Talk:Decomposition theorem of Beilinson, Bernstein and Deligne

How does the hard Lefschetz theorem induce the decomposition?

The article states:

This hard Lefschetz isomorphism induces canonical isomorphisms

$Rf_{*}(\mathbb {Q} ){\stackrel {\cong }{\to }}\bigoplus _{i=-d}^{d}R^{d+i}f_{*}(\mathbb {Q} )[-d-i].$

But I how does this follow? Does anyone know a reference?

Nielius (talk) 08:50, 8 March 2017 (UTC)

added a reference.Jakob.scholbach (talk) 20:40, 8 March 2017 (UTC)

Suggestions

1. There needs to be better examples showing what the decomposition theorem actually does. For example, showing the decomposition theorem for blow-ups, resolutions of singularities, and springer resolutions.
2. Computation of intersection cohomology of singular varieties
3. Statement and examples of perverse spectral sequence
4. The decomposition theorem should be stated for intersection cohomology sheaves. This is a much better description because it shows us how the simple objects decompose.

I agree with all your suggestions. Please go ahead, this is the encyclopaedia that everyone can edit! Jakob.scholbach (talk) 12:37, 10 June 2017 (UTC)