Algebraic stack

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 \mathcal{M}_{g,n} 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, 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 \mathcal{M}_{g,n} 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. One of the motivating examples of an algebraic stack is to consider a groupoid scheme (R,U,s,t,m) over a fixed scheme S . For example, if R = \mu_n\times_S\mathbb{A}^n_S (where \mu_n is the group scheme of roots of unity), U = \mathbb{A}^n_S , s = \text{pr}_U is the projection map, t is the group action \zeta_n \cdot (x_1,\ldots, x_n)=(\zeta_n x_1,\ldots,\zeta_n x_n) and m is the multiplication map m: (\mu_n\times_S \mathbb{A}^n_S)\times_{\mu_n\times_S \mathbb{A}^n_S} (\mu_n\times_S \mathbb{A}^n_S) \to \mu_n\times_S \mathbb{A}^n_S on \mu_n . Then, given an S -scheme \pi:X\to S , the groupoid scheme (R(X),U(X),s,t,m) forms a groupoid (where R,U are their associated functors). Moreover, this construction is functorial on (\mathrm{Sch}/S) forming a contravariant 2-functor (R(-),U(-),s,t,m): (\mathrm{Sch}/S)^\mathrm{op} \to \text{Cat} where \text{Cat} is the 2-category of small categories. Another way to view this is as a fibred category [U/R] \to (\mathrm{Sch}/S) through the Grothendieck construction. Getting the correct technical conditions, such as the Grothendieck topology on (\mathrm{Sch}/S) , gives the definition of an algebraic stack. For instance, in the associated groupoid of k -points for a field k , over the origin object 0 \in \mathbb{A}^n_S(k) there is the groupoid of automorphisms \mu_n(k) . However, in order to get an algebraic stack from [U/R] , and not just a stack, there are additional technical hypotheses required for [U/R] . The existence of an fppf scheme U \to S and a 1-morphism of fibered categories \mathcal{U} \to \mathcal{X} which is surjective and smooth depends on defining a smooth and surjective morphisms of fibered categories. Here \mathcal{U} is the algebraic stack from the representable functor h_U on h_U: (Sch/S)_{fppf}^{op} \to Sets upgraded to a category fibered in groupoids where the categories only have trivial morphisms. This means the set h_U(T) = \text{Hom}_{(Sch/S)_{fppf}}(T,U) is considered as a category, denoted h_\mathcal{U}(T) , with objects in h_U(T) as fppf morphisms f:T \to U and morphisms are the identity morphism. Hence h_{\mathcal{U}}:(Sch/S)_{fppf}^{op} \to Groupoids is a 2-functor of groupoids. Showing this 2-functor is a sheaf is the content of the 2-Yoneda lemma. Using the Grothendieck construction, there is an associated category fibered in groupoids denoted \mathcal{U} \to \mathcal{X} . Algebraic stacks, also known as Artin stacks, are by definition equipped with a smooth surjective atlas \mathcal{U} \to \mathcal{X} , where \mathcal{U} is the stack associated to some scheme U \to S . If the atlas \mathcal{U}\to \mathcal{X} is moreover étale, then \mathcal{X} is said to be a Deligne–Mumford stack. The subclass of Deligne-Mumford stacks is useful because it provides the correct setting for many natural stacks considered, such as the moduli stack of algebraic curves. In addition, they are strict enough that objects represented by points in Deligne-Mumford stacks do not have infinitesimal automorphisms . This is very important because infinitesimal automorphisms make studying the deformation theory of Artin stacks very difficult. For example, the deformation theory of the Artin stack BGL_n = [*/GL_n] , the moduli stack of rank n vector bundles, has infinitesimal automorphisms controlled partially by the Lie algebra \mathfrak{gl}_n . This leads to an infinite sequence of deformations and obstructions in general, which is one of the motivations for studying moduli of stable bundles. Only in the special case of the deformation theory of line bundles [*/GL_1] = [*/\mathbb{G}_m] is the deformation theory tractable, since the associated Lie algebra is abelian. Note that many stacks cannot be naturally represented as Deligne-Mumford stacks because it only allows for finite covers, or, algebraic stacks with finite covers. Note that because every Etale cover is flat and locally of finite presentation, algebraic stacks defined with the fppf-topology subsume this theory; but, it is still useful since many stacks found in nature are of this form, such as the moduli of curves \mathcal{M}_g . Also, the differential-geometric analogue of such stacks are called orbifolds. The Etale condition implies the 2-functor B\mu_n:(\mathrm{Sch}/S)^\text{op} \to \text{Cat} sending a scheme to its groupoid of \mu_n -torsors is representable as a stack over the Etale topology, but the Picard-stack B\mathbb{G}_m of \mathbb{G}_m -torsors (equivalently the category of line bundles) is not representable. Stacks of this form are representable as stacks over the fppf-topology. Another reason for considering the fppf-topology versus the etale topology is over characteristic p the Kummer sequence 0 \to \mu_p \to \mathbb{G}_m \to \mathbb{G}_m \to 0 is exact only as a sequence of fppf sheaves, but not as a sequence of etale sheaves. The structure sheaf of an algebraic stack is an object pulled back from a universal structure sheaf \mathcal{O} on the site (Sch/S)_{fppf} . This universal structure sheaf is defined as \mathcal{O}:(Sch/S)_{fppf}^{op} \to Rings, \text{ where } U/X \mapsto \Gamma(U,\mathcal{O}_U) and the associated structure sheaf on a category fibered in groupoids p:\mathcal{X} \to (Sch/S)_{fppf} is defined as \mathcal{O}_\mathcal{X} := p^{-1}\mathcal{O} where p^{-1} comes from the map of Grothendieck topologies. In particular, this means if x \in \text{Ob}(\mathcal{X}) lies over U , so p(x) = U , then \mathcal{O}_\mathcal{X}(x)=\Gamma(U,\mathcal{O}_U) . As a sanity check, it's worth comparing this to a category fibered in groupoids coming from an S -scheme X for various topologies. For example, if (\mathcal{X}_{Zar},\mathcal{O}_\mathcal{X}) = ((Sch/X)_{Zar}, \mathcal{O}_X) is a category fibered in groupoids over (Sch/S)_{fppf} , the structure sheaf for an open subscheme U \to X gives \mathcal{O}_\mathcal{X}(U) = \mathcal{O}_X(U) = \Gamma(U,\mathcal{O}_X) so this definition recovers the classic structure sheaf on a scheme. Moreover, for a quotient stack \mathcal{X} = [X/G] , the structure sheaf this just gives the G -invariant sections \mathcal{O}_{\mathcal{X}}(U) = \Gamma(U,u^*\mathcal{O}_X)^{G} for u:U\to X in (Sch/S)_{fppf} .

Text derived from https://en.wikipedia.org/w/index.php?oldid=1374366484 under CC BY-SA 4.0; modifications have been made.