K. Similarly, for every divisor D, we get It follows from the Riemann-Roch formula that this inclusion is an isomorphism. Corollary. For a connected curve X, the space H 1 (X, stl) is 1-dimensional. 3. The Serre Duality. Similarly one can establish the duality for (smooth, projective, and irreducible) n-dimensional varieties. We can find a sufficiently ample and smooth (by Bertini's theorem) divisor Y C X. Using th" Poincare residue, we get an exact sequence of sheaves I. n~- 1 ---+ 51 o.

Indeed, for a spaee eonsisting of a point, eh is just the dimension of a vector space. Note that deg ean also be thought of as a direet image homomorphism f*: A(X)Q ---+ A(Y)Q = Q. This leads to the following more general statement of the Riemann-Roeh theorem. Let f: X ---+ Y be a proper morphism of smooth varieties. For a sheaf F on X, we set fk(F) = 2:::( -1)qRq f*(F). We getan (additive) homomorphism fk: K(X)---+ K(Y). Now, the Riemann-Roeh formula deseribes the extend to whieh the Chern eharacter eh falls to eommute with direet images.

However, one can easily show that after a suitable twisting by O(m), we get a sheaf F(m) that has sufficiently many global sections, i. , there exists an epimorphism oN ---4 F(m). By twisting these sheaves, we see that any coherent sheaf F on IP' is a quotient sheaf of a sheaf of the form 0( -m)N. This fact is similar to a representation of a module as a quotient module of a free module, and it plays a similar role. Suppose we want to prove a general assertion concerning coherent sheaves. We verify it, first, for sheaves of the form O(m), next, for direct sums of such sheaves, and finally, using a resolution consisting of such sheaves, we extend the assertion to arbitrary coherent sheaves.

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