The Tangent Complex

R0

R₁ //R₀₁

is a pullback square wheneveroneof the mapsπ₀R₀→π₀R₀₁or π₀R₁→π₀R₀₁is surjective.

1.2 The Tangent Complex

LetX be an algebraic variety over the field C of complex numbers, and letx: SpecC →X be a point of X. Atangent vectortoX at the pointxis a dotted arrow rendering the diagram

SpecC ^x //

X

SpecC[]/(²) // 77 SpecC

commutative. The collection of tangent vectors toX at xcomprise a vector space TX,x, which we call the Zariski tangent spaceofXatx. IfOX,xdenotes the local ring ofXat the pointxandm⊆OX,xits maximal ideal, then there is a canonical isomorphism of vector spacesT_X,η'(m/m²)^∨.

The tangent space T_X,x is among the most basic and useful invariants one can use to study the local structure of an algebraic varietyX near a point x. Our goal in this section is to generalize the construction of T_X,x to the setting of an arbitrary formal moduli problem, in the sense of Definition 1.1.14. Let us identify X with its functor of points, given by X(A) = Hom(SpecA, X) for every C-algebra A (here the Hom is computed in the category of schemes overC). ThenTX,x can be described as the fiber of the map X(C[]/(²)) → X(C) over the point x ∈ X(C). Note that the commutative ring C[]/(²) is given by Ω^∞E, where E is the spectrum object of CAlg^aug_C appearing in Example 1.1.4. This suggests a possible generalization:

Definition 1.2.1. Let (Υ,{Eα}_α∈T) be a deformation context, and letY : Υ^sm→ S be a formal moduli problem. For eachα∈T, thetangent space ofY atαis the spaceY(Ω^∞Eα).

There is a somewhat unfortunate aspect to the terminology of Definition 1.2.1. By definition, a formal moduli problem Y is a S-valued functor, so the evaluation ofX on any object A ∈Υ^sm might be called a

“space”. The term “tangent space” in algebraic geometry has a different meaning: ifX is a complex algebraic variety with a base pointx, then we refer toT_X,x as the tangentspaceofX not because it is equipped with a topology, but because it has the structure of a vector space over C. In particular,T_X,x is equipped with an addition which is commutative and associative. Our next goal is to prove that this phenomenon is quite general: for any formal moduli problemY : Υ^sm→S, each tangent spaceY(Ω^∞Eα) ofY is an infinite loop space, and therefore equipped with a composition law which is commutative and associative up to coherent homotopy.

We begin by recalling some definitions.

Notation 1.2.2. LetCbe an∞-category which admits finite colimits andDan∞-category which admits finite limits. We say that a functorF :C→D isexcisiveif, for every pushout square

C01 //

C0

C1 //C inC, the diagram

F(C₀₁) //

F(C₀)

F(C1) //F(C)

is a pullback square inD. We say that F is strongly excisiveif it is excisive and carries initial objects ofC to final objects ofD.

We let S^fin∗ denote the∞-category of finite pointed spaces. For any ∞-categoryD which admits finite limits, we let Stab(D) denote the full subcategory of Fun(S^fin∗ ,D) spanned by the pointed excisive functors.

We recall that Stab(D) is an explicit model for the stabilization ofD; in particular, it is a stable∞-category (see∞-category, which is model for the stabilization ofD (see Corollary A.1.4.4.14). In particular, we can realize the∞-category Sp = Stab(S) of spectra as the full subcategory of Fun(S^fin∗ ,S) spanned by the strongly excisive functors.

Proposition 1.2.3. Let(Υ,{Eα}α∈T)be a deformation context. For eachα∈T, we identifyE_α∈Stab(Υ) with the corresponding functor S^fin_∗ →Υ. Then:

(1) For every map f : K → K⁰ of pointed finite spaces which induces a surjection π0K → π0K⁰, the induced map Eα(K)→Eα(K⁰)is a small morphism inΥ.

(2) For every pointed finite space K, the objectEα(K)∈Υis small.

Proof. We will prove (1); assertion (2) follows by applying (1) to the constant mapK→ ∗. Note that f is equivalent to a composition of maps

K=K0→K1→ · · · →Kn=K⁰,

where each K_i is obtained from K_i−1 by attaching a single cell of dimension n_i. Since π₀K surjects onto π₀K⁰, we may assume that eachn_i is positive. It follows that we have pushout diagrams of finite pointed spaces

K_i−1 //

Ki

∗ //Sⁿⁱ.

SinceEα is excisive, we obtain a pullback square

E_α(K_i−1) //

E_α(K_i)

∗ //ΩⁿⁱE_α,

so that each of the mapsEα(K_i−1)→Eα(Ki) is elementary.

It follows from Proposition 1.2.3 that if (Υ,{Eα}_α∈T) is a deformation context, then each Eα can be regarded as a functor from S^fin∗ to the full subcategory Υ^sm⊆Υ spanned by the small object. It therefore makes sense to composeEαwith any formal moduli problem.

Proposition 1.2.4. Let (Υ,{E_α}_α∈T) be a deformation context and let Y : Υ^sm →S be a formal moduli problem. For everyα∈T, the composite functor

S^fin∗ Eα

→Υ^sm→^Y S is strongly excisive.

Proof. It is obvious thatY ◦E_α carries initial objects ofS^fin∗ to contractible spaces. Suppose we are given a pushout diagram

K //

K⁰

L //L⁰ of pointed finite spaces; we wish to show that the diagramσ:

Y(E_α(K)) //

Y(E_α(K⁰))

Y(Eα(L)) //Y(Eα(L⁰))

is a pullback square inS. LetK₊⁰ denote the union of those connected components ofK⁰which meet the image of the mapK→K⁰. There is a retraction ofK⁰ ontoK₊⁰ , which carries the other connected components of K⁰ to the base point. DefineL⁰₊ and the retractionL⁰→L⁰₊ similarly. We have a commutative diagram of pointed finite spaces

K //

K⁰ //

K₊⁰

L //L⁰ //L⁰₊ where each square is a pushout, hence a diagram of spaces

Y(E_α(K)) //

Y(E_α(K⁰))

//Y(E_α(K⁰₊))

Y(E_α(L)) //Y(E_α(L⁰)) //Y(E_α(L⁰₊)).

To prove that the left square is a pullback diagrams, it will suffice to show that the right square and the outer rectangle are pullback diagrams. We may therefore reduce to the case where the mapπ0K→π0K⁰ is

surjective. Then the map π0L→π0L⁰ is surjective, so thatEα(L)→Eα(L⁰) is a small morphism in Υ^sm (Proposition 1.2.3). SinceEαis excisive, the diagram

Eα(K) //

Eα(K⁰)

Eα(L) //Eα(L⁰)

is a pullback square in Υ. Using the assumption thatY is a formal moduli problem, we deduce thatσis a pullback square of spaces.

Definition 1.2.5. Let (Υ,{Eα}α∈T) be a deformation context, and letY : Υ^sm→ S be a formal moduli problem. For eachα∈T, we letY(E_α) denote the composite functor

S^fin∗ E_α

→Υ^sm→^Y S.

We will view Y(Eα) as an object in the ∞-category Sp = Stab(S) of spectra, and refer toY(Eα) as the tangent complex toY atα.

Remark 1.2.6. In the situation of Definition 1.2.5, suppose thatT has a single element, so that{Eα}_α∈T = {E}for someE∈Stab(Υ) (this condition is satisfied in all of the main examples we will study in this paper).

In this case, we will omit mention of the index αand simply refer to Y(E) as the tangent complex to the formal moduli problemY.

Remark 1.2.7. LetY : Υ^sm→S be as in Definition 1.2.5. For every indexα, we can identify the tangent spaceY(Ω^∞Eα) atαwith the 0th space of the tangent complexY(Eα). More generally, there are canonical homotopy equivalences

Y(Ω^∞−nEα)'Ω^∞−nY(Eα) forn≥0.

Example 1.2.8. Let X = SpecR be an affine algebraic variety over the field Cof complex numbers, and suppose we are given a pointxofX (corresponding to an augmentation:R→Cof theC-algebraR). Then X determines a formal moduli problemX^∧: CAlg^sm_C →S, given by the formulaX^∧(A) = Map_CAlg^aug

C (R, A) (here we work in the deformation context (CAlg^aug_C ,{E}) of Example 1.1.4). Unwinding the definitions, we see that the tangent complex X^∧(E) can be identified with the spectrum MorModR(LR/C,C) classifying maps from the cotangent complex L_R/C into C (regarded as an R-module via the augmentation ). In particular, the homotopy groups ofX^∧(E) are given by

π_iX^∧(E)'(π_−i(k⊗RL_R/C))^∨.

It follows that π_iX^∧(E) vanishes fori > 0, and that π₀X^∧(E) is isomorphic to the Zariski tangent space (m/m²)^∨ofX at the pointx. IfX is smooth at the pointx, then the negative homotopy groups ofπ_iX^∧(E) vanish. In general, the homotopy groups π_iX^∧(E) encode information about the nature of the singularity of X at the pointx. One of our goals in this paper is to articulate a sense in which the tangent complex X^∧(E) encodescompleteinformation about the local structure ofX near the pointx.

Warning 1.2.9. The terminology of Definition 1.2.5 is potentially misleading. For a general deformation context (Υ,{E_α}_α∈T) and formal moduli problem Y : Υ^sm →S, the tangent complexesY(E_α) are merely spectra. Ifkis a field and (Υ,{E_α}_α∈T) = (CAlg^aug_k ,{E}) is the deformation context of Example 1.1.4, one can show that the tangent complexY(E) admits the structure of ak-module spectrum, and can therefore be identified with a chain complex of vector spaces over k. This observation motivates our use of the term

“tangent complex.” In the general case, it might be more appropriate to refer to Y(Eα) as a “tangent spectrum” to the formal moduli problemY.

The tangent complex of a formal moduli problem Y is a powerful invariant of Y. We close this section with a simple illustration:

Proposition 1.2.10. Let (Υ,{Eα}_α∈T) be a deformation context and let u:X →Y be a map of formal moduli problems. Suppose that u induces an equivalence of tangent complexes X(Eα) → Y(Eα) for each α∈T. Thenuis an equivalence.

Proof. Consider an arbitrary objectA∈Υ^sm, so that there exists a sequence of elementary morphisms A=A0→A1→ · · · →An' ∗

in Υ. We prove that the mapu(Ai) :X(Ai)→Y(Ai) is a homotopy equivalence using descending induction oni, the casei=nbeing trivial. Assume therefore thati < nand thatu(Ai+1) is a homotopy equivalence.

SinceAi→Ai+1 is elementary, we have a fiber sequence of maps u(A_i)→u(A_i+1)→u(Ω^∞−nE_α)

for some n > 0 and α ∈ T. To prove that u(Ai) is a homotopy equivalence, it suffices to show that u(Ω^∞−nEα) is a homotopy equivalence, which follows immediately from our assumption thatuinduces an equivalenceX(Eα)→Y(Eα).