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 $-\infty$ 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}.
Categories: math.AG