Solution Sheaves - Schemes and Deligne-Mumford Stacks

Schemes and Deligne-Mumford Stacks

1.4.3 Solution Sheaves

We now introduce some terminology which will be useful for the proof of Proposition 1.4.2.4.

Notation 1.4.3.1. Let pX,OXqbe a spectrally ringed 8-topos. For each E8-ringR, the construction pU PXq ÞÑ Map_CAlgpR,OXpUqq determines aS-valued sheaf on X. We let Sol_RpOXqdenote an object of X which represents this functor.

Example 1.4.3.2. Let R “ Sym^˚pSⁿq denote the free E8-ring on a single generator in degreen. IfOX is a sheaf ofE8-rings on an8-toposX, then there is a canonical equivalence Sol_RpOXq »Ω^8`nOX in the8-toposX.

Remark 1.4.3.3 (Functoriality in R). Let X be an 8-topos and let OX be a sheaf of E8-rings on X. Then the construction RÞÑ Sol_RpOXq is contravariantly functorial inR.

Moreover, it carries colimits (in the 8-category CAlg of E8-rings) to limits (in the8-topos X).

Remark 1.4.3.4 (Functoriality in OX). Let X be an 8-topos and let R be anE8-ring.

Then the construction OX ÞÑSol_RpOXqdetermines a functor Shv_CAlgpXq ÑShvpXqwhich preserves small limits.

Remark 1.4.3.5(Functoriality inX). Letf˚:X ÑYbe a geometric morphism of8-topoi and letR be anE8-ring. Then the diagram

Shv_CAlgpXq ^Sol^R ^//

f˚

Shv_CAlgpYq ^Sol^R ^//Y

commutes (up to canonical equivalence).

Remark 1.4.3.6. In the situation of Notation 1.4.3.1, suppose thatR is connective. Then for any object OX P Shv_CAlgpXq, the canonical map Sol_Rpτě0OXq Ñ Sol_RpOXq is an equivalence inX.

It follows from Remark 1.4.3.5 that for any geometric morphism of 8-topoi f˚:X ÑY and anyE8-ringR, there is a canonical natural transformation of functorsf^˚Sol_RÑSol_Rf^˚ from Shv_CAlgpYq toShv_CAlgpXq.

Lemma 1.4.3.7. LetR be a compact object ofCAlg. Then, for any geometric morphism of 8-topoif_˚ :X ÑY and any sheafOY of E8-rings onY, the canonical map f^˚Sol_RpOYq Ñ Sol_Rpf^˚OYq is an equivalence in X.

Proof. Let CĎCAlg_R be the full subcategory spanned by those objectsR for which the natural mapf^˚Sol_RÑSol_Rf^˚ is an equivalence of functors fromShv_CAlgpYqtoX. ThenC is closed under retracts, and it follows from Remark 1.4.3.3 (together with the left exactness off^˚) that C is closed under finite colimits in CAlg_R. To prove that C contains all compact objects of CAlg_R, it will suffice to show that it contains all free algebras of the form Sym^˚pSⁿq, which follows from Example 1.4.3.2.

Lemma 1.4.3.8. Let f :AÑB be an ´etale morphism between connectiveE8-rings, and let OX be a connective sheaf ofE8-rings on an 8-topos X. Then the diagram

Sol_BpOXq ^//

Sol_Bpπ₀OXq

Sol_ApOXq ^//Sol_Apπ₀OXq

is a pullback square inX.

Proof. Using Proposition B.1.1.3, we can choose a pushout diagram A₀ ^//

B0 //B

in CAlg, wheref₀ is ´etale and theE8-rings A₀ and B₀ are compact and connective. Using Remark 1.4.3.3, we can replacef by f0 and thereby reduce to the case whereA and B are compact. BecauseX is an 8-topos, we can choose a small 8-category C and a geometric morphism φ_˚ :X ÑPpCq which exhibits X as a left exact localization of PpCq. We then have equivalences

OX »φ^˚pτ_ě0φ_˚OXq π₀OX »φ^˚pπ₀φ_˚OXq.

Using Lemma 1.4.3.7, we can replace OX by τ_ě0pφ_˚OXq and thereby reduce to the case where X “PpCq is an8-category of presheaves. In this case, we are reduced to proving that for every object CPC, the diagram of mapping spaces

Map_CAlgpB,OXpCqq ^//

Map_CAlgpB, π₀OXpCqq

Map_CAlgpA,OXpCqq ^//Map_CAlgpA, π0OXpCqq is a pullback square, which follows from Theorem HA.7.5.4.2 .

1.4. SPECTRAL DELIGNE-MUMFORD STACKS 139 Lemma 1.4.3.9. paq Let pX,OXq be a spectrally ringed 8-topos. Then OX is strictly Henselian if and only if, for every E8-ring A and every faithfully flat ´etale morphism AÑś

1ďiďnA_i, the induced map ž

1ďiďn

Sol_A_ipOXq ÑSol_ApOXq

is an effective epimorphism inX.

pbq Letf :pX,OXq Ñ pY,OYqbe a morphism of spectrally ringed 8-topoi, where OX and OY are strictly Henselian. Then f is a morphism in 8Top^sHen_CAlg if and only if, for every ´etale morphism ofE8-rings AÑB, the associated diagram

f^˚Sol_BpOYq ^//

Sol_BpOXq

f^˚Sol_ApOYq ^//Sol_ApOXq.

Proof. We first prove the “only if” direction ofpaq. Assume thatOX is strictly Henselian and that we are given a collection of ´etale morphismstAÑAiu1ďiďn for which the induced mapAÑś

1ďiďnA_i is faithfully flat. We wish to prove that the induced map ž

1ďiďn

Sol_A_ipOXq ÑSol_ApOXq

is an effective epimorphism. Note that each of the maps AÑAi is flat, and therefore fits into a pushout square of E8-rings

τ_ě0A ^//

τ_ě0Ai

A ^//A_i.

Using Remark 1.4.3.3, we obtain a pullback diagram š

1ďiďnSol_A_ipOXq ^//

Sol_ApOXq

1ďiďnSol_τ_ě0_A_ipOXq ^//Sol_τ_ě0_ApOXq.

Consequently, to prove the the upper horizontal map in this diagram is an effective epimor-phism, it will suffice to show that the lower horizontal map is an effective epimorphism. We may therefore replace A byτ_ě0A (and each A_i withτ_ě0A_i) and thereby reduce to the case

where A is connective. In this case, there is no loss of generality in assuming that OX is connective (Remark 1.4.3.6). Lemma 1.4.3.8 now supplies a pullback diagram

where the bottom horizontal map is an effective epimorphism (between discrete objects of X) by virtue of our assumption thatπ₀OX is strictly Henselian.

We now prove the “if” direction of paq. Assume thatOX has the property described bypaq; we wish to show that for every commutative ring R and every finite collection of

etale morphisms tRÑR_iu1ďiďn for which the induced mapR Ñś

R_i is faithfully flat, the induced map >Sol_R_ipπ₀OXq ÑSol_Rpπ₀OXqis an effective epimorphism (between discrete objects ofX). Writing R as a direct limit of its finitely generated subrings, we may assume without loss of generality that R is finitely generated: that is, there exists a surjection of commutative rings P Ñ R, where P is a polynomial ring over Z. Using the structure theory of ´etale morphisms of commutative rings (see Proposition B.1.1.3), we can lift each R_i to an ´etale P-algebra P_i. Let U Ď SpecP be the union of the images of the maps SpecPi ÑSpecP. Since ´etale morphisms have open images, the setU is open with respect to the Zariski topology. It is clearly quasi-compact, so we can choose a collection of elements t₁, . . . , t_k PP for which U is covered by the open subsets SpecPrt^´1_j s ĎSpecP. Because the map RÑś

Ri is faithfully flat, the map SpecR ÑSpecP factors throughU. We may therefore choose coefficients c_j PR for which the sum c₁t₁` ¨ ¨ ¨ `c_kt_k “1 inR; here we abuse notation by identifying each t_j with its image inR. Using the surjectivity of the map P ÑR, we can lift each cj to an elementcj PP. Let u“c1t1` ¨ ¨ ¨ `cktk. Then the map Pru^´1s Ñś

P_iru^´1sis faithfully flat and the map P ÑR factors through Pru^´1s. We may therefore replaceR by Pru^´1sand thereby reduce to the case where the commutative ring R has the formZrx1, . . . , xmsru^´1sfor someuPZrx1, . . . , xms. epimorphisms. The mapρ is an effective epimorphism by virtue of our assumption thatOX

satisfies the condition described in paq. The mapρ¹ is an effective epimorphism because it is a pullback of themth power of the effective epimorphism Ω⁸OX Ñπ0OX. This completes the proof of paq.

1.4. SPECTRAL DELIGNE-MUMFORD STACKS 141

is a pullback square in X. As in the first part of the proof, it will suffice to prove this in the special case where Aand B are connective. Remark 1.4.3.6 then allows us to replace OX

andOY by their connective covers, and thereby reduce to the case where they are connective as well. In this case, we have a commutative diagram

f^˚Sol_BpOYq ^//

where the right square is a pullback by Lemma 1.4.3.8. It will therefore suffice to show that the outer rectangle is a pullback. This rectangle also appears in the commutative diagram

f^˚Sol_BpOYq ^//

where the left square is a pullback (Lemma 1.4.3.8). We are then reduced to showing that the right square is a pullback diagram (of discrete objects of X), which follows from Proposition 1.2.2.12.

For the converse, suppose that f satisfies the condition described inpbq; we claim that induced map f^˚OY Ñ OX is local. Fix an object X PX and a map e: X Ñ π0f^˚OY, and let e denote the composite map X Ñ^e π0f^˚OY Ñ π0OX. We must show that ife is invertible when regarded as an element of the commutative ringπ₀Map_XpX, π₀OXq, thene is invertible when regarded as an element of the commutative ringπ0Map_XpX, π0f^˚OYq.

This assertion is local X: we may therefore assume without loss of generality thatefactors through a mapX Ñf^˚OY. In this case, the desired result follows by inspecting the diagram

f^˚Sol_BpOYq ^//

in the special caseA“StxuandB“Stxurx^´1s; here the left square is a pullback by virtue of our hypothesis onf and the right square is a pullback by virtue of Lemma 1.4.3.8.

Proof of Proposition 1.4.2.4. Let R be an E8-ring and let O : CAlg^´^et_R Ñ CAlg be the forgetful functor. It follows from Theorem D.6.3.5 thatO is a CAlg-valued sheaf (with respect to the ´etale topology on CAlg^´^et_R). Note thatπ₀O can be identified with the sheafification of the presheaf of commutative rings given by the composite map CAlg^´^et_R Ñ^π⁰ CAlg^♥, which is the structure sheaf of the Deligne-Mumford stack Sp´etR. It follows from Proposition 1.2.3.2 thatπ₀O is strictly Henselian, so that the sheaf O is strictly Henselian.

Let X be an arbitrary 8-topos and let OX be a strictly Henselian CAlg-valued sheaf on X. The constructionA ÞÑ Sol_ApOXq determines a functor Sol_‚pOXq : CAlg^´^et_R Ñ X^op. Applying Proposition HTT.6.1.5.2 , we can identify the mapping space Map_8T_oppX,Shv^´^et_Rq with the full subcategory of FunpCAlg^´^et_A,Xq^»spanned by those functorsχ:pCAlg^´^et_Rq^op ÑX which satisfy the following conditions:

piq The functorχ preserves finite limits.

piiq For every faithfully flat ´etale morphism AÑś

1ďiďnAi in CAlg^´^et_R, the induced map

>χpA_iq ÑχpAqis an effective epimorphism in the8-toposX.

Ifχ is a functor satisfying these conditions and f_˚:X ÑShv^´^et_R is the associated geometric morphism, then we can identify the direct image f˚OX with the CAlg-valued sheaf on CAlg^´^et_R given by CAlg^´^et_R ÝÑ^χ X^op ÝÝÑ^O^X CAlg. We may therefore identify the mapping space over the pairpχ, φqcan be identified with the mapping space Map_FunppCAlg´et

Rq^op,Xqpχ,Sol⁰_‚pOXqq. It follows from Lemma 1.4.3.9 that this identification carries the homotopy fiber of the restriction

Map_8T_opsHen

CAlgppX,OXq,Sp´etRq ÑFunppCAlg^´^et_Rq^op,Xq^»ˆMap_CAlgpR,ΓpX,OXqq to the subspace of Map_FunppCAlg´et

Aq^op,Xqpχ,Sol⁰_‚pOXqqspanned by the equivalences. It follows that the fiber product

Map_8T_opsHen

CAlgppX,OXq,Sp´etRq ˆ_Map

CAlgpR,ΓpX,OXqqtφu

is either empty or contractible. We will complete the proof by explicitly constructing a point of this fiber product, given by a morphism of ringed8-topoipX,OXq ÑSp´etR. Let

1.4. SPECTRAL DELIGNE-MUMFORD STACKS 143 χ:pCAlg^´^et_Rq^op ÑX be the functorAÞÑSol⁰_ApOXq. The constructionAÞÑSol_ApOXqcarries colimits in CAlg to limits inX. It follows that the functorSol_‚pOXqpreserves pullbacks, so that Sol⁰_‚pOXq also preserves pullbacks. By construction, the functorSol⁰_‚pOXq carriesR to a final object of X, so thatSol⁰_‚pOXq preserves finite limits. SinceOX is strictly Henselian, Lemma 1.4.3.9 implies that the functor χ“Sol⁰_‚pOXq satisfies condition piiq above, and therefore determines a geometric morphism of 8-topoi f_˚ : X Ñ Shv^´^et_R. The preceding analysis shows that the projection mapSol⁰_‚pOXq ÑSol_‚pOXq determines a morphism of CAlg-valued sheavesα :O Ñf_˚OX, so that we can regardf “ pf_˚, αq as a morphism of spectrally ringed topoi from pX,OXq into Sp´etA. We will complete the proof by showing thatf is a morphism in the 8-category 8Top^sHen_CAlg. To this end, fix an object U PX and an elementxPπ₀Map_XpU, f^˚pπ₀Oqqwhose image in π₀Map_XpU, π₀OXq is invertible; we wish to show that x is invertible. For each object A PCAlg^´^et_R, let h^A PShv^´^et_R denote the sheaf corepresented by B. Since the objects h^A generate Shv^´^et_R under small colimits, we can therefore choose an effective epimorphism >h^A^α Ñ π₀O, which induces an effective epimorphism š

f^˚h^A^α Ñ f^˚π0O. Working locally on U, we may assume that the map x:U Ñf^˚π0O factors as a compositionU ÝÑ^ψ f^˚h^{A f}

˚y

ÝÝÑf^˚π0O, where ψ:U Ñf^˚h^A» Sol⁰_ApOXqclassifies an R-algebra morphism φ:AÑOXpUq, andy:h^AÑπ₀O is a map of discrete objects of Shv^´^et_R which we can identify with an element of the commutative ring π0OpAq “ π0A. Applying φ to y, we obtain an element φpyq P π0pOXpUqqwhose image in Map_XpU, π₀OXq is an equivalence. It follows that multiplication by φpyq induces an equivalence from OX |U to itself, so that φpyq is invertible in OX |U. Consequently, the mapψ admits an (essentially unique) lift to a map ψ:U ÑSol⁰_Ary´1spOXq »f^˚h^Ary^´1^s, so that xfactors as a compositionU ÝÑ^ψ f^˚h^Ary^´1^s ÝÝÑ^f^˚^y f^˚π0O, and is therefore invertible, as desired.

1.4.4 Spectral Deligne-Mumford Stacks

Im Dokument Spectral Algebraic Geometry (Under Construction!) (Seite 137-143)