# Subspace inclusion with non-vanishing higher direct images

I'm looking for concrete topological intuition for the derived pushforward.

Let $$f:X\to Y$$ be a continuous map. The derived pushforward $$\mathbf Rf_\ast$$ takes a sheaf $$F$$ to the sheafification of the cohomology presheaf $$V\mapsto \mathrm H^\bullet(f^{-1}V,F)$$. When $$f$$ is the identity, the sheafification is zero for $$n\geq 1$$.

The sheafification of a presheaf $$P$$ can be constructed by taking mapping $$PU$$ to the equivalence class of families of sections of $$P$$ defined on an open cover of $$U$$, where we identify families which coincide on sufficiently small open covers. A section $$s\in PU$$ is mapped to the equivalence class it represents.

Hence the fact the above sheafification is zero when $$f=1$$ expresses the fact sheaf cohomology is global in nature (every cocycle admits a cover on which it's zero).

Using the above construction of sheafification, a section in $$\mathbf Rf_\ast F(V)$$ is an equivalence class of a family of cocycles $$(\Gamma_i\in \mathrm H^\bullet(f^{-1}V_i,F))$$ where $$(V_i)\twoheadrightarrow V$$ is an open cover, and we identify families if they coincide on a sufficiently small preimage of an open cover.

The sheafification map is non-zero for $$n\geq 1$$ if there's a cocycle of $$F$$ on some preimage $$f^{-1}V$$ which is not killed by any preimage of an open cover of $$V$$.

When $$f:X\subset Y$$ is a subspace inclusion the above means there's a cocycle of $$F$$ on $$f^{-1}V=X\cap V$$ such which does not restrict to zero on any open nighborhood in $$f^{-1}V_0=X\cap V_0$$ in $$X$$ of some problematic point $$x_0\in X$$.

This can't happen for closed embeddings because their pushforward functor is exact.

Question 1. What's an instructive example of a subspace inclusion whose higher direct images are nonzero?

Question 2. For "what kind of maps" $$f$$ does one expect non-zero higher direct images non-zero? (Examples welcome.)

Lastly, I'd appreciate references with explicit topological examples the derived pushforward.

• Are you sure about locally closed embeddings? Because if $X$ is the spectrum of a local ring, then $\Gamma \colon \mathbf{Ab}(X) \to \mathbf{Ab}$ is exact since it agrees with taking the stalk at the closed point. But if $j_* \colon \mathbf{Ab}(U) \to \mathbf{Ab}(X)$ is exact as well, then global sections on $U$ is exact, which we know is not always true. Apr 21 '20 at 19:04
• @R.vanDobbendeBruyn, I corrected the question. Thanks! Apr 21 '20 at 19:07

You expect the higher direct images of $$f: X \to Y$$ to be nonzero if for arbitrarily small neighborhoods $$y_0 \in U$$ the space $$f^{-1}(U)$$ has non-vanishing higher cohomology. I.e. $$R^i f_* \mathbb Z$$ vanishes if and only if all of its stalks do.
For instance, think about the inclusion of $$\mathbb R^2 - 0$$ into $$\mathbb R^2$$. Taking $$y_0 = 0$$, you see that if $$U$$ is a small ball around $$y_0$$, then $$f^{-1}(U) = U - 0$$ is homotopy equivalent to a circle, which has $$H^1 = \mathbb Z$$
At every other point, the preimages of contractible neighborhoods are contractible. So, from looking at the stalks, we see that $$R^1f_* \mathbb Z$$ is $$i_* \mathbb Z$$ where $$i$$ is the inclusion of the origin.
The takeaway is, to understand $$R^i f_*$$ you need to understand the cohomology of preimages $$f^{-1}(U)$$ for $$U$$ small.