Algebraic stack

 ( Algebraic Curves ) 

In mathematics, an algebraic stack is a vast generalization of , 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 $\backslash mathcal\{M\}\_\{g,n\}$ 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 scheme. 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.

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Definition

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Motivation 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 $R\; =\; \backslash mu\_n\backslash times\_S\backslash mathbb\{A\}^n\_S$ (where $\backslash mu\_n$ is the group scheme of roots of unity), $U\; =\; \backslash mathbb\{A\}^n\_S$, $s\; =\; \backslash text\{pr\}\_U$ is the projection map, $t$ is the group action

$\backslash zeta\_n\; \backslash cdot\; (x\_1,\backslash ldots,\; x\_n)=(\backslash zeta\_n\; x\_1,\backslash ldots,\backslash zeta\_n\; x\_n)$

and $m$ is the multiplication map

$m:\; (\backslash mu\_n\backslash times\_S\; \backslash mathbb\{A\}^n\_S)\backslash times\_\{\backslash mu\_n\backslash times\_S\; \backslash mathbb\{A\}^n\_S\}\; (\backslash mu\_n\backslash times\_S\; \backslash mathbb\{A\}^n\_S)\; \backslash to\; \backslash mu\_n\backslash times\_S\; \backslash mathbb\{A\}^n\_S$

on $\backslash mu\_n$. Then, given an $S$-scheme $\backslash pi:X\backslash to\; S$, 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 $(\backslash mathrm\{Sch\}/S)$ forming a contravariant 2-functor

$(R(-),U(-),s,t,m):\; (\backslash mathrm\{Sch\}/S)^\backslash mathrm\{op\}\; \backslash to\; \backslash text\{Cat\}$

where $\backslash text\{Cat\}$ is the 2-category of Small category. Another way to view this is as a fibred category $U/R\; \backslash to\; (\backslash mathrm\{Sch\}/S)$ through the Grothendieck construction. Getting the correct technical conditions, such as the Grothendieck topology on $(\backslash mathrm\{Sch\}/S)$, gives the definition of an algebraic stack. For instance, in the associated groupoid of $k$-points for a field $k$, over the origin object $0\; \backslash in\; \backslash mathbb\{A\}^n\_S(k)$ there is the groupoid of automorphisms $\backslash mu\_n(k)$. However, in order to get an algebraic stack from $U/R$, and not just a stack, there are additional technical hypotheses required for $U/R$.

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Algebraic stacks It turns out using the Fppf topology (faithfully flat and locally of finite presentation) on $(\backslash mathrm\{Sch\}/S)$, denoted $(\backslash mathrm\{Sch\}/S)\_\{fppf\}$, forms the basis for defining algebraic stacks. Then, an algebraic stack is a fibered category

$p:\; \backslash mathcal\{X\}\; \backslash to\; (\backslash mathrm\{Sch\}/S)\_\{fppf\}$

such that

1. $\backslash mathcal\{X\}$ is a category fibered in groupoids, meaning the overcategory for some $\backslash pi:X\backslash to\; S$ is a groupoid
2. The diagonal map $\backslash Delta:\backslash mathcal\{X\}\; \backslash to\; \backslash mathcal\{X\}\backslash times\_S\backslash mathcal\{X\}$ of fibered categories is representable as algebraic spaces
3. There exists an $fppf$ scheme $U\; \backslash to\; S$ and an associated 1-morphism of fibered categories $\backslash mathcal\{U\}\; \backslash to\; \backslash mathcal\{X\}$ which is surjective and smooth called an atlas.

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Explanation of technical conditions

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Using the fppf topology First of all, the fppf-topology is used because it behaves well with respect to Descent theory. For example, if there are schemes $X,Y\; \backslash in\; \backslash operatorname\{Ob\}(\backslash mathrm\{Sch\}/S)$ and $X\; \backslash to\; Y$ can be refined to an fppf-cover of $Y$, if $X$ is flat, locally finite type, or locally of finite presentation, then $Y$ has this property. this kind of idea can be extended further by considering properties local either on the target or the source of a morphism $f:X\backslash to\; Y$. For a cover $\backslash \{X\_i\; \backslash to\; X\backslash \}\_\{i\; \backslash in\; I\}$ we say a property $\backslash mathcal\{P\}$ is local on the source if

$f:X\backslash to\; Y$ has $\backslash mathcal\{P\}$ if and only if each $X\_i\; \backslash to\; Y$ has $\backslash mathcal\{P\}$.

There is an analogous notion on the target called local on the target. This means given a cover $\backslash \{Y\_i\; \backslash to\; Y\; \backslash \}\_\{i\; \backslash in\; I\}$

$f:X\backslash to\; Y$ has $\backslash mathcal\{P\}$ if and only if each $X\backslash times\_YY\_i\; \backslash to\; Y\_i$ has $\backslash mathcal\{P\}$.

For the fppf topology, having an immersion is local on the target. In addition to the previous properties local on the source for the fppf topology, $f$ being universally open is also local on the source. Also, being locally Noetherian and Jacobson are local on the source and target for the fppf topology. This does not hold in the fpqc topology, making it not as "nice" in terms of technical properties. Even though this is true, using algebraic stacks over the fpqc topology still has its use, such as in chromatic homotopy theory. This is because the Moduli stack of formal group laws $\backslash mathcal\{M\}\_\{fg\}$ is an fpqc-algebraic stack^{pg 40}.

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Representable diagonal By definition, a 1-morphism $f:\backslash mathcal\{X\}\; \backslash to\; \backslash mathcal\{Y\}$ of categories fibered in groupoids is representable by algebraic spaces if for any fppf morphism $U\; \backslash to\; S$ of schemes and any 1-morphism $y:\; (Sch/U)\_\{fppf\}\; \backslash to\; \backslash mathcal\{Y\}$, the associated category fibered in groupoids

$(Sch/U)\_\{fppf\}\backslash times\_\{\backslash mathcal\{Y\}\}\; \backslash mathcal\{X\}$

is representable as an algebraic space, meaning there exists an algebraic space

$F:(Sch/S)^\{op\}\_\{fppf\}\; \backslash to\; Sets$

such that the associated fibered category $\backslash mathcal\{S\}\_F\; \backslash to\; (Sch/S)\_\{fppf\}$$Sets\; \backslash to\; Cat$ is the embedding sending a set $S$ to the category of objects $S$ and only identity morphisms. Then, the Grothendieck construction can be applied to give a category fibered in groupoids is equivalent to $(Sch/U)\_\{fppf\}\backslash times\_\{\backslash mathcal\{Y\}\}\; \backslash mathcal\{X\}$. There are a number of equivalent conditions for representability of the diagonal which help give intuition for this technical condition, but one of main motivations is the following: for a scheme $U$ and objects $x,\; y\; \backslash in\; \backslash operatorname\{Ob\}(\backslash mathcal\{X\}\_U)$ the sheaf $\backslash operatorname\{Isom\}(x,y)$ is representable as an algebraic space. In particular, the stabilizer group for any point on the stack $x\; :\; \backslash operatorname\{Spec\}(k)\; \backslash to\; \backslash mathcal\{X\}\_\{\backslash operatorname\{Spec\}(k)\}$ is representable as an algebraic space.

Another important equivalence of having a representable diagonal is the technical condition that the intersection of any two algebraic spaces in an algebraic stack is an algebraic space. Reformulated using fiber products

the representability of the diagonal is equivalent to $Y\; \backslash to\; \backslash mathcal\{X\}$ being representable for an algebraic space $Y$. This is because given morphisms $Y\; \backslash to\; \backslash mathcal\{X\},\; Z\; \backslash to\; \backslash mathcal\{X\}$ from algebraic spaces, they extend to maps $\backslash mathcal\{X\}\backslash times\backslash mathcal\{X\}$ from the diagonal map. There is an analogous statement for algebraic spaces which gives representability of a sheaf on $(F/S)\_\{fppf\}$ as an algebraic space.

Note that an analogous condition of representability of the diagonal holds for some formulations of higher stacks where the fiber product is an $(n-1)$-stack for an $n$-stack $\backslash mathcal\{X\}$.

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Surjective and smooth atlas

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2-Yoneda lemma The existence of an $fppf$ scheme $U\; \backslash to\; S$ and a 1-morphism of fibered categories $\backslash mathcal\{U\}\; \backslash to\; \backslash mathcal\{X\}$ which is surjective and smooth depends on defining a smooth and surjective morphisms of fibered categories. Here $\backslash mathcal\{U\}$ is the algebraic stack from the representable functor $h\_U$ on $h\_U:\; (Sch/S)\_\{fppf\}^\{op\}\; \backslash to\; Sets$ upgraded to a category fibered in groupoids where the categories only have trivial morphisms. This means the set

$h\_U(T)\; =\; \backslash text\{Hom\}\_\{(Sch/S)\_\{fppf\}\}(T,U)$

is considered as a category, denoted $h\_\backslash mathcal\{U\}(T)$, with objects in $h\_U(T)$ as $fppf$ morphisms

$f:T\; \backslash to\; U$

and morphisms are the identity morphism. Hence

$h\_\{\backslash mathcal\{U\}\}:(Sch/S)\_\{fppf\}^\{op\}\; \backslash to\; Groupoids$

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 $\backslash mathcal\{U\}\; \backslash to\; \backslash mathcal\{X\}$.

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Representable morphisms of categories fibered in groupoids To say this morphism $\backslash mathcal\{U\}\; \backslash to\; \backslash mathcal\{X\}$ is smooth or surjective, we have to introduce representable morphisms. A morphism $p:\backslash mathcal\{X\}\; \backslash to\; \backslash mathcal\{Y\}$ of categories fibered in groupoids over $(Sch/S)\_\{fppf\}$ is said to be representable if given an object $T\; \backslash to\; S$ in $(Sch/S)\_\{fppf\}$ and an object $t\; \backslash in\; \backslash text\{Ob\}(\backslash mathcal\{Y\}\_T)$ the 2-fibered product

$(Sch/T)\_\{fppf\}\backslash times\_\{t,\backslash mathcal\{Y\}\}\; \backslash mathcal\{X\}\_T$

is representable by a scheme. Then, we can say the morphism of categories fibered in groupoids $p$ is smooth and surjective if the associated morphism

$(Sch/T)\_\{fppf\}\backslash times\_\{t,\backslash mathcal\{Y\}\}\; \backslash mathcal\{X\}\_T\; \backslash to\; (Sch/T)\_\{fppf\}$

of schemes is smooth and surjective.

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Deligne–Mumford stacks Algebraic stacks, also known as Artin stacks, are by definition equipped with a smooth surjective atlas $\backslash mathcal\{U\}\; \backslash to\; \backslash mathcal\{X\}$, where $\backslash mathcal\{U\}$ is the stack associated to some scheme $U\; \backslash to\; S$. If the atlas $\backslash mathcal\{U\}\backslash to\; \backslash mathcal\{X\}$ is moreover étale, then $\backslash mathcal\{X\}$ 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 object 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 $\backslash mathfrak\{gl\}\_n$. 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 $*/GL\_1\; =\; */\backslash mathbb\{G\}\_m$ 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 $\backslash mathcal\{M\}\_g$. Also, the differential-geometric analogue of such stacks are called . The Etale condition implies the 2-functor

$B\backslash mu\_n:(\backslash mathrm\{Sch\}/S)^\backslash text\{op\}\; \backslash to\; \backslash text\{Cat\}$

sending a scheme to its groupoid of $\backslash mu\_n$-torsors is representable as a stack over the Etale topology, but the Picard-stack $B\backslash mathbb\{G\}\_m$ of $\backslash mathbb\{G\}\_m$-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

$0\; \backslash to\; \backslash mu\_p\; \backslash to\; \backslash mathbb\{G\}\_m\; \backslash to\; \backslash mathbb\{G\}\_m\; \backslash to\; 0$

is exact only as a sequence of fppf sheaves, but not as a sequence of etale sheaves.

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Defining algebraic stacks over other topologies Using other Grothendieck topologies on $(F/S)$ gives alternative theories of algebraic stacks which are either not general enough, or don't behave well with respect to exchanging properties from the base of a cover to the total space of a cover. It is useful to recall there is the following hierarchy of generalization

$\backslash text\{fpqc\}\; \backslash supset\; \backslash text\{fppf\}\; \backslash supset\; \backslash text\{smooth\}\; \backslash supset\; \backslash text\{etale\}\; \backslash supset\; \backslash text\{Zariski\}$

of big topologies on $(F/S)$.

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Structure sheaf The structure sheaf of an algebraic stack is an object pulled back from a universal structure sheaf $\backslash mathcal\{O\}$ on the site $(Sch/S)\_\{fppf\}$. This universal structure sheaf is defined as

$\backslash mathcal\{O\}:(Sch/S)\_\{fppf\}^\{op\}\; \backslash to\; Rings,\; \backslash text\{\; where\; \}\; U/X\; \backslash mapsto\; \backslash Gamma(U,\backslash mathcal\{O\}\_U)$

and the associated structure sheaf on a category fibered in groupoids

$p:\backslash mathcal\{X\}\; \backslash to\; (Sch/S)\_\{fppf\}$

is defined as

$\backslash mathcal\{O\}\_\backslash mathcal\{X\}\; :=\; p^\{-1\}\backslash mathcal\{O\}$

where $p^\{-1\}$ comes from the map of Grothendieck topologies. In particular, this means if $x\; \backslash in\; \backslash text\{Ob\}(\backslash mathcal\{X\})$ lies over $U$, so $p(x)\; =\; U$, then $\backslash mathcal\{O\}\_\backslash mathcal\{X\}(x)=\backslash Gamma(U,\backslash mathcal\{O\}\_U)$. 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

$(\backslash mathcal\{X\}\_\{Zar\},\backslash mathcal\{O\}\_\backslash mathcal\{X\})\; =\; ((Sch/X)\_\{Zar\},\; \backslash mathcal\{O\}\_X)$

is a category fibered in groupoids over $(Sch/S)\_\{fppf\}$, the structure sheaf for an open subscheme $U\; \backslash to\; X$ gives

$\backslash mathcal\{O\}\_\backslash mathcal\{X\}(U)\; =\; \backslash mathcal\{O\}\_X(U)\; =\; \backslash Gamma(U,\backslash mathcal\{O\}\_X)$

so this definition recovers the classic structure sheaf on a scheme. Moreover, for a quotient stack $\backslash mathcal\{X\}\; =\; X/G$, the structure sheaf this just gives the $G$-invariant sections

$\backslash mathcal\{O\}\_\{\backslash mathcal\{X\}\}(U)\; =\; \backslash Gamma(U,u^*\backslash mathcal\{O\}\_X)^\{G\}$

for $u:U\backslash to\; X$ in $(Sch/S)\_\{fppf\}$.

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Examples

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Classifying stacks Many classifying stacks for Algebraic group are algebraic stacks. In fact, for an algebraic group space $G$ over a scheme $S$ which is flat of finite presentation, the stack $BG$ is algebraic^{theorem 6.1}.

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See also

• Gerbe
• Chow group of a stack
• Cohomology of a stack
• Quotient stack
• Sheaf on an algebraic stack
• Toric stack
• Artin's criterion
• Pursuing Stacks
• Derived algebraic geometry

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External links

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Artin's Axioms

• - Look at "Axioms" and "Algebraic stacks"
• target="_blank" rel="nofollow"> Artin Algebraization and Quotient Stacks - Jarod Alper

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Papers

• Kai A. Behrend (2026). 9781470403720 . ISBN 9781470403720

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Applications

• (1973). 9783540065586 ISBN 9783540065586

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Other

• Examples of Stacks
• target="_blank" rel="nofollow"> Notes on algebraic stacks

Categories: Algebraic Curves, Moduli Theory, Algebraic Geometry

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