Math 233a Final Presentation: Serre Duality for Projective Spaces
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چکیده
Let V be a vector space of dimension n+1 over a field k, and consider the scheme X = PV ∼= Pk ∼= Proj(k[x0, . . . , xn]). Consider F a quasi-coherent sheaf over X. We can examine its Cech cohomology which coincides with its sheaf (nonetale) cohomology because X is Noetherian and separated (cf. Hartshorne, ch. III theorem 4.5.). Particularly, H(X,F) ∼= Γ(F , X). However Γ(F , X) ∼= HomX(OX ,F). Indeed, the identification is obtained as follows: if s ∈ Γ(F , X) we define φ : OX → F by φU : Γ(OX , U) → Γ(F , U) given by φU (x) = x · ResUs. Conversely, if φ : OX → F gets sent to φ(1X) ∈ Γ(F , X). Moreover this identification is clearly an isomorphism of k-vector spaces. We can therefore consider a k-bilinear map H(X,F) ⊗k H(X,F ′) → H(X,F ⊗X F ′) defined as follows for any s ∈ H(X,F) identified with φ : OX → F as above, we have the map F ′ ∼= OX ⊗X F ′ → F ⊗X F ′ induced by φ, and hence by functoriality of homology we have the map φs : H(X,F ′) → H(X,F ⊗X F ′) whence we have the map H(X,F) ⊗k H(X,F ′) → H(X,F ⊗X F ′) given by s ⊗ t → φs(t). Take now F = O(i) and F ′ = O(−n − 1 − i). Since H(X,O(−n− 1)) ∼= k, the above bilinear map reads
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