Spectral Schemes

1.5.3 Sober Topological Spaces . . . 162 1.5.4 Locales Associated to 8-Topoi . . . 165 1.6 The Functor of Points . . . . 167 1.6.1 The Case of a Spectrally Ringed Space . . . 167 1.6.2 Flat Descent . . . 168 1.6.3 The Functor of Points of a Spectral Scheme . . . 171 1.6.4 The Functor of Points of a Spectrally Ringed8-Topos . . . 173 1.6.5 The Spatial Case . . . 174 1.6.6 Comparison of Zariski and ´Etale Topologies . . . 179 1.6.7 Schematic Spectral Deligne-Mumford Stacks . . . 181 1.6.8 Spectral Deligne-Mumfordn-Stacks . . . 183

1.1 Spectral Schemes

1.1.1 Review of Scheme Theory

Our primary objective in this book is to develop the theory of spectral algebraic geometry:

a variant of algebraic geometry which uses structured ring spectra in the place of ordinary commutative rings. Before we begin this undertaking, it will be helpful to review some of the foundational definitions of the classical theory.

Definition 1.1.1.1. A ringed space is a pairpX,OXq, whereX is a topological space and OX is a sheaf of commutative rings on X. In this case, we will say that OX is thestructure sheaf of X. We will regard the collection of all ringed spaces as the objects of a category Top_CAlg♥, where a morphism frompX,OXq topY,OYqinTop_CAlg♥ consists of a pairpπ, φq, where π:X ÑY is a continuous map of topological spaces and φ:OY Ñπ˚OX is a map between sheaves of commutative rings onY.

Example 1.1.1.2. Let R be a commutative ring. We let |SpecR| denote the set of all prime ideals ofR. For every idealI ĎR, we set V_I “ tpP |SpecR|:I Ďpu, and refer toV_I as thevanishing locus ofI. We will regard|SpecR|as a topological space, where a subset of|SpecR|is closed if and only if it has the form VI for some idealI ĎR. We will refer to the resulting topology on |SpecR|as the Zariski topology.

For each element xPR, let U_x “ tpP |SpecR|:xRpu denote the complement of the vanishing locus of the principal ideal pxq. We will say that an open set U Ď |SpecR| is elementaryif it has the form U_x, for some elementxPR. The collection of elementary open sets forms a basis for the topology of|SpecR|.

The structure sheaf of|SpecR|is a sheaf of commutative rings O on|SpecR|with the following properties:

paq There is a ring homomorphism φ:RÑΓp|SpecR|;Oq “Op|SpecR|q.

pbq For each elementxPR, the composite mapRÑ^φ Op|SpecR|q ÑOpUxq carries x to an invertible element of OpUxq, and induces an isomorphism of commutative rings Rrx^´1s »OpUxq. In particular, the map α itself is an isomorphism.

Since the open sets U_x form a basis for the topology of R, property pbq determines the structure sheafO up to unique isomorphism, once the mapα:R ÑOp|SpecR|qhas been fixed (for the existence ofO, see Example 1.1.4.7 below).

Remark 1.1.1.3. It is customary to abuse notation by identifying the spectrum SpecR of a commutative ringR with its underlying topological space |SpecR|. However, we will avoid this abuse of notation for the time being.

Definition 1.1.1.4. LetpX,OXq be a ringed space. For every open subset U ĎX, we let OX|U denote the restriction ofOX to open subsets of U (which we regard as a sheaf of commutative rings on U). ThenpX,OX|Uq is itself a locally ringed space.

We say that pX,OXq is ascheme if, for every pointxPX, there exists an open subset U ĎX containing x such thatpU,OX|Uq is isomorphic to SpecR for some commutative ring R (in the category of ringed spaces). We say that a schemepX,OXq is affine if it is isomorphic (in the categoryTop_CAlg♥ of ringed spaces) to SpecR, for some commutative ring R.

1.1.2 Spectrally Ringed Spaces

Our goal in this section is to introduce an 8-categorical generalization of the theory of schemes, which we will refer to as the theory of spectral schemes. For this, we will need to work with topological spaces equipped with a sheaf of E8-rings, rather than a sheaf of ordinary commutative rings.

Definition 1.1.2.1. LetX be a topological space and letUpXqdenote the partially ordered set of all open subsets of X (which we will regard as a category). For any8-categoryC, a C-valued presheafon X is a functor F :UpXq^opÑC. We will say that aC-valued presheaf F is asheaf if it satisfies the following condition:

• LettUαube a collection of open subsets ofXhaving unionU, and letU¹“ tV PUpXq: pDαqrV ĎU_αsu. Then the functorF exhibits FpUq as a limit of the diagram F|_U^1op (in other words, F induces an equivalenceFpUq »lim

ÐÝ^VPU1FpVq in the 8-category C).

We letShv_CpXq denote the full subcategory of FunpUpXq^op,Cq spanned by those functors which are C-valued sheaves on X..

1.1. SPECTRAL SCHEMES 71 Construction 1.1.2.2. LetC be an8-category, and letπ :XÑY be a continuous map of topological spaces. Let F : UpXq^op Ñ C be a C-valued presheaf on X. Then we can define another C-valued presheaf pπ_˚Fq : UpYq^op Ñ C, given on objects by the formula pπ˚FqpUq “Fpπ^´1Uq. If F is a C-valued sheaf on X, then π˚F is a C-valued sheaf on Y. Moreover, the construction F ÞÑπ_˚F determines a functor π_˚:Shv_CpXq ÑShv_CpYq, given by precomposition with the map of partially ordered sets π^´1 :UpYq ÑUpXq. We will refer to π_˚ as thepushforward functor associated to π.

We can regard the construction XÞÑShv_CpXq^op as a functor from the categoryTop of topological spaces to the category of simplicial sets, which carries a continuous map map π:XÑY to the pushforward functor π_˚:Shv_CpXq^opÑShv_CpYq^op. We let Top_C denote the relative nerve of this functor, in the sense of Definition HTT.3.2.5.2 . ThenTop_C is an 8-category equipped with a coCartesian fibration Top_C ÑTop, having the property that each fiber Top_CˆToptXu is canonically isomorphic to the8-category Shv_CpXq^op.

Remark 1.1.2.3. Let C be an8-category. Then the objects of Top_C are given by pairs pX,OXq, where X is a topological space and OX is a C-valued sheaf on X. A morphism frompX,OXqtopY,OYq inTop_C is given by a pairpπ, αq, whereπ:X ÑY is a continuous map of topological spaces and α:OY Ñπ_˚OX is a morphism in the8-categoryShv_CpYq. Example 1.1.2.4. Let CAlg^♥ denote the category of commutative rings (regarded as an 8-category). Then Top_CAlg♥ is equivalent to the category of ringed spaces.

Definition 1.1.2.5. Let CAlg denote the8-category ofE8-rings. Aspectrally ringed space is a pairpX,OXq, whereX is a topological space and OX is a CAlg-valued sheaf onX. In this case, we will refer to OX as thestructure sheaf of X. We will refer to Top_CAlg as the 8-category of spectrally ringed spaces.

Notation 1.1.2.6. Let X be a topological space, and let F be a sheaf on X with values in the 8-category Sp of spectra. For each integer n, the construction U ÞÑ π_npFpUqq determines a presheaf of abelian groups on X. We let πnF denote the sheafification of this presheaf. If OX is a sheaf of E8-rings onX, then π0OX is a sheaf of commutative rings on X, and eachπ_nOX can be regarded as a sheaf ofπ₀OX-modules onX.

The construction pX,OXq ÞÑ pX, π0OXq determines a functor from the 8-category Top_CAlg of spectrally ringed spaces to the category of ringed spaces. We will refer to pX, π₀OXq as theunderlying ringed space ofpX,OXq.

Warning 1.1.2.7. LetpX,OXq be a spectrally ringed space. For every integernand every open setU ĎX, there is a canonical map of abelian groups

πnpOXpUqq Ñ pπnOXqpUq, which is generally not an isomorphism.

We are now ready to introduce our main objects of interest.

Definition 1.1.2.8. A spectral schemeis a spectrally ringed spacepX,OXq which satisfies the following conditions:

p1q The underlying ringed space pX, π0OXq is a scheme.

p2q Each of the sheavesπ_nOX is quasi-coherent (when viewed as a sheaf ofπ₀OX-modules on X).

p3q LetU be an open subset ofX for which the schemepU,pπ₀OXq|Uq is affine. Then, for each integern, the canonical mapπnpOXpUqq Ñ pπnOXqpUq is an isomorphism.

p4q The sheavesπ_nOX vanish whennă0.

Variant 1.1.2.9. We will say that a spectrally ringed space pX,OXq is a nonconnective spectral scheme if it satisfies conditions p1q,p2q, andp3q of Definition 1.1.2.8.

Remark 1.1.2.10. If pX,OXq is a nonconnective spectral scheme, then the ringed space pX, π0OXq is a scheme. We will refer topX, π0OXqas the underlying schemeofpX,OXq.

1.1.3 Digression: Hypercompleteness

We next show that conditionp3q of Definition 1.1.2.8 admits an alternate formulation, and is automatically satisfied in many cases of interest (see Corollary 1.1.3.6).

Definition 1.1.3.1. LetF be a spectrum-valued sheaf on a topological spaceX. We will say that F is hypercomplete if the functor U ÞÑ Ω⁸FpUq determines a hypercomplete object of the 8-topos Shv_SpXq (see §HTT.6.5.2 ).

Remark 1.1.3.2. Let X be a topological space. Then the collection of hypercomplete objects of Shv_SppXq is closed under small limits and under suspensions (see Proposition 1.3.3.3). Moreover, the condition that an objectF PShv_SppXq be hypercomplete can be tested locally on X (Corollary 1.3.3.8).

Remark 1.1.3.3. Let u : F Ñ F¹ be a morphism of hypercomplete spectrum-valued sheaves on a topological space X. Then u is an equivalence if and only if it induces an isomorphism πnF ÑπnF¹ (in the category of sheaves of abelian groups) for every integer n. The “only if” direction is obvious. To prove the converse, it suffices to show that the map u_n: Ω⁸pΣⁿFq ÑΩ⁸pΣⁿF¹q is an equivalence in the 8-toposShv_SpXq for every integer n. This is clear, sinceunis a morphism between hypercomplete objects of Shv_SpXq which induces an isomorphism on homotopy sheaves.

Proposition 1.1.3.4. Let pX,OXq be a spectrally ringed space satisfying the following conditions:

1.1. SPECTRAL SCHEMES 73