Applications of $h$-topology and $h$-descent

Question

This is a technical problem about applications of Grothendieck topologies. In some recent works, the technique of $h$-topology and $h$-descent is very useful, for an introduction see https://stacks.math.columbia.edu/tag/0ETQ.

However, it seems that such technique is not well-known (for example, there are very few discussions on MathOverflow). What are some good applications of such technique? Some examples are

• Original work by Voevodsky to study the homology of schemes.
• Beilinson's approach to p-adic Hodge theory.
• Bhatt and Scholze's work on projectivity of Witt affine Grassmanian.

And why is $h$-topology and $h$-descent technique useful ? In philosophy, it captures mainly the topological information of schemes (although it's not subcanonical). But I hope to understand more about technical advantages.

Answer 1

I guess a relevant part of the philosophy is that schemes are smooth locally in the $h$-topology. So, if we are in characteristic zero, we can use resolution of singularities to produce an $h$-hypercovering $U_\bullet\to X$ of a scheme $X$. Then, if we have something like a cohomology theory, which is defined on smooth schemes and satisfies $h$-descent, then a very natural way of extending it to arbitrary schemes $X$ is to evaluate it on $h$-hypercoverings $U_\bullet\to X$. This way, we can extend things from to schemes without changing anything on smooth schemes.

The other relevant point is that a lot of things we know and love on smooth schemes have $h$-descent. (Maybe this is a combination that we know many things that have étale descent and where we can control what happens in blowups.) For example, differential forms and the de Rham complex. The above strategy to take something on smooth schemes and extend via $h$-hypercoverings was used implicitly in Deligne's Theorie de Hodge III to define the mixed Hodge structure for a proper variety, way before the explicit definition of the $h$-topology. More recent examples of the use of the $h$-topology to deal with differential forms on non-smooth schemes can be found in the following papers:

• A. Huber and C. Jörder. Differential forms in the h-topology, Alg. Geom. 1(4), 449-478, (2014). https://arxiv.org/abs/1305.7361

• A. Huber, S. Kebekus and S. Kelly. Differential forms in positive characteristic avoiding resolution of singularities. Bull. Soc. Math. France 145 (2017), 305-343. https://arxiv.org/abs/1407.5786

• A. Huber. Differential forms in algebraic geometry - a new perspective in the singular case, Port. Math. 73 (2016), no. 4, 337–367.

Modifications of this are possible. Motivic cohomology or K-theory for smooth schemes don't have etale descent and therefore can't have h-descent. However, they satisfy cdh-descent, so for instance the definition of motivic cohomology for non-smooth schemes is as cdh-hypercohomology of a suitable cycle complex. In characteristic $p$, we don't have resolution of singularities, and there are modifications of $h$-related topologies where we can use alterations instead.