Derived categories of (coherent) sheaves of modules: exceptional images, gluing, and proper descent?

Question

I am interested in the properties of (the derived categories) of various categories of (coherent) sheaves of modules (over varieties). I would like to understand in what extent these properties are similar to those of (etale) constructible sheaves and what are the differences. I have looked at Derived categories of coherent sheaves: suggested references? but I still do not know the answers to the following questions: 1. Do the categories in question become 'very bad' if the base variety is not proper or is not smooth? 2. Are there (derived) exceptional images (i.e. $Rf^!$ and $Rf_!$) defined in this setting? 3. Could one 'glue' somehow (some version) of the derived category of sheaves of modules over $X$ from those over $Z$ and over $X\setminus Z$ ($Z$ is a closed subvariety of $X$)? 4. Does there exist a version of proper descent for this setting?

Any comments or references would be very welcome!

Answer 1

1. It depends what you mean by bad. The categories still do what they are meant to do even if the underlying variety is not proper or smooth. However, there are some subtleties. For instance, if you try to pushforward a coherent sheaf along a non-proper morphism, of course the result might only be quasicoherent. Likewise, if you try to pullback a coherent sheaf from a non-regular base, then the result might be unbounded (since you might have to take an infinitely long locally free resolution). However, perfect complexes (an intrinsically defined subcategory of $D^{b}_{Coh}(X)$ behave well under pullback.

2. Since you already have problems for pullback and pushforward, $Rf^!$ and $Rf_!$ could also not exist in non-proper or non-smooth settings. However, under fairly general hypotheses these functors will exist for the unbounded derived category of quasi-coherent sheaves. See for instance the Springer Lecture Notes of Lipman on Grothendieck duality and references therein.

3. Gluing (recollement) can be a bit of a problem depending on the codimension of $Z$. I think it is fine for $Z$ of codimension at least $2$. (Actually, see the below comment of t3suji pointing out that codimension doesn't help.) You always have a localization sequence $$D^{b}_{Z, Coh(X)}(X) \rightarrow D^{b}_{Coh(X)}(X) \rightarrow D^{b}(X \setminus Z).$$

However, $D^{b}_{Z, Coh(X)}(X) \rightarrow D^{b}_{Coh(X)}(X)$ might not have the desired adjoints to get a recollement and so glue. The right adjoint would be local cohomology, but these might not be coherent. ( However, you do have gluing for unbounded derived categories of quasi-coherent sheaves.

4. I don't know the answer to this, but probably reading Toen would help. I would guess that you have to work not with triangulated categories but with their natural dg enhancements, since morphisms don't glue in the triangulated categories. That is, already for the identity morphism, I think you have problems. For instance, take a short exact sequence of vector bundles that doesn't split, so the connecting homomorphism is non-trivial. However, the sequence splits locally, so the connecting homomorphism is locally trivial.

Answer 2

Briefly re 2 and 4: the Ind-completion IndDCoh has shriek pullbacks and star pushforwards. Moreover, it has what one might call "derived h-" descent with respect to shriek pullback. This includes "derived proper" descent -- note that derived is modifying the topology, not just the functors/categories; more on this below.

Just to clarify: by DCoh I mean the "bounded coherent" dg- or infinity-category, and by Ind the infinity-categorical Ind-completion (e.g., close up the dg-Yoneda image by filtered homotopy colimits). This is modelled by Krause's homotopy category of objectives, Positselski's coderived category, and variations. In particular, it will not coincide with the usual quasi-coherent derived category unless one imposes regularity: The usual version is IndPerf, so one needs Perf=DCoh. But the gain of this definition is shriek pullback without restricting to cohomologically bounded below complexes, and things like proper descent. (And both IndDCoh and IndPerf have flat descent..)

An important point is that one has to pass to derived schemes (not just complexes or dg-categories) to have any hope in the proper case. For non-flat proper maps, the descent condition will thus look very different from the classical versions: one needs to take derived fiber products in forming the Cech nerve.

An example giving a feeling for the previous paragraph: Consider the map from a point to a non-reduced point. It is a proper cover, and its classical and derived Cech nerves look very different! Also considering this example one sees that any sort of proper *hyper*descent will always be too much to ask for as it forces insensitivity to nilpotents.

I sketch some things like the above in an appendix to this preprint though a much better reference seems to now be available: here eg Section 7.2

Answer 3

This is just a slight addition to previous answers of Chris and Anatoly.

There are some very beautiful, more-recent-than-the-classical-references papers of Neeman and Lipman-Neeman on (roughly) the subject of question 2. Warning though: in the first (Neeman only) paper, $f^!$ means the right adjoint of $\mathbb{R}f_*$, which won't agree with the $f^!$ defined e.g. in Deligne's appendix to Hartshorne's "Residues and Duality" without a properness hypothesis. [By the way, this appendix is a classical reference.]

Also, as in Anatoly's answer, you shouldn't ask any questions in this area at the abelian level. :-) And indeed, without some additional hypotheses you will have to be more careful about which derived category you want to work in. EDIT: If you are interested mainly in a right adjoint to $\mathbb{R}f_*$ then quasicompact quasiseparated will do; this is the setting of Neeman/Lipman-Neeman. Thanks to Anatoly's help, though, I have finally noticed that the original post doesn't seem to indicate caring about this, so perhaps the observation is a useless one to make!