DELIGNE-MUMFORD STACKS 109 5 Deligne-Mumford Stacks as Functors

We now compare Definition 1.2.4.1 with the more traditional “functor-of-points” approach to the theory of Deligne-Mumford stacks.

Notation 1.2.5.1. Letτď1S denote the 8-category of 1-truncated spaces (in other words, the 8-category of groupoids). Let X “ pX,OXq be a Deligne-Mumford stack. We define a functorh_X : CAlg^♥ Ñτ_ď1S by the formula h_XpRq “Map_1TopsHen

CAlg♥pSp´etR,Xq. We will refer to hX as thefunctor of pointsof X.

Proposition 1.2.5.2. The construction XÞÑh_X determines a fully faithful embedding of 8-categories NpDMq ÑFunpCAlg^♥, τ_ď1Sq.

Proof of Proposition 1.2.5.2. Let Xand Ybe Deligne-Mumford stacks. Write X“ pX,OXq.

For each objectU PX, set X_U “ pX_{U,OX|Uq and consider the canonical map θ_U : Map_NpDMqpX_U,Yq ÑMap_FunpCAlg♥,τď1Sqph_XU, h_Yq.

Let us say that an objectU PX is goodifθ_U is a homotopy equivalence. We wish to show that the final object of X is good. In fact, we will show that every object ofX is good. The proof proceeds in several steps:

piq The construction U ÞÑ θ_U carries coproducts in X to products. Consequently, the collection of good objects ofX is closed under small coproducts.

piiq Suppose thatf :U₀ÑX is an effective epimorphism of X, and letU_‚ denote its ˇCech nerve. Thenθ_X can be identified with the totalization of the cosimplicial objectθ_U‚ in Funp∆¹,Sq. Consequently, if each of the objects Um is good, thenX is good.

piiiq Every affine object ofX is good (this follows from Yoneda’s lemma).

pivq Let f :XÑ Y be a monomorphism in X. IfY is affine, then X is good. To prove this, choose an effective epimorphism g:U0 ÑX, whereU0 is a coproduct of affine objectsU_α (see Remark 1.2.4.3). Let U_‚ be the ˇCech nerve ofg. By virtue ofpiiq, it will suffice to show that each U_m is good. Usingpiq, we are reduced to showing that each fiber productUα0ˆX ¨ ¨ ¨ ˆX Uαm is good. Sincef is a monomorphism, we have an equivalence

Uα0ˆX ¨ ¨ ¨ ˆX Uαm »Uα0ˆY ¨ ¨ ¨ ˆY Uαm.

It follows from Proposition 1.2.4.6 that this object is affine, so that the desired result follows frompiiiq.

pvq Let f :X ÑY be an arbitrary morphism in X. If Y is affine, then X is good. To prove this, choose an effective epimorphism g:U₀ ÑX, whereU₀ is a coproduct of affine objectsU_α (see Remark 1.2.4.3). LetU_‚ be the ˇCech nerve ofg. By virtue of piiq, it will suffice to show that eachUm is good. Usingpiq, we are reduced to showing that each fiber productU_α0ˆX¨ ¨ ¨ ˆXU_αm is good. This follows from pivq, since there exists a monomorphism

U_α0 ˆX ¨ ¨ ¨ ˆXU_αm ãÑU_α0ˆY ¨ ¨ ¨ ˆY U_αm whose codomain is affine by virtue of Proposition 1.2.4.6.

pviq Every objectX PX is good. To prove this, choose an effective epimorphismg:U0 ÑX, where U₀ is a coproduct of affine objects U_α (see Remark 1.2.4.3). Let U_‚ be the Cech nerve ofˇ g. By virtue of piiq, it will suffice to show that each Um is good. Using piq, we are reduced to showing that each fiber product Uα0 ˆX ¨ ¨ ¨ ˆX Uαm is good.

This follows from pvq, since the projection map U_α0ˆX ¨ ¨ ¨ ˆX U_αm ÑU_α0 has affine codomain.

For the remainder of this section, we will abuse notation by not distinguishing between a Deligne-Mumford stack and its image under the fully faithful embedding of Proposition 1.2.5.2 (in other words, we will identify a Deligne-Mumford stack X “ pX,OXq with the functor h_X).

Definition 1.2.5.3. Let R be a commutative ring and write Sp´etR“ pX,OXq. For each objectU PX “Shv_SetpCAlg^´et_Rq, we let Sp´et_URdenote the ringed topospX_{U,OX |Uq. Then Sp´et_UR is a Deligne-Mumford stack equipped with a canonical map Sp´et_URÑSp´etR.

If f : X Ñ Y is an arbitrary morphism in FunpCAlg^♥, τ_ď1Sq, we will say that f is representable and ´etale if, for every commutative ring R and every pointηPYpRq, the fiber product Sp´etRˆY X is equivalent to Sp´et_UpRq, for some objectU PShv_SetpCAlg^´et_Rq. Remark 1.2.5.4. In the situation of Definition 1.2.5.3, a mapf :XÑY is representable and ´etale if and only if, for each pointηPYpRq, the fiber product Sp´etRˆ_YXis representable by an algebraic space which is ´etale over Sp´etR. The veracity of this assertion requires that we adopt a slightly more general definition of algebraic space than the one given in [117].

Suppose that X “ Sp´et_UpRq for some sheaf U P Shv_SetpCAlg^´et_Rq. Choosing a set of sections η_αPUpR_αq which generate U, we obtain an effective epimorphism>Sp´etR_αÑX in the category of ´etale sheaves on (the opposite of) the category CAlg^´et_R. However, the maps vα: Sp´etRα ÑX need not be relatively representable by schemes. Nevertheless, the mapsv_α are relatively representable in the special case where there exists a monomorphism

1.2. DELIGNE-MUMFORD STACKS 111 XãÑSp´etA, for some ´etaleR-algebraA. In this case, each fiber product Sp´etBˆXSp´etRα

can be identified with the functor corepresented byBbAR_α.

In the general case, each fiber product X_α,β “Sp´etRαˆX Sp´etR_β is again ´etale over Sp´etR (in the sense of Definition 1.2.5.3), and admits a monomorphism

Xα,β ãÑSp´etRαˆSp´etRSp´etRβ »Sp´etpRαbRRβq.

It follows that eachX_α,β is a representable by an algebraic space in the sense of [117], so that the mapsv_α : Sp´etR_αÑX are relatively representable by algebraic spaces.

Remark 1.2.5.5. Let Y “ pY,OYq be a Deligne-Mumford stack, and suppose we are given a morphism f : X Ñ Y in FunpCAlg^♥, τď1Sq. Assume further thatX is a sheaf for the

´

etale topology. Then the following conditions are equivalent:

p1q The functorX is equivalent (as an object of the 8-category FunpCAlg^♥, τ_ď1Sq{Y) to a Deligne-Mumford stack of the form pY_{U,OY|Uq for some objectU PY.

p2q The mapf is representable and ´etale (in the sense of Definition 1.2.5.3).

The implication p1q ñ p2q follows immediately from the definitions. The converse follows by usingp2qto construct the object U PY locally, and then invoking the assumption that X is an ´etale sheaf.

Remark 1.2.5.6. Letf :XÑY be a morphism in FunpCAlg^♥, τ_ď1Sq, and suppose that both X and Y are sheaves for the ´etale topology. Thenf is ´etale and representable if and only if, for every Deligne-Mumford stackZ “ pZ,OZq equipped with a map Z ÑY, the fiber product Z ˆY X is equivalent to to pZ_{U,OZ|Uq for some object U P Z. The “if”

direction is obvious, and the converse follows from Remark 1.2.5.5. It follows from this characterization that the collection of representable ´etale morphisms (between ´etale sheaves) is closed under composition.

Proposition 1.2.5.7. Let φ : A Ñ B be a homomorphism of commutative rings. The following conditions are equivalent:

p1q The ring homomorphism φis ´etale (see §B.1).

p2q The induced map of Deligne-Mumford stacks Sp´etB Ñ Sp´etA is representable and

´etale (here we abuse terminology by identifying Sp´etA and Sp´etB with the functors they represent).

In other words, if A is a commutative ring, then an object U PShv_SetpCAlg^´et_Aqis affine if and only if U is corepresentable by an ´etaleA-algebraB.

Proof. The implication p1q ñ p2q follows immediately from the construction of Sp´etB. Conversely, suppose that p2q is satisfied. Write Sp´etA “ pX,OXq, so that Sp´etB » pX_{U,OX|Uq for some object U PX. For each ´etale A-algebra A¹, leth^A1 PX denote the functor corepresented by A¹. The objects h^A1 generate the topos X under colimits. We may therefore choose a collection of ´etale A-algebras tA_αuαPI and an effective epimorphism

>αPIh^Aα ÑU inX. Under the identification of X_{U with ShvSetpCAlg^´et_Bq, we can identify eachh^Aα with an objectVαPShv_SetpCAlg^´et_Bq. These objects cover the toposShv_SetpCAlg^´et_Bq. We may therefore choose a finite collection of indices α₁, α₂, . . . , α_n and a faithfully flat

´

etale map BÑś

1ďiďnBi such that each VαipBiq is nonempty. Set A¹ “ś

1ďiďnAαi and B¹ “ś

Bi, so that we morphisms of Deligne-Mumford stacks Sp´etB¹ÑSp´etA¹ ÑSp´etB ÑSp´etA,

induced by a sequence of ring homomorphismsAÑ^φ B ÑA¹ ÑB¹. It follows thatB¹ is a retract ofA¹bBB¹ in the category ofA-algebras. SinceB¹ is ´etale overB andA¹ is ´etale over A, A¹bBB¹ is ´etale over A, so thatB¹ is ´etale over A. Since the mapB ÑB¹ is ´etale and faithfully flat, we conclude that B is also ´etale overA (see Proposition B.1.4.1).

Definition 1.2.5.8. Let fα : Xα Ñ Y be a collection of representable ´etale morphisms in the 8-category FunpCAlg^♥, τ_ď1Sq. We will say that the set tf_αu is jointly surjective if, for every commutative ring R and every point η PYpRq, if we write Sp´etRˆY Xα as Sp´et_U

αR for some Uα P Shv_SetpCAlg^´et_Rq, then the objects Uα comprise a covering of the toposShv_SetpCAlg^´et_Rq.

We will say that a single representable ´etale morphismf :XÑY is surjectiveif the set tfu is jointly surjective.

Remark 1.2.5.5 admits the following converse:

Theorem 1.2.5.9. Let X : CAlg^♥ Ñ S_ď1 be a functor. The following conditions are equivalent:

p1q The functorX is (representable by) a Deligne-Mumford stack.

p2q The functorX is a sheaf for the ´etale topology, and there exists a jointly surjective col-lection of representable ´etale morphismsf_α :U_αÑX, where each U_α is (representable by) an affine Deligne-Mumford stack Sp´etRα.

p3q The functorX is a sheaf for the ´etale topology, and there exists a jointly surjective col-lection of representable ´etale morphismsfα :UαÑX, where each Uα is (representable by) a Deligne-Mumford stack.

p4q The functor X is a sheaf for the ´etale topology, and there exists a representable ´etale surjection f :U₀ ÑX, where U₀ is (representable by) a Deligne-Mumford stack.

1.2. DELIGNE-MUMFORD STACKS 113 Example 1.2.5.10. Let X : CAlg^♥ Ñ Set be a functor which is representable by a scheme. ThenX satisfies conditionp2q of Theorem 1.2.5.9 (in fact, we can take the maps u_α : SpecR_α ÑXto be any open covering of Xby affine schemes). Theorem 1.2.5.9 implies that the essential image of the functor NpDMqãÑFunpCAlg^♥, τď1Sqincludes all functors which are representable by schemes. We therefore obtain a fully faithful embedding from the category Sch of schemes to the 2-category DM of Deligne-Mumford stacks, given on affine schemes by SpecRÞÑSp´etR. Note that this embedding isnot given by the formula pX,OXq ÞÑ pShv_SetpXq,OXq, because the structure sheaf of a scheme is usually not strictly Henselian. Instead, the Deligne-Mumford stack associated to a scheme pX,OXq can be viewed as the classifying topos for strict Henselizations ofOX: see§1.6 for more details).

Proof of Theorem 1.2.5.9. Let X: CAlg^♥Ñτď1S be the functor represented by a Deligne-Mumford stackpX,OXq. Fix a commutative ringA and write Sp´etA“ pY,OYq. For every

´

etaleA-algebra A¹, let h^A1 PShv_SetpCAlg^´et_Aq denote the functor corepresented by A¹. Then the restriction ofX to CAlg^´et_A is given by the formula

XpA¹q “Map_DMppY_{hA1,OY|_hA1q,pX,OXq,

from which it follows easily that the restriction of X to CAlg^´et_A is an ´etale sheaf. It follows thatXis a sheaf for the ´etale topology, so that the implicationp1q ñ p2qfollows immediately from the definition of Deligne-Mumford stack.

The implication p2q ñ p3q is trivial, and the implication p3q ñ p4q follows by taking U₀ to be the coproduct of theU_α (in the 2-category of Deligne-Mumford stacks). We will complete the proof by showing thatp4q ñ p1q. LetU0 be a functor which is representable by a Deligne-Mumford stackpU₀,OU₀q, letf :U₀ ÑX be a representable ´etale surjection, and letU‚ be the ˇCech nerve of f (formed in the8-category FunpCAlg^♥, τď1Sq). Sincef is an effective epimorphism of ´etale sheaves, we can identifyX with the geometric realization ofU_‚ in the 8-category of τ_ď1S-valued ´etale sheaves on CAlg^♥. Each U_m admits a representable

´

etale map to U0, and is therefore representable by a Deligne-Mumford stack pU_m,OUmq.

We can view U_‚ as a simplicial object in the 2-category of Grothendieck topoi; let X denote its geometric realization (so that the objects ofX can be identified with sequences tXmPU_mumě0 which are compatible with one another under pullback). Since each of the mapsU_mÑU_n is ´etale, the collection of structure sheavestOUmumě0 can be identified with a commutative ring objectOX PX.

For each integerm, the inclusionrmsãÑ rm`1sdetermines a representable ´etale surjection mapU<sub>m`1</sub>ÑU_m. We may therefore choose an object V_mPU_m and an equivalence

pU_{m`1</sub>,OU<sub>m`1}q » pU_m{Vm,OU_m|Vmq.

The objects Vm are compatible under pullback, and therefore determine an objectV of the toposX. Since eachV_m covers the final object of U_m, the objectV covers the final object of

X. Moreover, we can identify pX_{V,OX|Vqwith the geometric realization of the simplicial ringed topos

pU_‚{V‚,OU_‚|V‚q » pU_{‚`1</sub>,OU<sub>‚`1}q,

which is equivalent to pU₀,OU0q. It follows from Remark 1.2.4.4 that pX,OXq is a Deligne-Mumford stack. LetX¹ : CAlg^♥Ñτ_ď1S denote the functor represented by pX,OXq. Let f¹ :U0 ÑX¹ be the canonical map. The natural isomorphismf^˚V »V0 shows that U‚ is the ˇCech nerve of the mapf¹, so that f¹ factors as a composition

U0

Ñ |Uf ‚| »XÑ^g X¹,

where g is a monomorphism. To complete the proof, it will suffice to show thatf¹ is an epimorphism of ´etale sheaves. This follows from the observation thatV covers the final object of the toposX.

1.2.6 Quasi-Coherent Sheaves on a Deligne-Mumford Stack

We close this section with a brief discussion of quasi-coherent sheaves on a Deligne-Mumford stack, which will play a role in§1.4:

Definition 1.2.6.1. Let pX,OXq be a Deligne-Mumford stack, and let F be a OX -module object of X. For each objectU PX, we let OXpUq denote the commutative ring Hom_XpU,OXq, andFpUq the module Hom_XpU,Fq. We will say thatF isquasi-coherent if the following condition is satisfied:

p˚q For every morphismU ÑV between affine object of X, the induced map OXpUq b_OXpVqFpVq ÑFpUq

is an isomorphism of modules over the commutative ringOXpUq.

Example 1.2.6.2. LetAbe a commutative ring and let write Sp´etA“ pShv_SetpCAlg^´et_Aq,Oq. Proposition 1.2.5.7 implies that a O-module F inShv_SetpCAlg^´et_Aqis quasi-coherent if and only if, for every morphismB ÑCof ´etaleA-algebras, the induced mapCb_BFpBq ÑFpCq is an equivalence. In other words, F is quasi-coherent if and only if there exists a (discrete) A-module M for which F is given by the formula FpBq “BbAM. Conversely, for any discrete A-module M, the theory of faithfully flat descent implies that the construction B ÞÑ B bAM determines a sheaf for the ´etale topology on CAlg^´et_A. We summarize the situation as follows: the category of quasi-coherent sheaves on Sp´etA is equivalent to the category of (discrete)A-modules.

Proposition 1.2.6.3. LetpX,OXq be a Deligne-Mumford stack, and letF be a OX-module object of X. The condition that F be quasi-coherent can be tested locally on X. More precisely, if there exists a covering of X by objects Uα such that each restriction F|Uα is a quasi-coherent sheaf onpX_{Uα,OX|Uαq, then F is quasi-coherent.