Deformations of Objects

X(A)

//Y(A)

X(B⁰) //X(B) //Y(B)

is (n−2)-Cartesian. Our assumption thatf is (n−2)-truncated guarantees that the right square is (n− 2)-Cartesian; it will therefore suffice to show that the outer square is (n−2)-Cartesian (Lemma 5.1.3). Using Lemma 5.1.3 again, we are reduced to showing that both the left and right squares in the diagram

X(A⁰) //

Y(A⁰)

//Y(A)

X(B⁰) //Y(B⁰) //Y(B)

are n-Cartesian. For the left square, this follows from our assumption thatf is (n−2)-truncated; for the right square, it follows from our assumption thatY is ann-proximate formal moduli problem.

We now complete the proof by showing that (1)⇒(3). Assume thatX is ann-proximate formal moduli problem; we wish to show that the mapX →LX is (n−2)-truncated. According to Lemma 5.1.14, it will suffice to show that the mapφ:X(Ω^∞−mEα)→LX(Ω^∞−mEα) is (n−2)-truncated for eachα∈Tand each m≥0. Using Lemma 5.1.12, we can identifyφwith the canonical mapX(Ω^∞−mEα)→Ω^∞−mL0(X◦Eα)), which is (n−2)-truncated by Lemma 5.1.13.

5.2 Deformations of Objects

Let X be an algebraic variety defined over a field k, and let E be an algebraic vector bundle on X. A first order deformation ofE is an algebraic vector bundle Eover the scheme X =X ×SpeckSpeck[]/(²), together with an isomorphismi^∗E→E(whereidenotes the closed immersionX ,→X). Standard arguments in deformation theory show that the collection of isomorphism classes of first-order deformations can be identified with the cohomology group H¹(X; End(E)), while the automorphism of each first order deformation ofEis given by H⁰(X; End(E)).

Our goal in this section is to place the above observations into a more general context.

(a) In the definition above, we can replace the ring of dual numbersk[]/(²) by an arbitraryR∈CAlg^sm_k to obtain a notion of a deformation of E over R. Let ObjDef_E(R) denote a classifying space for deformations ofEoverR. Then ObjDef_Ecan be regarded as a functor CAlg^sm_k →S. We will see below that this functor is a formal moduli problem.

(b) The isomorphisms

H⁰(X; End(E))'π₁ObjDef_E(k[]/(²)) H¹(X; End(E))'π₀ObjDef_E(k[]/(²))

follow from an identification of the cochain complexC^∗(X; End(E)) with the (shifted) tangent complex TObjDef_E[−1].

(c) The definition of ObjDef_E(R) does not require the commutativity ofR. Consequently, we can extend the domain of definition of ObjDef_Eto Alg^sm_k , and thereby regard ObjDef_Eas a formalE1-moduli prob-lem (see Definition 3.0.3). We will see that the identificationC^∗(X; End(E))'TObjDef_E[−1] is multi-plicative: that is, it can be regarded as an equivalence of nonunital E1-algebras (where TObjDef_E[−1]

is equipped with the nonunitalE1-algebra structure given by Remark 3.2.6).

(d) The definition of the formal moduli problem ObjDef_E depends only on the algebraic vector bundleE as an object of the stable ∞-category QCoh(X) of quasi-coherent sheaves on X. We will therefore consider the more general problem of deforming an object M of an arbitraryk-linear∞-categoryC. We begin by introducing some notation.

Construction 5.2.1. Let k be a field, let C be ak-linear∞-category, and let M ∈C be an object. We let RMod(C) denote the∞-category of pairs (A, MA) where A∈Alg_k and MA is a rightA-module object of C. The forgetful functor q: RMod(C)→Alg_k is a coCartesian fibration. We let RMod^coCart(C) denote the subcategory of RMod(C) spanned by theq-coCartesian morphisms, so thatqrestricts to a left fibration RMod^coCart(C) → Alg_k. We will abuse notation by identifying M with an object of RMod^coCart(C) (via the equivalence RModk(C)→C). We let Defor[M] = RMod^coCart(C)_/M. We will refer to Defor[M] as the

∞-category of deformations of M.

There is an evident forgetful functorθ: Defor[M]→Alg^aug_k . The fiber ofθover an augmentedk-algebra Acan be identified with the∞-category of pairs (MA, µ), whereMA∈RModA(C) and µ:MA⊗Ak→M is an equivalence inC. The mapθ is a left fibration, classified by a functor χ: Alg^aug_k →bS; herebSdenotes the∞-category of spaces which are not necessarily small.

LetCbe as in Construction 5.2.1. Proposition IX.7.4 implies that for every pullback diagram A⁰ //

A

B⁰ //B

in Alg_k, the induced functor RMod_A⁰(C)→RMod_A(C)×RMod_B(C)RMod_B⁰(C) is fully faithful. This imme-diately implies the following:

Proposition 5.2.2. Let k be a field, Cak-linear∞-category, M ∈C an object, and letχ: Alg^aug_k →bS be as in Construction 5.2.1. Then, for every pullback diagram

A⁰ //

A

B⁰ //B

in Alg^aug_k , the induced map χ(A⁰) → χ(A)×χ(B)χ(B⁰) induces a homotopy equivalent from χ(A⁰) to a summand ofχ(A)×χ(B)χ(B⁰).

Corollary 5.2.3. Letk be a field, Cak-linear ∞-category,M ∈C an object, and let χ: Alg^aug_k →bS be as in Construction 5.2.1. Then:

(1) The space χ(k)is contractible.

(2) Let V ∈Modk. Then the spaceχ(k⊕V)is essentially small.

(3) Let A∈Alg^aug_k be small. Then the spaceχ(A)is essentially small.

Proof. Assertion (1) is immediate from the definitions. To prove (2), we note that for eachA∈Alg^aug_k , the spaceχ(A) is locally small (when regarded as an∞-category). We have a pullback diagram

k⊕V //

k

k //k⊕V[1]

so that Proposition 5.2.2 guarantees thatχ(k⊕V) is a summand of Ωχ(k⊕V[1]), and therefore essentially small. We now prove (3). Assume thatAis small, so that there exists a finite sequence of maps

A'A0→A1→ · · · →An'k and pullback diagrams

Ai //

k

Ai+1 //k⊕k[mi].

Using (1), (2), and Proposition 5.2.2, we deduce that eachχ(Ai) is essentially small by descending induction oni.

Notation 5.2.4. Letk be a field, C a k-linear∞-category, and M ∈Can object. Let χ: Alg^aug_k →bS be the functor classifying the left fibration Defor[M]→Alg^aug_k of Construction 5.2.1. We let ObjDef_M denote the restriction ofχto the full subcategory Alg^sm_k ⊆Alg^aug_k spanned by the smallE1-algebras over k. Using Corollary 5.2.3, we can identify ObjDef_M with a functor from Alg^sm_k toS.

More informally: the functor ObjDef_M assigns to eachR∈Alg^sm_k a classifying space for pairs (M_R, µ), whereM_R∈RMod_R(C) andµ:M_R⊗Rk→M is an equivalence inC.

Combining Corollary 5.2.3 and Proposition 5.2.2, we obtain the following:

Corollary 5.2.5. Let k be a field, C a k-linear ∞-category, and M ∈ C an object. Then the functor ObjDef_M : Alg^sm_k →S is a1-proximate formal moduli problem (see Definition 5.1.5).

Notation 5.2.6. Letkbe a field, and letL: Fun(Alg^sm_k ,S)→Moduli⁽¹⁾_k denote a left adjoint to the inclusion.

IfCis ak-linear∞-category andM ∈Cis an object, we let ObjDef^∧_M denote the formalE1-moduli problem L(ObjDef_M). By construction, we have a natural transformation ObjDef_M → ObjDef^∧_M. It follows from Theorem 5.1.9 that this natural transformation exhibits ObjDef_M(R) as a summand of ObjDef^∧_M(R), for eachR∈Alg^sm_k . Moreover, ObjDef^∧_M is characterized up to equivalence by this property: see Remark 5.1.11.

Notation 5.2.7. Letkbe a field, letCbe ak-linear∞-category, and letM ∈Cbe an object. We let End(M) denote a classifying object for endomorphisms of M. That is, End(M) is an object of Modk equipped with a map a: M ⊗End(M)→ M in Chaving the following universal property: for every object V ∈ Modk, composition withainduces a homotopy equivalence

Map_Modk(V,End(M))'Map_C(M⊗V, M).

The existence of the object End(M) follows from Proposition A.4.2.1.33. Moreover, it follows from the results of§A.6.1.2 show that we can regard End(M) as anE1-algebra overk, andM as a right module over End(M). In what follows, it will be more convenient to view End(M) as a nonunital E1-algebra over k, which can be identified with the augmentation ideal in the augmentedE1-algebra k⊕End(M).

We can now formulate the main result of this section.

Theorem 5.2.8. Letkbe a field, letCbe ak-linear∞-category, letM ∈Cbe an object, and letΨ : Alg^aug_k → Moduli⁽¹⁾_k be the equivalence of∞-categories of Theorem 3.0.4. Then there is a canonical equivalenceΨ(k⊕ End(M))'ObjDef^∧_M.

Example 5.2.9. Letkbe a field and regard Mod_k as ak-linear∞-category. LetV be a finite-dimensional vector space over k, and define ObjDef_V as above. We will see below that ObjDef_V is a formalE1 moduli problem (Proposition 5.2.14), so that ObjDef_V |CAlg^sm_k is a formal moduli problem over k. Assume now thatk has characteristic zero, and let Φ : Liek →Modulik be the equivalence of Theorem 2.0.2. Combining Theorems 5.2.8 and 3.3.1, we deduce that ObjDef_V|CAlg^sm_k corresponds, under the equivalence Φ, to the matrix algebra End(V) (equipped with its usual Lie bracket).

Remark 5.2.10. Letk be a field, let C be ak-linear ∞-category, and let M ∈C be an object. For each R∈Alg^sm_k , Theorem A.6.3.4.1 yields a canonical homotopy equivalence

ObjDef_M(R) = RModR(C)×_C{M} 'Map_LinCatk(LModR,C)×_C^'{M}.

It follows that ObjDef_M(R) depends only on thek-linear∞-category LMod_R, together with the distinguished objectk∈LModR (given by the augmentationR→k), so that the constructionR7→ObjDef_M(R) enjoys some extra functoriality. This special feature of ObjDef_M(R) is reflected in the structure of the associated formal E1 moduli problem ObjDef^∧_M(R): according to Theorem 5.2.8, ObjDef^∧_M(R) has the form Ψ(A), where A ∈ Alg^aug_k is an augmented E1-algebra over k whose augmentation ideal mA ' End(M) is itself unital.

Theorem 5.2.8 is a consequence of a more precise assertion (Proposition 5.3.16) which describes the equivalence Ψ(k⊕End(M))'ObjDef^∧_M explicitly. Before we can formulate it, we need to introduce a bit more notation.

Construction 5.2.11. Letkbe a field, letCbe a k-linear∞-category, and letM ∈Cbe an object. We let λ:M⁽¹⁾→Alg^aug_k ×Alg^aug_k be the pairing of Construction 4.4.6, so that we can identify objects ofM⁽¹⁾with triples (A, B, ) whereA, B∈Alg_k and:A⊗kB→kis an augmentation. Given an object (A, B, )∈M⁽¹⁾ and an object (A, M_A, µ)∈Defor[M], we regardM_A⊗kB as an object of_BBMod_A⊗kB(C), so that

(M ⊗kB)⊗A⊗_kBk

can be identified with an object of LModB(C) whose image inCis given byMA⊗Ak'M. This construction determines a functor

Defor[M]×_Alg^aug

k M⁽¹⁾→LMod(C)×_C{M}

(see Corollary A.6.1.2.40). Let LMod^aug(C) denote the fiber product Alg^aug_k ×Alg_kLMod(C). The induced map Defor[M]×_Alg^aug

k M⁽¹⁾→Defor[M]×(LMod^aug(C)×_C{M}) factors as a composition Defor[M]×_Alg^aug

k M⁽¹⁾→ⁱ Me⁽¹⁾→^λ0 Defor[M]×(LMod^aug(C)×_C{M})

whereiis an equivalence of∞-categories andλ⁰ is a categorical fibration. It is not difficult to see thatλ⁰ is a left representable pairing of∞-categories, which induces a duality functor

D⁽¹⁾_M : Defor[M]^op→LMod^aug(C)×_C{M}.

Concretely, the functorD⁽¹⁾_M assigns to each object (A, MA, µ)∈Defor[M]^opthe object (D⁽1)(A), M), where we regardM as a leftD⁽¹⁾-module object ofCvia the equivalence

M 'M_A⊗_Ak'(M ⊗_kD⁽¹⁾(A))⊗_A⊗kD(1)(A)k.

Using Corollary A.6.1.2.40, we have a canonical equivalence of ∞-categories η : LMod(C)×_C{M} ' (Alg_k)_/End(M). Thenη induces an equivalence

LMod^aug(C)×_C{M} '(Alg^aug_k )/(k⊕End(M)).

Combining this equivalence with Construction 5.2.11, we obtain a diagram of∞-categories Defor[M]^op //

(Alg^aug_k )_/k⊕End(M)

(Alg^aug_k )^op //Alg^aug_k

which commutes up to canonical homotopy, where the vertical maps are right fibrations. This diagram determines a natural transformationβ : ObjDef_M →X, where X : Art⁽¹⁾ →Sis the functor given by the formulaX(A) = Map_Alg^aug

k (D⁽¹⁾(A), k⊕End(M))'Map_Alg

k(D⁽¹⁾(A),End(M)).

Theorem 5.2.8 is a consequence of Remark 5.1.11 together with the following result:

Proposition 5.2.12. Let k be a field, C ak-linear ∞-category, and M ∈ C an object. Then the natural transformationβ: ObjDef_M →X of Construction 5.2.11 exhibitsObjDef_M(R)as a summand ofX(R)for each R∈Alg^sm_k .

Proposition 5.2.12 asserts that β induces an equivalence of formal moduli problemsβ : ObjDef^∧_M →X. According to Proposition 1.2.10, it suffices to show that β induces an equivalence of tangent complexes.

Using the description of the tangent complex of ObjDef^∧_M supplied by Lemma 5.1.12, we are reduced to proving the following special case of Proposition 5.2.12:

Proposition 5.2.13. Letkbe a field,Cak-linear∞-category, and letM ∈Cbe an object. For eachm≥0, the natural transformationβ : ObjDef_M →X of Construction 5.2.11 induces a (−1)-truncated map

ObjDef_M(k⊕k[m])→Map_Alg^aug

k (D⁽¹⁾(k⊕k[m]), k⊕End(M))'Map_Alg

k(D⁽¹⁾(k⊕k[m]),End(M)).

Proof. We have a commutative diagram

ObjDef_M(k⊕k[m]) //

Map_Alg

k(D⁽¹⁾(k⊕k[m]),End(M))

Ω ObjDef_M(k⊕k[m+ 1]) ^θ //Ω Map_Alg

k(D⁽¹⁾(k⊕k[m+ 1]),End(M)),

where the left vertical map is (−1)-truncated by Corollary 5.2.5 and the right vertical map is a homotopy equivalence. It will therefore suffice to show thatθ is a homotopy equivalence. Let A=k⊕k[m+ 1] and let MA = M ⊗k A ∈ RModA(C). We can identify the domain of θ with the homotopy fiber of the map ξ: Map_RModA(C)(MA, MA)→Map_C)(M, M). We have a fiber sequence

M[m+ 1]→MA→M

in RModA(C), whereAacts onM via the augmentation mapA→k. It follows that the homotopy fiber of ξis given by

Map_RMod

A(C)(M_A, M[m+ 1]) ' Map_C(M_A⊗Ak, M[m+ 1]) ' Map_C(M, M[m+ 1]) ' Map_Mod

k(k[−m−1],End(M))

The mapθis induced by a morphismν :k[−m−1]→D⁽¹⁾(k⊕k[m]) in Mod_k. Let Free⁽¹⁾: Mod_k→Alg_k be a left adjoint to the forgetful functor, so thatνdetermines an augmentation (k⊕k[m])⊗kFree⁽¹⁾(k[−m−1])→ k. This pairing exhibits k⊕k[m] as a Koszul dual of Free⁽¹⁾(k[−m −1]), and therefore also exhibits Free⁽¹⁾(k[−m−1]) as a Koszul dual of k⊕k[m] (Theorem 3.1.14). It follows immediately that θ is a homotopy equivalence.

We conclude this section with a few observations concerning the discrepancy between the deformation functor ObjDef_M and the associated formal moduli problem ObjDef^∧_M introduced in Notation 5.2.6. Under favorable circumstances, one can show that these two functors are equivalent:

Proposition 5.2.14. Let kbe a field, letCbe a k-linear∞-category, and letM be an object. Assume that C admits a left complete t-structure and that M is connective. Then the functor ObjDef_M : Alg^sm_k →S of Notation 5.2.4 is a formal moduli problem.

Proof. LetA ∈ Alg^sm_k . We first show that if (M, µ)∈ ObjDef_M(A), then M ∈RModA(C)_≥0. Since A is small, we can choose a finite sequence of maps

A'A₀→A₁→ · · · →A_n'k and pullback diagrams

A_i //

k

Ai+1 //k⊕k[mi]

for some integersmi >0. We prove by descending induction onithat M⊗AAi belongs to RModA_i(C)_≥0. In this case i =n, this follows from our assumption that Mn is connective. If i < n, it follows from the inductive hypothesis since we have a fiber sequence

M[m_i−1]→M ⊗AA_i→M⊗AA_i+1 inC.

Proposition IX.7.6 implies that if

A⁰ //

A

f

B^{0 g} //B

is a pullback diagram in Alg^sm_k where the maps f andg induce surjections π0A→π0B ←π0B⁰, then the functor

RModA⁰(C)_≥0→RModA(C)_≥0×_RModB(C)≥0RModB⁰(C)_≥0 is an equivalence of∞-categories. It follows immediately that

ObjDef_M(A⁰) //

ObjDef_M(A)

ObjDef_M(B⁰) //ObjDef_M(B) is a pullback diagram inS.

Corollary 5.2.15. Letk be a field, letCbe ak-linear ∞-category which admits a left completet-structure, and let M ∈ C≥0. Then the natural transformation β : ObjDef_M → X of Construction 5.2.11 is an equivalence. In other words,ObjDef_M : Alg^sm_k →S is the formal moduli problem which corresponds, under the equivalence of Theorem 3.0.4, to the augmented algebrak⊕End(M).

Proof. Combine Theorem 5.2.8, Corollary 5.2.5, and Theorem 5.1.9.

Remark 5.2.16. Let k be a field. For R ∈ Alg^sm_k , let RMod^!_R denote the ∞-category of Ind-coherent right R-modules over R (see§3.4). One can show that RMod^!_R has the structure of a k-linear∞-category (depending functorially on R). For any k-linear ∞-category C, let RMod^!_R(C) denote the relative tensor product C⊗Mod_kRMod^!_R. The equivalence RMod^!_R ' LMod_D(1)(R) of Proposition 3.5.2 is k-linear, and (combined with Theorem A.6.3.4.6) determines an equivalence RMod^!_R(C) 'LMod_D(1)(R)(C). LetM ∈C and letX: Alg^sm_k →Sbe the formal moduli problem described in Construction 5.2.11. ThenX is given by the formula

X(R) = Map_Alg^aug

k (D⁽¹⁾(R), k⊕End(M)) ' Map_Alg

k(D⁽¹⁾(R),End(M)) ' LMod_D(1)(R)(C)×_C{M} ' RMod^!_R(C)×_C{M}.

In other words, the formal moduli problemX assigns to eachR∈Alg^sm_k a classifying space for pairs (M, α), whereM ∈RMod^!_R(C) andαis an equivalence ofM with the image ofM ⊗RkofM in the∞-categoryC. The (−1)-truncated map ObjDef_M ,→ObjDef^∧_M 'X is induced by fully faithful embedding RModR(C),→ RMod^!_R(C), which are in turn determined by the fully faithful embeddings RModR,→RMod^!_Rof Proposition 3.4.14. From this point of view, we can view Proposition 5.2.14 as a generalization of Proposition 3.4.18:

it asserts that the fully faithful embedding RModR(C),→RMod^!_R(C) induces an equivalence on connective objects (where we declare an object of RMod^!_R(C) to be connective if its image inCis connective).

Dans le document Derived Algebraic Geometry X: Formal Moduli Problems (Page 144-150)