Interpretation of hypercohomology

Question

Let $X$ be a scheme and $K \in D_{qc} (X)$. For simplicity assume that $K$ is concentrated only in cohomological degrees, that is $H^{-i} (K) = 0$ for $i>0$. My question is the following: is there any interpretation of $H^0, H^1$? The question has an affirmative answer if $K$ comes from $Sh(X)$. However, it's not clear to me if $K$ is a complex and not merely a sheaf. Of course, it can generally be anything as worse as we want: we can take $F[-n]$ for a sheaf $F$. However, I'm wondering if anything can be said when $K= T_0 \to T_1 \to \cdots$ where $T_0$ is in degree $0$. It's already interesting for me when $K = T_0 \to T_1.$

Answer 1

There are definitely a few ways to think about this, but one which helps generally is to think homotopically. That is, sheaves should be thought of as nice assignments of discrete data to your space, and sheaf cohomology has an interpretation as obstructions to taking sections (and obstructions to these obstructions, etc). The reason this is intuitive is that sections of “discrete” data are pretty straightforward, one can read things off at once from the sheaf condition.

In the derived world, one can view an object of a derived category of sheaves as still assigning data to open subsets, but now the data is “not discrete”. In particular, your stalks can be concentrated in multiple degrees, and now your sections can’t be computed as cleanly, since the “shape” of your sections might cause problems.

We can make this more intuitive with an example. Consider covering your space with two opens $U$ and $V$, with intersection $U\cap V$. Then Mayer Vietoris in hypercohomology (= homotopic global sections) says that one obtains a degree “$n$ section” over $X$ with two sections over $U$ and $V$ that agree on $U\cap V$, however this section isn’t unique, since really we are saying the two sections are homotopic over $U\cap V$, and the data of a homotopy of these two sections is an $n-1$ section over $U\cap V$.

This can be made precisely in the sense that sets can be interpreted as discrete spaces, but it’s abstract homotopy theory at play here, rather than just for spaces. This also clarifies where derived functors can come from, and why one needs to take projective/injective resolutions, since this is (co)fibrant replacement.

Answer 2

A simple and useful interpretation is $$ H^0(K) = \mathrm{Hom}(\mathcal{O}_X,K), \qquad H^1(K) = \mathrm{Hom}(\mathcal{O}_X,K[1]) = \mathrm{Hom}(\mathcal{O}_X[-1],K). $$