Stack (mathematics)

A category ${\displaystyle c}$ with a functor to a category ${\displaystyle C}$ is called a fibered category over ${\displaystyle C}$ if for any morphism ${\displaystyle F:X\to Y}$ in ${\displaystyle C}$ and any object ${\displaystyle y}$ of ${\displaystyle c}$ with image ${\displaystyle Y}$ (under the functor), there is a pullback ${\displaystyle f:x\to y}$ of ${\displaystyle y}$ by ${\displaystyle F}$. This means a morphism with image ${\displaystyle F}$ such that any morphism ${\displaystyle g:z\to y}$ with image ${\displaystyle G=F\circ H}$ can be factored as ${\displaystyle g=f\circ h}$ by a unique morphism ${\displaystyle h:z\to x}$ in ${\displaystyle c}$ such that the functor maps ${\displaystyle h}$ to ${\displaystyle H}$. The element ${\displaystyle x=F^{*}y}$ is called the pullback of ${\displaystyle y}$ along ${\displaystyle F}$ 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 the over category C/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. 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 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×_VV_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 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.

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 Y${\displaystyle \rightarrow }$ X of stacks is representable if, for every morphism S ${\displaystyle \rightarrow }$ 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 ${\displaystyle \Delta$ :{\mathfrak {X}}\to {\mathfrak {X}}\times {\mathfrak {X}}} is representable if and only if for any pair of morphisms of algebraic spaces ${\displaystyle X,Y\to {\mathfrak {X}}}$, their fiber product ${\displaystyle X\times _{\mathfrak {X}}Y}$ 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.

Since the inception of algebraic stacks it was expected that they are locally quotient stacks of the form ${\displaystyle [{\text{Spec}}(A)/G]}$ where ${\displaystyle G}$ is a linearly reductive algebraic group. This was recently proved to be the case:[1] given a quasi-separated algebraic stack ${\displaystyle {\mathfrak {X}}}$ locally of finite type over an algebraically closed field ${\displaystyle k}$ whose stabilizers are affine, and ${\displaystyle x\in {\mathfrak {X}}(k)}$ a smooth and closed point with linearly reductive stabilizer group ${\displaystyle G_{x}}$, there exists an etale cover of the GIT quotient ${\displaystyle (U,u)\to (N_{x}//G_{x},0)}$, where ${\displaystyle N_{x}=(J_{x}/J_{x}^{2})^{\vee }}$, such that the diagram

${\displaystyle {\begin{matrix}([W/G_{x}],w)&\to &([N_{x}/G_{x}],0)\\\downarrow &&\downarrow \\(U,u)&\to &(N_{x}//G_{x},0)\end{matrix}}}$

is cartesian, and there exists an etale morphism

${\displaystyle f:([W/G_{x}],w)\to ({\mathfrak {X}},x)}$

inducing an isomorphism of the stabilizer groups at ${\displaystyle w}$ and ${\displaystyle x}$.

• Every sheaf ${\displaystyle {\mathcal {F}}:C^{op}\to Sets}$ from a category ${\displaystyle C}$ with a Grothendieck topology can canonically be turned into a stack. For an object ${\displaystyle X\in {\text{Ob}}(C)}$, instead of a set ${\displaystyle {\mathcal {F}}(X)}$ there is a groupoid whose objects are the elements of ${\displaystyle {\mathcal {F}}(X)}$ and the arrows are the identity morphism.
• More concretely, let ${\displaystyle h}$ be a contravariant functor

${\displaystyle h:(Sch/S)^{op}\to Sets}$

Then, this functor determines the following category ${\displaystyle H}$

1. an object is a pair ${\displaystyle (X\to S,x)}$ consisting of a scheme ${\displaystyle X}$ in ${\displaystyle (Sch/S)^{op}}$ and an element ${\displaystyle x\in h(X)}$
2. a morphism ${\displaystyle (X\to S,x)\to (Y\to S,y)}$ consists of a morphism ${\displaystyle \phi :X\to Y}$ in ${\displaystyle (Sch/S)}$ such that ${\displaystyle h(\phi )(y)=x}$.

Via the forgetful functor ${\displaystyle p:H\to (Sch/S)}$, the category ${\displaystyle H}$ is a category fibered over ${\displaystyle (Sch/S)}$. For example, if ${\displaystyle X}$ is a scheme in ${\displaystyle (Sch/S)}$, then it determines the contravariant functor ${\displaystyle h=\operatorname {Hom} (-,X)}$ and the corresponding fibered category is the stack associated to X. Stacks (or prestacks) can be constructed as a variant of this construction. In fact, any scheme ${\displaystyle X}$ with a quasi-compact diagonal is an algebraic stack associated to the scheme ${\displaystyle X}$.

• A Group-stack.
• 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)

If ${\displaystyle X}$ is a scheme ${\displaystyle (Sch/S)}$ and ${\displaystyle G}$ is a smooth affine group scheme acting on ${\displaystyle X}$, then there is a quotient algebraic stack ${\displaystyle [X/G]}$,[2] taking a scheme ${\displaystyle Y\to S}$ to the groupoid of ${\displaystyle G}$-torsors over the ${\displaystyle S}$-scheme ${\displaystyle Y}$ with ${\displaystyle G}$-equivariant maps to ${\displaystyle X}$. Explicitly, given a space ${\displaystyle X}$ with a ${\displaystyle G}$-action, form the stack ${\displaystyle [X/G]}$ which (intuitively speaking) sends a space ${\displaystyle Y}$ to the groupoid of pullback diagrams

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

where ${\displaystyle \Phi }$ is a ${\displaystyle G}$-equivariant morphism of spaces and ${\displaystyle Z\to Y}$is a principal ${\displaystyle G}$-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 ${\displaystyle G}$-bundles.

A special case of this when X is a point gives the classifying stack BG of a smooth affine group scheme G: ${\displaystyle {\textbf {B}}G:=[pt/G].}$ It is named so since the category ${\displaystyle \mathbf {B} G(Y)}$, the fiber over Y, is precisely the category ${\displaystyle \operatorname {Bun} _{G}(Y)}$ of principal ${\displaystyle G}$-bundles over ${\displaystyle Y}$. Note that ${\displaystyle \operatorname {Bun} _{G}(Y)}$ itself can be considered as a stack, the moduli stack of principal G-bundles on Y.

An important subexample from this construction is ${\displaystyle \mathbf {B} GL_{n}}$ which is the moduli stack of principal ${\displaystyle GL_{n}}$-bundles. Since the data of a principal ${\displaystyle GL_{n}}$-bundle is equivalent to the data of a rank ${\displaystyle n}$ vector bundle, this is isomorphic to the moduli stack of rank ${\displaystyle n}$ vector bundles ${\displaystyle Vect_{n}}$.

The moduli stack of line bundles is ${\displaystyle B\mathbb {G} _{m}}$ since every line bundle is canonically isomorphic to a principal ${\displaystyle \mathbb {G} _{m}}$-bundle. Indeed, given a line bundle ${\displaystyle L}$ over a scheme ${\displaystyle S}$, the relative spec

${\displaystyle {\underline {\text{Spec}}}_{S}({\text{Sym}}_{S}(L^{\vee }))\to S}$

gives a geometric line bundle. By removing the image of the zero section, one obtains a principal ${\displaystyle \mathbb {G} _{m}}$-bundle. Conversely, from the representation ${\displaystyle id:\mathbb {G} _{m}\to {\text{Aut}}(\mathbb {A} ^{1})}$, the associated line bundle can be reconstructed.

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

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.

• 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 ${\displaystyle {\mathcal {M}}_{g}}$ defined as a universal family of smooth curves of given genus ${\displaystyle g}$ does not exist as an algebraic variety because in particular there are curves admitting nontrivial automorphisms. However there is a moduli stack ${\displaystyle {\mathcal {M}}_{g}}$ which is a good substitute for the non-existent fine moduli space of smooth genus ${\displaystyle g}$ curves. More generally there is a moduli stack ${\displaystyle {\mathcal {M}}_{g,n}}$ of genus ${\displaystyle g}$ curves with ${\displaystyle n}$ marked points. In general this is an algebraic stack, and is a Deligne–Mumford stack for ${\displaystyle g\geq 2}$ or ${\displaystyle g=1,n\geq 1}$ or ${\displaystyle g=0,n\geq 3}$ (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 ${\displaystyle g}$ and ${\displaystyle n}$) which is proper over Spec Z. For example, ${\displaystyle {\mathcal {M}}_{0}}$ is the classifying stack ${\displaystyle B{\text{PGL}}(2)}$ of the projective general linear group. (There is a subtlety in defining ${\displaystyle {\mathcal {M}}_{1}}$, as one has to use algebraic spaces rather than schemes to construct it.)

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 ${\displaystyle X}$ whose image represents a fixed cohomology class. These moduli spaces are denoted[3]

${\displaystyle {\overline {\mathcal {M}}}_{g,n}(X,\beta )}$

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

${\displaystyle {\overline {\mathcal {M}}}_{1,0}(\mathbb {P} ^{2},3[H])}$

has smooth curves parametrized by an open subset ${\displaystyle U\subset \mathbb {P} ^{9}=\mathbb {P} (\Gamma (\mathbb {P} ^{2},{\mathcal {O}}(3)))}$. On the boundary of the moduli space, where curves may degenerate to reducible curves, there is a substack parametrizing reducible curves with a genus ${\displaystyle 0}$ component and a genus ${\displaystyle 1}$ component intersecting at one point, and the map sends the genus ${\displaystyle 1}$ curve to a point. Since all such genus ${\displaystyle 1}$ curves are parametrized by ${\displaystyle U}$, and there is an additional ${\displaystyle 1}$ dimensional choice of where these curves intersect on the genus ${\displaystyle 1}$ curve, the boundary component has dimension ${\displaystyle 10}$.

• A Picard stack generalizes a Picard variety.
• The moduli stack of formal group laws classifies formal group laws.
• An ind-scheme such as an infinite projective space and a formal scheme is a stack.
• A moduli stack of shtukas is used in geometric Langlands program. (See also shtukas.)

Constructing weighted projective spaces involves taking the quotient variety of some ${\displaystyle \mathbb {A} ^{n+1}-\{0\}}$ by a ${\displaystyle \mathbb {G} _{m}}$-action. In particular, the action sends a tuple

${\displaystyle g\cdot (x_{0},\ldots ,x_{n})\mapsto (g^{a_{0}}x_{0},\ldots ,g^{a_{n}}x_{n})}$

and the quotient of this action gives the weighted projective space ${\displaystyle \mathbb {WP} (a_{0},\ldots ,a_{n})}$. Since this can instead be taken as a stack quotient, the weighted projective stack[4] ^{pg 30} is

${\displaystyle {\textbf {WP}}(a_{0},\ldots ,a_{n}):=[\mathbb {A} ^{n}-\{0\}/\mathbb {G} _{m}]}$

Taking the vanishing locus of a weighted polynomial in a line bundle ${\displaystyle f\in \Gamma ({\textbf {WP}}(a_{0},\ldots ,a_{n}),{\mathcal {O}}(a))}$ gives a stacky weighted projective variety.

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

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

which is generically etale. The stack quotient of the domain by ${\displaystyle \mu _{5}}$ gives a stacky ${\displaystyle \mathbb {P} ^{1}}$ with stacky points that have stabilizer group ${\displaystyle \mathbb {Z} /5}$ at the fifth roots of unity in the ${\displaystyle x/y}$-chart. This is because these are the points where the cover ramifies.

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 ${\displaystyle [\mathbb {G} _{m}/(\mathbb {Z} /2)]\to [\mathbb {A} ^{1}/(\mathbb {Z} /2)]}$.

• Behrend, Kai; Conrad, Brian; Edidin, Dan; Fulton, William; Fantechi, Barbara; Göttsche, Lothar; Kresch, Andrew (2006), Algebraic stacks, archived from the original on 2008-05-05
• Goméz, Tomás (1999), Algebraic stacks, arXiv:math/9911199, Bibcode:1999math.....11199G is an expository article describing the basics of stacks with examples.
• Edidin, Dan (2003), "What is... a Stack?" (PDF), Notices of the AMS, 50 (4): 458–459

• https://maths-people.anu.edu.au/~alperj/papers/stacks-guide.pdf
• http://stacks.math.columbia.edu/tag/03B0

• Artin, Michael (1974), "Versal deformations and algebraic stacks", Inventiones Mathematicae, 27 (3): 165–189, Bibcode:1974InMat..27..165A, doi:10.1007/BF01390174, ISSN 0020-9910, MR 0399094, S2CID 122887093
• Deligne, Pierre; Mumford, David (1969), "The irreducibility of the space of curves of given genus", Publications Mathématiques de l'IHÉS, 36 (36): 75–109, CiteSeerX 10.1.1.589.288, doi:10.1007/BF02684599, ISSN 1618-1913, MR 0262240, S2CID 16482150
• Fantechi, Barbara (2001), "Stacks for everybody" (PDF), European Congress of Mathematics Volume I, Progr. Math., vol. 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{{citation}}: CS1 maint: location missing publisher (link)
• Giraud, Jean (1971), Cohomologie non abélienne, Springer, ISBN 3-540-05307-7
• Gómez, Tomás L. (2001), "Algebraic stacks", Proceedings - Mathematical Sciences, 111 (1): 1–31, arXiv:math/9911199, doi:10.1007/BF02829538, MR 1818418, S2CID 373638
• 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, vol. 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 (1): 109–168, arXiv:math/0512097, doi:10.1007/s10240-008-0011-6, MR 2434692, S2CID 371801
• Mumford, David (1965), "Picard groups of moduli problems", in Schilling, O. F. G. (ed.), 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 (2016), Algebraic spaces and stacks, Colloquium Publications, vol. 62, American Mathematical Society, ISBN 978-1470427986
• Vistoli, Angelo (2005), "Grothendieck topologies, fibered categories and descent theory", Fundamental algebraic geometry, Math. Surveys Monogr., vol. 123, Providence, R.I.: Amer. Math. Soc., pp. 1–104, arXiv:math/0412512, Bibcode:2004math.....12512V, MR 2223406