# The Stacks Project

## Important

Before hacking away and spending enormous amounts of time on a project for the Stacks project, choose a smaller task, say something you can do in 5 minutes up to an hour. Email the result (usually the modified TeX file) to stacks.project@gmail.com and see what it feels like to donate some of your own work to a publicly maintained project. Having done this successfully you can try your hand at some more ambitious projects.

Also, it is very helpful if you try to keep to the coding style which is used throughout the TeX files.

## Tasks you can do while having a beer

1. Run any of the TeX files through a spell checker and correct any errors.
2. Find incompatible notation and correct it. The same mathematical object should be coded in the same way everywhere.
3. Read a random section and find small mathematical errors, such as arrows pointing the wrong way, wrong font, sign errors, etc. If they are small enough you can simply correct them. Otherwise, just email what's wrong.
4. Provide counterexamples for silly statements. For example, find a Noetherian ring which is not of finite type over a field, namely $\mathbb{Z}$. (Just to give you an idea.)
5. Basic notions. Write something basic about algebra, topology, fields, etc. which goes in an early part (and hasn't been written yet).

For many of these tasks the commenting system available on the website suffices. Just look up the tag and post a comment, we will deal with the actual change in the Stacks project.

## Tasks you can do while having tea

1. Provide missing proofs of easy statements which have been omitted. To find these do a case-insensitive search for the string omit in the text. If you hit on a omitted proof which you find too hard, then please report this.
2. Check for missing internal references. Generally speaking the goal is to refer to all of the previous lemmas, propositions, theorems used in a proof. Go through some of the proofs and check if previous results are used without referencing them.
3. Find mathematical mistakes.
4. Find superfluous assumptions.
5. Find missing assumptions.
6. Specific example of 2): Find all places where it is used that an étale morphism of schemes is locally quasi-finite and put in a reference to lemma-etale-locally-quasi-finite.

Again, many of these can be done using the commenting system.

## Tasks you can do while having coffee

1. Split longer proofs into pieces by finding intermediate results.
2. Find alternative proofs (but beware of creating circular arguments).
3. Write introductions, overviews of already existing material.
4. Add sections on your favorite topic. For example: You may be interested in curves. Start a chapter entitled Curves. For example you can provide atheorem saying that the category of curves (with dominant rational maps) over a field $k$ is equivalent to the category of finitely generated field extensions transcendence degree 1 over $k$.

What you see here is the current status of the file todo-list in the project.

1. We need somebody to add a bit more to the section on the snake lemma in algebra.tex. There should be a bit more about kernels, cokernels, pushouts, and fibre products of modules and abelian groups. As well we need a tiny bit about exact functors between module categories, or more generally some categories whose objects are abelian groups with a bit of extra structure (but do not formalize this notion because that is just awful). And then point out that the notions agree with those in categories.tex via the material in homology.tex (without proof, because it comes later).
2. Add more material on algebraic stacks.
3. Algebraic Spaces: It might be useful to list all the properties $P$ such that: $f$ has $P \Rightarrow \Delta_f$ has $P$. Then if $f$ is stable under base change, then $g \circ f$ and $g$ have $P \Rightarrow f$ has $P$. Notable exceptions are quasi-compact and finite type and this explains the relevance of qcqs and finite presentation.
4. For non-representable morphisms (of Artin stacks), one can define "unramified $=$ $R$-unramified" as "locally of finite type and diagonal etale" or as "locally of finite type and formally unramified" and "etale" as "locally of finite presentation, flat, and unramified". This looks like a circular definition but in each step we take the diagonal.
5. For stacks there is also a notion of "formally Deligne-Mumford". One gets the very pleasing list:
• DM $=$ formally DM
• R-unramified $=$ formally unramified + loc. of finite type
• etale $=$ formally etale + loc. of finite presentation
Here, the increasing finiteness hypothesis can be explained by the fact the diagonal of anything is locally of finite type and the diagonal of locally of finite type is locally of finite presentation. Also
• DM $\Leftrightarrow$ diagonal unramified
• unramified $\Rightarrow$ diagonal etale.
6. Define quasi-finite morphisms of algebraic stacks.
7. Limits of Schemes: Absloute Noetherian approximation. Add a second proof following Temkin's proof in [Relative RZ-spaces, section 1.1]. Look also at David Rydh's paper [Noetherian approximation of algebraic spaces and stacks]. In fact, using this method one gets a short proof of a more general approximation result: $X$, $S$ qc and qs schemes, then $X \to S$ can be approximated as affine and finite presentation and if $X \to S$ is of finite type then we can do closed immersion and finite presentation. The main point here will be to excise push-outs from the proof.
8. Introduce the notion: "pseudo-noetherian" (suggested by Brian Conrad) as a scheme/stack $X$ which is quasi-compact, quasi-separated and has the property that any quasi-coherent sheaf is the direct limit of finitely presented sheaves. David Rydh suggests: require that this holds on $X'$ for any finitely presented $X' \to X$ as this turns out to be quite useful. Examples of pseudo-noetherian stacks are noetherian stacks, qcqs algebraic spaces and qcqs stacks with quasi-finite diagonal.
9. Add the equivalence (for morphisms of algebraic spaces):
• radicial + loc. separated $\Leftrightarrow$ diagonal nil-immersion
Not known whether there exist radicial non-quasi-separated morphisms (necessarily not locally separated).
10. Also, for a stack (with algebraic points) one would have to interpret radicial'' as "there is exactly one point in every fiber and the residue field extension is inseparable". The definition of universally injective as $X(K) \to S(K)$ injective is not good for stacks (perhaps ok if we restrict to $K$ algebraically closed) unless we pass to the associated sheaf. Again we have:
• universally injective $\Leftrightarrow$ diagonal surjective
11. Write a chapter on push-outs in the stacks project. This may have been one of the essential parts of the first conception of EGA V (later moved to Chapter VI). The algebra/scheme part is worked out in detail by Ferrand "Conducteur, Descente et Pincement" and it generalizes to algebraic spaces (the correct level of generality).
12. Rewrite parts of the chapter on Chow homology and Chern classes in order to have intersections with supports where relevant. There should be explicit'' supports and not just of the order of saying that the product $D \cdot \alpha$ is supported in $Supp(D) \cap Supp(\alpha)$.
13. Chapter on Etale cohomology:
• Do a bit more on Galois cohomology to prove that vanishing of Brauer groups of all finite extensions implies cohomological dimension < 2
• Picard groups of curves: show that the n-torsion in the Picard group of a smooth projective curve over an algebraically closed field is isomorphic to (Z/nZ)^{2g}. Currently the proof uses unproven material about Jacobians and Abelian varieties.
• For the section "The trace formula" onwards the material has not yet been integrated with the rest of the Stacks project
14. Write sections on Brauer groups: for each case of algebra, schemes, spaces, stacks. We already have a bit of theory for Brauer groups of fields.
15. Start a chapter on noncommutative algebra. We already have a tiny bit in the chapter on differential graded algebra.
16. Keel and Mori (some of it is already there). Etc, etc. See also the chapter Desirables.
17. Put the following (suggested by David Rydh) in the stacks project: Using ZMT, one proves the fact that if $f : X \to Y$ is quasi-finite and separated then the subset $U$ of $y$'s such that $f$ restricted to $Spec(\mathcal{O}_{Y, y})$ is finite is open. This is almost a one-liner: The question is local so we can assume that $Y$ is affine. Take a ZMT factorization $X \to W \to Y$. Then $U = Y \ (image of W \ X)$. Indeed, the closure of a subset commutes with flat base change. In the same spirit, one easily shows (without ZMT) that if $f : X \to Y$ is quasi-affine then the subset of $y$'s such that $f$ restricted to $Spec(\mathcal{O}_{Y,y})$ is affine is open.
18. (Not completely sure this is correct. Haven't worked out all the details.) Show that if $G$ is a flat group scheme over an Artinian local ring $A$, and $G$ acts on the scheme $X$ over $A$ such that
1. the special fibre $X_0$ is a torsor under $G_0$, and
2. $A \subset \Gamma(X, O_X)$
then $X$ is a $G$-torsor over $S$. Generalize to $A$ just local with (locally?) nilpotent maximal ideal.
19. Show that if $(U, R, s, t, c)$ is a groupoid scheme with $U = Spec(k)$ and $s$, $t$ finite type, then $(U, R, s, t, c)$ is defined over a field $k_0$ which is a subfield of $k$ of finite index.
20. Redo the sections on syntomic ring maps using the material on Koszul sequences in rings.
21. Improve the chapter "Simplicial Methods" in the following way:
1. Distinguish more clearly between general material, material on (co)simplicial sets, and material on (co)simplicial objects in abelian categories. Maybe rearrange things so general material comes first?
2. Introduce the functor from semi-simplicial objects in an abelian category to simplicial objects in the same.
3. Prove the Dold-Kan correspondence directly using 2.
4. Introduce Eilenberg-Maclane objects, etc and explain the significance of these in view of Dold-Kan.
22. Split the chapter "Cohomology of Algebraic Stacks" into two:
1. A chapter discussing cohomology of a single O_X-module (analogous to the chapters "Cohomology of Schemes" and "Cohomology of Spaces", and
2. A chapter on derived categories of algebraic stacks.
When making the split make sure that the proof of QCoh forms a weak Serre subcategory of Mod(O_{lisse-etale}) goes into the first chapter and gets a proof which avoids using the functor Lg_!
23. Extend the definition of "numerical polynomial" (Tag 00JX) to multivariate numerical polynomials. In Kleiman's "Toward a numerical theory of ampleness" Chapter I, Section 1, the definition of "numerical polynomial" is given, and an equivalent characterization is given without proof in Chapter I, Section 2, Lemma 1.
24. Upgrade the chapter on intersection theory to work over arbitrary fields.

## Maintenance

Contact the maintainer at the email address above before attempting these!

1. Split algebra chapter in two (this is hard to do without messing up the tags system).
2. Improve the Makefile.
3. Clean up Python scripts.
4. Improve consistency of notation. Example: 'known' categories such as Sets, Groups, Sheaves, Abelian Sheaves etc are not named in a consistent manner.
5. Find people willing to mirror the project online, preferably in a very different geographical location. If you are interested and a major geek please contact via the email address above.