Introduction
This is basically just a list of things that we want to put in the stacks project. As we add material to the Stacks project continuously this is always somewhat behind the current state of the Stacks project. In fact, it may have been a mistake to try and list things we should add, because it seems impossible to keep it up to date.
Last updated: Thursday, August 31, 2017.
Conventions
We should have a chapter with a short list of conventions used in the document. This chapter already exists, see Conventions, Section 0003, but a lot more could be added there. Especially useful would be to find “hidden” conventions and tacit assumptions and put those there.
Sites and Topoi
We have a chapter on sites and sheaves, see Sites, Section 00V0. We have a chapter on ringed sites (and topoi) and modules on them, see Modules on Sites, Section 03A5. We have a chapter on cohomology in this setting, see Cohomology on Sites, Section 01FR. But a lot more could be added, especially in the chapter on cohomology.
Stacks
We have a chapter on (abstract) stacks, see Stacks, Section 0267. It would be nice if
improve the discussion on “stackyfication”,
give examples of stackyfication,
more examples in general,
improve the discussion of gerbes.
Example result which has not been added yet: Given a sheaf of abelian groups \(\mathcal{F}\) over \(\mathcal{C}\) the set of equivalence classes of gerbes banded by \(\mathcal{F}\) is bijective to \(H^2(\mathcal{C}, \mathcal{F})\).
Simplicial methods
We have a chapter on simplicial methods, see Simplicial, Section 0163. This has to be reviewed and improved. The discussion of the relationship between simplicial homotopy (also known as combinatorial homotopy) and Kan complexes should be improved upon. There is a chapter on simplicial spaces, see Simplicial Spaces, Section 09VJ. This chapter briefly discusses simplicial topological spaces, simplicial sites, and simplicial topoi. We can further develop “simplicial algebraic geometry” to discuss simplicial schemes (or simplicial algebraic spaces, or simplicial algebraic stacks) and treat geometric questions, their cohomology, etc.
Cohomology of schemes
There is already a chapter on cohomology of quasi-coherent sheaves, see Cohomology of Schemes, Section 01X7. We have a chapter discussing the derived category of quasi-coherent sheaves on a scheme, see Derived Categories of Schemes, Section 08CV. We have a chapter discussing duality for Noetherian schemes and relative duality for morphisms of schemes, see Duality for Schemes, Section 0DWF. We also have chapters on étale cohomology of schemes and on crystalline cohomology of schemes. But most of the material in these chapters is very basic and a lot more could/should be added there.
Deformation theory à la Schlessinger
We have a chapter on this material, see Formal Deformation Theory, Section 06G8. We have a chapter discussing examples of the general theory, see Deformation Problems, Section 0DVL. We have a chapter, see Deformation Theory, Section 08KX which discusses deformations of rings (and modules), deformations of ringed spaces (and sheaves of modules), deformations of ringed topoi (and sheaves of modules). In this chapter we use the naive cotangent complex to describe obstructions, first order deformations, and infinitesimal automorphisms. This material has found some applications to algebraicity of moduli stacks in later chapters. There is also a chapter discussing the full cotangent complex, see Cotangent, Section 08P6.
Definition of algebraic stacks
An algebraic stack is a stack in groupoids over the category of schemes with the fppf topology that has a diagonal representable by algebraic spaces and is the target of a surjective smooth morphism from a scheme. See Algebraic Stacks, Section 026N. A “Deligne-Mumford stack” is an algebraic stack for which there exists a scheme and a surjective étale morphism from that scheme to it as in the paper [DM] of Deligne and Mumford, see Algebraic Stacks, Definition 03YO. We will reserve the term “Artin stack” for a stack such as in the papers by Artin, see [ArtinI], [ArtinII], and [ArtinVersal]. A possible definition is that an Artin stack is an algebraic stack \(\mathcal{X}\) over a locally Noetherian scheme \(S\) such that \(\mathcal{X} \to S\) is locally of finite type1.
Examples of schemes, algebraic spaces, algebraic stacks
The Stacks project currently contains two chapters discussing moduli stacks and their properties, see Moduli Stacks, Section 0DLU and Moduli of Curves, Section 0DMH. Over time we intend to add more, for example:
\(\mathcal{A}_g\), i.e., principally polarized abelian schemes of genus \(g\),
\(\mathcal{A}_1 = \mathcal{M}_{1, 1}\), i.e., \(1\)-pointed smooth projective genus \(1\) curves,
\(\mathcal{M}_{g, n}\), i.e., smooth projective genus \(g\)-curves with \(n\) pairwise distinct labeled points,
\(\overline{\mathcal{M}}_{g, n}\), i.e., stable \(n\)-pointed nodal projective genus \(g\)-curves,
\(\SheafHom_S(\mathcal{X}, \mathcal{Y})\), moduli of morphisms (with suitable conditions on the stacks \(\mathcal{X}\), \(\mathcal{Y}\) and the base scheme \(S\)),
\(\textit{Bun}_G(X) = \SheafHom_S(X, BG)\), the stack of \(G\)-bundles of the geometric Langlands programme (with suitable conditions on the scheme \(X\), the group scheme \(G\), and the base scheme \(S\)),
\(\Picardstack_{\mathcal{X}/S}\), i.e., the Picard stack associated to an algebraic stack over a base scheme (or space).
More generally, the Stacks project is somewhat lacking in geometrically meaningful examples.
Properties of algebraic stacks
This is perhaps one of the easier projects to work on, as most of the basic theory is there now. Of course these things are really properties of morphisms of stacks. We can define singularities (up to smooth factors) etc. Prove that a connected normal stack is irreducible, etc.
Lisse étale site of an algebraic stack
This has been introduced in Cohomology of Stacks, Section 0786. An example to show that it is not functorial with respect to \(1\)-morphisms of algebraic stacks is discussed in Examples, Section 07BF. Of course a lot more could be said about this, but it turns out to be very useful to prove things using the “big” étale site as much as possible.
Things you always wanted to know but were afraid to ask
There are going to be lots of lemmas that you use over and over again that are useful but aren’t really mentioned specifically in the literature, or it isn’t easy to find references for. Bag of tricks.
Example: Given two groupoids in schemes \(R\Rightarrow U\) and \(R' \Rightarrow U'\) what does it mean to have a \(1\)-morphism \([U/R] \to [U'/R']\) purely in terms of groupoids in schemes.
Quasi-coherent sheaves on stacks
These are defined and discussed in the chapter Cohomology of Stacks, Section 073Q. Derived categories of modules are discussed in the chapter Derived Categories of Stacks, Section 08MX. A lot more could be added to these chapters.
Flat and smooth
Artin’s theorem that having a flat surjection from a scheme is a replacement for the smooth surjective condition. This is now available as Criteria for Representability, Theorem 06DC.
Artin’s representability theorem
This is discussed in the chapter Artin’s Axioms, Section 07T0. We also have an application, see Quot, Theorem 08WC. There should be a lot more applications and the chapter itself has to be cleaned up as well.
DM stacks are finitely covered by schemes
We already have the corresponding result for algebraic spaces, see Limits of Spaces, Section 0ACX. What is missing is the result for DM and quasi-DM stacks.
Martin Olsson’s paper on properness
This proves two notions of proper are the same. The first part of this is now available in the form of Chow’s lemma for algebraic stacks, see More on Morphisms of Stacks, Theorem 0CQ8. As a consequence we show that it suffices to use DVR’s in checking the valuative criterion for properness for algebraic stacks in certain cases, see More on Morphisms of Stacks, Section 0CQL.
Proper pushforward of coherent sheaves
We can start working on this now that we have Chow’s lemma for algebraic stacks, see previous section.
Keel and Mori
See [K-M]. Their result has been added in More on Morphisms of Stacks, Section 0DUK.
Add more here
Actually, no we should never have started this list as part of the Stacks project itself! There is a todo list somewhere else which is much easier to update.