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$ I have a particular $X,Y$ in mind (the inclusion of one compact complex manifold into another, each with even-degree cohomology) but I'm hoping phrases like "Postnikov tower", "cofibrant replacement …

In the context of coherent sheaves, very often the approach is to compute Euler characteristic, give some reason that higher sheaf cohomology vanishes, and conclude that one has computed $H^0$. … Serre gave a talk (around 1998) in which he explained that "It was once believed that the only good sheaf cohomology is dead sheaf cohomology" but "nowadays this is not the [politically] correct view." …

That computes T^m-equivariant cohomology, in which the GL(m)-equivariant sits as the S_m-invariant part. …

Pull back the equation $[W] = [H]^{codim\ W} \in \mathbb P^N$. This cohomological pullback is computable set-theoretically, as $[W\cap G]$, if the intersection is transverse.

In general for a Grassmannian $Gr_k(\mathbb C^n)$, the Schubert polynomials are Schur polynomials in $k$ variables, one for each partition $\lambda$ in a $k\times (n-k)$ rectangle. In this case $S_\la …

The Schubert classes on $G/P$ are the classes of the Schubert varieties, which are the closures of the Schubert cells, each of which contains a unique $T$-fixed point. The $T$-fixed points on $G/P$ ar …