Bolondi, Giorgio Seminatural cohomology and stability. (English) Zbl 0598.14014 C. R. Acad. Sci., Paris, Sér. I 301, 407-410 (1985). A reflexive sheaf F on \({\mathbb{P}}^ 3\) of rank 2 is said to have ”seminatural cohomology” if for every n with \(2n+c_ 1(F)+4\geq 0\), at most one group \(H^ i(F(n))\) is different from zero. R. Hartshorne and A. Hirschowitz have proved in Ann. Sci. Éc. Norm. Supér., IV. Sér. 15, 365-390 (1982; Zbl 0509.14015)] that a locally free sheaf with seminatural cohomology is necessarily stable; this is extended here to reflexive sheaves with the following exceptions. Normalise so that \(c_ 1=0\) or -1. Then the exceptional cases are \((c_ 1,c_ 2,c_ 3)=(0,1,2),\quad (0,2,6),\quad (0,3,10),\quad (-1,1,3).\) Sheaves in these cases are related to curves of degree 5, 6, 7, on quadrics or to cubic curves (but a precise classification is not given). Reviewer: C.T.C.Wall Cited in 1 Document MSC: 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010) 14D20 Algebraic moduli problems, moduli of vector bundles 18G20 Homological dimension (category-theoretic aspects) Keywords:stableness of reflexive sheaf; seminatural cohomology Citations:Zbl 0509.14015 PDFBibTeX XMLCite \textit{G. Bolondi}, C. R. Acad. Sci., Paris, Sér. I 301, 407--410 (1985; Zbl 0598.14014)