Where can I find a definition of $\underline{H}^p(X, \mathscr{F})$?

Let $$X$$ be a topological space and $$\mathscr{F}$$ a sheaf on $$X$$. In the paper Tropical cycle classes for non-archimedean spaces and weight decomposition of de Rham cohomology sheaves by Yifeng Liu, the notation $$\underline{H}^p(X, \mathscr{F})$$ is used. (More generally, $$\mathscr{F}$$ could be an object of the derived category of sheaves on $$X$$, e.g. some complex of sheaves.) For example we have the expression $$H^0(X, \underline{H}^p(X, i_{Z!} i_Z^! \mathscr{K}_X^p))$$ appearing right before Lemma 2.4 or $$H^0(U, \underline{H}^p(U, i_{Z!}i_Z^! \mathscr{D}_U^{p,\bullet}))$$ in the proof of Theorem 3.9.

It seems to be some kind of sheaf, but I cannot find its definition anywhere. Is it the sheaf associated to the presheaf $$U \mapsto H^p(U, \mathscr{F})$$? Where can I read some basic properties about this construction? I searched in the Stacks Project, in Hartshorne, Lei Fu's book on etale cohomology, Gelfand-Manin's homological algebra, but I couldn't find it.

• Milne, Etale Cohomology, p. 84, et seq.
– user483792
Jul 20, 2022 at 19:41

As indicated in the comments, the notation $$\def\HH{\underline{\rm H}}\def\H{{\rm H}}\HH^p(X,F)$$ is defined (for example) by Milne in Étale Cohomology as the $$p$$th right derived functor of the inclusion functor from the category of sheaves of abelian groups to the category of presheaves of abelian groups on the same site. This is also known as hypercohomology and is studied under this name by Grothendieck, although I could not locate this notation in his paper.
In modern terms, we can compute $$\HH^p(X,F)$$ by replacing $$F$$ with a locally quasi-isomorphic presheaf $$G$$ of chain complexes that satisfies homotopy descent, e.g., by fibrantly replacing in the local injective model structure on presheaves or sheaves of chain complexes, or the local projective model structure on presheaves of chain complexes. Then $$\HH^p(X,F)$$ is simply the $$p$$th cohomology group of $$G$$ computed in the abelian category of presheaves of abelian groups.
• Thanks for your answer! I'm still a bit confused. If I understand it correctly, hypercohomology generalizes sheaf cohomology from sheaves to complexes of sheaves, but for complexes concentrated in degree zero, it coincides with ordinary sheaf cohomology. So if I view $\underline{H}^p(X, F)$ as the presheaf sending $U$ to $p$-th hypercohomology of $F$ over $U$, I should have $H^0(X, \underline{H}^p(X, F)) = H^p(X, F)$ for purely formal reasons, no? But in the paper cited by me, this seems to be a non-trivial fact, holding only in certain situations: Jul 21, 2022 at 8:06
• Before Lemma 2.4 it says: “By Lemma 2.2(3) (vanishing of some cohomology groups) and the local to global spectral sequence, we obtain a canonical isomorphism $H^0(X, \underline{H}^p(X, i_{z!} i_Z^! \mathscr{K}_X^p)) \cong H^p(X, i_{Z!} i_Z^! \mathscr{K}_X^p)$”. Jul 21, 2022 at 8:08
• @JakobWerner: If H^0 refers to the ordinary sheaf cohomology, then it sheafifies its argument first. A priori this may produce a different answer. However, looking at Section 5 (Example 5.2) of the arXiv version (which is where the local-to-global spectral sequence is mentioned), he says explicitly that the $k$th hypercohomology group vanishes for $k<p$. In particular, the $p$th hypercohomology presheaf is a sheaf of abelian groups (and not just a presheaf), so its 0th sheaf cohomology coincides with its global sections. Jul 21, 2022 at 15:25