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Let $X$ be a smooth Fano manifold equipped with a `` nice '' $n$-blockscollection 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 weprove that $\mathcal {F}$ has regularity $-\infty$ in the sense of \cite{cm2}if ${Supp}(\mathcal {F})$ is finite, the converse being true under mildassumptions. The corresponding result is also true when $X$ has a geometriccollection in the sense of \cite{cm1}.

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