Rational cohomology of toric diagrams

Abstract

In this note, (rational) Betti numbers of homotopy colimits for toric diagrams and their classifying spaces are described in terms of sheaf cohomology over CW posets. We prove for any $T$-diagram $D$ over any CW poset that Cohen-Macaulayness (over $\mathbb{Q}$) of the $T$-action on $hocolimD$ is equivalent to acyclicity for a certain sheaf. The ordinary and bigraded Betti numbers are computed for skeletons of equivariantly formal spaces from this class (in particular, of compact smooth toric manifolds).

v4: 15 pages. Compactly supported cohomology replaced by singular cohomology. Minor changes, typos corrected. Comments are welcome!