Massey products \(\langle y,x,x,\ldots ,x,x,y\rangle\) in Galois cohomology via rational points
type: Item
instance of: scholarly article
title: Massey products \(\langle y,x,x,\ldots ,x,x,y\rangle\) in Galois cohomology via rational points
zbMATH Open document ID: 1380.55013
author: Kirsten Wickelgren
published in: Journal of Pure and Applied Algebra
publication date: 2017-03-17T00:00:00Z
full work available at URL: https://arxiv.org/abs/1601.06318
review text: A smooth curve \(X\) over a field \(k\) with a choice of basepoint can be embedded into its generalized Jacobian, Jac \(X\). Given a rational point of \(X\) there is a commutative diagram between Spec \(k\), \(X\), and Jac \(X\). Applying the functor \(\pi_1^{et}\) we obtain a diagram where we ask about the lifting of a map. For \(X= P_k^1\setminus\{0,1,\infty\}\) the obstructions for such a lifting are higher order Massey products. The whole study is motivated by Grothendieck's section conjecture, and Belyi's theorem suggests to study such \(X\). The main result is as follows. Theorem. Let \(x\) be an element of \(k^*\setminus \{1\}\). The \(n\)-th order Massey product \[ \langle (1-x)^{-1},x^{-1},\dots,x^{-1},(1-x)^{-1}\rangle \] contains \(0\) and \(-[f]\cup (1-x)^{-1}\) (where \(f\) is a certain cocycle). Let \(x\) be an element of \(k^*\). The \(n\)-th order Massey product \[ \langle -x^{-1},x^{-1},\dots,x^{-1},-x^{-1}\rangle \] contains \(0\) and \(-([f]\cup (-x)^{-1})\).
reviewed by: N. Kalinin
Mathematics Subject Classification ID: 55S30
Mathematics Subject Classification ID: 11S25
Mathematics Subject Classification ID: 14H30
zbMATH DE Number: 6695853
zbMATH Keywords: Massey product
zbMATH Keywords: Grothendieck section conjecture
zbMATH Keywords: Belyi theorem
zbMATH Keywords: Galois cohomology
zbMATH Keywords: lower central series
zbMATH Keywords: group cohomology
MaRDI profile type: Publication
OpenAlex ID: W2564037549
arXiv ID: 1601.06318
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DOI: 10.1016/J.JPAA.2016.12.027
description: scientific article