Here is a list of algebra projects that I eventually want to have written up for the stacks project. This list is a bit random, and I will edit it every now and then to add more items. Hopefully I’ll be able to take some off the list every now and then also. If you are interested in helping out with any of these, then it may be a good idea to email me so we can coordinate. It is not necessary that the first draft be complete, just having some kind of text with a few definitions, some lemmas, etc is already a good thing to have.
G: A bit about Galois groups of fields. (Also the infinite case.)
I: A bit about inertia and decomposition groups. (Not just local fields.)
Pi: A bit about “Galois groups of rings”, i.e., a bit about finite etale extensions of rings and how this is related to (unramified) Galois groups. (Lenstra’s notes.)
BR: Write about Brauer groups of fields. My favorite exposition of this material is a set of lectures by Serre in Seminaire Cartan, Applications algebriques de la cohomologie des groupes. II: theorie des algebres simples, exp. n. 5, 6, 7. (Search for Serre on Numdam.)
HO: Write up Gabbers proof of Br = Br’ for affine schemes, see Hoobler’s paper on this topic. This also leads to some nice material about K-theory of rings.
CI: Write about complete intersection rings. Introduce the notion of a complete intersection ring (for a Noetherian local ring using its completion and the Cohen structure theorem), and prove that if A —> B is a flat local homomorphism of Noetherian local rings, then B is CI if and only if A and B/m_AB are CI. This is a result of Avramov. Use it to show that the localization of a CI ring is CI.
BP: Bass’ result “Big projective modules are free”.
UFD: Regular rings are UFDs and related material.
P: Write about p-bases.
E: Write about excellent and quasi-excellent rings.
GND: General Neron desingularization.
JH: Artin’s “Joins of henselian rings”. You can generalize the main algebraic trick in this paper a bit. I don’t quite remember how or what though. Anybody?
DC: More introductory material on (unbounded) derived categories. Currently the focus in the discussion of derived categories (in the chapter on homology) is to quickly get to a point where you can start using them.
D(R): It would be useful to have a preliminary discussion of the derived category of the category of modules over a ring (before actually introducing it in general perhaps? not sure).
D: Duality (in algebra). Matlis duality. Local cohomology. Dualizing complexes. Finiteness theorem.
L: Definition and basic properties of the cotangent complex (not the Netherlander complex, but the full one, in the setting of ring maps).
HH: Introduction to Hochschild homology.
HA: Introduction to Hopf algebras, modules, comodules, etc.
Meta 0: Find examples and counter examples illustrating the results in the algebra chapter.
Meta 1: Clean up the beginning of the algebra chapter and put in some really basic stuff.
Meta 2: Find a more reasonable organization of the algebra chapter which however does not lead to vicious circles.
ZZ: and so on.
Projects which are done (of course exposition can always be improved upon…):
FG: Formal glueing. Bhargav sent me a write-up. See also his home page. Show that if A is a Noetherian ring and f ∈ A then A is somehow gotten by glueing A_f and A^* along (A^*)_f. Really what I mean is the corresponding result for the categories of modules. See Section 4.6 of this paper. You can find this in a bunch of locations in the literature for example, M. Artin, Algebraization of formal moduli II. Existence of modifications, Annals of Math. 91 (1970), pp. 88–135. OR D. Ferrand, M. Raynaud, Fibres formelles d’un anneau local noethérien, Annals Sci. Ecole Norm. Sup. (4) 3 (1970), pp. 295–311; especially: Appendix 308–311. OR L. Moret-Bailly, Un probleme de descente, Bull. Soc. Math. France 124 (1996), pp. 559–585.
I would be willing to write up a section on GND, but I’m not really sure how much of the prerequisite material is already covered in the stacks project. I’m currently reading through Swan’s paper on it, so just let me know.
I also fear that including the cotangent complex in any substantial way will require at least a pretty big extension of the section on simplicial methods and probably at least a small section on homotopical algebra à la Quillen. The advantage of this POV over the standard injective/projective resolution POV is that it can make things like derived categories and derived functors a lot less arbitrary. I would be willing to write up a section on homotopical algebra if you’d want it in the stacks project. The POV of Dwyer-Hirschhorn-Kan-Smith on homotopical algebra in terms of deformable functors and homotopical categories (which allow us to give an elementary description of homotopy limits, homotopy colimits, and homotopy Kan extensions, etc. in terms of a presentation for a simplicial category (by a presentation of a simplicial category, I mean a category C with distinguished weak equivalences W satisfying 2 of 6 and containing identities. These data “present” a simplicially enriched category, i.e. the hammock localization of C at W)
Also, one final question: What are you using unbounded chain complexes for? They introduce a substantial amount of headache (since you can’t use the Dold-Kan correspondence), but I can’t really see what you’re using them for here.
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