Schemes and Deligne-Mumford Stacks
1.4.7 Affine Spectral Deligne-Mumford Stacks
Let X“ pX,OXq be a nonconnective spectral Deligne-Mumford stack. We will say that X isaffineit is equivalent to Sp´etA, for someE8-ringA.
Remark 1.4.7.1. Arguing as in Remark 1.1.5.7, we see that the constructionAÞÑSp´etA determines a fully faithful embedding Sp´et : CAlgop ÑSpDMnc, whose essential image is the full subcategory of SpDMnc spanned by the affine nonconnective spectral Deligne-Mumford stacks.
Our main goal is to establish the following characterization of affine spectral Deligne-Mumford stacks:
Proposition 1.4.7.2. LetpX,OXq be a spectrally ringed8-topos which satisfies the follow-ing conditions:
paq Let X♥ denote the underlying topos of X. Then the ringed topos pX♥, π0OXq is equivalent toSp´etR for some commutative ring R.
pbq The8-toposX is 1-localic (that is, the natural geometric morphismX ÑShvSpX♥q » Shv´etR is an equivalence).
pcq For each integer n, the pπ0OXq-module πnOX is quasi-coherent (in the sense of Definition 1.2.6.1).
pdq The sheafOX is hypercomplete.
Then pX,OXq is equivalent to Sp´etA, for some E8-ringA.
Corollary 1.4.7.3. Let pX,OXq be a nonconnective spectral Deligne-Mumford stack. Then pX,OXq is affine if and only if the0-truncated spectral Deligne-Mumford stack pX, π0OXq is affine.
Corollary 1.4.7.4. Suppose we are given a commutative diagram of spectral Deligne-Mumford stacks
X f //
Y
Z,
and suppose that the underlying map XˆZτď0ZÑYˆZτď0Zis an equivalence. Then f is an equivalence.
Proof. The assertion is local on Y; we may therefore assume without loss of generality that Y“Sp´etAandZ»Sp´etR are affine. Our assumption guarantees thatf induces an equivalence of 0-truncations, so thatτď0X is affine. Applying Corollary 1.4.7.3, we deduce that X » Sp´etB is affine. Let K denote the cofiber of the map A Ñ B (formed in the 8-category ModR). We wish to prove thatK »0. Assume otherwise. SinceK is connective, there exists a smallest integer nsuch thatπnK is nontrivial. In this case, we have
πnK »Torπ00Rpπ0R, πnKq »πnpπ0RbRKq »πncofibpπ0RbRAÑπ0RbRBq »0, and obtain a contradiction.
The proof of Proposition 1.4.7.2 will require some preliminaries.
Lemma 1.4.7.5. Let X be an8-topos and let ně ´1 be an integer. Suppose we are given a collection of morphisms fα:UαÑX in X with the following properties:
piq Each of the morphisms fα is pn´1q-truncated.
piiq Each of the objects Uα isn-truncated.
piiiq The induced mapf :>αUαÑX is an effective epimorphism inX. Then X isn-truncated.
Proof. Without loss of generality, we may assume that X is given as a left exact localization ofPpCq “FunpCop,Sq, for some small8-categoryC. LetL:PpCq ÑX be a left adjoint to the inclusion. For each objectC PC, let X1pCq ĎXpCqdenote the union of those connected components which meet the image of one of the maps UαpCq ÑXpCq, so that we have an
1.4. SPECTRAL DELIGNE-MUMFORD STACKS 149 effective epimorphism f1 :>αUα ÑX1 in the8-topos PpCq. It follows frompiiiq that the functorLcarriesX1 toX. SinceLis left exact, it will suffice to prove thatX1 isn-truncated.
We may therefore replaceX byX1, and thereby reduce to the case whereX “PpCq is an 8-category of presheaves. Working objectwise, we may reduce to the case where C»∆0, so thatX is the 8-categoryS of spaces.
If n“ ´1, then either X is empty or one of the maps hα is an equivalence; in either case, we immediately conclude that X is p´1q-truncated. Suppose that ně0. We wish to prove thatπmpX, xq »qfor each integermąn and each base pointxPX. Using piiiq, we may assume that x“fαpxq for some point xPUα. In this case, condition piq implies that the map πmpUα, xq Ñ πmpX, xq is an isomorphism, and condition piiq implies that πmpUα, xq »0.
Remark 1.4.7.6. In the situation of Lemma 1.4.7.5, we can replacepiiq by the following apparently weaker condition:
pii1q The mapfα factors as a composition UαÑVαÑX, whereVα is n-truncated.
Indeed, if this condition is satisfied, thenUα can be realized as a retract ofUαˆX Vα. If fα satisfies condition piq, then the projection map UαˆX Vα Ñ Vα is pn´1q-truncated.
Since Vα is n-truncated, we conclude that UαˆX Vα is n-truncated, and therefore Uα is n-truncated.
Lemma 1.4.7.7. Let X be an 8-topos containing an object X and let ně0 be an integer.
Then:
p1q If the8-toposX ispn`1q-localic andX{X is n-localic, then the objectXis n-truncated.
p2q If the 8-toposX is n-localic andX isn-truncated, then the 8-topos X{X isn-localic.
p3q If the 8-topos X{X is pn`1q-localic and the object X is both n-truncated and 0-connective, thenX is pn`1q-localic.
Proof. We first provep1q. IfX ispn`1q-localic, then we can choose an effective epimorphism
>Vα Ñ X where each Vα is an n-truncated object of X. If X{X is n-localic, then we can choose effective epimorphisms>Uα,β ÑVα, where eachUα,β is anpn´1q-truncated object of X{X. Applying Remark 1.4.7.6, we conclude that X isn-truncated.
We now prove p2q. If X is n-localic, then we can write X as a topological localization of PpCq, for some smalln-categoryC(see the proof of Proposition HTT.6.4.5.7 ). Let us identify X with the corresponding subcategory ofPpCq. Then X{X is a topological localization of PpCq{X. According to Proposition HTT.6.4.5.9 , it will suffice to show that the 8-topos PpCq{X is n-localic. The presheaf X :Cop ÑS classifies a right fibration of 8-categories θ : rC Ñ C. Since X is n-truncated, the fibers of θ are n-truncated Kan complexes, so
that rC is also ann-category. We complete the proof by observing that there is a canonical equivalence of8-categoriesPpCq{X »PprCq.
We now prove p3q. Suppose that X is n-truncated and 0-connective and that X{X is pn`1q-localic. Letf˚ :X ÑY be a geometric morphism which exhibits Y as anpn`1q -localic reflection ofX. Then the associated pullback functor f˚ restricts to an equivalence τďnY Ñ τďnX. In particular, we can assume without loss of generality that X “ f˚Y for some n-truncated object Y PY. By construction, the pullback functorf˚ induces an equivalence of8-categoriespτďnYq{Y Ñ pτďnXq{X. Restricting to n-truncated objects on both sides, we see that f˚ induces an equivalence of 8-categories τďnpY{Yq ÑτďnpX{Xq.
It follows from p2q that Y{Y is pn`1q-localic and the 8-topos X{X is pn`1q-localic by assumption, so that f˚ induces an equivalenceY{Y ÑX{X.
We will show that the counit map v:f˚f˚ÑidX is an equivalence (that the unit map u: idY Ñf˚f˚ is an equivalence can be proven by the same argument). LetX1 be an object of X; we wish to show that the natural mapvX1 :f˚f˚X1ÑX1 is an equivalence. SinceX is 0-connective, it will suffice to show that
vX1ˆidX :pf˚f˚X1q ˆX ÑX1ˆX
is an equivalence. Unwinding the definitions, we see that this map factors as a composition pf˚f˚X1q ˆX » f˚pf˚X1ˆYq
» f˚f˚pX1ˆXq Ñ X1ˆX,
where the last map is an equivalence by virtue of our assumption that f˚ and f˚ induce mutually inverse equivalences betweenY{Y and X{X.
Proof of Proposition 1.4.7.2. We proceed as in the proof of Proposition 1.1.3.4. For each integern, we have a fiber sequence of spectrum-valued sheaves
ΣnpπnOXq ÑτďnOX Ñτďn´1OX,
where we abuse notation by identifying the sheaf of abelian groups πnOX with the cor-responding object in the heart of ShvSppXq. Passing to global sections and extracting homotopy groups, we obtain a long exact sequence
Hn´mpX{U;pπnOXq|Uq ÑπmpτďnOXqpUq Ñπmpτďn´1OXpUqq ÑHn´m`1pX{U;pπnOXq|Uq for each object U P X. Using assumptions paq, pbq, and pcq, we see that the groups HipX{U;pπnOXq|Uq vanish if U PX♥ is affine and ią0 (see §HTT.7.2.2 for a discussion of the cohomology of an8-topos, Remark HTT.7.2.2.17 for a comparison with the usual
1.4. SPECTRAL DELIGNE-MUMFORD STACKS 151 theory of sheaf cohomology, and§D.3 for a closely related discussion). SincepτďnOXqpUq andpτďn´1OXqpUqare n-truncated andpn´1q-truncated, respectively, we conclude that our long exact sequence degenerates to give isomorphisms
πmpτďnOXqpUq »
$
’’
&
’’
%
0 ifmąn
pπnOXqpUq ifm“n pτďn´1OXqpUq ifmăn.
when U PX♥ is affine.
Set O1X “lim
ÐÝnτďnOX PShvSppXq. We have an evident map u :OX ÑO1X, and the above calculation shows that this map induces an equivalence pπnOXqpUq ÑπnpO1XpUqq for every affine objectU PX♥. In particular,uinduces an isomorphism of sheaves πnOX Ñ πnO1X for every integern. SinceO1X is hypercomplete by construction, conditionpdqimplies that u is an equivalence. It follows that the canonical mapπnpOXpUqq Ñ pπnOXqpUq is an isomorphism for each affine objectU PX♥.
Set A“ΓpX;OXq. Then Proposition 1.4.2.4 supplies a map of spectrally ringed8-topoi f :pX,OXq ÑSp´etA. Conditionpaq implies that the final object of X is affine, so that the canonical map πnAÑΓpX♥;πnOXq is an isomorphism for every integer n. In particular, π0A can be identified with the ring of global sections of π0OX, so that paq supplies an equivalenceX♥ »ShvSetpCAlg´etπ
0Aq. Combining this observation withpbq, we deduce thatf induces an equivalence of the underlying8-topoi. Then f supplies a morphism of structure sheavesα :O Ñ f˚OX; we wish to show that this map is an equivalence. SinceShv´etA is generated under small colimits by corepresentable functors hB, we are reduced to proving thatα induces an equivalence
B»OphBq ÑOXpf˚hBq
for each object B P CAlg´etA. We will prove that for each integer n, the map πnB Ñ πnOXpf˚hBq is an isomorphism of abelian groups. Since f˚hB is an affine object of X♥, we can identify πnOXpf˚hBq with the abelian group HomX♥pf˚hB, πnOXq. Assumption pbq implies that πnOX is a quasi-coherent sheaf on the affine Deligne-Mumford stack pX♥, π0OXq, so that we can identify HomX♥pf˚hB, πnOXq with
π0Bbπ0AΓpX♥, πnOXq »π0Bbπ0AπnA»πnB.
We close this section with a remark about affine “opens” in an arbitrary spectral Deligne-Mumford stack.
Definition 1.4.7.8. LetpX,OXqbe a nonconnective spectral Deligne-Mumford stack. We will say that an object U PX is affineif the nonconnective spectral Deligne-Mumford stack pX{U,OX|Uq is affine.
Proposition 1.4.7.9. Let pX,OXq be an nonconnective spectral Deligne-Mumford stack, and let X0 be the full subcategory ofX spanned by the affine objects. Then X is generated by X0 under small colimits (in other words, X is the smallest full subcategory of itself which contains X0 and is closed under small colimits).
Proof. Let X1 ĎX be a full subcategory containingX0 and closed under small colimits. We wish to prove that X1 contains every object XPX. We first prove this under the additional assumption that there exists a morphismXÑY, whereY PX is affine. In this case, we can replace pX,OXq by pX{Y,OX|Yq, and thereby reduce to the case where pX,OXq »Sp´etR is affine. In this case, we can identifyX with Shv´etR. It follows thatX is generated under small colimits by corepresentable functors hA (whereA ranges over ´etaleR-algebras), each of which is affine.
We now treat the general case. Let 1denote the final object of X. Choose an effective epimorphism U “ >αUα Ñ 1, where each Uα is affine. Let U‚ be the ˇCech nerve of the map U Ñ1, so that |U‚| »1. Then X is the geometric realization of the simplicial object
|XˆU‚|. It will therefore suffice to show that each of the objectsXˆUn belongs toX1. Note that there exists a mapUnÑU, so that we can writeXˆUnas a coproduct of objects of the formXˆUnˆUUα. We conclude by observing that each of these objects admits a morphism to the affine objectUα PX, and therefore belongs to X1 by the first part of the proof.
1.4.8 A Recognition Criterion for Spectral Deligne-Mumford Stacks