The Functor of Points

and faithfully flat. Since C^p0q is a saturated sieve containing each Bi, it must also contain A.

Remark 1.5.4.7. Letφ:TopÑ 8Top be the functor of Remark 1.4.1.6, which carries a topological space X to the8-toposShvpXq. Then φfactors as a composition

TopÑ^U LocÑ 8^ι Top,

where ιis the fully faithful embedding of Remark 1.5.4.2, andU is the functor of Example 1.5.1.6. It follows thatφis fully faithful when restricted to sober topological spaces (Remark 1.5.3.6). Consequently, the functor φ : Top_CAlg Ñ 8Top_CAlg of Remark 1.4.1.6 is fully faithful when restricted to spectrally ringed spaces pX,OXq for which X is sober. In particular,φ is fully faithful when restricted to nonconnective spectral schemes (Corollary 1.5.3.8).

1.6 The Functor of Points

In classical algebraic geometry, we can often describe algebraic varieties (or schemes) as solutions to moduli problems. For example, the n-dimensional projective space Pⁿ can be characterized as follows: it is universal among schemes over which there is a line bundle L generated by pn`1q global sections. In particular, for any commutative ring A, the set HompSpecA,P<sup>n</sup>q can be identified with the set of isomorphism classes of pairs pL, η : A<sup>n`1</sup> Ñ Lq where L is an invertible A-module and η is a surjective A-module homomorphism (such a pair is determined up to unique isomorphism by the submodule kerpηq ĎA^n`1q.

More generally, any scheme X determines a covariant functor h_X from commutative rings to sets, given by the formula hXpAq “HompSpecA, Xq. We refer to hXpAq as the set ofA-valued points of X, and toh_X as thefunctor of points ofX. This functor determinesX up to canonical isomorphism. More precisely, the construction XÞÑh_X determines a fully faithful embedding from the category of schemes to the presheaf category FunpCAlg^♥,Setq. Consequently, it is possible to think of schemes as objects of FunpCAlg^♥,Setq, rather than the category of locally ringed spaces. This point of view is often valuable: it is sometimes easier to describe the functor represented by a schemeX than it is to provide an explicit construction ofX as a locally ringed space. Moreover, the “functor of points” perspective becomes essential when we wish to study more general algebro-geometric objects such as algebraic stacks.

1.6.1 The Case of a Spectrally Ringed Space We begin by associating a functor to each spectrally ringed space.

Definition 1.6.1.1. LetpX,OXq be a locally spectrally ringed space. We define a functor h^nc_X : CAlgÑS by the formula

h^nc_XpRq “Map_Toploc

CAlgpSpecR,pX,OXqq,

where Top^loc_CAlg denotes the8-category of locally spectrally ringed spaces (see Definition 1.1.5.3). We let h_X denote the restriction of h^nc_X to the full subcategory CAlg^cn Ď CAlg spanned by the connective E8-rings. We will refer to both h_X andh^nc_X as the functor of points ofpX,OXq.

Warning 1.6.1.2. The notation of Definition 1.6.1.1 is abusive: if pX,OXq is a locally spectrally ringed 8-topos, then the functorsh^nc_X andh_X depend on the structure sheafOX, and not only on the underlying topological spaceX.

1.6.2 Flat Descent Our first main result in this section can be stated as follows:

Theorem 1.6.2.1. Let pX,OXq be a locally spectrally ringed space. Then the functor h^nc_X : CAlg Ñ S is a hypercomplete sheaf with respect to the fpqc topology of Proposition B.6.1.3.

Since the construction Spec : CAlg^op ÑTop^loc_CAlg is fully faithful (Remark 1.1.5.7), the functor A ÞÑ h^nc_SpecA coincides with the Yoneda embedding CAlg^op Ñ FunpCAlg,Sq. We may therefore view Theorem D.6.3.5 (which asserts that the fpqc topology on CAlg^op is subcanonical) as a special case of Theorem 1.6.2.1. This observation does not supply a new proof of Theorem D.6.3.5, because Theorem D.6.3.5 is one of the main ingredients in our proof of Theorem 1.6.2.1. The other main ingredient is the compatibility of the Zariski topology with flat descent, which can be formulated more precisely as follows:

Proposition 1.6.2.2. For every E8-ringA, let UpAq be the collection set of open subsets of the topological space |SpecA|. Then AÞÑ UpAq determines a functor U : CAlgÑ Set.

This functor is a sheaf (of sets) with respect to the fpqc topology on CAlg^op.

Remark 1.6.2.3. The sheaf U : CAlg Ñ Set of Proposition 1.6.2.2 can be regarded as a discrete object in the 8-category of S-valued sheaves on CAlg^op. Consequently, it is automatically hypercomplete.

Proof of Proposition 1.6.2.2. We will show that the functor U : CAlgÑSet satisfies condi-tions p1q andp2q of Proposition A.3.3.1. To verifyp1q, we must show that for every finite collection of E8-rings tAiu1ďiďn, the map Upś

Aiq Ñ ś

UpAiq is bijective. This follows from the observation that there is a canonical homeomorphism|Specpś

iA_iq| » >i|SpecA_i|.

1.6. THE FUNCTOR OF POINTS 169 We now prove p2q. Letf :AÑB be a faithfully flat morphism ofE8-rings; we wish to prove that

UpAq ^//UpBq ^////UpBbABq

is an equalizer diagram in the category of sets. We can divide this assertion into two parts:

paq The map UpAq Ñ UpBq is injective. To prove this, we must show that an open subset U Ď |SpecA| is determined by its inverse image in |SpecB|. This is clear, since the assumption that A Ñ B is faithfully flat guarantees the induced map

|SpecB| Ñ |SpecA|is surjective.

pbq Letφ₀, φ₁ :|SpecBbAB| Ñ |SpecB|be the two projection maps. We claim that if Z Ď |SpecB|is a closed subset with φ^´1₀ Z“φ^´1₁ Z, thenZ “φ^´1V for some closed subset V Ď |SpecB|. Choose an ideal I Ďπ0B such that Z “ tp Ď π0B : I Ď pu, and let J “ f^´1I Ď π₀A. Set V “ tq Ď π₀A :J Ď qu. Then φ^´1V “ tp Ď π₀B : fpJqπ₀B Ď pu. To prove that φ^´1V “ Z, it suffices to show that fpJqπ₀B and I have the same nilradical. Let R denote the commutative ring π0A{J and R¹ the commutative ringπ₀B{J π₀B, and letI¹ denote the image of I in R¹. Then RÑR¹ is faithfully flat and the composite mapRÑR¹ÑR¹{I¹ is injective; we wish to prove that every elementxPI¹ is nilpotent. Sinceφ^´1₀ Z “φ^´1₁ Z, we deduce that the ideals I¹bRR¹ and R¹bRI¹ have the same radical in R¹bRR¹. Consequently, since xb1 belongs toI¹bRR¹, some powerxⁿb1 belongs to R¹bRI¹. It follows that the image ofxⁿ vanishes in R¹bRR¹{I¹. SinceR¹ is flat overR, the injectionR ÑR¹{I¹ induces an injectionR¹ ÑR¹bRR¹{I¹. It follows that xⁿ“0 inR¹, as desired.

Proposition 1.6.2.4. p1q The functorSpec : CAlg^op ÑTop^loc_CAlg preserves finite coprod-ucts.

p2q LetRbe anE8-ring, and letR^‚ be a cosemisimplicialE8-ring which is a hypercovering ofR with respect to the fpqc topology (see Definition A.5.7.1). ThenSpecR is a colimit of the diagram tSpecR^‚u in Top^loc_CAlg.

Proof. Let Top_CAlg denote the8-category of spectrally ringed spaces (Definition 1.1.2.5) and let Top denote the ordinary category of topological spaces and continuous maps, so that we have forgetful functors

Top^loc_CAlgãÑ^j Top_CAlgÑ^q Top. We will deduce assertionp1q from the following three claims:

p1¹q The functorq˝j˝Spec : CAlg^opÑTop preserves finite coproducts.

p1²q The functorj˝Spec : CAlg^op ÑTop_CAlg carries finite coproducts toq-coproducts.

p1³q The functor Spec : CAlg^op ÑTop^loc_CAlg carries finite coproducts toj-coproducts.

To prove these claims, let tR_iu1ďiďn be a finite collection of E8-rings having product R. Let Xi “ |SpecRi| and let X “ |SpecR|, so that we can write SpecRi “ pXi,OXiq and SpecR “ pX,OXq. For each index i, let φ_i : X_i Ñ X denote the map induced by the projection R ÑR_i. Assertion p1¹q was established as part of Proposition 1.6.2.2. By virtue of Proposition HTT.4.3.1.9 , assertionp1²q is equivalent to the requirement that the canonical mapOX Ñś

ipφ_iq˚OXi is an equivalence of CAlg-valuedX. Note that X has a basis of open sets of the form U_f “ tpĂπ₀R:f Rpu, where f “ pf₁, . . . , f_nq ranges over the elements ofπ0R»π0R1ˆ ¨ ¨ ¨ ˆπ0Rn. Since this basis is stable under finite intersections, it suffices to observe that the canonical map

Rrf^´1s »OXpU_fq Ñ p

źpφiq˚OXiqpU_fq »

źOXipU_f ˆX Xiq »

źRirf_i^´1s is an equivalence ofE8-rings.

Unwinding the definitions, we can formulate assertion p1³q as follows: a morphism g : pX,OXq Ñ pY,OYq in Top_CAlg belongs to Top^loc_CAlg if and only if, for 1 ď iď n, the induced map gi:pXi,OXiq Ñ pY,OYq belongs toTop^loc_CAlg. This follows immediately from the definitions, sinceOXi can be identified with the restrictionOX|Xi.

We now prove p2q. Let R^‚ :∆_s,` ÑCAlg_R be an fpqc hypercovering of R “ R^´1 in the8-category CAlg^op. Reasoning as above, we are reduced to proving the following three assertions:

p2¹q The compositionq˝j˝Spec˝R^‚ is a colimit diagram in the8-categoryTop.

p2²q The compositionj˝Spec˝R^‚ is aq-colimit diagram in the8-categoryTop_CAlg. p2³q The composition Spec˝R^‚ is aj-colimit diagram in the8-category Top^loc_CAlg.

By virtue of p1¹q and Proposition A.5.7.2, assertion p2¹q is equivalent to the requirement that the functor

q˝j˝Spec : CAlgÑTop^op

is a hypercomplete sheaf with respect to the fpqc topology. Because Top is an ordinary category, it will suffice to show thatq˝j˝Spec is a sheaf with respect to the fpqc topology, which follows from Proposition 1.6.2.2. We now prove p2²q. Let X“ |SpecR|, so that we can write SpecR“ pX,OXq. For every nonnegative integernletX_n“ |SpecRⁿ|and write SpecRⁿ “ pXn,OXnq. Let Fⁿ denote the pushforward of OXn along the canonical map XnÑX. ThenF^‚ is a cosemisimplicial object in the8-categoryShv_CAlgpXq. By virtue of Proposition HTT.4.3.1.9 , condition 23²qis equivalent to the requirement that the canonical

1.6. THE FUNCTOR OF POINTS 171