where B, called the base locus, is the intersection of the divisors of zero of nonzero sections in V (see Linear system of divisors#A map determined by a linear system for the construction of the map). Cohomology of coherent sheaves Let X be a projective scheme over a field (or, more generally over a Noetherian ring A). Cohomology of coherent sheaves on X satisfies the following important theorems due to Serre: is a finite-dimensional k-vector space for any p. There exists an integer (depending on ; see also Castelnuovo–Mumford regularity) such that for all and p > 0, where is the twisting with a power of a very ample line bundle These results are proven reducing to the case using the isomorphism where in the right-hand side is viewed as a sheaf on the projective space by extension by zero. The result then follows by a direct computation for n any integer, and for arbitrary reduces to this case without much difficulty. As a corollary to 1. above, if f is a projective morphism from a noetherian scheme to a noetherian ring, then the higher direct image is coherent. The same result holds for proper morphisms f, as can be shown with the aid of Chow's lemma. Sheaf cohomology groups Hi on a noetherian topological space vanish for i strictly greater than the dimension of the space. Thus the quantity, called the Euler characteristic of , is a well-defined integer (for X projective). One can then show for some polynomial P over rational numbers. Applying this procedure to the structure sheaf , one recovers the Hilbert polynomial of X. In particular, if X is irreducible and has dimension r, the arithmetic genus of X is given by which is manifestly intrinsic; i.e., independent of the embedding. The arithmetic genus of a hypersurface of degree d is in . In particular, a smooth curve of degree d in has arithmetic genus . This is the genus formula. Smooth projective varieties Let X be a smooth projective variety where all of its irreducible components have dimension n. In this situation, the canonical sheaf ωX, defined as the sheaf of Kähler differentials of top degree (i.e., algebraic n-forms), is a line bundle. Serre duality Serre duality states that for any locally free sheaf on X, where the superscript prime refers to the dual space and is the dual sheaf of . A generalization