The purpose of this
note is twofold. Part I consists of an example of an algebraic scheme which
is the union of two closed, quasi-projective subscheme, but which is not
itself quasiprojective. The main result of Part II is a structure theorem for
coherent sheaves over divisorial schemes and, as an application, the proof that
Theorem 2 of Borel-Serre’s paper “Le Théorèm de Riemann-Roch”, which is
stated only for quasiprojective, nonsingular schemes, can be extended to
arbitrary nonsingular schemes. (See the Remark on page 108 of the mentioned
paper.)
