Ring-theoretic properties of certain Hecke algebras
type: Item
instance of: scholarly article
title: Ring-theoretic properties of certain Hecke algebras
zbMATH Open document ID: 0823.11030
DOI: 10.2307/2118560
published in: Annals of Mathematics. Second Series
publication date: 1995-08-10T00:00:00Z
review text: This paper provides a key fact needed in the previous paper by \textit{A. Wiles} [Ann. Math. (2) 141, No. 3, 443--551 (1995; Zbl 0823.11029); namely that (certain) Hecke algebras are complete intersections. The key ideas are a computation of Euler characteristics (Tate-Poitou) and the introduction of auxiliary primes \(p\equiv 1\bmod p^ n\) such that the projective limit (as \(n\to \infty\)) of the corresponding Hecke algebras is a power series ring. To see this one estimates its minimal number of generators (say \(d\)) and then shows that its dimension is at least \(d+1\). To derive the assertion one uses that the original Hecke algebra can be defined by dividing this projective limit by \(d\) equations. It should be noted that this treats only the minimal case, where the condition ``minimal'' is used in the computation of Euler characteristics. An appendix explains a remark of the reviewer which allows one to simplify some arguments. The authors were probably too exhausted to find this additional shortcut.
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 11F11
Mathematics Subject Classification ID: 11D41
Mathematics Subject Classification ID: 13C40
Mathematics Subject Classification ID: 14M10
Mathematics Subject Classification ID: 14H52
zbMATH DE Number: 783778
zbMATH Keywords: Galois representations
zbMATH Keywords: Fermat's last theorem
zbMATH Keywords: Hecke algebras
zbMATH Keywords: complete intersections
zbMATH Keywords: Euler characteristics
Wikidata QID: Q18227964
MaRDI profile type: Publication
OpenAlex ID: W2016499826
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author: Richard Taylor
author: Andrew Wiles
reviewed by: Gerd Faltings
description: scientific article; zbMATH DE number 783778