Stack (mathematics)

In mathematics a stack or 2-sheaf is, roughly speaking, a sheaf that takes values in categories rather than sets. Stacks are used to formalise some of the main constructions of descent theory, and to construct fine moduli stacks when fine moduli spaces do not exist.

Descent theory is concerned with generalisations of situations where isomorphic, compatible geometrical objects (such as vector bundles on topological spaces) can be "glued together" within a restriction of the topological basis. In a more general set-up the restrictions are replaced with pullbacks; fibred categories then make a good framework to discuss the possibility of such gluing. The intuitive meaning of a stack is that it is a fibred category such that "all possible gluings work". The specification of gluings requires a definition of coverings with regard to which the gluings can be considered. It turns out that the general language for describing these coverings is that of a Grothendieck topology. Thus a stack is formally given as a fibred category over another base category, where the base has a Grothendieck topology and where the fibred category satisfies a few axioms that ensure existence and uniqueness of certain gluings with respect to the Grothendieck topology.

Overview

Stacks are the underlying structure of algebraic stacks (also called Artin stacks) and Deligne–Mumford stacks, which generalize schemes and algebraic spaces and which are particularly useful in studying moduli spaces. There are inclusions: schemes ⊆ algebraic spaces ⊆ Deligne–Mumford stacks ⊆ algebraic stacks (Artin stacks) ⊆ stacks.

Edidin (2003) and Fantechi (2001) give a brief introductory accounts of stacks, Gómez (2001), Olsson (2007) and Vistoli (2005) give more detailed introductions, and Laumon & Moret-Bailly (2000) describes the more advanced theory.

Motivation and history

La conclusion pratique à laquelle je suis arrivé dès maintenant, c'est que chaque fois que en vertu de mes critères, une variété de modules (ou plutôt, un schéma de modules) pour la classification des variations (globales, ou infinitésimales) de certaines structures (variétés complètes non singulières, fibrés vectoriels, etc.) ne peut exister, malgré de bonnes hypothèses de platitude, propreté, et non singularité éventuellement, la raison en est seulement l'existence d'automorphismes de la structure qui empêche la technique de descente de marcher.

Grothendieck's letter to Serre, 1959 Nov 5.

The concept of stacks has its origin in the definition of effective descent data in Grothendieck (1959). In a 1959 letter to Serre, Grothendieck observed that a fundamental obstruction to constructing good moduli spaces is the existence of automorphisms. A major motivation for stacks is that if a moduli space for some problem does not exist because of the existence of automorphisms, it may still be possible to construct a moduli stack.

Mumford (1965) studied the Picard group of the moduli stack of elliptic curves, before stacks had been defined. Stacks were first defined by Giraud (1966, 1971), and the term "stack" was introduced by Deligne & Mumford (1969) for the original French term "champ" meaning "field". In this paper they also introduced Deligne–Mumford stacks, which they called algebraic stacks, though the term "algebraic stack" now usually refers to the more general Artin stacks introduced by Artin (1974).

When defining quotients of schemes by group actions, it is often impossible for the quotient to be a scheme and still satisfy desirable properties for a quotient. For example, if a few points have non-trivial stabilisers, then the categorical quotient will not exist among schemes.

In the same way, moduli spaces of curves, vector bundles, or other geometric objects are often best defined as stacks instead of schemes. Constructions of moduli spaces often proceed by first constructing a larger space parametrizing the objects in question, and then quotienting by group action to account for objects with automorphisms which have been overcounted.

Definitions

A category c with a functor to a category C is called a fibered category over C if for any morphism F from X to Y in C and any object y of c with image Y, there is a pullback f:x →y of y by F. This means a morphism with image F such that any morphism g:z→y with image G=FH can be factored as g=fh by a unique morphism h from z to x with image H. The element x=F*y is called the pullback of y along F and is unique up to canonical isomorphism.

The category c is called a prestack over a category C with a Grothendieck topology if it is fibered over C and for any object U of C and objects x, y of c with image U, the functor from objects over U to sets taking F:V→U to Hom(F*x,F*y) is a sheaf. This terminology is not consistent with the terminology for sheaves: prestacks are the analogues of separated presheaves rather than presheaves.

The category c is called a stack over the category C with a Grothendieck topology if it is a prestack over C and every descent datum is effective. A descent datum consists roughly of a covering of an object V of C by a family V_i, elements x_i in the fiber over V_i, and morphisms f_ji between the restrictions of x_i and x_j to V_ij=V_i×_UV_j satisfying the compatibility condition f_ki = f_kjf_ji. The descent datum is called effective if the elements x_i are essentially the pullbacks of an element x with image U.

A stack is called a stack in groupoids or a (2,1)-sheaf if it is also fibered in groupoids, meaning that its fibers (the inverse images of objects of C) are groupoids. Some authors use the word "stack" to refer to the more restrictive notion of a stack in groupoids.

An algebraic stack or Artin stack is a stack in groupoids X over the etale site such that the diagonal map of X is representable and there exists a smooth surjection from (the stack associated to) a scheme to X. A morphism Y $\rightarrow$ X of stacks is representable if, for every morphism S $\rightarrow$ X from (the stack associated to) a scheme to X, the fiber product Y ×_X S is isomorphic to (the stack associated to) an algebraic space. The fiber product of stacks is defined using the usual universal property, and changing the requirement that diagrams commute to the requirement that they 2-commute. See also morphism of algebraic stacks for further information.

The motivation behind the representability of the diagonal is the following: the diagonal morphism $\Delta :{\mathfrak {X}}\to {\mathfrak {X}}\times {\mathfrak {X}}$ is representable if and only if for any pair of morphisms of algebraic spaces $X,Y\to {\mathfrak {X}}$ , their fiber product $X\times _{\mathfrak {X}}Y$ is representable.

A Deligne–Mumford stack is an algebraic stack X such that there is an étale surjection from a scheme to X. Roughly speaking, Deligne–Mumford stacks can be thought of as algebraic stacks whose objects have no infinitesimal automorphisms.

Examples

If the fibers of a stack are sets (meaning categories whose only morphisms are identity maps) then the stack is essentially the same as a sheaf of sets. This shows that a stack is a sort of generalization of a sheaf, taking values in arbitrary categories rather than sets.
More concretely, let h be a contravariant functor from the category C of all schemes over some fixed base scheme to the category of sets. Then it determines the following category H:

an object is a pair (S, s) consisting of a scheme S in C and an element s of the set h(S)
a morphism (S, s) → (T, t) consists of a morphism φ: S → T in C such that h(φ)(t)= s.

Via the forgetful functor p: H → C, the category H is a category fibered over C. For example, if X is a scheme in C, then it determines the contravariant functor

h=\operatorname {Hom} (-,X)

and the corresponding fibered category is the stack associated to X. Stacks (or prestacks) can be constructed as a variant of this construction.

Any scheme with quasi-compact diagonal is an algebraic stack (or more precisely represents one) since every scheme has an action by the trivial group.
The moduli stack of vector bundles: the category of vector bundles V→S is a stack over the category of topological spaces S. A morphism from V→S to W→T consists of continuous maps from S to T and from V to W (linear on fibers) such that the obvious square commutes. The condition that this is a fibered category follows because one can take pullbacks of vector bundles over continuous maps of topological spaces, and the condition that a descent datum is effective follows because one can construct a vector bundle over a space by gluing together vector bundles on elements of an open cover.
The stack of quasi-coherent sheaves on schemes (with respect to the fpqc-topology and weaker topologies)
The stack of affine schemes on a base scheme (again with respect to the fpqc topology or a weaker one)
Mumford (1965) studied the moduli stack M_1,1 of elliptic curves, and showed that its Picard group is cyclic of order 12. For elliptic curves over the complex numbers the corresponding stack is similar to a quotient of the upper half-plane by the action of the modular group.
The moduli space of algebraic curves M_g defined as a universal family of smooth curves of given genus g does not exist as an algebraic variety because in particular there are curves admitting nontrivial automorphisms. However there is a moduli stack M_g which is a good substitute for the non-existent fine moduli space of smooth genus g curves. More generally there is a moduli stack M_g,n of genus g curves with n marked points. In general this is an algebraic stack, and is a Deligne–Mumford stack for g≥2 or g=1,n>0 or g=0, n≥3 (in other words when the automorphism groups of the curves are finite). This moduli stack has a completion consisting of the moduli stack of stable curves (for given g and n) which is proper over Spec Z. For example, M₀ is the classifying stack BPGL(2) of the projective general linear group. (There is a subtlety in defining M₁, as one has to use algebraic spaces rather than schemes to construct it.)
A gerbe is a stack in groupoids; for example the trivial gerbe that assigns to each scheme the principal G-bundles over the scheme, for some group G.
If Y is a scheme and G is a smooth affine group scheme acting on Y, then there is a quotient algebraic stack Y/G, taking a scheme T to the groupoid of G-torsors over T with G-equivariant maps to Y. Explicitly, given a space $X$ with a $G$ -action, form the stack $[X/G]$ which (intuitively speaking) sends a space $Y$ to the groupoid of pullback diagrams

[X/G](Y)={\begin{Bmatrix}Z&{\xrightarrow {\Phi }}&X\\\downarrow &&\downarrow \\Y&{\xrightarrow {\phi }}&[X/G]\end{Bmatrix}}

where

\Phi

is a

G

-equivariant morphism of spaces and

Z\to Y

is a principal

G

-bundle. The morphisms in this category are just morphisms of diagrams where the arrows on the right-hand side are equal and the arrows on the left-hand side are morphisms of principal

G

-bundles.

A special case of this when Y is a point gives the classifying stack BG of a smooth affine group scheme G: ${\textbf {B}}G:=[pt/G].$ It is named so since the category $\mathbf {B} G(X)$ , the fiber over X, is precisely the category $\operatorname {Bun} _{G}(X)$ of principal $G$ -bundles over $X$ . Note that $\operatorname {Bun} _{G}(X)$ itself can be considered as a stack, the moduli stack of principal G-bundles on X.
An important subexample from this construction is $\mathbf {B} GL_{n}$ which is the moduli stack of principal $GL_{n}$ -bundles. Since the data of a principal $GL_{n}$ -bundle is equivalent to the data of a rank $n$ vector bundle, this is isomorphic to the moduli stack of rank $n$ vector bundles $Vect_{n}$ .
An example of a stack which is not globally a quotient stack is the disjoint union of two quotient stacks which have non-equal quotienting group $G$ ; e.g. consider $\mathbf {B} \mathbb {Z} /2\coprod \mathbf {B} S_{3}$ .
Stacky curves, or orbicurves, can be constructed by taking the stack quotient of a morphism of curves by the monodromy group of the cover over the generic points. For example, take a projective morphism

{\text{Proj}}(\mathbb {C} [x,y,z]/(x^{5}+y^{5}+z^{5}))\to {\text{Proj}}(\mathbb {C} [x,y])

which is generically etale. The stack quotient of the domain gives a stacky

\mathbb {P} ^{1}

with stacky points that have stabilizer group

\mathbb {Z} /5

at the fifth roots of unity in the

x/y

-chart since these are the points where the cover ramifies.

If A is a quasi-coherent sheaf of algebras in an algebraic stack X over a scheme S, then there is a stack Spec(A) generalizing the construction of the spectrum Spec(A) of a commutative ring A. An object of Spec(A) is given by an S-scheme T, an object x of X(T), and a morphism of sheaves of algebras from x*(A) to the coordinate ring O(T) of T.
If A is a quasi-coherent sheaf of graded algebras in an algebraic stack X over a scheme S, then there is a stack Proj(A) generalizing the construction of the projective scheme Proj(A) of a graded ring A.
The moduli stack of formal group laws classifies formal group laws.
A Picard stack generalizes a Picard variety.
An ind-scheme such as an infinite projective space and a formal scheme is a stack.

Quasi-coherent sheaves on algebraic stacks

On an algebraic stack one can construct a category of quasi-coherent sheaves similar to the category of quasi-coherent sheaves over a scheme.

A quasi-coherent sheaf is roughly one that looks locally like the sheaf of a module over a ring. The first problem is to decide what one means by "locally": this involves the choice of a Grothendieck topology, and there are many possible choices for this, all of which have some problems and none of which seem completely satisfactory. The Grothendieck topology should be strong enough so that the stack is locally affine in this topology: schemes are locally affine in the Zariski topology so this is a good choice for schemes as Serre discovered, algebraic spaces and Deligne–Mumford stacks are locally affine in the etale topology so one usually uses the etale topology for these, while algebraic stacks are locally affine in the smooth topology so one can use the smooth topology in this case. For general algebraic stacks the etale topology does not have enough open sets: for example, if G is a smooth connected group then the only etale covers of the classifying stack BG are unions of copies of BG, which are not enough to give the right theory of quasicoherent sheaves.

Instead of using the smooth topology for algebraic stacks one often uses a modification of it called the Lis-Et topology (short for Lisse-Etale: lisse is the French term for smooth), which has the same open sets as the smooth topology but the open covers are given by etale rather than smooth maps. This usually seems to lead to an equivalent category of quasi-coherent sheaves, but is easier to use: for example it is easier to compare with the etale topology on algebraic spaces. The Lis-Et topology has a subtle technical problem: a morphism between stacks does not in general give a morphism between the corresponding topoi. (The problem is that while one can construct a pair of adjoint functors f_*, f*, as needed for a geometric morphism of topoi, the functor f* is not left exact in general. This problem is notorious for having caused some errors in published papers and books.^[1]) This means that constructing the pullback of a quasicoherent sheaf under a morphism of stacks requires some extra effort.

It is also possible to use finer topologies. Most reasonable "sufficiently large" Grothendieck topologies seem to lead to equivalent categories of quasi-coherent sheaves, but the larger a topology is the harder it is to handle, so one generally prefers to use smaller topologies as long as they have enough open sets. For example, the big fppf topology leads to essentially the same category of quasi-coherent sheaves as the Lis-Et topology, but has a subtle problem: the natural embedding of quasi-coherent sheaves into O_X modules in this topology is not exact (it does not preserve kernels in general).

Other types of stack

Differentiable stacks and topological stacks are defined in a way similar to algebraic stacks, except that the underlying category of affine schemes is replaced by the category of smooth manifolds or topological spaces.

More generally one can define the notion of an n-sheaf or n–1 stack, which is roughly a sort of sheaf taking values in n–1 categories. There are several inequivalent ways of doing this. 1-sheaves are the same as sheaves, and 2-sheaves are the same as stacks. They are called higher stacks.

A very similar and analogous extension is to develop the stack theory on non-discrete objects (i.e., a space is really a spectrum in algebraic topology). The resulting stacky objects are called derived stacks (or spectral stacks). Jacob Lurie's under-construction book Spectral Algebraic Geometry studies a generalization which he calls a spectral Deligne–Mumford stack. By definition, it is a ringed ∞-topos that is étale-locally the étale spectrum of an E_∞-ring (this notion subsumes that of a derived scheme, at least in characteristic zero.)

Set-theoretical problems

There are some minor set theoretical problems with the usual foundation of the theory of stacks, because stacks are often defined as certain functors to the category of sets and are therefore not sets. There are several ways to deal with this problem:

One can work with Grothendieck universes: a stack is then a functor between classes of some fixed Grothendieck universe, so these classes and the stacks are sets in a larger Grothendieck universe. The drawback of this approach is that one has to assume the existence of enough Grothendieck universes, which is essentially a large cardinal axiom.
One can define stacks as functors to the set of sets of sufficiently large rank, and keep careful track of the ranks of the various sets one uses. The problem with this is that it involves some additional rather tiresome bookkeeping.
One can use reflection principles from set theory stating that one can find set models of any finite fragment of the axioms of ZFC to show that one can automatically find sets that are sufficiently close approximations to the universe of all sets.
One can simply ignore the problem. This is the approach taken by many authors.

Notes

↑ See, for example, Olsson, Martin (2007), Sheaves on Artin Stacks.

References

Pedagogical

Behrend, Kai; Conrad, Brian; Edidin, Dan; Fulton, William; Fantechi, Barbara; Göttsche, Lothar; Kresch, Andrew (2006), Algebraic stacks
Edidin, Dan (2003), "What is... a Stack?" (PDF), Notices of the AMS, 50 (4): 458–459

Guides to the Literature

Reference

Artin, Michael (1974), "Versal deformations and algebraic stacks", Inventiones Mathematicae, 27: 165–189, ISSN 0020-9910, MR 0399094, doi:10.1007/BF01390174
Deligne, Pierre; Mumford, David (1969), "The irreducibility of the space of curves of given genus", Publications Mathématiques de l'IHÉS (36): 75–109, ISSN 1618-1913, MR 0262240

Fantechi, Barbara (2001), "Stacks for everybody" (PDF), European Congress of Mathematics Volume I, Progr. Math., 201, Basel: Birkhäuser, pp. 349–359, ISBN 3-7643-6417-3, MR 1905329
Giraud, Jean (1964), "Méthode de la descente", Société Mathématique de France. Bulletin. Supplément. Mémoire, 2: viii+150, MR 0190142
Giraud, Jean (1966), Cohomologie non abélienne de degré 2, thesis, Paris
Giraud, Jean (1971), Cohomologie non abélienne, Springer, ISBN 3-540-05307-7
Gómez, Tomás L. (2001), "Algebraic stacks", Indian Academy of Sciences. Proceedings. Mathematical Sciences, 111 (1): 1–31, MR 1818418, arXiv:math/9911199 , doi:10.1007/BF02829538
Grothendieck, Alexander (1959). "Technique de descente et théorèmes d'existence en géométrie algébrique. I. Généralités. Descente par morphismes fidèlement plats". Séminaire Bourbaki. 5 (Exposé 190).
Laumon, Gérard; Moret-Bailly, Laurent (2000), Champs algébriques, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 39, Berlin, New York: Springer-Verlag, ISBN 978-3-540-65761-3, MR 1771927 Unfortunately this book uses the incorrect assertion that morphisms of algebraic stacks induce morphisms of lisse-étale topoi. Some of these errors were fixed by Olsson (2007).
Laszlo, Yves; Olsson, Martin (2008), "The six operations for sheaves on Artin stacks. I. Finite coefficients", Institut des Hautes Études Scientifiques. Publications Mathématiques, 107 (107): 109–168, MR 2434692, doi:10.1007/s10240-008-0011-6
Mumford, David (1965), "Picard groups of moduli problems", in Schilling, O. F. G., Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963), New York: Harper & Row, pp. 33–81, MR 0201443
Olsson, Martin Christian (2007), Geraschenko, Anton, ed., Course notes for Math 274: Stacks (PDF)
Olsson, Martin (2007), "Sheaves on Artin stacks", Journal für die reine und angewandte Mathematik, 603 (603): 55–112, MR 2312554, doi:10.1515/CRELLE.2007.012
Olsson, Martin (2016), Algebraic spaces and stacks, Colloquium Publications, 62, American Mathematical Society, ISBN 978-1470427986
Vistoli, Angelo (2005), "Grothendieck topologies, fibered categories and descent theory", Fundamental algebraic geometry, Math. Surveys Monogr., 123, Providence, R.I.: Amer. Math. Soc., pp. 1–104, MR 2223406, arXiv:math/0412512

External links

stack in nLab
descent in nLab
de Jong, Aise Johan, Stacks Project
Fulton, William, What is a stack?, MSRI video lecture and notes
Toën, Bertrand (2007), Cours de Master 2 : Champs algébriques (2006-2007)
"Good introductory references on algebraic stacks?"

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