# Cohomology groups for singular non-compact variety with paving by affine spaces

Suppose $$X$$ is a complex algebraic variety that is paved by affines. We take the most general definition, which is that $$X$$ has a filtration $$0=X_0 \subset X_1 \subset \cdots \subset X_n=X$$, each $$X_i$$ a closed subvariety of $$X$$, such that $$X_i\setminus X_{i-1}$$ is a disjoint union of a finite number of copies of affine spaces $$\mathbb{C}^{d_{ij}}$$ (possibly of different dimensions). (See Optimal definition of "paving by affine spaces"? for a discussion of the different possible definitions.)

If you know all the dimensions $$d_{ij}$$, then you know the ranks of all the Borel-Moore homology groups (aka homology with compact support) of $$X$$. When $$X$$ is nonsingular, this gives the ranks of the (ordinary) cohomology groups by Poincare duality, and when $$X$$ is compact, this gives the ranks of the cohomology groups by the universal coefficient theorem because homology equals Borel-Moore homology.

Is there a way to get the ranks of the singular cohomology groups out of the $$d_{ij}$$ when $$X$$ is neither compact nor nonsingular? Is it even true that the total rank of cohomology is the number of "cells"?

I am interested in the case where $$X$$ is reducible, but you can assume that the affine paving is compatible with the decomposition into irreducible components and that I have the data of which "cells" in the paving belong to which irreducible components (which includes the data of the dimension of each component).