Decomposition theorem

In mathematics, especially algebraic geometry the decomposition theorem is a set of results concerning the cohomology of algebraic varieties.

The first case of the decomposition theorem arises via the hard Lefschetz theorem which gives isomorphisms, for a smooth proper map ${\displaystyle f:X\to Y}$ of relative dimension d between two projective varieties

Here ${\displaystyle \eta }$ is the fundamental class of a hyperplane section, ${\displaystyle f_{*}}$ is the direct image (pushforward) and ${\displaystyle R^{n}f_{*}}$ is the n-th derived functor of the direct image. This derived functor measures the n-th cohomologies of ${\displaystyle f^{-1}(U)}$, for ${\displaystyle U\subset Y}$. In fact, the particular case when Y is a point, amounts to the isomorphism

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