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).
Add more material on algebraic stacks.
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.
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.
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.
Define quasi-finite morphisms of algebraic stacks.
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.
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.
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).
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
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).
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 $Supp(D) \cap Supp(\alpha)$.
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
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.
Start a chapter on noncommutative algebra. We already have a tiny bit in the
chapter on differential graded algebra.
Keel and Mori (some of it is already there). Etc, etc. See also the
chapter Desirables.
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.
(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.
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.
Redo the sections on syntomic ring maps using the material on Koszul
sequences in rings.
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.
5. Say something about derived functors of non-additive functors?
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_!
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.
Upgrade the chapter on intersection theory to work over arbitrary fields.