Title: Chern classes and cohomology for rank 2 reflexive sheaves on P<sup>3</sup>
Abstract: The paper shows that, if F is a nonsplit rank 2 reflexive sheaf on P 3 , then the knowledge of the numbers d n = h 2 (F(n)) -h\F(n)) gives an explicit algorithm to compute the Chern classes C\, Cι, c?> and the dimensions h°(F(n)), for all n (in particular the first integer a such that the sheaf F(a) has some nonzero section).If the sheaf is a vector bundle it is also proved that the knowledge of the numerical sequence {h ι (F(n))} together with the first Chern class gives all the information as above.In some special cases, i.e. when h x (F(n)) Φ 0 for at most three values of n, an algorithm is also produced to compute the first Chern class from the sequence {h ι (F(n))} .Vector bundles with natural cohomology are also discussed.It must be remarked that, if one knows not only the dimensions h ι (F(n)), for all n, but also the whole structure of the Rao-module Q)H ι (F(n)), then the first Chern class C\ is uniquely determined (as it is shown in a paper by P. Rao).