Properties of The Complex IC(X)
The complex ICp(X) has the following properties
- On the complement of some closed set of codimension 2, we have
- is 0 for i+m≠ 0, and for i=−m the groups form the constant local system C
- is 0 for i + m < 0
- If i > 0 then is zero except on a set of codimension at least a for the smallest a with p(a) ≥ m − i
- If i>0 then is zero except on a set of codimension at least a for the smallest a with q(a) ≥ (i)
As usual, q is the complementary perversity to p. Moreover the complex is uniquely characterized by these conditions, up to isomorphism in the derived category. The conditions do not depend on the choice of stratification, so this shows that intersection cohomology does not depend on the choice of stratification either.
Verdier duality takes ICp to ICq shifted by n = dim(X) in the derived category.
Read more about this topic: Intersection Homology
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