Selected introductory topics in derived algebraicgeometry

Gabriele VezzosiUniversita di Firenze, Italy

Etats de la RechercheIMT Toulouse - June 2017

Plan of the talk

1 Affine derived geometryHomotopical algebra of dg-modules and cdga’s (char 0)Cotangent complexDerived commutative algebra: finiteness, flat, smooth, etale,....

2 Derived stacks

3 Derived algebraic stacks

4 Lurie’s Representability Theorem

5 DAG explains classical deformation theory

dg-modules and dg-algebras I

k : commutative Q-algebra

dgmod≤0(k): symmetric ⊗ ∞-category obtained by inverting (in the∞-categorical sense) qisos in dgmod≤0(k).Note that dgmod≤0(k) is a model category with W = qisos andFib = surjections in deg < 0, endowed with a symmetric ⊗ structurecompatible with the model structure (“a symm ⊗ model category”).Same for dgmod(k) (fibrations := surjective in any degree).

dAlg(k) := CAlg(dgmod≤0(k)): ∞-category of non-positively gradeddifferential graded k-algebras =: derived k-algebras.dAlg(k) := CAlg(dgmod≤0(k)) is a model category with W =qisos andFib= surjections in deg < 0: this uses that k is a Q-algebra]

∞-adjunction:

dgmod≤0(k)Symk //

dAlg≤0(k)Uoo

(induced by a Quillen adjunction on models).

dg-modules and dg-algebras II

Explicit mapping spaces in dAlg(k):

Ω•n:= algebraic de Rham complex of k[t0, t1, · · · , tn]/(∑

i ti − 1) over k .

[n] 7→ Ω•n is a simplicial object in dgmod(k).

Let B ∈ dAlg(k):[n] 7→ τ≤0(Ω•n ⊗k B) is a simplicial object in dAlg(k).

For any pair (A,B) in dAlg(k), there is an equivalence of of spaces(simplicial sets)

MapdAlg(k)(A,B) ' ([n] 7→ Homcdga≤0

k(QkA, τ≤0(Ω•n ⊗k B)))

dg-modules and dg-algebras III

Warning exercises.

Exercise. Let char(k) = p. Show that CAlg(dgmod(k)) cannot have amodel structure with W=qisos and Fib= degreewise surjections.

[Hint: 1) Exhibit a map f : A→ B in CAlg(dgmod(k)) with the followingproperty: ∃i , α ∈ H i (B) such that αp is not in the image of H ip(f ).2) Prove that no such f can be factored as Fib (W ∩ Cof ).]

Exercise. Let char(k) = 0. Show that the obvious ∞-functorCAlg(dgmod(k))→ Alg(dgmod(k)) is not fully faithful.

dg-modules and dg-algebras IV

A ∈ dgmod(A): symmetric ⊗ ∞-category of (unbounded) dg-modulesover A

CAlg(dgmod(A)) ' A/dAlg(k)

f : A→ B ∈ dAlg(k) =⇒

dgmod(B)f ∗=(−)⊗AB//

dgmod(A)f∗oo

(∞-adjunction). This is an equivalence of ∞-categories if f is anequivalence in dAlg(k).

Derivations and cotangent complex

Let f : A→ B in dAlg(k). For any M ∈ dgmod≤0(B), let B ⊕M indAlg(k)trivial square zero extension of B by M (i.e. B acts on itself and on M as given,and M ·M = 0). B ⊕M is naturally an A-algebra and it has a (projection) mapprB : B ⊕M → B of A-algebras.

Derivations

The space DerA(B,M) of derivations from B to M over A is the mapping spaceMapA/dAlg(k)/B(B,B ⊕M). Or, equivalently,DerA(B,M) = fib(prB,∗ : MapA/dAlg(k)(B,B ⊕M)→ MapA/dAlg(k)(B,B) ; idB).

We have the following derived analog of the existence of Kahler differentials.

Relative cotangent complex

The ∞-functor M 7→ DerA(B,M) is corepresentable, i.e. ∃LB/A ∈ dgmod≤0(B)and a canonical equivalence DerA(B,−) ' Mapdgmod(B)(LB/A,−).

Pf. Let QAB ' B a cofibrant replacement of B in A/dAlg(k). ThenLB/A := Ω1

QAB/A⊗QAB B works. 2

Properties I

(1) Given a commutative square

Au′ //

A′

B

u// B ′

in dAlg(k) =⇒ induced map

LB/A → u∗LB′/A′ ⇔ LB/A ⊗B B ′ → LB′/A′ .

In particular ∃ canonical map LB/A → LH0(B)/H0(A).If B = S ,A = R are discrete, the canonical map H0(LS/R)→ Ω1

S/R is anisomorphism.(1’) If the square in (1) is a pushout, then the map LB/A ⊗B B ′ → LB′/A′ is anequivalence.(2) If C → A→ B in dAlg(k) =⇒ fiber sequence

LA/C ⊗A B → LB/C → LB/A

in dgmod(B) (first map as in (1) via u′ = idC , u : A→ B, second map as in (1)with u = idB and u′ = (C → A)).

Properties II

(3) If

Au′ //

A′

B u

// B ′

is a pushout in dAlg(k), then we have a fiber sequence

LA/k ⊗A B ′ → LB/k ⊗B B ′ ⊕ LA′/k ⊗A B ′ → LB′/k

in dgmod(B ′).

Exercise. Compute LA/k for A := k[x1, · · · , xn]/(f ). [Hint: Koszul]

Postnikov tower

Definition. Let A ∈ dAlg(k). A Postnikov tower for A is a sequence Pn(A)n≥0in A/dAlg(k), and of maps

· · · // Pn(A)πn // Pn−1(A)

πn−1 // · · · // P1(A)π1 // P0(A)

in A/dAlg(k) with the following properties:(1) Pn(A) is n-truncated (i.e. H−i (Pn(A)) = 0 for i > n)(2) H j(A)→ H j(Pn(A)) is an iso, for all 0 ≤ j ≤ n.(3) For any n-truncated B ∈ dAlg(k), the mapMapdAlg(k)(Pn(A),B)→ MapdAlg(k)(A,B) induced by A→ Pn(A), is anequivalence of spaces.

Proposition. A Postnikov tower is unique in the ∞-categoryFun(Nop,A/dAlg(k)). Moreover, the canonical map A→ limn≥0 Pn(A) is anequivalence in dAlg(k).

Exercise. Construct a Postnikov tower such that P0(A) = A0/d(A−1),P1(A) = (A−1/d(A−2)→ A0), etc.

The cotangent complex controls Postnikov towers

Proposition

Let

· · · // Pn(A)πn // Pn−1(A)

πn−1 // · · · // P1(A)π1 // P0(A)

be a Postnikov tower for A. For any n, there exists a mapφn : LPn(A)/A → H−n−1(A)[n + 1] such that the following square is cartesian inA/dAlg(k)

Pn+1(A) //

πn

Pn(A)

d0

Pn(A)

dφn

// Pn(A)⊕ H−n−1(A)[n + 1]

,

where dφn is the derivation corresponding to φn via

π0(MapA/dAlg(k)/Pn(A)(Pn(A),Pn(A)⊕ H−n−1(A)[n + 1])) '

' HomHo(dgmod(Pn(A))(LPn(A)/A,H−n−1(A)[n + 1]),

The cotangent complex controls Postnikov towers

Corollary

A map A→ B in dAlg(k) is an equivalence iff(i) H0(A)→ H0(B) is an isomorphism (of discrete k-algebras), and(ii) LB/A ' 0.

This is the first of several examples of the following general principle:

“derived algebraic geometry = algebraic geometry (of the truncation) +deformation theory”.

Finiteness

Finitely presented map

(classical) Define R → S of discrete comm. k-algebras finitely presented ifHomR/Alg(k)(S ,−) commutes with filtered colimits.

(derived) Define A→ B in dAlg(k) is finitely presented ifMapA/dAlg(k)(B,−) commutes with filtered colimits.

Proposition: A→ B in dAlg(k) is fp iff (i) H0(A)→ H0(B) is classically fp,and (ii) LB/A is a perfect B-dg module (i.e. is dualizable in(dgmod(B),⊗B)).

So for discrete rings derived fp does not imply classical fp. But classical fp+ lci =⇒ derived fp.

Flatness

Flatness

(classical) Define R → S of discrete comm. k-algebras flat if(−)⊗R S : mod(R)→ mod(S) preserves pullbacks (⇔ preserves kernels).

(derived) Define A→ B in dAlg(k) flat if(−)⊗A B : dgmod≤0(A)→ dgmod≤0(B) preserves pullbacks.

So, for discrete rings, classical flat ⇔ derived flat.

Formally etale & formally smooth

Formally etale & formally smooth

(classical) Define R → S of discrete comm. k-algebras formally etale ifτ≥−1LS/R ' 0.

(derived) Define A→ B in dAlg(k) formally etale if LB/A ' 0.

(classical) Define R → S of discrete comm. k-algebras formally smooth ifHomD(S)(τ≥−1LS/R ,M) = 0 for any M ∈ dgmod≤0(S) s.t. H0(M) = 0.

(derived) Define A→ B in dAlg(k) formally smooth ifHomD≤0(B)(LB/A,M) = 0 for any S ∈ dgmod≤0(B) s.t. H0(M) = 0.

Exercise. Show that the above definition of classical formally smooth (etale)coincides with the usual (EGA one: by lifting property). [Hint formally smoothcase: show that f : R → S is EGA-formally smooth iff τ≥−1Lf ' P[0] with Pprojective over S .]

Rmk. τ≥−1 necessary here ! : ∃f : k → S , k field s.t. f is formally etale (henceformally smooth) but H−2(Lf ) 6= 0.

Etale, formally smooth, Zariski

Etale, smooth, Zariski

(classical) Define R → S of discrete comm. k-algebras etale (resp. smooth)if it is finitely presented and formally etale (resp. formally smooth).

(derived) Define A→ B in dAlg(k) etale (resp. smooth) if if it is finitelypresented and formally etale (resp. formally smooth).

(classical) Define R → S of discrete comm. k-algebras a Zariski openimmersion if it is flat, finitely presented and the product map S ⊗R S → S isan iso.

(classical) Define A→ B in dAlg(k) a Zariski open immersion if it is flat,finitely presented and the product map B ⊗A B → B is an equivalence.

Alternative characterization

Strongness

Define A→ B in dAlg(k) strong if the canonical mapH0(B)⊗H0(A) H i (A)→ H i (B) is an iso for all i ’s.

Theorem: A→ B in dAlg(k) is flat (resp. smooth, etale, a Zariski openimmersion) iff it is strong and H0(A)→ H0(B) is classically flat (resp.smooth, etale, a Zariski open immersion).

Corollary: H0 induces an equivalence of etale or Zariski sites of A and ofH0(A), for any A ∈ dAlg(k) [This Cor. is slightly misplaced here, sorry.]

Exercise. Assume the Theorem above. Show that f : A→ B in dAlg(k) issmooth iff the following conditions hold:(i) f is finitely presented; (ii) f is flat; (iii) B is a perfect B ⊗A B-dg module.

Etale topology and derived stacks

Etale topology

A family A→ Aii of maps in dAlg(k) is an etale covering family if(i) each A→ Ai is etale;(ii) the family H0(A)→ H0(Ai )i is a classical etale covering family (ofdiscrete commutative k-algebras).This defines the etale topology on the ∞-category dAff(k) := dAlg(k)op

(i.e., by definition, a Grothendieck topology on Ho(dAff(k))).

The ∞-category Sh(dAff(k), etale) of sheaves of spaces on this ∞-site, isdenoted by dSt(k) and called the ∞-category of derived stacks over k.

A functor F : dAff(k)op → Spaces is a derived stack iff for any A ∈ dAlg(k)and any etale hypercover A→ B•, the induced map F (A)→ lim F (B•) isan equivalence of spaces (etale hyperdescent).

Rmk. Cech nerves of etale coverings are special etale hypercovers. One may alsorequire to have descent just for these Cech etale hypercovers: we obtain, ingeneral, different categories of derived stacks. However, the full subcategories oftruncated stacks are equivalent.

Derived stacks: basics

adjunction

St(k)i //

dSt(k)t0oo

between derived and underived stacks (for the etale topology). t0 is thetruncation functor, and is determined by the propertyt0(Spec A) = Spec H0(A), and the fact that it preserves limits (being rightadjoint). i is fully faithful.

dSt(k) is cartesian closed, i.e. there are internal Hom’s, denoted asMAPdSt(k)(F ,G ) ∈ dSt(k), and equivalences of spaces

MapdSt(k)(F × G ,H) ' MapdSt(k)(F ,MAP(G ,H)).

i does not preserve pullbacks nor internal Hom’s (a nice feature)

If F ∈ dSt(k), we define QCoh(F ) := limSpec A→F dgmod(A) (limit insidek-linear stable, symmetric ⊗ ∞-categories). This is called the ∞-category ofquasi-coherent complexes on F .

Derived stacks: cotangent complex

F ∈ dSt(k), x : S = Spec A→ F , M ∈ dgmod≤0(A).Projection map pr : A⊕M → A.

Cotangent complex

We say that F has a cotangent complex at x if the ∞-functor

dgmod≤0(A)→ Spaces

M 7→ fib(F (pr) : F (A⊕M)→ F (A); x)

is corepresentable. A corepresentative is denoted by LF ,x .

We say that F has a (global) cotangent complex if ∃ LF ∈ QCoh(F ), suchthat for any x : S = Spec A→ F , x∗LF is a cotangent complex for F at x

Derived schemes I

A map f : F → G in dSt(k) is defined to be an epimorphism (resp. amonomorphism) if the induced map π0(F )→ π0(G ) of sheaves of sets on(Ho(dAff (k)), et) is an epimorphism (resp. if the diagonal ∆f : F → F ×G F isan equivalence).

Derived schemes

A map u : F → S = Spec A of derived stacks is a Zariski open immersion ifit is a monomorphism and there is a family pi : Spec Ai → Spec A ofZariski open immersions such that each pi factors as u qi , whereqi : Spec Ai → F , and

∐i qi :

∐i Spec Ai → Spec F is an epimorphism.

A morphism F → G of derived stack is a Zariski open immersion if for anyS → G with S affine, the induced map F ×G S → S is a Zariski openimmersion (as defined in the previous point).

A derived stack F is a derived scheme if there exists a familySpec Ai → Fi of Zariski open immersions, such that the induced map∐

i Spec Ai → F is an epimorphism. Such a family is called a Zariski atlasfor F .

Derived schemes II

A morphism of derived schemes f : X → Y is smooth (flat, etale) if thereare Zariski atlases Ui → Xi∈I , Vj → Y and, for any i ∈ I there isj(i) ∈ J and a commutative diagram

Ui

fi,j(i) //

Vj(i)

X

f// Y

such that fi ,j(i) is smooth (flat, etale) between derived rings.

Derived algebraic stacks I

0-algebraic

A derived stack F is 0-algebraic if it is (equivalent to) a derivedscheme.

A morphism of derived stacks F → G is 0-representable if for anyS → G where S is 0-algebraic, the pullback S ×G F is 0-algebraic.

a 0-representable morphism of derived stacks F → G is smooth if forany S → G where S is 0-algebraic, the morphism of derived schemesS ×G F → S is smooth.

Derived algebraic stacks II

n-algebraic

Let n > 0 and suppose we have defined the notions of derived (n − 1)-algebraicstack, of (n − 1)-representable morphism (between arbitrary derived stacks), andof smooth (n − 1)-representable morphism (between arbitrary derived stacks).Then

A derived stack F is n-algebraic if there exists a smooth (n − 1)-algebraicmorphism p : U → F of derived stacks, where U is 0-algebraic and p is anepimorphism. Such a p is called a smooth n-atlas for F .

A morphism of derived stacks F → G is n-representable if for any S → Gwhere S is 0-algebraic, the pullback S ×G F is n-algebraic.

a n-representable morphism of derived stacks F → G is smooth (flat, etale)if for any S → G where S is 0-algebraic, there exists a smooth n-atlasU → S ×G F , such that the composite U → S , between 0-algebraic stacks,is smooth (flat, etale).

A derived stack is algebraic if it is m-algebraic for some m ≥ 0.

Derived algebraic stacks III

Exercise. If F is n-algebraic then the diagonal map F → F × F is(n − 1)-representable (not easy).

Some properties.

• The full sub-∞-category dStalg (k) ⊂ dSt(k) of algebraic stacks is closed underpullbacks and disjoint unions.• Representable morphisms are stable under composition and arbitrary pullbacks.• If F → G is a smooth epimorphism of derived stacks, then F is algebraic iff Gis algebraic.• A non-derived stack X is algebraic iff the derived stack i(X ) is algebraic.• If F is a derived algebraic stack, then its truncation t0(F ) is an algebraic(underived) stack.• If F is a derived algebraic stack and t0(F ) is m-truncated, then for anyn-truncated A ∈ dAlg(k), the space F (A) is (n + m)-truncated.• If F → G is a flat morphism of derived algebraic stacks, then: G underived (i.e.' i(X )) =⇒ F underived.

Derived algebraic stacks IV

Some properties (cont’d.)

• A (n − 1)-representable morphism is n-representable.• A (n− 1)- representable smooth (resp. etale, flat, Zariski open immersion) is n-representable smooth (resp. etale, flat, Zariski open immersion).• If F → G is a map of derived stack, G is n-algebraic and ∃ a a smooth atlasS → G such that S ×G F is n-algebraic, then F is n-algebraic (i.e. beingn-algebraic is smooth-local on the target).• A map A→ B in dAlg(k) is almost finitely presented (afp) if H0(A)→ H0(B)is (classically) finitely presented. Flat afp covers define a topology (afpf ) ondAff(k). We can replace (et) by (afpf), and smooth (atlases) by flat atlases. Weget another subcategory dStafpf ,alg (k) of derived stacks which are algebraic forthis new pair (afpf, flat). A deep result of B. Toen says that dStafpf ,alg (k) =dStalg (k).• Derived algebraic stacks admitting etale atlases are called derivedDeligne-Mumford stacks.

Representability theorem I

Representability Theorem (Lurie)

Let k be an excellent noetherian commutative ring (noetherian G-ringshould suffice). A derived stack F over k is algebraic and locally of finitepresentation over k iff the following conditions hold:

∃n ≥ 0 such that F (R) is n-truncated for any classical commutativering R.

For any derived k-algebra A, the natural Postnikov tower mapF (A)→ limn F (A≤n) is an equivalence.

F commutes with filtered colimits of derived k-algebras

F has a cotangent complex

F is complete i.e. for any classical complete local noetherian k-algebra(R,m), the canonical map F (R)→ limn F (R/mn) is an equivalence.

Rmk. There is also a version where the base k is a derived base ring(Lurie, SAG).

Representability theorem II

Definition. A derived stack F is infinitesimally cartesian if for anyA ∈ dAlg(k), any M ∈ dgmod≤0(A), s.t. H0(M) = 0, and any derivationd from A to M, the image by F of the pullback

A⊕d ΩM //

A

d

Atriv

// A⊕M

is a pullback.

Representability theorem III

“Easy” Representability Theorem

Let k be a noetherian commutative ring. A derived stack F over k isalgebraic and locally of finite presentation over k iff the followingconditions hold:

The truncation t0(F ) is an (underived) algebraic n-stack.

For any derived k-algebra A, the natural Postnikov tower mapF (A)→ limn F (A≤n) is an equivalence.

F has an obstruction theory i.e. it is infinitesimally cartesian and hasa cotangent complex.

Rmk. The excellence or G -ring condition appears only if we want tofurther unzip the first condition in this Theorem.

Exercise

Exercise. Let X/k be a flat and proper underived scheme and Y /k anunderived smooth scheme. LetMAPdSt(k)(X ,Y ) : A 7−→ MapdSt(k)(X × Spec A,Y ) the internal mappingderived stack.

Show that t0(MAPdSt(k)(X ,Y )) is the classical scheme of morphismsfrom X to Y

Let MAPdSt(k)(X ,Y ) X ×MAPdSt(k)(X ,Y )poo ev // Y .

Show that T := p∗ev∗(TY /k [0]) is a tangent complex forMAPdSt(k)(X ,Y ).

Apply Lurie’s representability theorem (maybe in the easy case), todeduce that MAPdSt(k)(X ,Y ) is a derived geometric stack (actually aderived scheme).

Use Grothendieck-Serre duality to compute the cotangent complex ofMAPdSt(k)(X ,Y ).

DAG and classical deformation theory

Derived deformation theory (:= deformation theory in DAG) fills the ’gaps’in classical deformation theory (k = C here).Given aclassical moduli problem:F : commalgC −→ Grpds : R 7→ Y → Spec R , proper & smooth Fixing a C-point ξ = (f : X → SpecC) ∈ F (C), we get aFormal moduli problem :Fξ(A) := hofiber(F (augment) : F (A)→ F (C); ξ), i.e.

Fξ : ArtinC −→ GrpdsA 7→ Y → Spec A , proper & smooth + iso X ' Y ×A CClassical deformation theory:

1 Fξ(C[t]/tn+1) groupoid of infinitesimal n-th order deformations of ξ

2 if ξ1 ∈ Fξ(C[ε] = C[t]/t2), then AutFξ(C[ε])

(ξ1) ' H0(X ,TX )

3 π0(Fξ(C[ε])) ' H1(X ,TX )

4 If ξ1 ∈ Fξ(C[ε]), ∃ obs(ξ1) ∈ H2(X ,TX ) which vanishes iff ξ1 extends

to a 2nd order deformation ξ2 ∈ Fξ(C[t]/t3).

DAG and classical deformation theory

Items 1-3 above are satisfying. Much less about item 4.

Critique of obstructions:

1 what is the deformation theoretic interpretation of the wholeH2(X ,TX ) ?

2 how to determine the subspace of obstructions inside H2(X ,TX ) ?

These questions

are important classically: often H2(X ,TX ) 6= 0 butobstructions = 0 (e.g. X smooth hypersurface in P3

C of degree≥ 6);

have no general answers inside classical deformation theory

Let us see how how derived algebraic geometry answers to both.

DAG and classical deformation theory

Extend the functor F : commalgC −→ Grpds as follows:

Derived Moduli problem (derived stack): RF : cdgaC −→ SSetsA• 7→ Nerve( (dSch/RSpec A•)sm, proper, equivalences )

then RF (R) ' F (R) for R ∈ commalgC → cdgaC (since a derived schemeY mapping smoothly to Spec R is underived : Exercise)

Derived formal moduli problem (formal derived stack):

RF ξ := RF ×SpecC ξ : dgArtinC −→ sSets

RF ξ(A•) := hofiber(RF (A•)→ RF (C); ξ)

where:dgArtinC := A• ∈ cdgaC |H0(A•) ∈ ArtinC,

H i (A•) of finite type over H0(A•), H i (A•) = 0, i >> 0

DAG and classical deformation theory

Anwer to Question 1: what is the deformation theoretic interpretation ofthe whole H2(X ,TX ) ?

Proposition

There is a canonical isomorphism π0(RF ξ(C⊕ C[1])) ' H2(X ,TX )

I.e. H2(X ,TX ) classifies derived deformations over RSpec(C⊕ C[1]) !(derived deformations := deformations over a derived base).

This also explains why classical deformation could not answer thisquestion: it only deals with deformations over underived bases.

Rmk. This same answer to Question 1 was obtained by M. Manetti, whoonly worked formally at ξ (i.e. without having the global derived stackRF ). Thanks to B. Fantechi for pointing this out.

Derived def-theory explains classical def-theory

Anwer to Question 2: how to determine the subspace of obstructionsinside H2(X ,TX ) ?

Lemma

The following (obvious) diagram is h-cartesian

C[t]/t3 //

C[ε] = C[t]/t2

C // C⊕ C[1]

Proof: Exercise (or see Porta- V., arXiv:1310.3573).

Observe: both C→ C⊕ C[1] and C[t]/t2 → C⊕ C[1] are surjective onH0, with a nilpotent kernel.

DAG and classical deformation theory

Since RF is a derived Artin stack, it is infinitesimally cohesive (homotopyversion of Schlessinger condition), so

RF (C[t]/t3) = F (C[t]/t3) //

RF (C[ε]) = F (C[ε])

RF (C) = F (C) // RF (C⊕ C[1])

is h-cartesian; this whole diagram maps to F (C), and the h-fibers at ξ yield

Fξ(C[t]/t3) //

Fξ(C[ε])

pt // RF ξ(C⊕ C[1])

h-cartesian (formal consequence) of pointed simplicial sets.

DAG and classical deformation theory

Hence, get an exact sequence of vector spaces

π0(Fξ(C[t]/t3)) // π0(Fξ(C[ε]))obs // π0(RF ξ(C⊕ C[1])) ' H2(X ,TX )

Therefore : a 1st order deformation ξ1 ∈ π0(Fξ(C[ε])) of ξ, extends to a

2nd order deformation ξ2 ∈ π0(Fξ(C[t]/t3)) iff the image of ξ1 vanishes inH2(X ,TX ).So, Question 2 : how to determine the subspace of obstructions insideH2(X ,TX ) ?Answer: The subspace of obstructions is the image of the map obs above.

So, in particular, classical obstructions are derived deformations (:=deformations over a derived base).

Exercise: extend this argument to all higher orders infinitesimaldeformations.