Search Results

Classically, the functor $f^!$ is indeed not a right adjoint in general. Clausen and I have recently found a way to make it a right adjoint in general, by enlarging the category of modules to that of …

Let me highlight some quite unexpected (to me) geometric applications of the recent work on integral $p$-adic Hodge theory. Namely, prismatic cohomology, along with a $p$-adic form of the Riemann--Hil …

Your supposed example works indeed. More generally, I think whenever the étale part is not of locally constant height one will run into problems. Here's a proof that the $p$-divisible group $G$ of the …

Let me add there is now a reference for the claims in Dan Peterson's answer, namely Marco Volpe has worked out the Topological $6$-functor formalism. I also gave some (brief) account of this in Lectur …

Good question! We've been trying to figure this out as we went along, but so far unsuccessfully. Some more precise points: For many (but definitely not all) applications to geometry over the real num …

I'm sure there are easier and better ways to think about this, but here's how I like to think about it. Work on the big pro-etale site on all schemes, which maps to the pro-etale site of a point, $\pi …

I don't know much about this stuff, so instead of answering the question, I try to formulate more precise questions, in the hope someone else will take up these questions: One of the main reason to l …

Flatness in analytic geometry is an interesting question! As Dustin says, it comes with several important caveats. First, open immersions may not be flat even in the weakest sense of the word. Here is …

Classical Grothendieck-Messing theory relates deformations of $p$-divisible groups to lifts of the Hodge filtration (if the ideal defining the nilpotent immersion is equipped with a PD-structure). If …

I have finally found some time to write up the $6$-functor formalisms in coherent cohomology (a la Gaitsgory--Rozenblyum) and for $D$-modules, see Lecture 8 and its appendix. A short synopsis is that …

This is really just an elaboration of Emerton's comment: You should read Mark Kisins' review of Faltings's paper "Almost etale extensions". But I wanted to elaborate: Faltings regards the almost puri …

Somehow that question slipped my radar, sorry! The truth is that shamefully I'm not able to say much, as I don't have a strong knowledge of resolution of singularities. But at least so far, the flow o …

Isn't the dualizing complex defined in general in the proper case by taking applying the right adjoint of $\pi_\ast$ to $k$? That's what I'll take as the definition anyways. The Gorenstein property ju …

The question is local on $X$, so we may assume that $\mathcal E$ is finite free, of rank $n$, say. In that case, as also SashaP points out, the question amounts to the question whether $\mathbb A^n_X$ …

This is a sequel to the question Accumulation of algebraic subvarieties: Near one subvariety there are many others (?) . Let $Y$ be some projective variety, over $\mathbb{C}$. Let $X\subset Y$ be som …