a commutative algebraic group over~$K$,
199we can take $A=C(K)$, and then $A^G=C(k)$.
200
201\section{Cohomology}
202We now define the cohomology groups
203   $H^r(G,A)$,
204for $r\in\Z$.
205Abstractly, these are the right derived functors of the left exact functor
206$$\left\{G\text{-modules}\right\}\rightarrow\left\{\text{abelian groups}\right\}$$
207that sends $A\mapsto A^G$.
208Since $A^G=\Hom_{{\Z}[G]}({\Z},A)$,
209we have a canonical isomorphism
210 $$H^r(G,A)=\Ext^r_{{\Z}[G]}({\Z},A).$$
211
212More concretely, the cohomology groups $H^r(G,A)$ can be computed
213using the standard cochain complex'' (see,
214e.g., \cite[pg.~96]{cassels-frohlich}). Let
215      $$C^r(G,A):=\Maps(G^r,A);$$
216an element of $C^r(G,A)$ is a function~$f$ of~$r$ variables in~$G$,
217    $$f(\sigma_1,\dots,\sigma_r)\in A,$$
218and is called an \emp{$r$-cochain}.
219(If, in addition,~$A$ and~$G$ have a topological structure, then we
221There is a sequence
222$$\cdots \ra 0 \ra 0\rightarrow C^0(G,A)\stackrel{\delta}{\rightarrow} 223C^1(G,A)\stackrel{\delta}{\rightarrow} 224C^2(G,A)\stackrel{\delta}{\rightarrow}\cdots$$
225Here $C^0(G,A)=A$, since an element~$f$ of $C^0(G,A)$ is given by
226the single element $f_0(\bullet)\in A$,
227its value at the unique element $\bullet\in G^0$.
228The maps~$\delta$ are defined by
229$$\begin{array}{lll} 230(\delta f_0)(\sigma)& 231=&\sigma f_0(\bullet) - f_0(\bullet),\\ 232(\delta f_1)(\sigma,\tau)&=& 233\sigma f_1(\tau)-f_1(\sigma\tau)+f_1(\sigma),\\ 234(\delta f_2)(\sigma,\tau,\rho)&=& 235\sigma f_2(\tau,\rho)-f_2(\sigma\tau,\rho) 236+f_2(\sigma,\tau\rho)-f_2(\sigma,\tau),\\ 237\end{array} 238$$
239and so on.  Note that $\delta\circ\delta=0$.
240The cohomology groups are given by
241  $$H^r(G,A)=\ker\delta/\im\delta\subset C^r(G,A)/\im\delta.$$
242Cocycles are elements of the kernel of~$\delta$, and coboundaries are elements
243 of the image of~$\delta$.
244We have
245$$\begin{array}{lll} 246H^0(G,A)&=&A^G,\\ 247H^1(G,A)&=&\frac{\text{crossed-homomorphisms}} 248{\text{principal crossed-homomorphisms}}\\ 249&=&\Hom(G,A), \text{ if action is trivial},\\ 250H^2(G,A)&=&\text{classes of factor sets''}.\\ 251\end{array} 252$$
253
254The groups $H^2(G,A)$ and $H^1(G,A)$ arise in many situations.
255Perhaps the simplest is their connection with group extensions
256and their automorphisms.
257Given a $G$-module~$A$, suppose~${\mathcal G}$
258is a group extension of $G$ by $A$, that is,
259$\mathcal G$ is a group which contains~$A$ as
260a normal subgroup such that ${\mathcal G}/A\cong G$,
261where the given action of~$G$ on~$A$ is the same as the
262conjugation action induced by this isomorphism.
263Construct a $2$-cocycle  $a_{\sigma,\tau}$ as follows.
264For each element $\sigma\in G$, let
265$u_\sigma\in{\mathcal G}$ be a coset representative
266corresponding to~$\sigma$.
267Then
268${\mathcal G} = \coprod_{\sigma} Au_\sigma,$
269i.e.,
270every element of $\mathcal G$
271is uniquely of the form $au_{\sigma}$.
272Thus
273$$u_\sigma u_\tau = a_{\sigma,\tau}u_{\sigma\tau}$$
274for some $a_{\sigma,\tau}\in A$.  The
275map $(\sigma,\tau)\mapsto a_{\sigma,\tau}$
276is a $2$-cocycle.
277
278
279\begin{exercise}
280Using the associative law, check that $a_{\sigma,\tau}$ is a
281$2$-cocycle, and if~${\mathcal G}'$ is another extension of~$G$ by~$A$,
282then there is an isomorphism ${\mathcal G}'\cong{\mathcal G}$ that
283induces the identity on~$A$ and~$G$ if and only if the corresponding
284$2$-cocycles differ by a coboundary.
285\end{exercise}
286
287\begin{exercise}
288Conversely, show that every 2-cocycle arises in this way.
289For example, in the trival case, if $a_{\sigma,\tau}=1$
290for every~$\sigma$ and~$\tau$,
291then we can take~${\mathcal G}$ to be the semidirect product
292$G\ltimes A$.
293\end{exercise}
294
295Therefore we may view $H^2(G,A)$ as the group of isomorphism classes of
296extensions of~$G$ by~$A$ with a given action of~$G$ on~$A$.
297% (Note our terminology: we call~$\G$ an
298% extension of~$G$ by~$A$ rather than an extension of~$A$
299% by~$G$.)
300
301\begin{exercise}
302Show that an automorphism of~$\G$ that induces the identity on~$A$
303and on $G=\G/A$ is of the form $au_\sigma\mapsto ab_\sigma u_\sigma$
304with $\sigma\mapsto b_\sigma$ a $1$-cocycle, and it is an inner
305automorphism induced by an element of~$A$ if and only
306if $\sigma\mapsto b_\sigma$ is a coboundary.
307\end{exercise}
308
309\subsection{Examples}
310\label{brau}
311Given a finite Galois extension $K/k$, and
312a commutative algebraic group~$C$ over~$k$, the
313following notation is frequently used:
314     $$H^r(K/k,C):=H^r(\Gal(K/k),C(K)).$$
315We have $H^0(K/k,C)=C(k)$ as above, and
316\begin{align*}
317H^1(K/k,{\bG}_m)&=H^1(K/k,K^*)=0,\\
318H^2(K/k,{\bG}_m)&=\Br(K/k)\subset \Br(k)
319\end{align*}
320The first equality is Hilbert's Theorem 90.
321In the second equality  $\Br(k)$ is the Brauer group of~$k$;
322this is the
323group of equivalence classes of central simple algebras with
324center~$k$ that are finite dimensional over~$k$;
325two such algebras are equivalent if they are matrix
326algebras over $k$-isomorphic division algebras.
327
328The map from $H^2(K/k,{\bG}_m)$ to
329$\Br(K/k)$ is defined as follows.
330Given a $2$-cocycle $a_{\sigma,\tau}$,
331define a central simple algebra over~$k$ by
332${\mathcal A}=\oplus K {u_\sigma}$, which is a
333vector spaces over~$K$ with a basis
334$\{u_\sigma\}$ indexed by the elements $\sigma\in{}G$.
335Multiplication is defined by the same rules
336as for group extensions above (with $A=K^*$),
337extended linearly.
338
339\subsection{Characterization of $H^r(G,-)$}
340\label{subsec:char}
341For fixed $G$ and varying $A$ the groups $H^r(G,A)$ have the
342following fundamental properties:
343\begin{enumerate}
344\item $H^0(G,A)=A^G$.
345\item $H^r(G,-)$ is a functor
346     $$\left\{G\text{-modules}\right\} 347 \rightarrow \left\{\text{abelian groups}\right\}.$$
348\item Each short exact sequence
349   $$0\rightarrow A' \rightarrow A\rightarrow A''\rightarrow 0$$
350   gives rise to connecting homomorphisms (see below)
351    $$\delta:H^r(G,A'')\rightarrow H^{r+1}(G,A')$$
352   from which we get a long exact sequence of cohomology groups,
353   functorial in short exact sequences in the natural sense.
354\item If~$A$ is induced'' or injective'', then
355      $H^r(G,A)=0$ for all $r\not=0$.
356\end{enumerate}
357These properties characterize the sequence of functors
358$H^i$ equipped with the
359$\delta$'s uniquely, up to unique isomorphism.
360
361For $c\in H^r(G,A'')$, define $\delta(c)$ as follows.
362Let $c_1:G^r\rightarrow A''$ be a cocycle representing~$c$.
363Lift~$c_1$ to any map (cochain)
364$c_2:G^r\rightarrow A$.
365Since $\delta(c_1)=0$, the map $\delta(c_2):G^{r+1}\rightarrow A$ has
366image in $A'$, so defines a map
367$\delta(c_2):G^{r+1}\rightarrow A'$,
368and thus represents
369a class $\delta(c)\in{}H^{r+1}(G,A')$.
370
371For an infinite Galois extension, one uses cocycles that come by
372inflation from finite Galois subextensions.
373This amounts to using continuous cochains, where continuous means
374with respect to  the Krull topology on~$G$
375and the discrete topology on~$A$.
376
377Abstracting this situation leads to the notion of the cohomology
378of a profinite group~$G$ (i.e., a projective limit, in the category of
379topological groups, of finite groups~$G_i$) operating continuously on a
380discrete module~$A$. Without loss of generality the~$G_i$ can be taken to
381be the quotients~$G/U$ of~$G$ by its open normal subgroups~$U$, and
382then~$A$ is the union of its subgroups $A^U$. The cohomology groups
383$H^r(G,A)$ computed with continuous cochains are direct limits, relative
384to the inflation maps (see Section~\ref{sec:infres}),
385of the cohomology groups $H^r(G/U,A^U)$
386of the finite quotients, because the continuous cochain complex
387$C^*(G,A)$ is the direct limit of the complexes $C^*(G/U,A^U)$. Also, it is
388easy to see that the groups $\{H^r(G,-)\}_r$ are characterized by
389$\delta$-functoriality on the category of {\em discrete} $G$-modules.
390
391
392\section{Kummer theory}
393Let $k^{\sep}$ be a separable closure of a field~$k$,
394and put $G_k=\Gal(k^{\sep}/k)$.
395Let $m\ge 1$ be an integer, and assume that the image
396of~$m$ in~$k$ is nonzero.
397Associated to the exact sequence
398$$0\longrightarrow \mu_m\longrightarrow (k^{\sep})^{*}\stackrel{m}{\longrightarrow} 399(k^{\sep})^{*}\longrightarrow 0,$$
400we have a long exact sequence
401$$\xymatrix{ 4020\ar[r] 403&{\mu_m\cap k} 404\ar[r] 405& k^{*} 406\ar[r]^m 407&k^{*} 408\arr[d]d[][dlll]d[][dll]\\ 409&H^1(G_k,\mu_m)\ar[r] 410&H^1(G_k,(k^{\sep})^{*})=0,&{}\\ 411} 412$$
413where the last equality is
414by Hilbert's Theorem 90.
415Thus $H^1(G_k,\mu_m)\isom k^*/(k^*)^m.$
416
417Now assume that the group of $m$th roots of unity $\mu_m$ is contained
418in~$k$. Then
419\begin{align*}
420H^1(G_k,\mu_m)&=\Hom_{\cont}(G_k,\mu_m),\\
421\intertext{so}
422k^{*}/(k^{*})^m&\cong \Hom_{\cont}(G_k,\mu_m).
423\end{align*}
424Using duality, this isomorphism describes the finite abelian
425extensions of~$k$ whose Galois group is killed by~$m$.
426For example, consider a Galois extension $K/k$ such that
427$G=\Gal(K/k)$
428is a finite abelian group that is killed by~$m$.
429Since~$G$ is a quotient of $G_k=\Gal(k^{\sep}/k)$,
430we have a diagram
431$$\xymatrix{ 432k^*/(k^{*})^m 433\ar[rr]^{\cong}&&\Hom_{\cont}(G_k,\mu_m)\\ 434B\ar[rr]^{\cong} \ar[u] 435 &&{\widehat{G}:=\Hom(G,\mu_m),} \ar[u]\\ 436}$$
437where~$B$ is the subgroup of $k^*/(k^{*})^m$
438corresponding to $\widehat{G}$ under the isomorphism.
439\begin{exercise}
440Show that
441$$K=k(\sqrt[m]{B})=k(\{\sqrt[m]{b} \mid b\in B\}),$$
442and $[K:k]=\# B$.
443\end{exercise}
444The case when~$G$ cyclic is the crucial step in showing that
445a polynomial with solvable Galois group can be solved by
447
448For the rest of this section, we assume that~$k$ is a number field
449and continue to assume that~$k$ contains~$\mu_m$.
450Let~$S$ be a finite set of primes of~$k$ including all divisors
451of~$m$ and large enough so that the ring $\O_S$ of $S$-integers
452of~$k$ is a principal ideal ring.
453
454\begin{exercise}
455Show that the extension
456$K(\sqrt[m]{B})$ above is unramified outside~$S$ if and
457only if $B\subset U_S {k^*}^m/{k^*}^m \isom U_S/U_S^m$,
458where $U_S=\O_S^*$ is the
459group of $S$-units of~$k$.
460\end{exercise}
461
462\begin{exercise}\label{ex:finite}
463Let $k_S$ be the maximal extension of~$k$ which is unramified
464outside~$S$, and let $G_S=\Gal(k_S/k)$. Then
465$\Hom_{\cont}(G_S,\mu_m)=U_S/U_S^m$.
466It follows that $\Hom_{\cont}(G_S,\mu_m)$ is finite,
467because $U_S$ is finitely generated.
468\end{exercise}
469
470Now let~$E$ be an elliptic curve over~$k$.
471The $m$-torsion points of~$E$ over~$\overline{k}$ form a group
472$E_m=E_m(\overline{k})\ncisom (\Z/m\Z)^2.$  Suppose
473that, in addition to the conditions above,~$S$ also contain
474the places at which~$E$ has bad reduction.  Then it is
475a fact that $E(k_S)$ is divisible by~$m$, so we have an exact
476sequence
477 $$0 \ra E_m \ra E(k_S) \xrightarrow{m} E(k_S)\ra 0.$$
478 Taking cohomology we obtain an exact sequence
479 $$0 \ra E(k)/m E(k) \ra H^1(k_S/k,E_m) 480 \ra H^1(k_S/k,E)_m\ra0,$$
481where the subscript~$m$ means elements killed by~$m$.
482Thus, to prove that $E(k)/m E(k)$ is finite (the
483weak Mordell-Weil theorem''), it suffices to show that
484$H^1(k_S/k,E_m)$ is finite.  Let $k'=k(E_m)$ be the extension of~$k$
485obtained by adjoining the coordinates of the points of order~$m$.  Then
486$k'/k$ is finite and unramified outside~$S$.
487Hence $H^1(k'/k,E_m)$ is finite, and the exact
488inflation-restriction sequence (see Section~\ref{sec:infres})
489$$0 \ra H^1(k'/k,E_m)\ra H^1(k_S/k,E_m) 490 \ra H^1(k_S/k',E_m)$$
491shows that it suffices
492to prove $H^1(k_S/k,E_m)$ is
493finite when $k=k'$.  But then
494$$H^1(k_S/k,E_m) \isom \Hom_{\cont}(G_S,E_m) 495 \isom \Hom_{\cont}(G_S,\mu_m)^2$$
496is finite by Exercise~\ref{ex:finite}.
497
498\begin{exercise}
499Take $k=\Q$ and let~$E$ be the elliptic curve $y^2=x^3-x$.
500Let $m=2$ and $S=\{2\}$, $U_S=\langle -1, 2\rangle$, and
501show $(E(\Q):2 E(\Q))\leq 16$. (In fact, $E(\Q)=E_2$ is of order~$4$,
502killed by~$2$, but to show that we need to
503examine what happens over~$\R$ and over~$\Q_2$, not just use the
504lack of ramification at the other places.)
505\end{exercise}
506
507\begin{exercise}
508Suppose $S'=S\union\{P_1,P_2,\ldots,P_t\}$ is obtained
509by adding~$t$ new primes to~$S$.  Then
510$U_{S'}\isom U_S\cross\Z^t$.  Hence
511$H^1(k_{S'}/k,E)_m\isom H^1(k_S/k,E)\cross(\Z/m\Z)^{2t}$.
512Hence $H^1(k,E)$ contains an infinite number of independent elements of
513order~$m$.  Hilbert Theorem 90 is far from true for~$E$.
514\end{exercise}
515
516\section{Functor of pairs $(G,A)$}
517A {\em morphism of pairs}
518$(G,A)\mapsto(G',A')$
519is given by a pair of maps~$\phi$ and~$f$,
520$$\xymatrix{ 521G&G'\ar[l]_{\phi}}\>\>\>\text{and}\>\>\> 522\xymatrix{ 523A_\phi\ar[r]^{f}&A'}, 524$$
525where~$\phi$ is a group homomorphism, and~$f$ is a homomorphism of
526$G'$-modules, and
527$A_\phi$ means~$A$ with the~$G'$ action induced by~$\phi$.
528A morphism of pairs induces a map
529$$H^r(G,A)\ra H^r(G',A')$$
530got by composing the map
531$H^r(G,A)\rightarrow H^r(G',A_{\phi})$
532induced by~$\phi$
533with the map
534$H^r(G',A_\phi)\ra H^r(G',A')$
535induced by~$f$.
536We thus consider $H^r(G,A)$ as a functor of pairs $(G,A)$.
537
538If~$G'$ is a subgroup of~$G$ then there are maps
539$$\xymatrix{ 540H^r(G,A)\[email protected]/^/[rrr]^{\text{restriction}} 541&&&H^r(G',A).\[email protected]/^/[lll]^{\text{corestriction}} 542}$$
543Here the corestriction map (also called the transfer map'')
544is defined only if the index $[G:G']$ is finite.
545
546When $r=0$ the corestriction map is the trace or norm:
547$$\xymatrix{ 548K\[email protected]{-}[dr]^{G'}\ar[dd]^G\\ 549&K'\[email protected]{-}[dl]\\ 550k}\>\>\phantom{bigspace}\>\> 551\xymatrix{ 552{A^G}\ar[r]^{\res} & {A^{G'}}\[email protected]/^/[l]^{\cores}\\ 553{\displaystyle{\sum_{g\in\{\text{coset reps for G/G'}\}} ga}} 554 &{\begin{array}{c}a.\\ {}\\ \end{array}}\[email protected]{|->}@<-1ex>[l]\\ 555}$$
556
557\begin{corollary}
558If~$G$ is of finite cardinality~$m$, then
559$$mH^r(G,A)=0 \text{ for } r\not=0.$$
560\end{corollary}
561\begin{proof}
562Letting $G'=\{1\}$,
563we have
564$$\text{(corestriction)}\circ\text{(restriction)} = [G:G'] = [G:\{1\}]=m.$$
565Since
566$H^r(\{1\},A)=0$ for $r\neq 0$,
567this composition is~$0$, as claimed.
568\end{proof}
569
570\begin{exercise}
571Restriction to a $p$-Sylow subgroup is injective
572on the $p$-primary component of $H^r(G,A)$.
573\end{exercise}
574
575\section{The Shafarevich group}
576Let~$k$ be a number field,~$\nu$ a place of~$k$,
577and $k_\nu$ the completion of~$k$ at $\nu$.
578Let $\knubar$ be an algebraic closure of $k_\nu$ and
579let~$\kbar$ be the algebraic closure of~$k$ in~$\knubar$.
580These four fields are illustrated in the following diagram.
581$$\xymatrix{ 582&{\overline{k}_\nu}\[email protected]{-}[dr]^{G_\nu}\[email protected]{-}[dl]\\ 583{\overline{k}} && {k_\nu}\\ 584&{k}\[email protected]{-}[ul]^{G_k}\[email protected]{-}[ur]\\ 585}$$
586Let $E$ be an elliptic curve over $k$.  We have
587natural morphisms of pairs
588$$(G_k,E(\kbar))\ra(G_\nu,E(\knubar)),$$
589for each place~$\nu$,
590hence a homomorphism
591$$H^1(k,E)\rightarrow \prod_\nu H^1(k_\nu, E),$$
592where the product is taken over all places of~$k$.
593The kernel of this map is the Shafarevich group
594$\Sha(k,E)$, which is conjectured
595to be finite.
596
597If you can prove that~$\Sha$ is finite, then you will be famous,
598and you will have shown that the descent algorithm to compute
599the Mordell-Weil group, which seems to work in practice, will always work.
600Until 1986, there was no single instance where it was known that~$\Sha$
601was finite!  Now much is known for $k=\Q$ if the
602rank of $E(\Q)$ is~$0$ or~$1$; see ~\cite{kolyvagin} and~\cite{rubin} for
603results in this direction.
604Almost nothing is known for higher ranks.
605
606\section{The inflation-restriction sequence}
607\label{sec:infres}
608Recall that a morphism of pairs
609$$(G,A)\rightarrow (G',A')$$
610is a map $G'\rightarrow G$
611and a $G'$-homomorphism $A\rightarrow A'$,
612where $G'$ acts on~$A$ via $G'\rightarrow G$.
613In particular, we can take~$G'$ to be a subgroup~$H$ of~$G$.
614Here are three special instances of the above map:
615$$\begin{array}{lll} 6161)&\text{restriction}& H^r(G,A)\rightarrow H^r(H,A)\\ 6172)&\text{inflation}& H^r(G/H,A^H)\rightarrow H^r(G,A)\\ 618 &&(\text{for } H\triangleleft G,\> G\rightarrow G/H,\> A^H\subset A) \\ 6193)&\text{conjugation}& H^r(H,A)\stackrel{\tilde{\sigma}}{\rightarrow} 620 H^r(\sigma H\sigma^{-1},A), \>\sigma\in G\\ 621 &&(\text{for } \sigma h\sigma^{-1}\mapsto h\text{ and } 622 a \mapsto \sigma a)\\ 623\end{array}$$
624
625\begin{theorem}
626\label{conjth}
627If $\sigma\in H$, then the conjugation map $\tilde{\sigma}$ is
628the identity.
629\end{theorem}
630
631\begin{exercise}
632Given a commutative algebraic group $C$ defined over $k$ one sometimes
633uses the notation $H^r(k,C):=H^r(k^{\sep}/k,C)$, where $k$
634is a separable algebraic closure of $k$.  Show that this makes
635sense, in the sense that if $k_1^s$ and $k_2^s$ are two separable closures of $k$,
636then the isomorphism
637   $H^r(\Gal(k_1^s/k),C(k_1^s))\isom 638 H^r(\Gal(k_2^s/k),C(k_2^s))$
639induced by a $k$-isomorphism $\vphi:k_1^s\ra k_2^s$
640is independent of the choice of $\vphi$.
641\end{exercise}
642
643\begin{theorem}
644If~$H$ is a normal subgroup of~$G$, then
645there is a Hochschild-Serre'' spectral sequence
646    $$E_2^{rs}=H^r(G/H,H^s(H,A))\Rightarrow H^{r+s}(G,A)$$
647\end{theorem}
648By Theorem~\ref{conjth},~$G$ acts on $H^r(H,A)$ and~$H$
649acts trivially, so this makes sense.
650(The profinite case follows immediately from the
651 finite one by direct limit; cf. the end of Section~\ref{subsec:char}.)
652The low dimensional corner of the spectral sequence
653can be pictured as follows.
654$$\xymatrix{ 655&{E^{02}}\[email protected]{-}[d]\[email protected]{-}[dr]\\ 656&E^{01}\[email protected]{-}[d]\[email protected]{-}[r]\[email protected]{-}[dr]&E^{11}\[email protected]{-}[d]\[email protected]{-}[dr]&{}\\ 657{}&E^{00}\[email protected]{-}[r]&E^{10}\[email protected]{-}[r]&{E^{20}}\\ 658}$$
659Inflation and restriction are edge homomorphisms'' in the
660spectral sequence.
661The lower left corner pictured above gives the obvious isomorphism
662$A^G\isom (A^H)^{G/H}$, and the exact sequence
663$$\xymatrix{ 6640\ar[r] 665&H^1(G/H,A^H)\ar[r]^{\infff} 666&H^1(G,A)\ar[r]^{\res} 667&H^1(H,A)^{G/H}\arr[d]d[][dlll]_dd[][dll]\\ 668&H^2(G/H,A^H)\ar[r]^{\inf} 669&H^2(G,A).&{}\\ 670} 671$$
672The map~$d$ is the transgression'' and is induced by
673$d_2:E_2^{01}\ra E_2^{20}$.
674
675\begin{exercise}\mbox{}\vspace{-3ex}\newline
676\begin{enumerate}
677\item Show that this last sequence, or at least the first line,
678is exact by using standard $1$-cocycles.
679
680\item If $H^1(H,A)=0$, so that $E_2^{r1}=0$ for all~$r$, then
681the sequence obtained by increasing the superscripts on the
682$H$'s by~$1$ is exact.
683\end{enumerate}
684\end{exercise}
685
686Consider a subfield~$K$ of $k^{\sep}$ that is Galois over~$k$, and let~$C$
687be a commutative algebraic group over~$k$.
688$$\xymatrix{ 689&{k^{\sep}}\[email protected]{-}[d]\[email protected]{-}@/^1pc/[d]^{G_K}\[email protected]{-}@/_1pc/[dd]_{G_k}\\ 690&{K}\[email protected]{-}[d]\\ 691{C}\[email protected]{-}[r]&{k.}\\ 692} 693$$
694The inflation-restriction sequence is
695$$\xymatrix{ 6960\ar[r] 697&H^1(K/k,C(K))\ar[r] 698&H^1(k,C(k^{\sep}))\ar[r] 699&H^1(K,C(k^{\sep}))^{\Gal(K/k)}\arr[d]d[][dlll]d[][dll]\\ 700&H^2(K/k,C(K))\ar[r]&H^2(k,C(k^{\sep})). 701&{}\\ 702} 703$$
704If $C={\bG}_m$, then
705$H^1(K,C(k^{\sep}))=0$, and there is an inflation-restriction
706sequence with $(1,2)$ replaced by $(2,3)$:
707$$\xymatrix{ 7080\ar[r] 709&H^2(K/k,K^*)\ar[r] 710&H^2(k,(k^{\sep})^*))\ar[r] 711&H^2(K,(k^{\sep})^*))^{\Gal(K/k)}\arr[d]d[][dlll]d[][dll]\\ 712&H^3(K/k,K^*)\ar[r] &{H^3(k,(k^{\sep})^*)}.&{}\\ 713} 714$$
715An element $\alpha\in H^2(K,(k^{\sep})^*)^{\Gal(K/k)}$
716represents a central simple algebra~$A$ over~$K$
717which is isomorphic to all of its conjugates by $\Gal(K/k)$.  As
718the diagram indicates, the image $\alpha$ in $H^3(K/k,K^*)$ is the
719obstruction'' whose vanishing is the necessary and sufficient
720condition for such an algebra~$A$ to come by base extension from an
721algebra over~$k$.
722
723\section{Cup products}
724
725\subsection{$G$-pairing}
726
727If $A$, $A'$, and~$B$ are $G$-modules, then
728$$A\times A'\stackrel{b}{\rightarrow}B$$
729is a \emp{$G$-pairing}
730if it is bi-additive, and respects the action of $G$:
731$$b(\sigma a,\sigma a')=\sigma b(a,a').$$
732Such a pairing induces a map $\tilde{b}$
733$$\cup:H^r(G,A)\times H^s(G,A')\stackrel{\tilde{b}}{\rightarrow} 734H^{r+s}(G,B),$$
735as follows:
736given cochains~$f$ and~$f'$, one defines
737(for a given $b$) a cochain $f \cup f'$ by
738$$(f\cup f')(\sigma_1,\dots,\sigma_{r+s})= 739b(f(\sigma_1,\dots,\sigma_r),\sigma_1\dots\sigma_r 740 f'(\sigma_{r+1},\dots,\sigma_{r+s})),$$
741and checks the rule
742$$\delta(f\cup f')=\delta f\cup f' + (-1)^rf\cup \delta f'.$$
743If $\delta f=\delta f'=0$, then also
744$\delta(f\cup f')=0$; i.e., if~$f$ and~$f'$ are cocycles, so
745is $f\cup f'$.
746Similarly one checks that the cohomology class of $f\cup f'$
747depends only on the classes of~$f$ and~$f'$.
748Thus we obtain the desired pairing $\tilde{b}$.
749
750If $r=0$ and $a\in A^G$ is fixed, then $a'\mapsto b(a,a')$
751defines a $G$-homomorphism $\vphi_a:A'\ra B$,
752and  $\alpha'\mapsto a\cup \alpha'$ is the map
753$H^r(G,A')\ra H^r(G,B)$ induced by $\vphi_a$.
754
755If~$H$ is a subgroup of~$G$, and
756$\alpha\in H^r(G,A)$ and $\beta\in H^s(H,A')$,
757then we can form
758   $$\res (\alpha) \cup \beta\in H^{r+s}(H,B).$$
759Suppose that the index of~$H$ in~$G$ is finite, so that
760corestriction is defined; then one can show that
761$$\cores(\res (\alpha) \cup \beta)= 762\alpha \cup \cores (\beta)\in H^{r+s}(G,B).$$
763
764\subsection{Duality for finite modules}
765If~$A$ and~$B$ are $G$-modules, we make the group
766$\Hom_\Z(A,B)$ into a $G$-module by
767defining $(\sigma f)(a)=\sigma(f(\sigma^{-1}a)).$
768Note then that $\Hom_G(A,B)=(\Hom_\Z(A,B))^G$. Also,
769the obvious pairing $A\cross \Hom_\Z(A,B)\ra B$
770is a $G$-pairing.  The canonical map
771% I CAN'T REMEMBER THE RIGHT WAY TO DO THIS!
772 $$(*)\qquad\qquad\qquad\qquad\qquad 773 A \ra \Hom_\Z(\Hom_\Z(A,B))\hspace{1.6in}$$
774is a $G$-homomorphism.  In case $A$ is finite, killed by~$m$,
775and~$B$ has a unique cyclic subgroup of order~$m$,
776the map ($*$) is an isomorphism; one can thus recover~$A$
777from its dual'' $\Hom_\Z(A,B)$ which has the same
778order as~$A$.
779There are two especially important such duals for finite~$A$.
780\begin{itemize}
781\item The \emp{Pontrjagin Dual} of~$A$ is
782     $\Hom_\Z(A,\Q/\Z)$; this equals $\Hom_\Z(A,\Z/m\Z)$ if $mA=0$.
783\item The \emp{Cartier Dual} of~$A$ is
784    $\Hom_\Z(A,\mu(k^{\sep}))$; this equals $\Hom_\Z(A,\mu_m(k^{\sep}))$
785    if $mA=0$.
786\end{itemize}
787
788In the Pontrjagin case,~$G$ is an arbitrary profinite group
789and acts trivially on $\Q/\Z$.  Taking limits, this duality extends to a
790perfect duality (i.e., an anti-equivalence of categories) between
791discrete abelian torsion groups and profinite abelian groups.
792
793In the Cartier case, $G=\Gal(k^{\sep}/k)$ or some quotient thereof,
794and $m\neq 0$ in~$k$. (The Cartier dual of a $p$-group in characteristic~$p$
795is a \emp{group scheme}, not just a Galois module.)
796If~$E$ is an elliptic curve over~$k$ and the image of~$m$ in~$k$ is nonzero, the
797\emp{Weil pairing} $E_m(k^{\sep})\cross E_m(k^{\sep})\ra\mu_m$
798identifies $E_m$ with its Cartier dual.
799
800
801\section{Local fields}
802
803Let~$k$ be a local field, i.e., the field of fractions
804of a complete discrete valuation ring
805with finite residue field~$F$.  Let~$K$
806be a finite extension of~$k$.
807
808Fundamental facts:
809\begin{align*}
811H^2(K/k,K^{*})&={\Z}/[K:k]{\Z}\\
812H^2(k,{\bG}_m)&=\Br(k)={\Q}/{\Z} \\
813\end{align*}
814
815The equality $\Br(k)=\Q/{\Z}$
816is given canonically, by the Hasse invariant, as follows:
817The group $\Br(k)$ is the Brauer group, defined in \S\ref{brau}.
818Consider the inflation-restriction sequence for
819$H^2(-,\Gm)$ in the tower of fields
820$$\[email protected]=1.2pc{{\kbar}\[email protected]{-}[d]\\ 821 {k^{\ur}}\[email protected]{-}[d]\\ 822 {k}}$$
823where $k^{\ur}$ is the maximal unramified extension of~$k$.
824Since every central division algebra over a local field
825has an unramified splitting field, we have
826$\Br(k^{\ur})=0$, and hence an isomorphism
827$$\Br(k) \isom H^2(k^{\ur}/k,\Gm) 828 =H^2(\Frob^{\widehat{\Z}},(k^{\ur})^{*}).$$
829Using the exact sequence
830$$\xymatrix{ 8310\ar[r] 832& U(k^{\ur})\ar[r] 833&(k^{\ur})^*\ar[rr]^{\text{valuation}}&{} 834&{\Z}\ar[r] &0}$$
835and the fact that the unit group of an unramified extension has
836trivial cohomology in dimension $\neq 0$,
837we find that we can replace
838$(k^{\ur})^*$ by $\Z$, and hence
839$$\Br(k)\isom H^2(\hat{\Z},\Z) 840 =H^1(\hat{\Z},\Q/\Z)=\Q/\Z;$$
841the middle equality comes from the short exact sequence
842  $$0 \ra \Z \ra \Q \ra \Q/\Z\ra 0$$
843and the fact that~$\Q$, being uniquely divisible,
844has trivial cohomology in nonzero dimensions.
845The resulting map
846    $$\Br(k)\rightarrow{\Q}/{\Z}$$
847is the called the Hasse invariant.
848
849\begin{theorem}\label{thmdual}
850Let~$A$ be a finite $G_k$-module of
851order prime to the characteristic of~$k$.  Let
852$$A^*=\Hom(A,{\bG}_m)=\Hom(A,\mu(\overline{k}))$$
853be the Cartier dual of~$A$.
854Then the $G$-pairing
855$$A\times A^*\rightarrow \overline{k}^{*}$$
856induces a pairing
857$$H^r(k,A)\times 858H^{2-r}(k,A^*)\rightarrow 859H^2(G_k,(k^{\sep})^{*})=\Br(k)={\Q}/{\Z}.$$
860This is a perfect pairing of finite groups, for
861all $r\in\Z$.  It is nontrivial only if $r=0,1,2$, since
862for $r\ge 3$,
863$$H^r(k,A)=0\>\>\text{ for all }A,$$
864i.e., the cohomological dimension of a
865non-archimedean local field is~$2$.''
866\end{theorem}
867
868\emp{Example.} By Kummer theory, we have
869$$k^{*}/(k^{*})^m=H^1(k,\mu_m(\overline{k})).$$
870Thus there is a perfect pairing
871$$\[email protected]=.3pc{ 872H^1(k,{\Z}/m{\Z}) 873 &{\times } 874 &H^1(k,\mu_m({\overline{k}}))\ar[rrr] 875 &&&{\Q}/{\Z}\\ 876\\ 877\\ 878{\Hom(G_k,{\Z}/m{\Z})}\[email protected]{=}[uuu] 879 &{\times} & {k^{*}/(k^{*})^m}\[email protected]{=}[uuu]\\ 880}$$
881The left hand equality is because the action is trivial.
882Conclusion:
883$$G_k^{\ab}/(G_k^{\ab})^m \isom k^*/(k^*)^m.$$
884
885
886Taking the limit gives
887\emp{Artin reciprocity}:
888$$\xymatrix{ 889k^{*}\[email protected]{^{(}->}[r]&G_k^{ab} 890}; 891$$
892the image is dense.
893
894Let $E/k$ be an elliptic curve.
895In some sense,
896$$E=\Ext^1(E,{\bG}_m)$$
897in the category of algebraic groups.
898There is a pairing
899$$H^r(k,E)\times H^s(k,E)\rightarrow H^{r+s+1}(k,{\bG}_m).$$
900For example, taking $r=0$ and $s=1$, we have the following theorem.
901\begin{theorem}\label{thmpontrjagin}
902Let~$E$ be an elliptic curve over a non-archimedean local field~$k$,
903then we have the following
904perfect pairing between Pontrjagin duals.
905$$\[email protected]=.3pc{ 906H^0(k,E)&\times &H^1(k,E)\ar[rrr] &&&{H^2(k,\bG_m)={\Q}/{\Z}}\\ 907\\ 908\\ 909 {\begin{array}{c}E(k)\\ \text{profinite}\end{array}}\[email protected]{=}[uuu] 910 &{\begin{array}{c} \times \\ \end{array}} 911&{\begin{array}{c} H^1(k,E) \\ 912 \text{discrete, torsion}\end{array}}\[email protected]{=}[uuu] 913} 914$$
915\end{theorem}
916\begin{proof}[Sketch of Proof]
917We use the Weil pairing.
918Letting~$D$ denote Pontrjagin dual'',
919we have a diagram
920$$\xymatrix{ 9210\ar[r]&E(k)/mE(k)\ar[r]\ar[d]&H^1(k,E_m)\ar[r]\ar[d]&H^1(k,E)_m\ar[r]\ar[d]&0\\ 9220\ar[r]&H^1(k,E)_m^D\ar[r]&H^1(k,E_m)^D\ar[r] 923 &(E(k)/mE(k))^D\ar[r]&0}$$
924The rows are exact.  The top one from the Kummer sequence, and the
925bottom is the dual of the top one.  The middle vertical
926arrow is an isomorphism by Theorem~\ref{thmdual}.
927The outside vertical arrows are induced by the pairing
928of Theorem~\ref{thmpontrjagin}.
929The diagram commutes, so they are also isomorphisms,
930and Theorem~\ref{thmpontrjagin} follows by passage
931to the limit with more and more divisible~$m$.
932\end{proof}
933
934It was in trying to prove Theorem~\ref{thmpontrjagin} that I was
935led to Theorem~\ref{thmdual} in the late 1950's.  Of course
936the fundamental facts'' and the Artin isomorphism are a
937much older story.
938
939\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
940\begin{thebibliography}{1}
941
942\bibitem{cassels-frohlich}
943J.\thinspace{}W.\thinspace{}S. Cassels and A.~\protect{Fr{\"o}hlich} (eds.),
944  \emph{Algebraic number theory}, London, Academic Press Inc. [Harcourt Brace
945  Jovanovich Publishers], 1986, Reprint of the 1967 original.
946
947\bibitem{haberland}
948K.~Haberland, \emph{Galois cohomology of algebraic number fields}, VEB
949  Deutscher Verlag der Wissenschaften, Berlin, 1978, With two appendices by
950  Helmut Koch and Thomas Zink.
951
952\bibitem{kolyvagin}
953V.\thinspace{}A. Kolyvagin, \emph{On the {M}ordell-{W}eil group and the
954  {S}hafarevich-{T}ate group of modular elliptic curves}, Proceedings of the
955  International Congress of Mathematicians, Vol.\ I, II (Kyoto, 1990) (Tokyo),
956  Math. Soc. Japan, 1991, pp.~429--436.
957
958\bibitem{rubin}
959K.~Rubin, \emph{The main conjectures'' of {I}wasawa theory for imaginary
960  quadratic fields}, Invent. Math. \textbf{103} (1991), no.~1, 25--68.
961
962\bibitem{serre:local}
963J-P. Serre, \emph{Local fields}, Springer-Verlag, New York, 1979, Translated
964  from the French by Marvin Jay Greenberg.
965
966\bibitem{serre:gc}
967\bysame, \emph{Galois cohomology}, Springer-Verlag, Berlin, 1997, Translated
968  from the French by Patrick Ion and revised by the author.
969
970\end{thebibliography}
971
972\end{document}
973