In this section, we introduce a general axiomatic paradigm for the study of deformation theory. Let us begin by outlining the basic idea. We are ultimately interesting in studying some class of algebro-geometric objects (such as schemes, or algebraic stacks, or their spectral analogues). Using the functor of points philosophy, we will identify these geometric objects with functors X : Υ → S, where Υ denotes some ∞-category of test objects. The main example of interest (which we will study in detail in §2) is the case where Υ to be the ∞-category CAlgaugk = (CAlgk)/k of augmented E∞-algebras over a field k of characteristic zero. In any case, we will always assume that Υ contains a final object ∗; we can then define a point of a functor X : Υ→Sto be a point of the spaceX(∗). Our goal is to introduce some techniques for studying aformal neighborhood of X around a chosen pointη ∈X(∗). This formal neighborhood should encode information about the homotopy fiber products X(A)×X(∗){η} for every object A∈Υ which is sufficiently “close” to the final object∗. In order to make this idea precise, we need to introduce some terminology.
Notation 1.1.1. If Υ is a presentable∞-category, we let Υ∗ denote the∞-category of pointed objects of Υ (Definition T.7.2.2.1) and Stab(Υ) the stabilization of Υ (Definition A.1.4.4.1). Then Stab(Υ) can be described as a homotopy limit of the tower of∞-categories
· · · →Υ∗
→Ω Υ∗
→Ω Υ∗→ · · ·
In particular, we have forgetful functors Ω∞−n∗ : Stab(Υ) → Υ∗ for every integern ∈ Z. We let Ω∞−n : Stab(Υ)→Υ denote the composition of Ω∞−n∗ with the forgetful functor Υ∗→Υ.
Remark 1.1.2. We can describe Stab(Υ) explicitly as the∞-category of strongly excisive functors from Sfin∗ to Υ, where Sfin∗ is the ∞-category of pointed finite spaces (see Corollary A.1.4.4.14; we will return to this description of Stab(Υ) in§1.2).
Definition 1.1.3. Adeformation contextis a pair (Υ,{Eα}α∈T), where Υ is a presentable∞-category and {Eα}α∈T is a set of objects of the stabilization Stab(Υ).
Example 1.1.4. Let k be an E∞-ring, and let Υ = CAlgaugk = (CAlgk)/k denote the ∞-category of augmented E∞-algebras over k. Using Theorem A.7.3.5.14, we can identify Stab(Υ) with the ∞-category Modk of k-module spectra. Let E ∈ Stab(Υ) be the object which corresponds to k ∈ Modk under this identification, so that for every integernwe can identify Ω∞−nE with the square-zero extensionk⊕k[n] of k. Then the pair (CAlgaugk ,{E}) is a deformation context.
Definition 1.1.5. Let (Υ,{Eα}α∈T) be a deformation context. We will say that a morphismφ:A0 →Ain Υ iselementaryif there exists an indexα∈T, an integern >0, and a pullback diagram
A0
φ //∗
φ0
A //Ω∞−nEα.
Hereφ0 corresponds the image Ω∞−n∗ Eα in the∞-category of pointed objects Υ∗.
Example 1.1.6. Letkbe a field and let (Υ,{E}) be the deformation context described in Example 1.1.4.
Suppose thatφ:A0→Ais a map between connective objects of Υ = CAlgaugk . Using Theorem A.7.4.1.26, we deduce thatφis elementary if and only if the following conditions are satisfied:
(a) There exists an integer n ≥ 0 and an equivalence fib(φ) ' k[n] in the ∞-category ModA0 (here we regardkas an object of ModA0 via the augmentation mapA0→k).
(b) Ifn= 0, then the multiplication map π0fib(φ)⊗π0fib(φ)→φ0fib(φ) vanishes.
If (a) is satisfied forn= 0, then we can choose a generatorxforπ0fib(φ) having imagex∈π0A0. Condition (b0) is automatic if x= 0. Ifa6= 0, then the mapπ0fib(φ)→π0A0 is injective, so condition (b) is equivalent to the requirement thatx2= 0 inπ0A0.
Remark 1.1.7. Let (Υ,{Eα}α∈T) be a deformation context, and suppose we are given an object A∈Υ.
Every elementary map A0 → A in Υ is given by the fiber of a map A → Ω∞−nEα for some n > 0 and some α∈T. It follows that the collection of equivalence classes of elementary mapsA0 →Ais bounded in cardinality.
Definition 1.1.8. Let (Υ,{Eα}α∈T) be a deformation context. We will say that a morphismφ:A0 →A in Υ issmallif it can be written as a composition of finitely many elementary morphismsA0'A0→A1→
· · · →An 'A. We will say that an objectA∈Υ issmall if the mapA→ ∗(which is uniquely determined up to homotopy) is small. We let Υsm denote the full subcategory of Υ spanned by the small objects.
Example 1.1.9. Let (Υ,{Eα}α∈T) be a deformation context. For every integer n ≥ 0 and every index α∈T, we have a pullback diagram
Ω∞−nEα //
∗
∗ //Ω∞−n−1Eα.
It follows that the left vertical map is elementary. In particular, Ω∞−nEαis a small object of Υ.
Remark 1.1.10. Let (Υ,{Eα}α∈T) be a deformation context. It follows from Remark 1.1.7 that the subcategory Υsm⊆Υ is essentially small.
Proposition 1.1.11. Let kbe a field and let (Υ,{E})be the deformation context of Example 1.1.4. Then an objectA∈Υ = CAlgaugk is small (in the sense of Definition 1.1.8) if and only if the following conditions are satisfied:
(1) The homotopy groups πnAvanish for n <0andn0.
(2) Each homotopy group πnAis finite-dimensional as a vector space overk.
(3) The commutative ring π0A is local with maximal idealk, and the canonical mapk →(π0A)/m is an isomorphism.
Proof. Suppose first thatAis small, so that there there exists a finite sequence of maps A=A0→A1→ · · · →An'k
where eachAi is a square-zero extension ofAi+1 byk[mi], for someni≥0. We prove that eachAi satisfies conditions (1), (2), and (3) using descending induction oni. The casei=nis obvious, so let us assume that i < nand thatAi+1 is known to satisfy conditions (1), (2), and (3). We have a fiber sequence ofk-module spectra
k[mi]→Ai→Ai+1
which immediately implies that Ai satisfies (1) and (2). The map φ : π0Ai → π0Ai+1 is surjective and ker(φ)2= 0, from which it follows immediately thatπ0Ai is local.
Now suppose thatAsatisfies conditions (1), (2), and (3). We will prove thatAis small by induction on the dimension of thek-vector spaceπ∗A. Letnbe the largest integer for whichπnAdoes not vanish. We first treat the casen= 0. We will abuse notation by identifyingA with the underlying commutative ringπ0A.
Condition (3) implies that Ais a local ring; letm denote its maximal ideal. SinceA is a finite dimensional algebra over k, we havemi+1'0 for i0. Chooseias small as possible. If i= 0, thenm'0 andA'k, in which case there is nothing to prove. Otherwise, we can choose a nonzero element x∈mi ⊆m. Let A0 denote the quotient ringA/(x). It follows from Example 1.1.6 that the quotient mapA→A0 is elementary.
SinceA0 is small by the inductive hypothesis, we conclude thatA is small.
Now suppose thatn >0 and letM =πnA, so thatM has the structure of a module over the ringπ0A.
Letm⊆π0Abe as above, and letibe the least integer such thatmi+1M '0. Letx∈miM and letM0 be the quotient ofM byx, so that we have an exact sequence
0→k→x M →M0 →0
of modules overπ0A. We will abuse notation by viewing this sequence as a fiber sequence ofA00-modules, whereA00=τ≤n−1A. It follows from Theorem A.7.4.1.26 that there is a pullback diagram
A //
k
A00 //k⊕M[n+ 1].
Set A0 =A00×k⊕M0[n+1]k. Then A 'A0×k⊕k[n+1]k so we have an elementary map A → A0. Using the inductive hypothesis we deduce thatA0 is small, so thatAis also small.
Remark 1.1.12. Let k be a field and suppose that A ∈ CAlgk satisfies conditions (1), (2), and (3) of Proposition 1.1.11. Then the mapping space MapCAlg
k(A, k) is contractible. In particular,Acan be promoted (in an essentially unique way) to a small object of Υ = CAlgaugk . Moreover, the forgetful functor CAlgaugk → CAlgk is fully faithful when restricted to the full subcategory Υsm⊆Υ. We will denote the essential image of this restriction by CAlgsmk . We refer to CAlgsmk as the∞-category of smallE∞-algebras overk.
Remark 1.1.13. Let (Υ,{Eα}α∈T) be a deformation context. Then the collection of small morphisms in Υ is closed under composition. In particular, ifφ:A0 →Ais small andAis small, thenA0 is also small. In particular, if there exists a pullback diagram
B0 //
A0
φ
B //A whereB is small andφis small, then B0 is also small.
We are now ready to introduce the main objects of study in this paper.
Definition 1.1.14. Let (Υ,{Eα}α∈T) be a deformation context. A formal moduli problem is a functor X : Υsm→Ssatisfying the following pair of conditions:
(a) The spaceX(∗) is contractible (here∗denotes a final object of Υ).
(b) Letσ:
A0
//B0
φ
A //B
be a diagram in Υsm. Ifσis a pullback diagram andφis small, thenX(σ) is a pullback diagram inS.
We let ModuliΥ denote the full subcategory of Fun(Υsm,S) spanned by the formal moduli problems. We will refer to ModuliΥas the ∞-category of formal moduli problems.
Condition (b) of Definition 1.1.14 has a number of equivalent formulations:
Proposition 1.1.15. Let (Υ,{Eα}α∈T)be a deformation context, and let X : Υsm→S be a functor. The following conditions are equivalent:
(1) Let σ:
A0
//B0
φ
A //B
be a diagram inΥsm. Ifσis a pullback diagram and φis small, thenX(σ)is a pullback diagram inS. (2) Let σbe as in(1). Ifσis a pullback diagram andφis elementary, thenX(σ)is a pullback diagram in
S.
(3) Let σbe as in(1). Ifσis a pullback diagram andφis the base point morphism∗ →Ω∞−nEαfor some α∈T andn >0, thenX(σ) is a pullback diagram inS.
Proof. The implications (a)⇒(b)⇒(c) are clear. The reverse implications follow from Lemma T.4.4.2.1.
Example 1.1.16. Let (Υ,{Eα}α∈T) be a deformation context, and letA∈Υ be an object. Let Spec(A) : Υsm→Sbe the functor corepresented byA, which is given on small objects of Υ by the formula Spec(A)(B) = MapΥ(A, B). Then Spec(A) is a formal moduli problem. Moreover, the constructionA 7→Spec(A) deter-mines a functor Spec : Υop→ModuliΥ.
Remark 1.1.17. Let (Υ,{Eα}α∈T) be a deformation context. The∞-category Υsm⊆Υ is essentially small.
It follows from Lemmas T.5.5.4.19 and T.5.5.4.18 that the∞-category ModuliΥis an accessible localization of the∞-category Fun(Υsm,S); in particular, the∞-category ModuliΥ is presentable.
Remark 1.1.18. Let (Υ,{Eα}α∈T) be a deformation context, and let X : Υsm → S be a functor which satisfies the equivalent conditions of Proposition 1.1.15. For every point η ∈ X(∗), define a functor Xη : Υsm→Sby the formulaXη(A) =X(A)×X(∗){η}. ThenXη is a formal moduli problem. We may therefore identifyX as a family of formal moduli problems parametrized by the spaceX(∗). Consequently, condition (a) of Definition 1.1.14 should be regarded as a harmless simplifying assumption.
In the special case where (Υ,{Eα}α∈T) is the deformation context of Example 1.1.4, Definition 1.1.14 agrees with Definition 0.0.8. This is an immediate consequence of the following result:
Proposition 1.1.19. Let kbe a field and let X : CAlgsmk →S be a functor. Then conditions (1),(2), and (3)of Proposition 1.1.15 are equivalent to the following:
(∗) For every pullback diagram
R //
R0
R1 //R01
in CAlgsmC for which the underlying mapsπ0R0→π0R01←π0R1 are surjective, the diagram X(R) //
X(R0)
X(R1) //X(R01)
is a pullback square.
The proof of Proposition 1.1.19 will require the following elaboration on Proposition 1.1.11:
Lemma 1.1.20. Let k be a field and let f : A → B be a morphism in CAlgsmk . Then f is small (when regarded as a morphism inCAlgaugk ) if and only if it induces a surjection of commutative ringsπ0A→π0B.
Proof. LetK be the fiber off, regarded as anA-module. Ifπ0A→π0B is surjective, thenKis connective.
We will prove thatf is small by induction on the dimension of the graded vector spaceπ∗K. If this dimension is zero, thenK'0 andf is an equivalence. Assume therefore thatπ∗K6= 0, and letnbe the smallest integer such thatπnK6= 0. Letmdenote the maximal ideal ofπ0A. Thenmis nilpotent, som(πnK)6=πnKand we can choose a map ofπ0A-modulesφ:πnK→k. According to Theorem A.7.4.3.1, we have (2n+1)-connective mapK⊗AB →LB/A[−1]. In particular, we have an isomorphismπn+1LB/A'Torπ00A(π0B, πnK) so that φ determines a map LB/A →k[n+ 1]. We can interpret this map as a derivationB → B⊕k[n+ 1]; let B0=B×B⊕k[n+1]k. Thenf factors as a composition
A f
0
→B0 f
00
→B.
Since the map f00 is elementary, it will suffice to show that f0 is small, which follows from the inductive hypothesis.
Proof of Proposition 1.1.19. The implication (∗) ⇒ (3) is obvious, and the implication (1) ⇒ (∗) follows from Lemma 1.1.20.
Remark 1.1.21. The proof of Proposition 1.1.19 shows that condition (∗) is equivalent to the stronger condition that the diagram
R //
R0
R1 //R01
is a pullback square wheneveroneof the mapsπ0R0→π0R01or π0R1→π0R01is surjective.