Editing Algebraic stack (section)

=== Deligne–Mumford stacks ===
Algebraic stacks, also known as '''Artin stacks''', are by definition equipped with a smooth surjective atlas <math>\mathcal{U} \to \mathcal{X}</math>, where <math>\mathcal{U}</math> is the stack associated to some scheme <math>U \to S</math>. If the atlas <math>\mathcal{U}\to \mathcal{X}</math> is moreover étale, then <math>\mathcal{X}</math> 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 <u>points in Deligne-Mumford stacks do not have [[infinitesimal]] automorphisms</u>. 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 <math>BGL_n = [*/GL_n]</math>, the moduli stack of rank <math>n</math> vector bundles, has infinitesimal automorphisms controlled partially by the [[Lie algebra]] <math>\mathfrak{gl}_n</math>. This leads to an infinite sequence of deformations and obstructions in general, which is one of the motivations for studying [[Moduli space of stable bundles|moduli of stable bundles]]. Only in the special case of the [[deformation theory of line bundles]] <math>[*/GL_1] = [*/\mathbb{G}_m]</math> is the deformation theory tractable, since the associated Lie algebra is [[Abelian Lie algebra|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 algebraic curves|moduli of curves]] <math>\mathcal{M}_g</math>. Also, the differential-geometric analogue of such stacks are called [[orbifold]]s. The Etale condition implies the 2-functor<blockquote><math>B\mu_n:(\mathrm{Sch}/S)^\text{op} \to \text{Cat}</math></blockquote>sending a scheme to its groupoid of <math>\mu_n</math>-[[Torsor (algebraic geometry)|torsors]] is representable as a stack over the Etale topology, but the Picard-stack <math>B\mathbb{G}_m</math> of <math>\mathbb{G}_m</math>-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 <math>p</math> the [[Kummer sequence]]<blockquote><math>0 \to \mu_p \to \mathbb{G}_m \to \mathbb{G}_m \to 0</math></blockquote>is exact only as a sequence of fppf sheaves, but not as a sequence of etale sheaves.