Schemes and Deligne-Mumford Stacks
1.2. DELIGNE-MUMFORD STACKS 93
• The objects of Hom1Top
CAlg♥ppX,OXq,pY,OYqqare pairs pf˚, αq, wheref˚ :X Ñ Y is a geometric morphism of topoi (in other words, f˚ is a functor which admits a left adjoint f˚ which preserves finite limits) andα :OY Ñf˚OX is a morphism of commutative ring objects ofY.
• A morphism frompf˚, αqto pf˚1, α1q in Hom1Top
CAlg♥ppX,OXq,pY,OYqqis a natural transformation of functorsβ :f˚1 Ñf˚ for which the diagram
OY α1
{{
α
##f1˚OX
βpOXq //f˚OX
commutes.
We will regard the collection of all ringed topoi as a (strict) 2-category 1TopCAlg♥, with the categories of morphisms defined above and the evident composition law.
Example 1.2.1.2. Let X be a topological space and let OX be a sheaf of commutative rings on X. Then we can regardOX as a commutative ring object of the toposShvSetpXq of set-valued sheaves onX. The pairpShvSetpXq,OXq is a ringed topos.
Notation 1.2.1.3. Let pX,OXq be a ringed topos, and let U PX be an object. We let OX |U denote the product U ˆOX, which we view as a commutative ring object of the toposX{U. ThenpX{U,OX |Uqis another ringed topos, equipped with an evident morphism pX{U,OX|Uq Ñ pX,OXq.
We now review what it means for a commutative ring object of a topos X to belocal.
Let R be a commutative ring. For every element r P R, we let prq denote the principal ideal generated byr. If r is not a unit, then prq ‰R, so (by Zorn’s lemma)prq is contained in a maximal ideal mĂR. We say that R is localif it contains a unique maximal ideal mR. In this case, the above reasoning shows that mR can be described as the collection of non-invertible elements of R. The ringR is local if and only if the collection of non-units R´Rˆ forms an ideal inR. SinceR´Rˆ is clearly closed under multiplication by elements ofR, this is equivalent to the requirement thatR´Rˆ is an additive subgroup ofR. That is,R is local if and only if the following pair of conditions is satisfied:
paq The element 0 belongs toR´Rˆ. In other words, 0 is not a unit inR: this is equivalent to the requirement that 0‰1 inR.
pbq Ifr, r1PR´Rˆ, thenr`r1 PR´Rˆ. Equivalently, ifr`r1 is a unit, then eitherr or r1 is a unit. This is equivalent to the following apparently weaker condition: if sPR, then eithersor 1´sis a unit in R(to see this, takes“ r`rr 1, so thatsis invertible if and only if r is invertible and 1´s» r`rr11 is invertible if and only if r1 is invertible).
If R and R1 are local commutative rings, then we say that a ring homomorphism f :R ÑR1 islocalif it carriesmR intomR1: that is, if an element xPR is invertible if and only if its imagefpxq PR1 is invertible.
All of these notions admit generalizations to the setting of commutative ring objects of an arbitrary Grothendieck topos:
Definition 1.2.1.4. Let X be a topos with final object 1, and let O be a commutative ring object of X. Let Oˆ denote the group object ofX given by the units ofO, so that we have a pullback diagram
Oˆ //
OˆO
m
1 1 //O where m denotes the multiplication onO,
We will say that O is localif the following conditions are satisfied:
paq The sheafOX is locally nontrivial. That is, if 0 :1ÑO denotes the zero section ofO, then the fiber product 1ˆOOˆ is an initial object ofX.
pbq Lete:OˆãÑO denote the inclusion map. Then the mapseand 1´edetermine an effective epimorphismOˆ>Oˆ ÑO in the topos X.
If α:O ÑO1 is a map between commutative ring objects ofX, then we say that α is localif the diagram
Oˆ //
O1ˆ
O //O1
is a pullback square inX.
We let 1ToplocCAlg♥ denote the subcategory of 1TopCAlg♥ whose objects are ringed topoi pX,OXq where OX is local, and whose morphisms mapspf˚, αq :pX,OXq Ñ pY,OYq for which α classifies a local mapf˚OY ÑOX. We will refer to 1Toploc
CAlg♥ as the 2-category of locally ringed topoi.
Example 1.2.1.5. Let pX,OXq be a ringed space. Then OX is local (in the sense of Definition 1.2.1.4) if and only ifpX,OXq is a locally ringed space (in the sense of Definition 1.1.5.1): that is, if and only if each stalk OX,x is a local ring. Moreover, if pX,OXq and pY,OYq are locally ringed spaces, then a map of ringed spaces pX,OXq Ñ pY,OYq is a map of locally ringed spaces (in the sense of Definition 1.1.5.1) if and only if the induced map of ringed topoipShvSetpXq,OXq Ñ pShvSetpYq,OYqis local (in the sense of Definition 1.2.1.4).
1.2. DELIGNE-MUMFORD STACKS 95 Remark 1.2.1.6. Let O be a commutative ring object of a topos X, and letU “1ˆOOˆ be the fiber product appearing in conditionpaq of Definition 1.2.1.4. ThenU is a subobject of the final object 1 P X and is maximal among those subobjects V Ď 1 for which the restrictionO|V is trivial. If f˚ :Y ÑX is a geometric morphism of topoi, then:
piq The pullback f˚O is trivial (as a commutative ring object of Y) if and only if the geometric morphism f˚ factors through the open immersion of topoi X{U Ñ X determined byU.
piiq The pullbackf˚Osatisfies conditionpaqof Definition 1.2.1.4 if and only if the geometric morphismf˚ factors through the closed subtopos ofX complementary to U.
Remark 1.2.1.7. LetX be a topos and suppose we are given a commutative diagram O1
β
O
α
>>
γ //O2
of commutative ring objects ofX. Then:
paq Ifα and β are local, thenγ is local.
pbq Ifβ and γ are local, thenα is local.
pcq Ifα and γ are local and α is an effective epimorphism, thenβ is local.
Proposition 1.2.1.8. Let X be a topos and let α : O Ñ O1 be a morphism between commutative ring objects ofX. Then:
p1q If O1 is local and α is local, then O is local.
p2q If O is local and α is an effective epimorphism, then the following conditions are equivalent:
paq The commutative ring object O1 is local.
pbq The commutative ring object O1 satisfies conditionpaq of Definition 1.2.1.4.
pcq The morphism α is local.
Proof. Consider first the commutative diagram σ: Oˆ>Oˆ //
v
O1ˆ>O1ˆ
v1
O α //O1
where the vertical maps are defined as conditionpbq of Definition 1.2.1.4. Ifα is local, then σ is a pullback square. If, in addition, the commutative ring object O1 is local, then v1 is an effective epimorphism. It follows thatv is also an effective epimorphism. Since 1ˆO1O1ˆ is an initial object of X, the existence of a morphism
1ˆOOˆÑ1ˆO Oˆ
shows that1ˆOOˆ is also initial in X. This completest the proof ofp1q.
To prove p2q, assume that O is local and that α is an effective epimorphism. The implication paq ñ pbq is trivial, and the implication pbq ñ paq follows by inspecting the commutative diagram σ (if v and α are effective epimorphisms, it follows that v1 is an effective epmorphism as well). We next prove that pcq ñ pbq. Assume thatα is local. Then the induced map
β :1ˆOOˆÑ1ˆO1O1ˆ
is a pullback ofα, and therefore an effective epimorphism. Our assumption thatO is local guarantees that the domain ofβ is an initial object ofX. It follows that the codomain of β is also an initial object ofX, so that assertion pbq is satisfied.
We now complete the proof by showing that pbq implies pcq. Fix an objectX PX and a morphismf :XÑO, which we regard as an element of the commutative ring HomXpX,Oq. LetαX : HomXpX,Oq ÑHomXpX,O1q be the homomorphism of commutative rings deter-mined byα. We wish to show that ifαXpfqis invertible, thenf is invertible. Letg:XÑO1 be a multiplicative inverse ofαXpfq in the commutative ring HomXpX,O1q. Sinceα is an effective epimorphism, we can (after passing to a covering of X) assume without loss of generality thatg“αXpgq for some g:XÑO. Since O is local, we can (after passing to a further covering ofX) assume that eitherf g or 1´f gis invertible in the commutative ring HomXpX,Oq. In the first case, we conclude that f is invertible as desired. In the second case, it follows that αXp1´f gq “0 is invertible in the commutative ring HomXpX,O1q, so that condition pbq guarantees thatX is an initial object of X. In this case, HomXpX,Oq is the zero ring (so thatf is tautologically invertible).
1.2.2 Strictly Henselian Rings in a Topos
To develop the theory of Deligne-Mumford stacks, we will need to work with ringed topoi satisfying stronger locality requirements, related to the ´etale topology rather than the Zariski topology.
Notation 1.2.2.1. Let X be a topos, let OX be a commutative ring object of X, and let CAlg♥ denote the category of commutative rings. For every commutative ring R, let SolRpOXqdenote an object of X having the following universal property: for every object
1.2. DELIGNE-MUMFORD STACKS 97