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 counter examples for silly statements. For example, find a Noetherian ring which is not of finite type over a field, namely 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). (6) Find ocurrences of \coprod and \amalg and consistently have the following A \amalg B \coprod_{i\in I} A_i A \amalg \coprod_{i\in I} B_i i.e., use \amalg if there are only two and \coprod if there are more. The last one doesn't look good, but \coprod \coprod_i is even worse! 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 etale morphism of schemes is locally quasi-finite and put in a reference to Lemma lemma-etale-locally-quasi-finite. 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 a theorem saying that the category of curves (with dominant rational maps) over a field k is equivalent to the category of finitely generated field extensions of transcendence degree 1 over k. (5) ``opp'' ---> ``op'' for opposite category? MORE DIFFICULT TASKS -------------------- (1) Add more material on algebraic stacks. (2) Algebraic Spaces: It might be useful to list all the properties P such that: f has P => \Delta_f has P. Then if f is stable under base change, then gf and g has P => f has P. Notable exceptions are quasi-compact and finite type and this explains the relevance of qcqs and finite presentation. (3) 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. (4) 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 <=> diagonal unramified and unramified => diagonal etale. (5) 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 -> S can be approximated as affine and finite presentation and if X -> 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. (6) 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' -> 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. (7) The approximation results generalizes some of the foundational results, e.g. in Chevalley's theorem on affineness we can let the finite morphism be integral (and also let the target be an algebraic space). This is done for schemes but not yet done for algebraic spaces. (8) Add the equivalence (for morphisms of algebraic spaces): radicial + loc. separated <=> diagonal nil-immersion Not known whether there exist radicial non-quasi-separated morphisms (necessarily not locally separated). (9) 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)->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 <=> diagonal surjective (10) A related note is that points of any quasi-separated stack (i.e., stack with quasi-compact and quasi-separated diagonal) are algebraic. This is due to David Rydh. (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) In the chapter on Chow homology and Chern classes, maximize the use of Lemma lemma-secondary-ramification (Tag 02QJ). In particular, prove a version of Lemma lemma-commutativity-effective-Cartier (Tag 02TF) with supports. This will lead to a proof of commutativity of intersecting with effective Cartier divisors, which does not use blowing up Section section-blowing-up-lemmas (Tag 02SY) at all. (13) 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). (14) Chapter on Etale cohomology: Integrate with the rest of the project. This means finding references to unproven results, removing any mention of the fpqc site (although fpqc coverings can be used), etc. Either remove the last few sections or find a way of clearly indicating that they are `further topics''. (15) Write sections on Brauer groups: for each case of algebra, schemes, spaces, stacks. (16) Related to (15): Start a chapter on noncommutative algebra. (17) Done. (18) Artin's theorem on representability. (19) Done. (20) Keel and Mori (some of it is already there). (21) Etc, etc. See also the chapter Desirables. (22) Put the following (suggested by David Rydh) in the stacks project: Using ZMT, one proves the fact that if f:X->Y is quasi-finite and separated then the subset U of y's such that f restricted to Spec(O_{Y,y}) is finite is open. This is almost a one-liner: pf: The question is local so we can assume that Y is affine. Take a ZMT factorization X->W->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->Y is quasi-affine then the subset of y's such that f restricted to Spec(O_{Y,y}) is affine is open. (23) Prove an algebraic space is a scheme if and only if its reduction is a scheme. (24) Done. (25) Add material on connected component of identity of group schemes. (26) Done. (27) (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 (27.1) the special fibre X_0 is a torsor under G_0, and (27.2) A \subset \Gamma(X, O_X) then X is a G-torsor over S. Generalize to A just local with (locally?) nilpotent maximal ideal. (28) 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. (29) Introduce the notion of a complete intersection ring, and prove some of its basic properties, for example that if A ---> B is a flat local homomorphism of Noetherian local rings, then B is C.I. if and only if A and B/m_AB are C.I. (30) In the section "Ample invertible sheaves and cohomology" of the chapter on cohomology of coherent sheaves actually apply the results to ample invertible sheaves. (31) Redo the sections on syntomic ring maps using the material on Koszul sequences in rings. (32) Improve the chapter "Simplicial Methods" in the following way: (32.1) 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? (32.2) Introduce the functor from semi-simplicial objects in an abelian category to simplicial objects in the same. (32.3) Prove the Dold-Kan correspondence directly using (32.2). (32.4) Introduce Eilenberg-Maclane objects, etc and explain the significance of these in view of Dold-Kan. (32.5) Say something about derived functors of non-additive functors? MAINTENANCE ------------ CONTACT THE MAINTAINER AT THE EMAIL ADDRESS ABOVE BEFORE ATTEMPTING THESE: (0) Split algebra chapter in two (this is hard to do without messing up the tags system). (1) Improve Makefile. (2) Clean up python scripts. (3) Prettyfy the website. (4) Improve consistency of notation. Example: ``Known'' categories such as Sets, Groups, Sheaves, Abelian Sheaves etc are not named in a consistent manner. (5) Setup and run a mailing list. (6) Setup and run a bug system; mainly for feature requests. (7) Setup and run a sign off system, where collaborators can sign off on results in the stacks project, i.e., saying ``I declare this is true'', and where in addition we can put links to similar results in the literature. (8) 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. This is related to (1), (2) and (3) above. (9) Instead of (5), (6), (7) have a system for visitors of the website to leave comments, which are archived and visible (as the comments left on a blog for example). Some of these can be labeled as bugs, some as feature requests, some as declarations of correctness, etc.