We pursue the original strategy of my paper at JIMJ and we give a new
criterion so that the localization of the cohomology of KHT Shimura variety is
free. Precisely let $G$ be a similitude group with signatures
$(1,d-1),(0,d),\cdots,(0,d)$ and $V_{\xi,\overline{\mathbb Z}_l}$ a local
system associated to a fixed algebraic representation $\xi$ of $G(\mathbb Q)$.
Consider a system $\mathfrak m$ of Hecke eigenvalues appearing in the free
quotient of the cohomology group in middle degree of the Shimura variety $Sh_K$
associated to $G$ and with coefficients in $V_{\xi,\overline{\mathbb Z}_l}$.
Then if the modulo $l$ galoisian representation $\overline \rho_{\mathfrak m}$
is irreducible of dimension $d$, and essentially if $l \geq d+1$, the
localization at $\mathfrak m$ of every cohomology group of $Sh_K$ with
coefficients in $V_{\xi,\overline{\mathbb Z}_l}$, is free.