Sheaf Theory
Deligne's formula for intersection cohomology states that
where ICp(X) is a certain complex of sheaves on X (considered as an element of the derived category, so the cohomology on the right means the hypercohomology of the complex). The complex ICp(X) is given by starting with the constant sheaf on the open set X−Xn−2 and repeatedly extending it to larger open sets X−Xn−k and then truncating it in the derived category; more precisely it is given by Deligne's formula
where τ≤p is a truncation functor in the derived category, and ik is the inclusion of X−Xn−k into X−Xn−k−1 and CX−Xn−2 is the constant sheaf on X−Xn−2. (Warning: there is more than one convention for the way that the perversity enters Deligne's construction: the numbers p(k)−n are sometimes written as p(k).)
By replacing the constant sheaf on X−Xn−2 with a local system, one can use Deligne's formula to define intersection cohomology with coefficients in a local system.
Read more about this topic: Intersection Homology
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