The second syzygy of the trivial $G$-module, and an equivariant main conjecture

Abstract

For a finite abelian $p$-extension $K/k$ of totally real fields and the cyclotomic $\mathbb{Z}_{p}$-extension $K_{\infty}/K$, we prove a strong version of an equivariant Iwasawa main conjecture by determining completely the Fitting ideal over $\mathbb{Z}_{p}[[\mathop{\rm Gal}\nolimits(K_{\infty}/k)]]$ of the classical Iwasawa module $X_{K_{\infty},S}$, which is the Galois group of the maximal pro-$p$ abelian $S$-ramified extension of $K_{\infty}$, where $S$ contains all ramified primes in $K_{\infty}/k$. To do this, we prove a conjecture which was proposed in a previous paper by the first and second author, concerning the minors of a relation matrix for the second syzygy module of the trivial module $\mathbb{Z}$ over a suitable group ring.