One of the main annoyances in the theory of adic spaces is that, for a given Huber pair , the structure presheaf on is not always a sheaf. One usually remedies this by restricting attention to various classes of Huber rings, e.g. strongly Noetherian Tate rings, perfectoid Tate rings, Noetherian adic rings, etc. However, the following class of rings doesn’t seem to be addressed in the literature:
Definition. Let be an adic ring with finitely generated ideal of definition . We say is strongly Noetherian outside if, for all , the scheme is Noetherian.
Here denotes the usual ring of restricted power series. Note that if is a Tate ring and is any couple of definition, then is strongly Noetherian if and only if is strongly Noetherian outside . I should also point out that the condition of being strongly Noetherian outside is already considered in a very interesting paper of Fujiwara-Gabber-Kato; they use the terminology “topologically universally rigid-Noetherian”, but I prefer my terminology on account of the previous sentence. Anyway, the following conjecture seems reasonable:
Conjecture. If is strongly Noetherian outside , the structure presheaf on is a sheaf.
This implies that any strongly Noetherian Tate ring is sheafy (which of course is already known), but it also implies e.g. that if is topologically finitely presented over for some nonarchimedean field , then is sheafy. Sheafiness in the latter situation is known when is discretely valued, but to the best of my knowledge it’s open for general .
I’m sure this conjecture is within reach, and maybe it’s an easy exercise for experts on sheafiness (*cough* Kiran *cough*). Note that FGK already proved some interesting consequences of this definition, which are probably relevant to proving this conjecture. Precisely, they show that if is strongly Noetherian outside , then:
1. The -power-torsion submodule of any finitely generated -module is killed by a power of .
2. If is any inclusion of -modules, with finitely generated, then the subspace topology on induced by the -adic topology on coincides with the -adic topology on .
Let be some immersion of varieties over a separably closed field. Everyone knows that the intermediate extension functor on perverse sheaves (say with coefficients in ) is pretty great: it’s totally canonical, it commutes with Verdier duality, it preserves irreducibility, it preserves monic and epic maps of perverse sheaves, etc.
Recently I noticed that if is any map of varieties, with smooth and irreducible, there’s still a natural functor which commutes with Verdier duality. To define this functor, note that for any and any , there is a natural map , obtained by adjunction from the composite map (the first isomorphism here is the projection formula). Since is smooth, the dualizing complex is just , so then . Thus we get a natural map .
Next, note that the complex is concentrated in degrees , and in the lowest of these degrees it’s just the constant sheaf, i.e. . In particular, there is a canonical map . Shifting by and tensoring with gives a map . Putting things together, we get a natural map . Set so after shifting this becomes a natural map
This shifting has the advantage that Verdier duality exchanges the functors and on , and one can check that the Verdier dual of identifies with .
Definition. The functor sends any to the image of the map .
Here of course denotes the zeroth perverse cohomology sheaf.
Exercise. Show that .
It might be interesting to compute this functor in some other examples. Note that it can be quite stupid: if is a closed immersion (with ) and is already supported on , then . On the other hand, if is smooth and surjective, then is faithful.