Blocks of the category of smooth l-modular representations of GL(n,F) and its inner forms: reduction to level-0

Let G be an inner form of a general linear group over a non-archimedean locally compact field of residue characteristic p, let R be an algebraically closed field of characteristic different from p and let R_R(G) be the category of smooth representations of G over R. In this paper, we prove that a block (indecomposable summand) of R_R(G) is equivalent to a level-0 block (a block in which every object has non-zero invariant vectors for the pro-p-radical of a maximal compact open subgroup) of R_R(G'), where G' is a direct product of groups of the same type of G