bundles is a motivating factor for introducing moduli stacks, which encode the automorphisms groups as part of the data. We will synthesize the approaches fromSection 0.3on moduli groupoids andSection 0.4on moduli functors.
0.7.1
Specifying a moduli stack
To define a moduli stack, we need to specify 1. families of objects;
2. how two families of objects are isomorphic; and 3. and how families pull back under morphisms.
Notice the difference from specifying a moduli functor (Section 0.4.2) is that rather than specifyingwhentwo families are isomorphic, we specifyhow.
To specify a moduli stack in the algebro-geometric setting, we need to specify for each schemeT agroupoidFamT of families of objects overT. As a natural
generalization of functors to sets, we could consider assignments
This presents the technical difficulty of considering functors between the category of schemes and the ‘category’ of groupoids. Morphisms of groupoids are functors but there are also morphisms of functors (i.e. natural transformations) which we call2-morphisms. This leads to a ‘2-category’ of groupoids.
What is actually involved in defining such an assignment F? In addition to defining the groupoids FamT over each schemeT, we need pullback functors
f∗: FamT → FamS for each morphism f: S → T. But what should be the
compatibility for a composition S −→f T −→g U of schemes? Well, there should be an isomorphism of functors (i.e. a 2-morphism) µf,g: (f∗◦g∗)
∼
→ (g◦f)∗. Should the isomorphisms µf,g satisfy a compatibility condition under triples
S−→f T −→g U −→h V? Yes, but we won’t spell it out here (although we encourage the reader to work it out). Altogether this leads to the concept of a pseudo- functor(see [SP,Tag 003N]). We will take another approach however in specifying prestacks that avoids specifying such compatibility data.
0.7.2
Motivating the definition of a prestack
Instead of trying to define an assignmentT 7→FamT, we will build one massive
categoryXencoding all of the groupoids FamT which will live over the category
Sch of schemes. Loosely speaking, the objects ofXwill be a familyaof objects over a schemeS, i.e. a∈FamS. Ifa∈FamS andb∈FamT, a morphisma→b
inXwill be a morphismf:S →T together with an isomorphism a→∼ f∗b. Aprestack over Sch is a categoryXtogether with functorp:X→Sch, which we visualize as X p a α // _ b _ Sch S f //T
where the lower case lettersa, bare objects inXand the upper case lettersS, T are objects in Sch. We say thatais overSandα:a→bis overf:S→T. Moreover, we need to require certain natural axioms to hold for X−→p Sch. This will be given in full later but vaguely we need to require the existence and uniqueness of pullbacks: given a mapS→T and objectb∈XoverT, there should exist an arrowa−→α b overf satisfying a suitable universal property. See?? for a precise definition.
Given a scheme S, thefiber categoryX(S) is the category of objects overS
whose morphisms are over idS. IfXis built from the groupoids FamS as above,
then the fiber categoryX(S) = FamS.
Example 0.7.1 (Viewing a moduli functor as a moduli prestack). A moduli functorF: Sch→Sets can be encoded as a moduli prestack as follows: we define the category XF of pairs (S, a) where S is a scheme and a ∈ F(S). A map
(S0, a)→(S, a) is a mapf:S0 →S such thata0 =f∗a, wheref∗ is convenient
shorthand forF(f) :F(S)→F(S0). Observe that the fiber categoriesXF(S) are
equivalent (even equal) to the setF(S).
Example 0.7.2 (Moduli prestack of smooth curves). We define the moduli prestack of smooth curvesas the categoryMg of families of smooth curvesC→S
(C→S) is the data of mapsα:C0→Cand f:S0 →S such that the diagram C0 α // C S0 f //S is cartesian.
Example 0.7.3 (Moduli prestack of vector bundles). LetC be a fixed smooth, connected and projective curve overC, and fix integersr≥0 andd. We define the
moduli prestack of vector bundles onCas the categoryMC,r,dof pairs (E, S) where
Sis a scheme andEis a vector bundle onCS =C×CS together with the functor
p:MC,r,d→Sch/C,(E, S)7→S. A map (E0, S0)→(E, S) consists of a map of
schemesf:S0 →S together with a mapE→(id×f)∗E0 ofOCS-modules whose adjoint is an isomorphism (i.e. for any choice of pull back (id×f)∗E, the adjoint map (id×f)∗E→E0 is an isomorphism). Note that a map (E0, S)→(E, S) over
the identity map idS consists simply of an isomorphismE0→E.
Remark 0.7.4. We have formulated morphisms using the adjoint because the pull back is only defined up to isomorphism while the pushforward is canonical. If we were to instead parameterize the total spaces of vector bundles (i.e. A(E) rather thanE), then a morphism (V0, S0)→(V, S) would consist of morphisms
α:V0→V andf: S0→Ssuch thatV0 →V×CSCS0 is an isomorphism of vector bundles.
0.7.3
Motivating the definition of a stack
A stack is to a prestack as a sheaf is to a presheaf. The concept could not be more intuitive: we require that objects and morphisms glue uniquely.
Example 0.7.5 (Moduli stack of sheaves over a point). Define the categoryX over Sch of pairs (E, S) whereE is a sheaf of abelian groups on a schemeS, and the functorp:X→Sch given by (E, S)7→S. A map (E0, S0)→(E, S) inXis a map of schemesf:S0→S together with a mapE→f∗E0 ofOS0-modules whose adjoint is an isomorphism.
You already know that morphisms of sheaves glue [Har77, Exercise II.1.15]: letEandF be sheaves on schemesS andT, and letf:S →T be a map. If{Si}
is a Zariski-open cover ofS, then giving a morphismα: (E, S)→(F, T) is the same data as giving morphismsαi: (E|Si, Si)→(F, T) such thatαi|Sij =αj|Sij.
You also know how sheaves themselves glue [Har77, Exercise II.1.22]—it is more complicated than gluing morphisms since sheaves have automorphisms and given two sheaves, we prefer to say that they are isomorphic rather than equal. If {Si} is a Zariski-open cover of a scheme S, then giving a sheaf E on S is equivalent to giving a sheafEi onSi and isomorphisms φij:Ei|Sij →Ej|Sij such thatφik=φjk◦φij on the triple intersectionSijk.
In an identical way, we could have considered the moduli stack ofO-modules, quasi-coherent sheaves or vector bundles.
The definition of a stack simply axiomitizes these two natural gluing concepts; it is postponed until??.
Exercise 0.7.6. Convince yourself thatExamples 0.7.2 and0.7.3satisfying the same gluing axioms. (See also????.)
0.7.4
Motivating the definition of an algebraic stack
There are functorsF: Sch→Sets that are sheaves when restricted to the Zariski topology on any schemeT but that are not necessarily representable by schemes; see for instance ????. In a similar way, there are prestacksX that are stacks but that are not sufficiently algebro-geometric. If we wish to bring our algebraic geometry toolkit (e.g. coherent sheaves, commutative algebra, cohomology, ...) to study stacks in a similar way that we study schemes, we must impose an algebraicity condition.
The condition we impose on a stack to be algebraic is very natural. Recall that a functorF: Sch→Sets is representable by a scheme if and only if there is a Zariski-open cover{Ui ⊂F}such thatUiis an affine scheme. Similarly, we will
say that a stackX→Sch isalgebraicif
• there is a smooth cover{Ui→X}where eachUi is an affine scheme.
To make this precise, we need to define what it means for{Ui→X}to be a smooth
cover. Just like in the definition of Zariski-open cover (Definition 0.4.17(3)), we require that for every morphismT →Xfrom a schemeT, the fiber product (fiber products of prestacks will be formally introduced in§??)Ui×XT is representable
(by an algebraic space) such thatF
iUi×XT → T is a smooth and surjective
morphism. See??for the precise definition of an algebraic stack.
Constructing a smooth cover of a given moduli stack is a geometric problem inherent to the moduli problem. It can often be solved by ridigifying the moduli problem by parameterizing additional information. This concept is best absorbed in examples.
Example 0.7.7 (Moduli stack of elliptic curves). An elliptic curve (E, p) over C
is embedded intoP2 viaOE(3p) such thatE is defined by a Weierstrass equation
y2z=x(x−z)(x−λz) for someλ6= 0,1 [Har77, Prop. 4.6]. LetU =A1\ {0,1}
with coordinate λ. The family E ⊂ U ×P2 of elliptic curves defined by the
Weierstrass equation gives a smooth (even ´etale) coverU →M1,1.
Example 0.7.8 (Moduli stack of smooth curves). For any smooth, connected and projective curveCof genusg≥2, the third tensor powerωC⊗3 is very ample and gives an embeddingC ,→P(H0(C, ωc⊗3))∼=P5g−6. There is a Hilbert scheme
H parameterizing closed subschemes of P5g−6 with the same Hilbert polynomial
asC ⊂P5g−6, and there is a locally closed subschemeH0 ⊂H parameterizing
smooth subschemes such thatωC⊗3 ∼=OC(1). The universal subscheme over H0
yields a smooth coverH0→Mg.
Example 0.7.9(Moduli stack of vector bundles). For any vector bundle E of rankrand degreedon a smooth, connected and projective curveC, the twistE(m) is globally generated for sufficiently largem. TakingN =h0(C, E(m)), we can
viewE as a quotientOC(−m)N E. There is a Quot schemeQmparameterizing
quotientsOC(−m)N π
F with the same Hilbert polynomial asE and a locally closed subschemeQ0m⊂Qparameterizing quotients where E is a vector bundle and such that the induced map H0(π⊗O
C(m)) :CN → H0(C, E(m)) is an
isomorphism. The universal quotient overQ0mdefines a smooth mapQ0m→MC,r,d
0.7.5
Deligne–Mumford stacks and algebraic spaces
ADeligne–Mumford stackcan be defined in two equivalent ways:
• a stackXsuch that there exists an ´etale (rather than smooth) cover{Ui→X}
by schemes; or
• an algebraic stack such that all automorphisms groups of field-valued points are finite and reduced.
The moduli stacksMg andMg are Deligne–Mumford forg≥2, butMC,r,d is
not. Similarly, analgebraic spacecan be defined in two equivalent ways:
• a sheaf (i.e. a contravariant functorF: Sch→Sets that is a sheaf in the big ´
etale topology) such that there exists an ´etale cover{Ui→F}by schemes;
or
• an algebraic stack such that all automorphisms groups of field-valued points are trivial.
In other words, an algebraic space is an algebraic stack without any stackiness.
Table 1: Schemes, algebraic spaces, Deligne–Mumford stacks, and algebraic stacks are obtained by gluing affine schemes in certain topologies
Algebro-geometric space Type of object Obtained by gluing
Schemes sheaf affine schemes in the
Zariski topology Algebraic spaces sheaf affine schemes in the
´
etale topology
Deligne–Mumford stacks stack affine schemes in the ´
etale topology
Algebraic stacks stack affine schemes in the smooth topology
Example 0.7.10(Quotients by finite groups). Quotients by free actions of finite groups exist as algebraic spaces! See??.