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

For a finite abelian p-extension K/k of totally real fields and the cyclotomic ℤ_p-extension K_∞ /K, we prove a strong version of an equi-variant Iwasawa main conjecture by determining completely the Fitting ideal over ℤ_p [[Gal(K_∞ /k)]] of the classical Iwasawa module X_K∞,S, which is the Galois group of the maximal pro-p abelian S-ramified extension of K_∞, where S contains all ramified primes in K_∞ /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 ℤ over a suitable group ring.