Zariski’s Main Theorem
نویسنده
چکیده
By base change, i′n is also a closed embedding, hence affine. We have the unit map F → in∗i ′∗ n (F ). Applying Rf∗ gives a map Rf∗(F ) → Rf∗i ′ n∗i ′∗ n (F ). Since i ′ n is affine, Ri′n∗ ≃ i ′ n∗. By Leray’s spectral sequence, Rf∗i ′ n∗ ≃ R(f ◦ i ′ n)∗ ≃ R(in ◦ fn)∗ ≃ in∗Rfn∗. Applying H, we have a map Rf∗(F ) → R f∗i ′ n∗i ′∗ n (F ) ≃ in∗R fn∗(i ′∗ n (F )). Applying in∗i ∗ n to both sides and using i ∗ nin∗ ≃ id, we have a map in∗i ∗ nR f∗(F ) → R f∗(Fn) where Fn := i ′ n∗i ′∗ n (F ) is coherent on X . Since f is proper, R f∗(F ), R f∗(Fn) are coherent sheaves on Y for all k ≥ 0. With a slight abuse of notation, we can write the above map more intuitively as
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