Are there any good introductory texts on algebraic stacks?
I have found some readable half-finsished texts on the net, but the authors always seem to give up before they are finished. I have also browsed through FGA explained (Fantechi et al.). Although I find the level good, it is somewhat incomplete and I would want to see more basic examples. Unfortunately I don't read french.
[Math] Good introductory references on algebraic stacks
ag.algebraic-geometryreference-requeststacks
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First of all, Kevin is being quite modest in his comment above: his paper
Buzzard, Kevin. Integral models of certain Shimura curves. Duke Math. J. 87 (1997), no. 3, 591--612.
contains many basic results on integral models of Shimura curves over totally real fields, and is widely cited by workers in the field: 22 citations on MathSciNet. The most recent is a paper of mine:
Clark, Pete L. On the Hasse principle for Shimura curves. Israel J. Math. 171 (2009), 349--365.
http://alpha.math.uga.edu/~pete/plclarkarxiv7.pdf
Section 3 of this paper spends 2-3 pages summarizing results on the structure of the canonical integral model of a Shimura curve over $\mathbb{Q}$ (with applications to the existence of local points). From the introduction to this paper:
"This result [something about local points] follows readily enough from a description of their [certain Shimura curves over Q] integral canonical models. Unfortunately I know of no unique, complete reference for this material. I have myself written first (my 2003 Harvard thesis) and second (notes from a 2005 ISM course in Montreal) approximations of such a work, and in so doing I have come to respect the difficulty of this expository problem."
I wrote that about three years ago, and I still feel that way today. Here are the documents:
is my thesis. "Chapter 0" is an exposition on Shimura curves: it is about 50 pages long.
- For my (incomplete) lecture notes from 2005, go to
http://alpha.math.uga.edu/~pete/expositions2012.html
and scroll down to "Shimura Curves". There are 12 files there, totalling 106 pages [perhaps I should also compile them into a single file]. On the other hand, the title of the course was Shimura Varieties, and although I don't so much as attempt to give the definition of a general Shimura variety, some of the discussion includes other PEL-type Shimura varieties like Hilbert and Siegel moduli space. These notes do not entirely supercede my thesis: each contains some material that the other omits.
When I applied for an NSF grant 3 years ago, I mentioned that if I got the grant, as part of my larger impact I would write a book on Shimura curves. Three years later I have written up some new material (as yet unreleased) but am wishing that I had not said that so directly: I would need at least a full semester off to make real progress (partly, of course, to better understand much of the material).
Let me explain the scope of the problem as follows: there does not even exist a single, reasonably comprehensive reference on the arithmetic geometry of the classical modular curves (i.e., $X_0(N)$ and such). This would-be bible of modular curves ought to contain most of the material from Shimura's book (260 pages) and the book of Katz and Mazur Arithmetic Moduli of Elliptic Curves (514 pages). These two books don't mess around and have little overlap, so you get a lower bound of, say, 700 pages that way.
Conversely, I claim that there is some reasonable topology on the arithmetic geometry of modular curves whose compactification is the theory of Shimura curves. The reason is that in many cases there are several ways to establish a result about modular curves, and "the right one" generalizes to Shimura curves with little trouble. (For example, to define the rational canonical model for classical modular curves, one could use the theory of Fourier expansions at the cusps -- which won't generalize -- or the theory of moduli spaces -- which generalizes immediately. Better yet is to use Shimura's theory of special points, which nowadays you need to know anyway to study Heegner point constructions.) Most of the remainder concerns quaternion arithmetic, which, while technical, is nowadays well understood and worked out.
$\def\Spec{\mathop{\rm Spec}} \def\R{{\bf R}} \def\Ep{{\rm E}^+} \def\L{{\rm L}} \def\EpL{\Ep\L}$ One can argue that an object of the right category of spaces in measure theory is not a set equipped with a σ-algebra of measurable sets, but rather a set $S$ equipped with a σ-algebra $M$ of measurable sets and a σ-ideal $N$ of negligible sets, i.e., sets of measure 0. The reason for this is that you can hardly state any theorem of measure theory or probability theory without referring to sets of measure 0. However, objects of this category contain less data than the usual measured spaces, because they are not equipped with a measure. Therefore I prefer to call them enhanced measurable spaces, since they are measurable spaces enhanced with a σ-ideal of negligible sets. A morphism of enhanced measurable spaces $(S,M,N)→(T,P,Q)$ is a map $S\to T$ such that the preimage of every element of $P$ is a union of an element of $M$ and a subset of an element of $N$ and the preimage of every element of $Q$ is a subset of an element of $N$.
Irving Segal proved in “Equivalences of measure spaces” (see also Kelley's “Decomposition and representation theorems in measure theory”) that for an enhanced measurable space $(S,M,N)$ that admits a faithful measure (meaning $μ(A)=0$ if and only if $A∈N$) the following properties are equivalent.
- The Boolean algebra $M/N$ of equivalence classes of measurable sets is complete;
- The space of equivalence classes of all bounded (or unbounded) real-valued functions on $S$ modulo equality almost everywhere is Dedekind-complete;
- The Radon-Nikodym theorem is true for $(S,M,N)$;
- The Riesz representation theorem is true for $(S,M,N)$ (the dual of $\L^1$ is isomorphic to $\L^∞$);
- Equivalence classes of bounded functions on $S$ form a von Neumann algebra (alias W*-algebra).
An enhanced measurable space that satisfies these conditions (including the existence of a faithful measure) is called localizable. This theorem tells us that if we want to prove anything nontrivial about measurable spaces, we better restrict ourselves to localizable enhanced measurable spaces. We also have a nice illustration of the claim I made in the first paragraph: none of these statements would be true without identifying objects that differ on a set of measure 0. For example, take a nonmeasurable set $G$ and a family of singleton subsets of $G$ indexed by themselves. This family of measurable sets does not have a supremum in the Boolean algebra of measurable sets, thus disproving a naive version of (1).
But restricting to localizable enhanced measurable spaces does not eliminate all the pathologies: one must further restrict to the so-called compact and strictly localizable enhanced measurable spaces, and use a coarser equivalence relation on measurable maps: $f$ and $g$ are weakly equal almost everywhere if for any measurable subset $B$ of the codomain the symmetric difference $f^*B⊕g^*B$ of preimages of $B$ under $f$ and $g$ is a negligible subset of the domain. (For codomains like real numbers this equivalence relation coincides with equality almost everywhere.)
An enhanced measurable space is strictly localizable if it splits as a coproduct (disjoint union) of σ-finite (meaning there is a faithful finite measure) enhanced measurable spaces. An enhanced measurable space $(X,M,N)$ is (Marczewski) compact if there is a compact class $K⊂M$ such that for any $m∈M∖N$ there is $k∈K∖N$ such that $k⊂m$. Here a compact class is a collection $K⊂2^X$ of subsets of $X$ such that for any $K'⊂K$ the following finite intersection property holds: if for any finite $K''⊂K'$ we have $⋂K''≠∅$, then also $⋂K'≠∅$.
The best argument for such restrictions is the following Gelfand-type duality theorem for commutative von Neumann algebras.
Theorem. The following 5 categories are equivalent.
- The category of compact strictly localizable enhanced measurable spaces with measurable maps modulo weak equality almost everywhere.
- The category of hyperstonean topological spaces and open continuous maps.
- The category of hyperstonean locales and open maps.
- The category of measurable locales (and arbitrary maps of locales).
- The opposite category of commutative von Neumann algebras and normal (alias ultraweakly continuous) unital *-homomorphisms.
I actually prefer to work with the opposite category of the category of commutative von Neumann algebras, or with the category of measurable locales. The reason for this is that the point-set definition of a measurable space exhibits immediate connections only (perhaps) to descriptive set theory, and with additional effort to Boolean algebras, whereas the description in terms of operator algebras or locales immediately connects measure theory to other areas of the central core of mathematics (noncommutative geometry, algebraic geometry, complex geometry, differential geometry, topos theory, etc.).
Additionally, note how the fourth category (measurable locales) is a full subcategory of the category of locales. Roughly, the latter can be seen as a slight enlargement of the usual category of topological spaces, for which all the usual theorems of general topology continue to hold (e.g., Tychonoff, Urysohn, Tietze, various results about paracompact and uniform spaces, etc.). In particular, there is a fully faithful functor from sober topological spaces (which includes all Hausdorff spaces) to locales. This functor is not surjective, i.e., there are nonspatial locales that do not come from topological spaces. As it turns out, all measurable locales (excluding discrete ones) are nonspatial. Thus, measure theory is part of (pointfree) general topology, in the strictest sense possible.
The non-point-set languages (2–5) are also easier to use in practice. Let me illustrate this statement with just one example: when we try to define measurable bundles of Hilbert spaces on a compact strictly localizable enhanced measurable space in a point-set way, we run into all sorts of problems if the fibers can be nonseparable, and I do not know how to fix this problem in the point-set framework. On the other hand, in the algebraic framework we can simply say that a bundle of Hilbert spaces is a Hilbert W*-module over the corresponding von Neumann algebra.
Categorical properties of von Neumann algebras (hence of compact strictly localizable enhanced measurable spaces) were investigated by Guichardet in “Sur la catégorie des algèbres de von Neumann”. Let me mention some of his results, translated in the language of enhanced measurable spaces. The category of compact strictly localizable enhanced measurable spaces admits equalizers and coequalizers, arbitrary coproducts, hence also arbitrary colimits. It also admits products (and hence arbitrary limits), although they are quite different from what one might think. For example, the product of two real lines is not $\R^2$ with the two obvious projections. The product contains $\R^2$, but it also has a lot of other stuff, for example, the diagonal of $\R^2$, which is needed to satisfy the universal property for the two identity maps on $\R$. The more intuitive product of measurable spaces ($\R\times\R=\R^2$) corresponds to the spatial tensor product of von Neumann algebras and forms a part of a symmetric monoidal structure on the category of measurable spaces. See Guichardet's paper for other categorical properties (monoidal structures on measurable spaces, flatness, existence of filtered limits, etc.).
Another property worthy of mentioning is that the category of commutative von Neumann algebras is a locally presentable category, which immediately allows one to use the adjoint functor theorem to construct commutative von Neumann algebras (hence enhanced measurable spaces) via their representable functors.
Finally let me mention pushforward and pullback properties of measures on enhanced measurable spaces. I will talk about more general case of $\L^p$-spaces instead of just measures (i.e., $\L^1$-spaces). For the sake of convenience, denote $\L_p(M)=\L^{1/p}(M)$, where $M$ is an enhanced measurable space. Here $p$ can be an arbitrary complex number with a nonnegative real part. We do not need a measure on $M$ to define $\L_p(M)$. For instance, $\L_0$ is the space of all bounded functions (i.e., the commutative von Neumann algebra corresponding to $M$), $\L_1$ is the space of finite complex-valued measures (the dual of $\L_0$ in the ultraweak topology), and $\L_{1/2}$ is the Hilbert space of half-densities. I will also talk about extended positive part $\EpL_p$ of $\L_p$ for real $p$. In particular, $\EpL_1$ is the space of all (not necessarily finite) positive measures on $M$.
Pushforward for $\L_p$-spaces. Suppose we have a morphism of enhanced measurable spaces $M\to N$. If $p=1$, then we have a canonical map $\L_1(M)\to\L_1(N)$, which just the dual of $\L_0(N)→\L_0(M)$ in the ultraweak topology. Geometrically, this is the fiberwise integration map. If $p≠1$, then we only have a pushforward map of the extended positive parts, namely, $\EpL_p(M)→\EpL_p(N)$, which is nonadditive unless $p=1$. Geometrically, this is the fiberwise $\L_p$-norm. Thus $\L_1$ is a functor from the category of enhanced measurable spaces to the category of Banach spaces and $\EpL_p$ is a functor to the category of “positive homogeneous $p$-cones”. The pushforward map preserves the trace on $\L_1$ and hence sends a probability measure to a probability measure.
To define pullbacks of $\L_p$-spaces (in particular, $\L_1$-spaces) one needs to pass to a different category of enhanced measurable spaces. In the algebraic language, if we have two commutative von Neumann algebras $A$ and $B$, then a morphism from $A$ to $B$ is a usual morphism of commutative von Neumann algebras $f\colon A\to B$ together with an operator valued weight $T\colon\Ep(B)\to\Ep(A)$ associated to $f$. Here $\Ep(A)$ denotes the extended positive part of $A$. (Think of positive functions on $\Spec A$ that can take infinite values.) Geometrically, this is a morphism $\Spec f\colon\Spec B\to\Spec A$ between the corresponding enhanced measurable spaces and a choice of measure on each fiber of $\Spec f$. Now we have a canonical additive map $\EpL_p(\Spec A)\to\EpL_p(\Spec B)$, which makes $\EpL_p$ into a contravariant functor from the category of enhanced measurable spaces and measurable maps equipped with a fiberwise measure to the category of “positive homogeneous additive cones”.
If we want to have a pullback of $\L_p$-spaces themselves and not just their extended positive parts, we need to replace operator valued weights in the above definition by finite complex-valued operator valued weights $T\colon B\to A$ (think of a fiberwise finite complex-valued measure). Then $\L_p$ becomes a functor from the category of enhanced measurable spaces to the category of Banach spaces (if the real part of $p$ is at most $1$) or quasi-Banach spaces (if the real part of $p$ is greater than $1$). Here $p$ is an arbitrary complex number with a nonnegative real part. Notice that for $p=0$ we get the original map $f\colon A\to B$ and in this (and only this) case we do not need $T$.
Finally, if we restrict ourselves to an even smaller subcategory defined by the additional condition $T(1)=1$ (i.e., $T$ is a conditional expectation; think of a fiberwise probability measure), then the pullback map preserves the trace on $\L_1$ and in this case the pullback of a probability measure is a probability measure.
There is also a smooth analog of the theory described above. The category of enhanced measurable spaces and their morphisms is replaced by the category of smooth manifolds and submersions, $\L_p$-spaces are replaced by bundles of $p$-densities, operator valued weights are replaced by sections of the bundle of relative 1-densities, the integration map on 1-densities is defined via Poincaré duality (to avoid any dependence on measure theory) etc. There is a forgetful functor that sends a smooth manifold to its underlying enhanced measurable space.
Of course, the story does not end here, there are many other interesting topics to consider: products of measurable spaces, the difference between Borel and Lebesgue measurability, conditional expectations, etc. An index of my writings on this topic is available.
Best Answer
Anton live-texed notes to Martin's Olsson's course on stacks a few years ago. They are online here. Olsson's notes have been published as:
Algebraic Spaces and Stacks, M. Olsson, AMS Colloquium Publications, volume 62, 2016. ISBN 978-1-4704-2798-6
My general advice is to learn algebraic spaces first. The point is that the new things you need to learn for stacks fall into two categories (which are mostly disjoint): 1) making local, functorial, and non-topological definitions (e.g. what it means for a morphism to be smooth or flat or locally finitely presented) and 2) 2-categorical stuff (e.g. what is a 2-fiber product). You don't need to do things 2-categorically for algebraic spaces, so it makes sense to learn them first. I believe it really clarifies things to learn these separately.
Also, the formal notion of a stack is a generalization of functor. If you are not used to thinking of schemes functorially (e.g. as a functor from rings^op to sets) it will be hard to wrap your head around the notion of a stack. the The intermediate step of learning to think about geometry in terms of functors of points is crucial.
Knutson's book Algebraic Spaces is very good for the EGA-style content, and its introduction will point you to many nice applications of algebraic spaces that are worth learning and will motivate you to learn the EGA-style stuff. Laumon and Moret-Bailly's Champs Algébriques is nice and contains more theorems that just the EGA style stuff.
Its hard to point you any other particular reference without knowing what your goal in learning stacks is.