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 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 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 ⊆ stacks.
and give a brief introductory accounts of stacks,, and give more detailed introductions, and describes the more advanced theory.

Motivation and history

The concept of stacks has its origin in the definition of effective descent data in.
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.
studied the Picard group of the moduli stack of elliptic curves, before stacks had been defined. Stacks were first defined by , and the term "stack" was introduced by 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.
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

Abstract stacks

A category with a functor to a category is called a fibered category over if for any morphism in and any object of with image , there is a pullback of by. This means a morphism with image such that any morphism with image can be factored as by a unique morphism in such that the functor maps to. The element is called the pullback of along 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 the over category C/U to sets taking F:V→U to Hom is a sheaf. This terminology is not consistent with the terminology for sheaves: prestacks are the analogues of separated presheaves rather than presheaves. Some authors require this as a property of stacks, rather than of prestacks.
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 Vi, elements xi in the fiber over Vi, and morphisms fji between the restrictions of xi and xj to Vij=Vi×VVj satisfying the compatibility condition fki = fkjfji. The descent datum is called effective if the elements xi are essentially the pullbacks of an element x with image V.
A stack is called a stack in groupoids or a -sheaf if it is also fibered in groupoids, meaning that its fibers are groupoids. Some authors use the word "stack" to refer to the more restrictive notion of a stack in groupoids.

Algebraic stacks

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 a scheme to X.
A morphism Y X of stacks is representable if, for every morphism S X from a scheme to X, the fiber product Y ×X S is isomorphic 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 is representable if and only if for any pair of morphisms of algebraic spaces, their fiber product 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.

Local structure of algebraic stacks

Since the inception of algebraic stacks it was expected that they are locally quotient stacks of the form where is a linearly reductive algebraic group. This was recently proved to be the case: given a quasi-separated algebraic stack locally of finite type over an algebraically closed field whose stabilizers are affine, and a smooth and closed point with linearly reductive stabilizer group, there exists an etale cover of the GIT quotient, where, such that the diagram
is cartesian, and there exists an etale morphism
inducing an isomorphism of the stabilizer groups at and.

Examples

Elementary examples

Stack quotients

If is a scheme and is a smooth affine group scheme acting on, then there is a quotient algebraic stack, taking a scheme to the groupoid of -torsors over the -scheme with -equivariant maps to. Explicitly, given a space with a -action, form the stack which sends a space to the groupoid of pullback diagrams
where is a -equivariant morphism of spaces and is a principal -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 -bundles.

Classifying stacks

A special case of this when X is a point gives the classifying stack BG of a smooth affine group scheme G: It is named so since the category, the fiber over Y, is precisely the category of principal -bundles over. Note that itself can be considered as a stack, the moduli stack of principal G-bundles on Y.
An important subexample from this construction is which is the moduli stack of principal -bundles. Since the data of a principal -bundle is equivalent to the data of a rank vector bundle, this is isomorphic to the moduli stack of rank vector bundles.
Moduli stack of line bundles
The moduli stack of line bundles is since every line bundle is canonically isomorphic to a principal -bundle. Given a line bundle the relative spec
gives a geometric line bundle. After remove the zero section, there is an associated -bundle. Conversely, from the representation, the associated line bundle can be reconstructed.

Gerbes

A gerbe is a stack in groupoids which always has a nonempty category. for example the trivial gerbe that assigns to each scheme the groupoid of principal -bundles over the scheme, for some group.

Relative spec and proj

If A is a quasi-coherent sheaf of algebras in an algebraic stack X over a scheme S, then there is a stack Spec generalizing the construction of the spectrum Spec of a commutative ring A. An object of Spec is given by an S-scheme T, an object x of X, and a morphism of sheaves of algebras from x* to the coordinate ring O 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 generalizing the construction of the projective scheme Proj of a graded ring A.

Moduli stacks

Moduli of curves

Another widely studied class of moduli spaces are the Kontsevich moduli spaces parameterizing the space of stable maps between curves of a fixed genus to a fixed space whose image represents a fixed cohomology class. These moduli spaces are denoted
and can have wild behavior, such as being reducible stacks whose components are non-equal dimension. For example, the moduli stack
has smooth curves parametrized by an open subset. On the boundary of the moduli space, where curves may degenerate to reducible curves, there is a substack parametrizing reducible curves with a genus component and a genus component intersecting at one point, and the map sends the genus curve to a point. Since all such genus curves are parametrized by, and there is an additional dimensional choice of where these curves intersect on the genus curve, the boundary component has dimension.

Other moduli stacks

Weighted projective stacks

Constructing weighted projective spaces involves taking the quotient variety of some by a -action. In particular, the action sends a tuple
and the quotient of this action gives the weighted projective space. Since this can instead be taken as a stack quotient, the weighted projective stack pg 30 is
Taking the vanishing locus of a weighted polynomial in a line bundle gives a stacky weighted projective variety.

Stacky curves

s, 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
which is generically etale. The stack quotient of the domain by gives a stacky with stacky points that have stabilizer group at the fifth roots of unity in the -chart. This is because these are the points where the cover ramifies.

Non-affine stack

An example of a non-affine stack is given by the half-line with two stacky origins. This can be constructed as the colimit of two inclusion of.

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, 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. 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 OX modules in this topology is not exact.

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. The resulting stacky objects are called derived 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

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: