Title: Blocks of the category of smooth $\ell$-modular representations of $GL(n,F)$ and its inner forms: reduction to level-$0$,
Abstract: 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
$\mathscr{R}_R(G)$ be the category of smooth representations of $G$ over $R$.
In this paper, we prove that a block (indecomposable summand) of
$\mathscr{R}_R(G)$ 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 $\mathscr{R}_R(G')$, where $G'$ is a direct product
of groups of the same type of $G$.