(Non)formality of cochains of absolute Galois groups

[posic]

Dear Kirsten,

I am now visiting Ben Gurion University of the Negev in Be'er Sheva,
and Ido Efrat here kindly told me about your paper with
Michael Hopkins about splitting varieties for triple Massey products
http://www.math.harvard.edu/~kwickelg/papers/MPSV.pdf
I didn't yet have the time to read it properly, but there is one question
asked there which I think I know the answer to.

You are asking whether the cochain DG-algebra of an absolute Galois
group of a field F with the coefficients Z/2 is formal. The answer is,
I believe, "not in general", for the following reason.

Given a pro-finite group G and a prime number l, one can consider
the coalgebra Z/l(G) of Z/l-valued locally constant functions on G.
Then the cochain DG-algebra C^*(G,Z/l) of the group G with the constant
coefficients Z/l is otherwise known as the cobar-construction of
the coaugmented coalgebra Z/l(G).

Let G_l denote the maximal quotient pro-l-group of the pro-finite group G.
Then the coaugmented coalgebra Z/l(G_l) is conilpotent (i.e., in its
quotient coalgebra by the constant functions, which is a coalgebra
without counit, every element is killed by the iterated comultiplication
map of high enough order). There is an old paper by Sasha Vishik and
me discussing these concepts, http://arxiv.org/abs/alg-geom/9507010

It follows that the coalgebra Z/l(G_l) can be recovered from
the quasi-isomorphism class of the DG-algebra C^*(G_l,Z/l). In fact,
simply applying the bar-construction to the augmented DG-algebra
C^*(G_l,Z/l) produces a DG-coalgebra quasi-isomorphic to Z/l(G_l).
Unlike the cobar-construction, the bar-construction always takes
quasi-isomorphic DG-algebras to quasi-isomorphic DG-coalgebras.
This is all rather straightforward homological algebra; see. e.g.,
subsection 6.10 of my AMS Memoir "Two kinds of derived categories..."
http://arxiv.org/abs/0905.2621

In the case of the absolute Galois group G of a field F containing
a primitive l-root of unity, denoting by K the kernel of the surjective
map G \to G_l, one discovers that the obvious vanishing of H^1(K,Z/l)
implies H^*(K,Z/l) = 0 in all the degrees by the Milnor-Bloch-Kato
conjecture (Rost-Voevodsky theorem). Hence H^*(G,Z/l) = H^*(G_l,Z/l)
and the DG-algebras C^*(G,Z/l) and C^*(G_l,Z/l) are quasi-isomorphic
(perhaps there are simper ways to prove this).

Hence, for an absolute Galois group G of a field F as above,
the bar-construction of the cochain DG-algebra C^*(G,Z/l) computes
the coalgebra Z/l(G_l). This means that if the cochain DG-algebra
is formal, then the coalgebra Z/l(G_l) admits a positive grading
making it a quadratic (and, actually, Koszul) graded coalgebra.

Basically, the dual topological algebra to Z/l(G_l) is the quotient
algebra of the algebra of noncommutative formal power series
(in, generally speaking, an infinite number of variables) by
the closed ideal generated by some relations. If the cochain algebra
is formal, one can make these relations homogeneous by changing
the generatiors (replacing them with new ones that are some formal
power expressions in terms of the original ones).

Very simple examples show that this cannot be done in general.
The following one is taken from subsection 9.11 of my paper in
Moscow Math. Journal-2011, http://arxiv.org/abs/1006.4343 (where
you can find some further discussion).

Let F be a finite extension of F_p((z)) or Z_p containing a square
root of -1 (where p is an odd prime). So F is just a local field.
Then the group G_2 is the semidirect product of two copies of
Z_2, one of them acting nontrivially in the other one.

Its cohomology algebra H^*(G_2,Z/2), however, is simply
the exterior algebra in two variables. So if the cochain DG-algebra
C^*(G_l,Z/2) or C^*(G,Z/2) were formal, the group coalgeba Z/2(G_2)
would have to be isomorphic to the cocommutative symmetric
coalgebra in two variables, which is not the case (as the group
G_2 is not abelian).

With best wishes,
Leonid

Comments

[posic.livejournal.com]

Однако! Оно не только есть на Архиве, чего я умудрился не приметить:

http://arxiv.org/abs/1210.4964

но и уже выходит (почитай, вышло) из печати

http://www.sciencedirect.com/science/article/pii/S0022404914001625 , http://dx.doi.org/10.1016/j.jpaa.2014.06.006

Мне всегда казалось, что топологи бывают очень наивны, со своей наивной верой в формальность -- и вот, действительно.

[posic.livejournal.com]

Впрочем, надо признать, что контрпримера к вопросу Хопкинса-Викельгрен, являющегося полем, содержащим алгебраически замкнутое подполе (или просто все корни из единицы порядков, являющихся степенями l) я до сих пор никакого не могу привести. Хотя думаю, что ответ отрицательный и для таких полей тоже.