Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 126 / 2 / i

Table of contents
- Solution i

Solution

For an affine morphism

f : X \to Y

and a quasi-coherent sheaf

F

, every inverse image of an affine open

V \subseteq Y

is affine. Higher cohomology of a quasi-coherent sheaf on an affine scheme vanishes, so

(R^{q} f_{*} F) ∣_{V} = H^{q} (f^{- 1} V, F) = 0 (q > 0) .

The Leray spectral sequence therefore has only its zeroth row, and its edge maps give

H^{p} (X, F) ≃ H^{p} (Y, f_{*} F) (p \geq 0) .

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