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2
votes
1answer
144 views

the sixth morphism in the long exact sequence associated to the Hochschild-Serre spectral sequence

The long exact sequence associated to the Hochschild-Serre spectral sequence for extension of groups $1 \to H \to G \to G/H \to 1$ is $$ \begin{array}[t]{lll} 1 \to & H^1(G/H, A^{G/H}) ...
3
votes
2answers
319 views

Galois cohomology of the field of Laurent series

Let $k$ a separably closed field. Do we have that $k((t))$ is of cohomological dimension one?
2
votes
0answers
88 views

do commutative groups torsors have a point in an Abelian extension of the base field?

Let $A$ be a principal homogeneous space for a commutative algebraic group defined over a field $k$ that contains all roots of unity. Is it true that $A$ has a $K$-point for an extension $K \supset k$ ...
1
vote
1answer
154 views

gluing gerbes over a spectrum of a field

A theorem of Giraud says that gerbes over a scheme $X$ bounded by a sheaf of Abelian groups $A$ are classified by elements of the etale cohomology group $H^2(X,A)$. Similar statements hold in other ...
4
votes
2answers
330 views

Projectives in the category of discrete G-modules

If $G$ is a profinite group, then the category $Mod(G)$ of discrete $G$-modules has sufficiently many injectives (Neukirch, Schmidt, Wingberg: Cohomology of Number Fields, 2.6.5). Since the cited ...
2
votes
0answers
264 views

Where does the name Euler System come from?

I've recently been reading about Euler systems, and was curious where the name comes from. In particular, while the notion of an Euler system is still not rigorously defined, does the idea resemble ...
2
votes
0answers
316 views

galois cohomology over finite field

Let $X$ a smooth projective geometrically connected curve over a finite field $k$. Let $J$ a smooth commutative group scheme over $X$ and $F$ the function field of $X$. Do we have a formula to ...
3
votes
1answer
293 views

Langlands Paper on representations of abelian algebraic groups

I have been working through Langlands paper which you can see here http://www.sunsite.ubc.ca/DigitalMathArchive/Langlands/pdf/AbelianAlg-ps.pdf and I can understand why one of his maps is obvious and ...
1
vote
1answer
154 views

action of automorphisms on the Galois cohomology of the function field of a variety

Let $C$ be a (quasi-projective) variety over an algebraically closed field $k$ and let $k(C)$ be its field of rational functions. Then for any birational map $\sigma: C \dashrightarrow C$ there is an ...
1
vote
1answer
179 views

Is the number of twists of a curve with a section in a given field finite

Let $X$ be a smooth projective geometrically connected curve over a number field $k$ of genus $g\geq 2$. Is the number of twists of $X$ always infinite? (The answer is no, because there aren't any ...
0
votes
1answer
228 views

surjectivity of rational points induced by surjective map from affine space

Let $k$ be a local field of char $0$ (which is the case I concern). Let $V$ be a variety defined over $k$ and let $f: \mathbb A^n\to V$ be a surjective map (over the algebraic closure of $k$) ...
1
vote
1answer
462 views

Cohomology of Groups at Gregory Berhuy's Book

Regarding Gregory Berhuy's book "An Introduction to Galois Cohomology and its Applications": The book defined a cohomology sets for non-abelian $G$-groups. Let $A$ be a $G$-group, we define a ...
3
votes
1answer
213 views

What's the minimum number of generators for the wild inertia?

Suppose $K$ is a finite extension of $\mathbb{Q}_p$ and $K^{nr}$ the maximal unramified extension of $K$ in some fixed algebraic closure. Let $G_K$ be the absolute Galois group of $K$ and let $I_w$ be ...
12
votes
1answer
722 views

Explicit Bijection between Central Simple Algebras and twists of $\mathbb P^n$

The automorphism group of the algebra of $n$-dimensional matrices over a field $K$ is $PGL_n(K)$. The automorphism group of $n-1$-dimensional projective space over $K$ is also $PGL_n(K)$. Therefore, ...
1
vote
1answer
235 views

Biprincipal spaces and their composition, following Serre's “Galois Cohomology”

When reviewing my notes and Serre's book "Galois Cohomology" Chapter 5 dealing with non-abelian group cohomology, I realized that I don't fully understand the concept of biprincipal spaces such as ...
1
vote
1answer
372 views

Isomorphism between the set of classes of Principal Homogeneous spaces and non-Abelian H^1(G,A) cohomology

Let A be a G-group, i.e. a set on which G acts on, has a group structure and satisfies $^s(xy)=^s x ^s y$ for all $x,y \in A \ , s \in G$. A homogeneous principal space P is a non-empty G-set on which ...
16
votes
1answer
746 views

Leopoldt's conjecture and cup-products

Among the many equivalent formulations of Leopoldt's conjecture, this one is probably the shortest: For any number field $K$, prime number $p$, finite set $S$ of primes of $K$ containing the primes ...
4
votes
2answers
622 views

Galois cohomology H^1(Q_p, Z_p(2)) = 0?

For Tate twists Z_p(2), which is defined by the projective limit of \mu_{p^m}(2) over all m>0, I would like to calculate H^1(Q_p, Z_p(2)). I guess this is zero, but cannot prove it. Is it possible ...
1
vote
0answers
153 views

Galois cohomology of generic points of formal completions (of components of a hypercovering of the subvariety): examples or general statements?

Let $Y$ be a closed smooth subvariety in a (smooth) affine variety $X$. What can one say about the etale cohomology of the generic points of the formal completion of $X$ along $Y$ i.e. about the ...
0
votes
3answers
437 views

Brauer group of complete DVR

Let $A$ be a complete discrete valuation ring with fraction field $K$ and perfect residue field $\kappa$. Let $K_{nr}$ be the maximal unramified extension of $K$ and let $A_{nr}$ be its ring of ...
1
vote
1answer
282 views

a question on continuity of $G$-module for a profinite group $G$

I have seen the following statment somewhere, for example in Appendix B2 on Silverman's book "The Arithmetic of Elliptic Curves" : Let $M$ be an abelian group with discrete topology and $G$ be a ...
0
votes
0answers
247 views

Ordered Cech(-like) complexes that compute etale cohomology (of fields!)

It is well known (cf. Equivalence of ordered and unordered cech cohomology. ) that for 'usual' topologies one can compute the cohomology of sheaves either using unordered Cech complexes or ordered ...
5
votes
1answer
309 views

Equality of Galois modules

Let $k$ be a number field. Let $M$ be a (continuous) $\text{Gal}(\overline{k}/k)$-module. One can define two subgroups of the Galois cohomology group $H^i(k,M)$: the group of elements of $H^i(k,M)$ ...
6
votes
1answer
1k views

p-adic Hodge theory for varieties defined over \C _p ?

I have a question on p-adic Hodge theory: When e.g. $X$ is a smooth proper scheme over a finite extension $K$ of $\mathbf{Q}_{p}$ then e.g. one variant of $p$-adic Hodge theory says that there is a ...
2
votes
1answer
329 views

Additive form of Hilbert 90 for schemes?

First, I am by no means well-versed on cohomology so I apologize if this is too elementary. I have been going through some basics of etale cohomology, with my ultimate goal being an understanding of ...
0
votes
1answer
407 views

Poitou-Tate dualities for Galois representations into power series rings?

Suppose $K$ is a finite extension of $\mathbf{Q}_p$, $A=K[[T_1,\dots,T_n]]$, $V$ a finite-rank free $A$-module, and $\rho:G_{\mathbf{Q}} \to \mathrm{GL}(V)$ a continuous Galois representation. Are ...
3
votes
2answers
340 views

Why does $H^1(G_p/I_p,\mathbb{F}(\delta\epsilon^{-1}))$ vanish?

For a finite field $\mathbb{F}$ of char $p$ $\geq 2$, let $f$ be a normalised eigenform of weight $k\geq 2$ that is ordinary at $p$ and $\overline{\rho}_f$ be the mod $p$ Gal representation attached ...
9
votes
1answer
684 views

Galois descent for K-groups (or for étale cohomology groups)

Let $F/K$ be a Galois extension of number fields with Galois group $G$. Let $\mathcal{O}_F$ and $\mathcal{O}_K$ be the associated rings of integers, and let $n\geq 1$. When is $$ ...
23
votes
1answer
2k views

How is etale cohomology of integer rings related to Galois cohomology?

In the paper of Bloch and Kato in the Grothendieck Festschrift, and some other papers relating to the Bloch-Kato conjecture and the ETNC, the cohomology groups $H^i_{\mathrm{et}}(\operatorname{Spec} ...
4
votes
1answer
810 views

Galois cohomology groups given by étale cohomology

What are cases when Galois cohomology groups are given by étale cohomology? Example: $S = Spec(K)$ the spectrum of a field, $F \in Sh(K)$, then $H^p(K, F) = H^p(G_K, F_{\bar{K}})$. What if $G = ...
12
votes
2answers
859 views

Compatibility of Bloch-Kato and Beilinson-Bloch

Suppose $V/K$ is a smooth projective variety. Let $\mathrm{Ch}^{j}(V)_{0}$ be the group of codimension-$j$ homologically trivial $K$-rational cycles on $V$, modulo rational equivalence. A conjecture ...
12
votes
1answer
613 views

Is there a canonical height on the Weil-Chatelet group?

Jon Hanke and I were just chatting and realized we didn't know the answer to the following question. If E is an elliptic curve over a number field, is there in any sense a "canonical height" on the ...
7
votes
1answer
355 views

Is H^2(W_p,C^times) well-known?

Let $W_p$ be the Weil group of $\mathbf{Q}_p$. What is the Galois cohomology group $H^2(W_p,\mathbf{C}^{\times} )$ (with trivial action)? Is it zero, or something huge and complicated? (This group ...
5
votes
1answer
980 views

What kind of structures allow Galois descent?

EDIT: Question solved. Let me explain what I mean. The classical formulation of Galois descent, e. g. in Crawley-Boevey's "Cohomology and central simple algebras", uses the following notion: ...
6
votes
0answers
321 views

Real approximation for homogeneous spaces of linear algebraic groups

Let $X$ be a smooth geometrically integral variety over $\mathbf{Q}$, having a $\mathbf{Q}$-point. We say that $X$ has the real approximation property if $X(\mathbf{Q})$ is dense in $X(\mathbf{R})$. ...
23
votes
1answer
2k views

Iwasawa main conjectures vs Bloch-Kato conjectures

Let $p$ be a prime, $K$ be a number field, $S$ a finite set of finite places of $K$ containing the set $S_p$ of places above $p$ and the places at infinity, $G:=G_{K,S}$ the Galois group of the ...
22
votes
1answer
964 views

Does the everywhere unramified extension of Q(mu_37) of degree 37 grow into a Z_37-extension?

Let $p=37$. Since $p$ divides the numerator of $B_{32}$, by Ribet's proof of the converse of Herbrand's theorem, we know that the class group of ${\bf Q}(\mu_p)$ has size divisible by $p$. More ...
3
votes
2answers
680 views

Restriction of scalars of simple algebraic groups

I'm trying to understand the following basic property of the restriction of scalars: Given an absolutely simple algebraic groups $G$ defined over a number field $k$, are there at most finitely (up-to ...
1
vote
0answers
172 views

twisted forms of a given group embedded in a second group?

Consider the following question about forms of a given group that are embedded in a fixed group. Fix for simplicity $k$ a perfect field, and $H\subsetneq G$ a pair of connected reductive $k$-groups, ...
1
vote
1answer
604 views

Is the direct limit of Weil restriction of an elliptic curve a scheme?

In a discussion today on the Shafarevich-Tate group of an elliptic curve, the following structure and question came up. I will abuse many notations and be very vague about some things, but am very ...
11
votes
0answers
643 views

What is the Shafarevich-Tate group of GL(2)?

Let $K$ be a number field, $\bar{K}$ a fixed algebraic closure, $G_K$ the absolute galois group, $O_\bar{K}$ the ring of integers of the algebraic closure, $\mathfrak{p}$ runs over prime ideals and ...
26
votes
3answers
2k views

Why aren't there more classifying spaces in number theory?

Much of modern algebraic number theory can be phrased in the framework of group cohomology. (Okay, this is a bit of a stretch -- much of the part of algebraic number theory that I'm interested ...
5
votes
1answer
807 views

Galois groups via cohomology

I would like to know about references for the following result (point 3): Let $K/k$ be a normal extension (I am interested in number fields, but everything should work in fields of characteristic ...
14
votes
1answer
2k views

Are all Galois cohomology groups also étale cohomology groups?

Let $K$ be a field and $K^s$ a separable closure of $K$, and let $\mathcal{F}$ be a sheaf on $\mathrm{Spec}(K)$ (in the étale topology). By Grothendieck's Galois Theory, we have the isomorphism ...
11
votes
2answers
795 views

Hilbert 90 for algebras

Let $L\diagup K$ be a Galois extension of fields satisfying $\left[L:K\right] < \infty$. Let $B$ be a finite-dimensional (as a $K$-vector space) $K$-algebra. Then, the Galois group $G$ of $L\diagup ...