Intersection Homology - Sheaf Theory

Sheaf Theory

Deligne's formula for intersection cohomology states that

where IC_p(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 IC_p(X) is given by starting with the constant sheaf on the open set X−X_n−2 and repeatedly extending it to larger open sets X−X_n−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 i_k is the inclusion of X−X_n−k into X−X_n−k−1 and C_X−Xn−2 is the constant sheaf on X−X_n−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−X_n−2 with a local system, one can use Deligne's formula to define intersection cohomology with coefficients in a local system.