Stack (mathematics)

Last updated

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.

Contents

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, but it will exist as a stack.

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 (under the functor), 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:VU 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. 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 (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.

Algebraic stacks

An algebraic stack or Artin stack is a stack in groupoids X over the fppf 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 YX of stacks is representable if, for every morphism SX 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 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: [1] 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

Then, this functor determines the following category
  1. an object is a pair consisting of a scheme in and an element
  2. a morphism consists of a morphism in such that .
Via the forgetful functor , the category is a category fibered over . For example, if is a scheme in , then it determines the contravariant functor and the corresponding fibered category is the stack associated to X. Stacks (or prestacks) can be constructed as a variant of this construction. In fact, any scheme with a quasi-compact diagonal is an algebraic stack associated to the scheme.

Stacks of objects

Constructions with stacks

Stack quotients

If is a scheme and is a smooth affine group scheme acting on , then there is a quotient algebraic stack , [2] 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 (intuitively speaking) 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. Indeed, given a line bundle over a scheme , the relative spec

gives a geometric line bundle. By removing the image of the zero section, one obtains a principal -bundle. Conversely, from the representation , the associated line bundle can be reconstructed.

Gerbes

A gerbe is a stack in groupoids that is locally nonempty, 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(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.

Moduli stacks

Moduli of curves

  • Mumford (1965) studied the moduli stack M1,1 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 defined as a universal family of smooth curves of given genus does not exist as an algebraic variety because in particular there are curves admitting nontrivial automorphisms. However there is a moduli stack , which is a good substitute for the non-existent fine moduli space of smooth genus curves. More generally there is a moduli stack of genus curves with marked points. In general this is an algebraic stack, and is a Deligne–Mumford stack for or or (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 and ), which is proper over Spec Z. For example, is the classifying stack of the projective general linear group. (There is a subtlety in defining , as one has to use algebraic spaces rather than schemes to construct it.)

Kontsevich moduli spaces

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 [3]

and can have wild behavior, such as being reducible stacks whose components are non-equal dimension. For example, [3] 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

Geometric 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 [4] pg 30 is

Taking the vanishing locus of a weighted polynomial in a line bundle gives a stacky weighted projective variety.

Stacky curves

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

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.[ citation needed ]

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 (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. [5] ) 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 (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 that 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:

See also

Notes

  1. Alper, Jarod; Hall, Jack; Rydh, David (2020). "A Luna étale slice theorem for algebraic stacks". Annals of Mathematics. 191 (3): 675–738. doi:10.4007/annals.2020.191.3.1. hdl: 10150/641331 . ISSN   0003-486X. JSTOR   10.4007/annals.2020.191.3.1. S2CID   3225788.
  2. Heinloth, Jochen (January 29, 2009), "Lectures on the Moduli Stack of Vector Bundles on a Curve", Affine Flag Manifolds and Principal Bundles, Basel: Springer Basel (published 2010), pp. 123–153, doi:10.1007/978-3-0346-0288-4_4, ISBN   978-3-0346-0287-7
  3. 1 2 Massarenti, Alez. "Moduli of Stable Maps, Gromov-Witten Invariants, and Quantum Cohomology" (PDF). pp. 1–4. Archived (PDF) from the original on 2018-01-23.
  4. Fantechi, Barbara; Mann, Etienne; Nironi, Fabio (2009-09-22). "Smooth toric DM stacks". arXiv: 0708.1254 [math.AG].
  5. See, for example, Olsson, Martin (2007). "Sheaves on Artin stacks". Journal für die reine und angewandte Mathematik . 2007 (603): 55–112. doi:10.1515/CRELLE.2007.012. MR   2312554. S2CID   15445962.

Related Research Articles

In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category C that makes the objects of C act like the open sets of a topological space. A category together with a choice of Grothendieck topology is called a site.

In mathematics, a sheaf is a tool for systematically tracking data attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could be the ring of continuous functions defined on that open set. Such data are well behaved in that they can be restricted to smaller open sets, and also the data assigned to an open set are equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original open set.

In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It is also a fundamental tool in the field of operator algebras. It can be seen as the study of certain kinds of invariants of large matrices.

In mathematics, in particular algebraic geometry, a moduli space is a geometric space whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such spaces frequently arise as solutions to classification problems: If one can show that a collection of interesting objects can be given the structure of a geometric space, then one can parametrize such objects by introducing coordinates on the resulting space. In this context, the term "modulus" is used synonymously with "parameter"; moduli spaces were first understood as spaces of parameters rather than as spaces of objects. A variant of moduli spaces is formal moduli. Bernhard Riemann first used the term "moduli" in 1857.

In mathematics, specifically algebraic geometry, a scheme is a structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities and allowing "varieties" defined over any commutative ring.

In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures. Étale cohomology theory can be used to construct ℓ-adic cohomology, which is an example of a Weil cohomology theory in algebraic geometry. This has many applications, such as the proof of the Weil conjectures and the construction of representations of finite groups of Lie type.

In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with reference to a sheaf of rings that codifies this geometric information.

In mathematics, an algebraic stack is a vast generalization of algebraic spaces, or schemes, which are foundational for studying moduli theory. Many moduli spaces are constructed using techniques specific to algebraic stacks, such as Artin's representability theorem, which is used to construct the moduli space of pointed algebraic curves and the moduli stack of elliptic curves. Originally, they were introduced by Alexander Grothendieck to keep track of automorphisms on moduli spaces, a technique which allows for treating these moduli spaces as if their underlying schemes or algebraic spaces are smooth. After Grothendieck developed the general theory of descent, and Giraud the general theory of stacks, the notion of algebraic stacks was defined by Michael Artin.

In mathematics, a gerbe is a construct in homological algebra and topology. Gerbes were introduced by Jean Giraud following ideas of Alexandre Grothendieck as a tool for non-commutative cohomology in degree 2. They can be seen as an analogue of fibre bundles where the fibre is the classifying stack of a group. Gerbes provide a convenient, if highly abstract, language for dealing with many types of deformation questions especially in modern algebraic geometry. In addition, special cases of gerbes have been used more recently in differential topology and differential geometry to give alternative descriptions to certain cohomology classes and additional structures attached to them.

<span class="mw-page-title-main">Grothendieck–Riemann–Roch theorem</span>

In mathematics, specifically in algebraic geometry, the Grothendieck–Riemann–Roch theorem is a far-reaching result on coherent cohomology. It is a generalisation of the Hirzebruch–Riemann–Roch theorem, about complex manifolds, which is itself a generalisation of the classical Riemann–Roch theorem for line bundles on compact Riemann surfaces.

Fibred categories are abstract entities in mathematics used to provide a general framework for descent theory. They formalise the various situations in geometry and algebra in which inverse images of objects such as vector bundles can be defined. As an example, for each topological space there is the category of vector bundles on the space, and for every continuous map from a topological space X to another topological space Y is associated the pullback functor taking bundles on Y to bundles on X. Fibred categories formalise the system consisting of these categories and inverse image functors. Similar setups appear in various guises in mathematics, in particular in algebraic geometry, which is the context in which fibred categories originally appeared. Fibered categories are used to define stacks, which are fibered categories with "descent". Fibrations also play an important role in categorical semantics of type theory, and in particular that of dependent type theories.

In mathematics, algebraic spaces form a generalization of the schemes of algebraic geometry, introduced by Michael Artin for use in deformation theory. Intuitively, schemes are given by gluing together affine schemes using the Zariski topology, while algebraic spaces are given by gluing together affine schemes using the finer étale topology. Alternatively one can think of schemes as being locally isomorphic to affine schemes in the Zariski topology, while algebraic spaces are locally isomorphic to affine schemes in the étale topology.

The étale or algebraic fundamental group is an analogue in algebraic geometry, for schemes, of the usual fundamental group of topological spaces.

In algebraic geometry, the Nisnevich topology, sometimes called the completely decomposed topology, is a Grothendieck topology on the category of schemes which has been used in algebraic K-theory, A¹ homotopy theory, and the theory of motives. It was originally introduced by Yevsey Nisnevich, who was motivated by the theory of adeles.

In mathematics, more specifically sheaf theory, a branch of topology and algebraic geometry, the exceptional inverse image functor is the fourth and most sophisticated in a series of image functors for sheaves. It is needed to express Verdier duality in its most general form.

In mathematics, a topos is a category that behaves like the category of sheaves of sets on a topological space. Topoi behave much like the category of sets and possess a notion of localization; they are a direct generalization of point-set topology. The Grothendieck topoi find applications in algebraic geometry; the more general elementary topoi are used in logic.

In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaf cohomology is a technique for producing functions with specified properties. Many geometric questions can be formulated as questions about the existence of sections of line bundles or of more general coherent sheaves; such sections can be viewed as generalized functions. Cohomology provides computable tools for producing sections, or explaining why they do not exist. It also provides invariants to distinguish one algebraic variety from another.

This is a glossary of algebraic geometry.

In algebraic geometry, Behrend's trace formula is a generalization of the Grothendieck–Lefschetz trace formula to a smooth algebraic stack over a finite field conjectured in 1993 and proven in 2003 by Kai Behrend. Unlike the classical one, the formula counts points in the "stacky way"; it takes into account the presence of nontrivial automorphisms.

In algebraic geometry, a torsor or a principal bundle is an analogue of a principal bundle in algebraic topology. Because there are few open sets in Zariski topology, it is more common to consider torsors in étale topology or some other flat topologies. The notion also generalizes a Galois extension in abstract algebra. Though other notions of torsors are known in more general context this article will focus on torsors over schemes, the original setting where torsors have been thought for. The word torsor comes from the French torseur. They are indeed widely discussed, for instance, in Michel Demazure's and Pierre Gabriel's famous book Groupes algébriques, Tome I.

References

Pedagogical

Guides to the literature

References

Further reading