Algebraic Geometry II: Cohomology of Algebraic Varieties. by V. I. Danilov (auth.), I. R. Shafarevich (eds.) PDF

Thus H 1 (X,Ox) is not trivialand even infinite-dimensional. 5. Cohomological Dimension. A consequence of the theorem on affine coverings is the vanishing of the cohomology Hq for large q. Precisely, if a scheme X can be covered by n open affine charts, then Hq(X, F) = 0 for q?. n. In particular, the cohomology Hq of an arbitrary sheaf on lP'n are trivial for q > n. On the other hand, in the next section, we will see that there are sheaves F on lP'n with Hn(JP>n, F) I 0. Since an arbitrary n-dimensional projective variety X admits a covering by n + 1 affine charts, Hq(X, F) = 0 for q > n = dimX.

1. The Riemann-Roch Theorem for Curves. Tobegin with we consider a classical problem. Let X be a smooth projective curve over an algebraically closed field, and D = LPEX np[P] a divisor on X. The Riemann problern is to describe H 0 (X, Ox(D)). The elements ofthis space arerational functions f on X with restrictions on the order of zeros and poles, namely: ordp(f) 2 -np for every point P E X. As we know from the preceding section, the space H 0 (X, O(D)) and its dimension depend considerably on the divisor D.