Comparing direct limit and inverse limit of even $K$-groups in multiple ${\mathbb{Z}}_p$-extensions

Lim, Meng Fai

Journal de théorie des nombres de Bordeaux, Tome 36 (2024) no. 2, pp. 537-555

Iwasawa first established a duality relating the direct limit and the inverse limit of class groups in a ${\mathbb{Z}}_p$-extension, and this result has recently been extended to multiple ${\mathbb{Z}}_p$-extensions by many authors. In this paper, we establish an analogous duality for the direct limit and the inverse limit of higher even $K$-groups in a ${\mathbb{Z}}_p^d$-extension. We then give some examples where the direct limit may or may not vanish.

Iwasawa a établi une dualité entre la limite directe et la limite inverse des groupes des classes dans une ${\mathbb{Z}}_p$-extension. Récemment, ce résultat a été généralisé au cas de ${\mathbb{Z}}_p$-extensions multiples par de nombreux auteurs. Dans cet article, nous établissons une dualité analogue entre la limite directe et la limite inverse des $K$-groupes en degrés pairs dans une ${\mathbb{Z}}_p^d$-extension. Nous donnons ensuite quelques exemples où la limite directe peut être nulle ou pas.

Classification : 11R23, 11R70, 11S25
Keywords: Even $K$-groups, $\mathbb{Z}_p^d$-extension.

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