$n$-blocks collections on Fano manifolds and sheaves with regularity $-\mathrm{\infty }$

Let $X$ be a smooth Fano manifold equipped with a `` nice ‘’ $n$-blocks collection in the sense of \cite{cm2} and $\mathcal{F}$ a coherent sheaf on $X$. Assume that $X$ is Fano and that all blocks are coherent sheaves. Here we prove that $\mathcal{F}$ has regularity $-\mathrm{\infty }$ in the sense of \cite{cm2} if $Supp(\mathcal{F})$ is finite, the converse being true under mild assumptions. The corresponding result is also true when $X$ has a geometric collection in the sense of \cite{cm1}.