Cohomology of coherent sheaves on projective schemes

Let X be a projective scheme of dimension n over a an algebraically closed field k and let OX denote its structure sheaf. Let F be any coherent sheaf of OX-modules. In this essay, we wish to compute the sheaf cohomology Hi(X,F), for i ∈ Z. We will see the Serre’s duality which asserts that the k-finite dimensional vector space Hn−i is dual to Exti(F,ωX), for i ≥ 0, where ωX is the canonical sheaf on X. Applying this result to a smooth projective curves of genus g, we will be able to prove the Riemann–Roch theorem.