Spectral Deligne-Mumford Stacks

ÑFunpX^op,C₀q^LÑ⁰ Shv_C0pXq,

where G¹ is given by composition withg. It follows that Gadmits a left adjointF, which can be described as the composition

ShvpCqÐ^L FunpX^op,Cq ĚFunpX^op,C₀q ĚShv_C0pXq.

To complete the proof, it suffices to show that F is fully faithful. In other words, we wish to show that for every objectF PShv_C0pXq, the unit mapF Ñ pG˝FqpFqis an equivalence.

In other words, we wish to show that the map α : F Ñ LF becomes an equivalence after applying the functor G¹. Since G¹pFq »F andG¹pLFqbelong to ShvC₀pXq, this is equivalent to the requirement thatG¹pαqis anL0-equivalence in the8-category FunpX^op,C₀q. This follows from p3q and Lemma 1.3.4.4, sinceα is anL-equivalence in FunpX^op,Cq.

1.4 Spectral Deligne-Mumford Stacks

In §1.1 and §1.2 we introduced two different generalizations of the notion of scheme:

the notion of spectral scheme (Definition 1.1.2.8) and the notion of Deligne-Mumford stack (Definition 1.2.4.1). These two generalizations serve rather different purposes:

• The 8-category SpSch of spectral schemes can viewed roughly as a “left-derived”

version of the category of schemes. More precisely, though the category Sch and the 8-category SpSch both admit fiber products, the inclusion Sch ãÑ SpSch doesnot preserve fiber products. If f :X ÑY andf¹ :X¹ ÑY are morphisms of schemes, then we can form a fiber product pZ,OZq of X and X¹ over Y in the 8-category SpSch, whose underlying ordinary scheme pZ, π₀OZq is the fiber productXˆY X¹ in the category of schemes. However, the structure sheaf OZ need not be 0-truncated, so thatpZ,OZq need not be an ordinary scheme. This is not a bug, but a feature: the sheaves π_nOZ carry useful geometric information which can detect (and help correct for) the failure of the maps f and f¹ to be transversal with respect to one another.

• The collection of Deligne-Mumford stacks is organized into a 2-category DM which can be regarded as a “right-derived” enlargement of the category of schemes. More precisely, there is a fully faithful embedding SchãÑC which is not compatible with certain very basic colimit constructions, such as passage to quotients under the action

1.4. SPECTRAL DELIGNE-MUMFORD STACKS 133 of a finite group. IfX is a scheme equipped with an action of a finite group G, then one can consider either the quotient X{Gin the category of schemes (which exists under mild hypotheses onX), or the stack-theoretic quotientX{{Gin the 2-category C. The usual quotient X{G can be recovered as the “coarse moduli space” of the Deligne-Mumford stack X{{G, but they are generally not the same unless G acts freely on X. Once again, this should be regarded as a feature rather than a bug:

the stack-theoretic quotient X{{G carries useful geometric information about the subgroups ofGwhich stabilize points of X; this information is forgotten when passing to the usual quotientX{G.

For some purposes, it is useful to enlarge the category of schemes simultaneously in both of these directions. To accomplish this, we will introduce the notion of a spectral Deligne-Mumford stack(Definition 1.4.4.2). Roughly speaking, the definition of a spectral Deligne-Mumford stacks is obtained by modifying the definition of a scheme pX,OXq in three different ways:

piq For every topological space X, the8-categoryShvpXq of sheaves of spaces on X is an8-topos. Moreover, if the topological spaceX issober(that is, if every irreducible closed subset ofX has a unique generic point), then we can recover X from ShvpXq: the points xPX can be identified with isomorphism classes of geometric morphisms x^˚ : ShvpXq ÑS, and open subsets of X can be identified with subobjects of the unit object 1PShvpXq. In other words, the space X and the8-topos ShvpXq are interchangable: either one canonically determines the other. The situation described above can be summarized by saying that we can regard the theory of 8-topoi as a generalizationof the classical theory of topological spaces (more precisely, of the theory of sober topological spaces). For this reason, we opt to dispense with topological spaces altogether and work instead with a general 8-topos X.

piiq In place of the sheaf OX of commutative rings on X, we consider an arbitrary sheaf OX of E8-rings onX.

piiiq In place of the requirement that pX,OXq be locally isomorphic to the spectrum of a commutative ring, we require that pX,OXq be locally equivalent to Sp´etA, where A is an E8-ring and Sp´etA denotes its spectrum with respect to the ´etale topology (see Proposition 1.4.2.4).

1.4.1 Spectrally Ringed 8-Topoi We begin with a discussion of CAlg-valued rings on 8-topoi.

Definition 1.4.1.1. Aspectrally ringed 8-topos is a pairpX,Oq, whereX is an 8-topos and O PShv_CAlgpXq is a sheaf ofE8-rings onX.

Remark 1.4.1.2. LetX“ pX,Oq be a spectrally ringed 8-topos. We will often refer to O as the structure sheaf ofX. We will often denote the structure sheafO byOX orOX (the latter notation is convenient when we wish to distinguish between spectrally ringed8-topoi having the same underlying8-topos).

Construction 1.4.1.3. Precomposition with a geometric morphism of8-topoif^˚:X ÑY induces a pushforward functor f˚ : Shv_CAlgpXq Ñ Shv_CAlgpYq. We may therefore view the construction X ÞÑ Shv_CAlgpXq^op as determining a functor Shv_CAlg : 8Top Ñ yCat₈, where 8Top denotes the 8-category of 8-topoi. This functor classifies a coCartesian fibration 8Top_CAlg Ñ 8Top. More informally, the objects of 8Top_CAlg are spectrally ringed8-topoipX,OXq, and a morphism frompX,OXqtopY,OYqin 8Top_CAlg is given by a pairpf^˚, φq, wheref_˚:X ÑY is a geometric morphism of8-topoi andφ:OY Ñf_˚OX

is a morphism of sheaves ofE8-rings onY. We will refer to8Top_CAlg as the8-category of spectrally ringed 8-topoi.

Remark 1.4.1.4. LetX be an8-topos and letX^♥ denote the underlying topos ofX. For any sheaf of E8-rings OX on X, we can regard π₀OX as a commutative ring object of X^♥. We will refer to pX^♥, π₀OXq as the underlying ringed toposof pX,OXq. The construction

pX,OXq ÞÑ pX^♥, π0OXq

determines a functor from the homotopy 2-category of8Top_CAlgto the 2-category 1Top_CAlg♥

of ringed topoi.

Remark 1.4.1.5. Let pX,OXq be a ringed topos, and let Shv_SpXq denote the 1-localic 8-topos associated to X (see §HTT.6.4.5 ). Remark 1.3.5.6 supplies an equivalence from the category of commutative ring objects of X to the 8-category of connective 0-truncated sheaves of E8-rings onShv_SpXq. We letO denote the image of OX under this equivalence.

Then pShv_SpXq,Oq is a spectrally ringed8-topos. Moreover, the construction pX,OXq ÞÑ pShv_SpXq,Oq determines a fully faithful embedding from the 8-category of ringed topoi (obtained from the 2-category 1Top_CAlg♥ by discarding noninvertible 2-morphisms) to the 8-category8Top_CAlgof spectrally ringed 8-topoi. The essential image of this fully faithful embedding consists of those spectrally ringed8-topoipY,OYqwhere Y is 1-localic and the structure sheaf OY is connective and 0-truncated.

Remark 1.4.1.6. For every topological spaceX, Example 1.3.1.2 supplies an equivalence of 8-categories Shv_CAlgpShv_SpXqq Ñ Shv_CAlgpXq, which depends functorially on X. It follows that there is a commutative diagram of8-categories

Top_CAlg ^{φ //}

8Top_CAlg

Top ^{φ //}8Top,

1.4. SPECTRAL DELIGNE-MUMFORD STACKS 135