TY - JOUR
T1 - Generalized Bockstein maps and Massey products
AU - Lam, Yeuk Hay Joshua
AU - Liu, Yuan
AU - Sharifi, Romyar
AU - Wake, Preston
AU - Wang, Jiuya
N1 - This paper arose out of a project of the first, second and fifth authors at the 2018 Arizona Winter School on Iwasawa theory that was proposed by the third and led by the third and fourth authors. We would like to thank the AWS for the stimulating environment that enabled our collaboration. We would also like to thank Nguyễn Duy Tân for helpful comments on an earlier draft of this article, as well as the anonymous referee for a number of suggestions that helped us to improve the exposition. The second author’s research was partially supported by the National Science Foundation under Grant No. DMS-2200541. The third author’s research was supported in part by the National Science Foundation under Grant No. DMS-2101889. The fourth author’s research was supported in part by the National Science Foundation under Grant No. DMS-1901867. The fifth author was partially supported by a Foerster-Berstein Fellowship at Duke University and the National Science Foundation under Grant No. DMS-2201346.
PY - 2023/1/23
Y1 - 2023/1/23
N2 - Given a profinite group G of finite p-cohomological dimension and a pro-p quotient H of G by a closed normal subgroup N, we study the filtration on the Iwasawa cohomology of N by powers of the augmentation ideal in the group algebra of H. We show that the graded pieces are related to the cohomology of G via analogues of Bockstein maps for the powers of the augmentation ideal. For certain groups H, we relate the values of these generalized Bockstein maps to Massey products relative to a restricted class of defining systems depending on H. We apply our study to prove lower bounds on the p-ranks of class groups of certain nonabelian extensions of and to give a new proof of the vanishing of Massey triple products in Galois cohomology.
AB - Given a profinite group G of finite p-cohomological dimension and a pro-p quotient H of G by a closed normal subgroup N, we study the filtration on the Iwasawa cohomology of N by powers of the augmentation ideal in the group algebra of H. We show that the graded pieces are related to the cohomology of G via analogues of Bockstein maps for the powers of the augmentation ideal. For certain groups H, we relate the values of these generalized Bockstein maps to Massey products relative to a restricted class of defining systems depending on H. We apply our study to prove lower bounds on the p-ranks of class groups of certain nonabelian extensions of and to give a new proof of the vanishing of Massey triple products in Galois cohomology.
KW - 20J05 20J06 12G05 11R23 11R34
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U2 - 10.1017/fms.2022.103
DO - 10.1017/fms.2022.103
M3 - Article
AN - SCOPUS:85146267887
SN - 2050-5094
VL - 11
JO - Forum of Mathematics, Sigma
JF - Forum of Mathematics, Sigma
M1 - A1
ER -