In Wiles' proof of ``$R = {\Bbb T}$", if we associate the extra primes the set of which is denoted $D$, there is a famous result in the following$\colon$
Theorem(De Shalit's Lemma). Let ${\Bbb T}_N$ is the reduced Hecke algebra associated to $X_0(N)$. Then, ${\Bbb T}_N \otimes_{\Bbb Z}{\Bbb Z}_p $ is a free ${\Bbb Z}_p[\Delta_D]$-module, where $\Delta_D$ is the maximal $p$-primary quotient of $({\Bbb Z}/N)^*$ with $N = \underset{p \in D}{\prod} p$.
I am looking for the generalisation of this result to totally real cases, i.e. Hilbert modular forms. K.Fujiwara seems not have published his paper, so when one works with Hecke algebras associated to Shimura curve $C$ over totally real field $F$, one needs to know the proof of the similitude of De Shalit's Lemma.
In the case of Modular curve, the problem is more or less reduced to the group cohomology, and via Shapiro's lemma Free group provides its cohomology as free ${\Bbb Z}[\Delta_D]$-module. This comes from the fact that the $1$-cocycle is determined by the value of elements of generators of the free group.
I am wondering whether the proof goes in the similar manner in the case of Shimura curve $C$ over a totally real number field.
