[PDF] Derived Algebraic Geometry X: Formal Moduli Problems

View - Download Derived Algebraic Geometry X: Formal Moduli Problems

Previous PDF

Next PDF

Derived Algebraic Geometry X: Formal Moduli Problems

November 5, 2011

Contents

1 Deformation Theories: Axiomatic Approach 8

1.1 Formal Moduli Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.2 The Tangent Complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.3 Deformation Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.4 Digression: The Small Object Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.5 Smooth Hypercoverings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2 Moduli Problems for Commutative Algebras 30

2.1 Dierential Graded Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.2 Homology and Cohomology of Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.3 Koszul Duality for Dierential Graded Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . 42

2.4 Quasi-Coherent Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3 Moduli Problems for Associative Algebras 58

3.1 Koszul Duality for Associative Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.2 Formal Moduli Problems for Associative Algebras . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.3 Comparison of Commutative and Associative Deformation Theories . . . . . . . . . . . . . . 72

3.4 Quasi-Coherent and Ind-Coherent Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.5 Koszul Duality for Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4 Moduli Problems forEn-Algebras 89

4.1 CoconnectiveEn-Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.2 Twisted Arrow1-Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

4.3 The Bar Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

4.4 Koszul Duality forEn-Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

4.5 Deformation Theory ofEn-Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

5 Examples of Moduli Problems 137

5.1 Approximations to Formal Moduli Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

5.2 Deformations of Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

5.3 Deformations of Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

1

Introduction

The following thesis plays a central role in deformation theory: () IfXis a moduli space over a eldkof characteristic zero, then a formal neighborhood of any point x2Xis controlled by a dierential graded Lie algebra. This idea was developed in unpublished work of Deligne, Drinfeld, and Feigin, and has powerfully in uenced

subsequent contributions of Hinich, Kontsevich-Soibelman, Manetti, Pridham, and many others. One of our

main goals in this paper is to give a precise formulation (and proof) of (), using the language of higher

category theory. The rst step in formulating () is to decide exactly what we mean by a moduli space. For simplicity,

let us work for now over the eldCof complex numbers. We will adopt Grothendieck's \functor of points"

philosophy, and identify an algebro-geometric objectX(for example, a scheme) with the functorR7! X(R) = Hom(SpecR;X). This suggests a very general denition: Denition 0.0.1.Aclassical moduli problemis a functorX: RingC!Set, where RingCdenotes the category of commutativeC-algebras andSet denotes the category of sets.

Unfortunately, Denition 0.0.1 is not adequate for the needs of this paper. First of all, Denition 0.0.1

requires that the functorXtake values in the category of sets. In many applications, we would like to consider

functorsXwhich assign to each commutative ringRsome collection of geometric objects parametrized by the ane scheme SpecR. In such cases, it is important to keep track of automorphism groups. Example 0.0.2.For every commutativeC-algebraR, letX(R) denote the category of elliptic curves E!SpecR(morphisms in the categoryX(R) are given by isomorphisms of elliptic curves). ThenF determines a functor from Ring Cto Gpd, where Gpd denotes the 2-category ofgroupoids. In this case,X determines an underlying set-valued functor, which assigns to each commutative ringRthe set0X(R) of isomorphism classes of elliptic curves overR. However, the groupoid-valued functorX: RingC!Gpd is much better behaved than the set-valued functor0X: RingC!Set. For example, the functorXsatises descent (with respect to the at topology on the category of commutative rings), while the functor0Xdoes not: two elliptic curves which are locally isomorphic need not be globally isomorphic. Because the functorXof Example 0.0.2 is notSet-valued, it cannot be represented by a scheme. How-

ever, it is nevertheless a reasonable geometric object: it is representable by a Deligne-Mumford stack. To

accommodate Example 0.0.2, we would like to adjust Denition 0.0.1 to allow groupoid-valued functors. Variant 0.0.3.LetCbe an1-category. AC-valued classical moduli problemis a functor N(RingC)!C.

Here Ring

Cdenotes the category of commutative algebras over the eldCof complex numbers.

Remark 0.0.4.We recover Denition 0.0.1 as a special case of Variant 0.0.3, by takingCto be (the nerve

of) the category of sets. In practice, we will be most interested in the special case whereCis the1-category

Sof spaces.

The next step in formulating () is to decide what we mean by a formal neighborhood of a pointxin a moduli spaceX. Suppose, for example, thatX= SpecAis an ane algebraic variety over the eldCof complex numbers. Then a closed pointx2Xis determined by aC-algebra homomorphism:A!C, which is determined a choice of maximal idealm= ker()A. One can dene theformal completionofX at the pointxto be the functorX^: Ring!Set given by the formula X ^(R) =ff2X(R) :f(SpecR) fxg SpecAg: In other words,X^(R) is the collection of commutative ring homomorphisms:A!Rhaving the property

thatcarries each element ofmto a nilpotent element ofR. Sincemis nitely generated, this is equivalent

to the condition thatannihilatesmnfor some integern0, so that the image ofis a quotient ofAby somem-primary ideal. 2 Denition 0.0.5.LetRbe a commutative algebra over the eldCof complex numbers. We will say that

Ris alocal Artinianif it is nite dimensional as aC-vector space and has a unique maximal idealmR. The

collection of local ArtinianC-algebras forms a category, which we will denote by RingartC. The above analysis shows that ifXis an ane algebraic variety overCcontaining a pointx, then the formal completionX^can be recovered from its values on local ArtinianC-algebras. This motivates the following denition: Denition 0.0.6.LetCbe an1-category. AC-valued classical formal moduli problemis a functor

N(Ring

artC)!C. IfXis aSet-valued classical moduli problem and we are given a point2X(C), we can dene a Set-valued classical formal moduli problemX^by the formula X ^(R) =X(R)X(R=mR)fg: We will refer toX^as thecompletion ofXat the point. IfXis Gpd-valued, the same formula determines a Gpd-valued classical formal moduli problemX^(here we take a homotopy ber product of the relevant groupoids). Example 0.0.7.For every commutativeC-algebraR, letX(R) denote the groupoid whose objects are smooth properR-schemes and whose morphisms are isomorphisms ofR-schemes. Suppose we are given a point2X(C), corresponding to smooth and proper algebraic varietyZoverC. The formal completion X ^assigns to every local ArtinianC-algebraRthe groupoidX^(R) ofdeformations overZoverR: that is, smooth proper morphismsf:Z!SpecRwhich t into a pullback diagram Z //Z

SpecC//SpecR:

Example 0.0.7 is a typical example of the kind of formal moduli problem we would like to study. Let us

summarize some well-known facts about the functorX^: (a) The functorX^carries the ringC[]=(2) to the groupoid of rst-order deformations of the variety Z. Every rst order deformation ofZhas an automorphism group which is canonically isomorphic to H

0(Z;TZ), whereTZdenotes the tangent bundle ofZ.

(b) The collection of isomorphism classes of rst order deformations ofZcan be canonically identied with

the cohomology group H

1(Z;TZ).

(c) To every rst order deformation1ofZ, we can assign an obstruction class2H2(Z;TZ) which vanishes if and only if1extends to a second-order deformation22X^(C[]=(3)).

Assertion (a) and (b) are very satisfying: they provide a concrete geometric interpretations of certain

cohomology groups, and (b) can be given a conceptual proof using the interpretation of H1as classifying

torsors. By contrast, (c) is often proven by an ad-hoc argument which uses the local triviality of the rst

order deformation to extend locally, and then realizes the obstruction as a cocycle representing the (possible)

inability to globalize this extension. This argument is computational rather than conceptual, and it does

not furnish a geometric interpretation of the entire cohomology group H

2(Z;TZ).

Let us now sketch an explanation for (c) using the language of spectral algebraic geometry, which does

not share these defects. The key observation is that we can enlarge the category on which the functorX

is dened. IfRis a connectiveE1-algebra overC, we can deneX(R) to be the underlying1-groupoid of the1-category of spectral schemes which are proper and smooth overR. IfRis equipped with an 3 augmentation:R!C, we letX^(R) denote the ber productX(R)X(C)fg, which we can think of as a classifying space fordeformations ofZover SpecR. In the special case whereRis a discrete local

ArtinianC-algebra, we recover the groupoid-valued functor described in Example 0.0.7. However, we can

obtain more information by evaluating the functorX^onE1-algebras overCwhich are not discrete. For

example, letC[] denote the square-zero extensionCC[1]. One can show that there is a canonical bijection

H

2(Z;TZ)'0X^(C[]). We can regard this as an analogue of (c): it gives a description of cohomology

group H

2(Z;TZ) as the set of isomorphism classes of rst order deformations ofZto the \nonclassical"

commutative ringC[]. The interpretation of obstructions as elements of H

2(X;TX) can now be obtained as follows. The ordinary

commutative ringC[]=(3) is a square-zero extension ofC[]=(2) by the idealC2, and therefore ts into a pullback diagram ofE1-rings

C[]=(3)//C[]=(2)

C //C[]: InxIX.9, we saw that this pullback square determines a pullback square of spaces

X(C[]=(3))//X(C[]=(2))

X(C)//X(C[]);

and therefore a ber sequence of spaces X ^(C[]=(3))!X^(C[]=(2))!X^(C[]): In particular, every rst-order deformation1ofZdetermines an element of0X^(C[])'H2(Z;TZ), which vanishes precisely when1can be lifted to a second order deformation ofZ.

The analysis that we have just provided in Example 0.0.7 cannot be carried out for an arbitrary classical

formal moduli problem (in the sense of Denition 0.0.6): it depends crucially on the fact that the functor

X ^could be dened onE1-rings which are not assumed to be discrete. This motivates another variant of

Denition 0.0.1:

Denition 0.0.8.Let CAlgsmCdenote the1-category of smallE1-algebras overC(see Proposition 1.1.11). Aformal moduli problemoverCis a functorX: CAlgsmC!Swhich satises the following pair of conditions: (1) The spaceX(C) is contractible. (2) For every pullback diagram R //R 0 R 1//R 01 in CAlg smCfor which the underlying maps0R0!0R01 0R1are surjective, the diagram

X(R)//X(R0)

X(R1)//X(R01)

is a pullback square. 4 Remark 0.0.9.Let CAlgcnCdenote the1-category of connectiveE1-algebras over the eldCof complex numbers, and letX: CAlgcnC!Sbe a functor. Given a pointx2X(C), we dene theformal completion ofXat the pointxto be the functorX^: CAlgsmC!Sgiven by the formulaX^(R) =X(R)X(C)fxg.

The functorX^automatically satises condition (1) of Denition 0.0.8. Condition (2) is not automatic, but

holds whenever the functorXis dened in a suciently \geometric" way. To see this, let us imagine that there exists some1-category of geometric objectsCwith the following properties: (a) To every objectA2CAlgcnCwe can assign an object SpecA2C, which is contravariantly functorial in A. (b) There exists an objectX2Cwhich representsX, in the sense thatX(A)'HomC(SpecA;X) for every smallC-algebraA. To verify thatX^satises condition (2) of Denition 0.0.8, it suces to show that when:R0!R01and

0:R1!R01are maps of smallE1algebras overCwhich induce surjections0R0!0R01 0R1, then

the diagram

SpecR01//SpecR1

SpecR0//Spec(R1R01R0)

is a pushout square inC. This assumption expresses the idea that Spec(R0R01R1) should be obtained by \gluing" SpecR0and SpecR1together along the common closed subobject SpecR01. Example 0.0.10.LetCdenote the1-category StkCof spectral Deligne-Mumford stacks overC. The construction Spec et: CAlgcnC!Csatises the gluing condition described in Remark 0.0.9 (Corollary IX.6.5). It follows that every spectral Deligne-Mumford stackXoverCequipped with a base pointx: SpecetC!X determines a formal moduli problemX^: CAlgsmC!S, given by the formula X ^(R) = MapStkC(SpecetR;X)MapStkC(SpecetC;X)fxg: We refer toX^as theformal completion ofXat the pointx. Remark 0.0.11.LetX: CAlgsmC!Sbe a formal moduli problem. ThenXdetermines a functorX: hCAlg smC!Set between ordinary categories (here hCAlgsmCdenotes the homotopy category of CAlgsmC),

given by the formulaX(A) =0X(A). It follows from condition (2) of Denition 0.0.8 that if we are given

maps of smallE1-algebrasA!B A0which induce surjections0A!0B 0A0, then the induced mapX(ABA0)!X(A)X(B)X(A0)

is a surjection of sets. There is a substantial literature on set-valued moduli functors of this type; see, for

example, [50] and [33]. Warning 0.0.12.IfXis a formal moduli problem overC, thenXdetermines a classical formal moduli

problem (with values in the1-categoryS) simply by restricting the functorXto the subcategory of CAlgsmCconsisting of ordinary local ArtinianC-algebras (which are precisely the discrete objects of CAlgsmC).

IfX= (X;O) is a spectral Deligne-Mumford stack overCequipped with a point: SpecC!XandX is dened as in Example 0.0.10, then the restrictionX0=XjN(RingartC) depends only on the pair (X;0O). In particular, the functorXcannot be recovered fromX0. In general, if we are given a classical formal moduli problemX0: N(RingartC)!S, there may or may not exist a formal moduli problemXsuch thatX0=XjN(RingartC). Moreover, ifXexists, then it need not be

unique. Nevertheless, classical formal moduli problemsX0which arise naturally are often equipped with a

natural extensionX: CAlgsmC!S(as in our elaboration of Example 0.0.7). 5 Theorem 0.0.13.LetModulidenote the full subcategory ofFun(CAlgsmC;S)spanned by the formal moduli problems, and letLiedg Cdenote the category of dierential graded Lie algebras overC(seex2.1). Then there is a functor : N(Liedg

C)!Moduli

with the following universal property: for any1-categoryC, composition withinduces a fully faithful embeddingFun(Moduli;C)!Fun(N(Liedg C);C), whose essential image is the collection of all functorsF: N(Lie dg C)!Cwhich carry quasi-isomorphisms of dierential graded Lie algebras to equivalences inC.

Remark 0.0.14.An equivalent version of Theorem 0.0.13 has been proven by Pridham; we refer the reader

to [54] for details. Remark 0.0.15.LetWbe the collection of all quasi-isomorphisms in the category Liedg

C, and let Liedg

C[W1] denote the1-category obtained from N(Liedg C) by formally inverting the morphisms inW. Theorem 0.0.13 asserts that there is an equivalence of1-categories Liedg

C[W1]'Moduli. In particular, every dierential

graded Lie algebra overCdetermines a formal moduli problem, and two dierential graded Lie algebras g andg0determine equivalent formal moduli problems if and only if they can be joined by a chain of quasi-isomorphisms.

Theorem 0.0.13 articulates a sense in which the theories of commutative algebras and Lie algebras are

closely related. In concrete terms, this relationship is controlled by theChevalley-Eilenbergfunctor, which

associates to a dierential graded Lie algebraga cochain complex of vector spacesC(g). The cohomol- ogy of this cochain complex is theLie algebra cohomologyof the Lie algebrag, and is endowed with a

commutative multiplication. In fact, this multiplication is dened at the level of cochains: the construction

g

7!C(g) determines a functorCfrom the (opposite of) the category Liedg

Cof dierential graded Lie

algebras overCto the category CAlgdg Cof commutative dierential graded algebras overC. This functor

carries quasi-isomorphisms to quasi-isomorphisms, and therefore induces a functor between1-categories

: Liedg

C[W1]op!CAlgdg

C[W01];

whereWis the collection of quasi-isomorphisms in Liedg

C(as in Remark 0.0.15) andW0is the collection of

quasi-isomorphisms in CAlg dg

C(here the1-category CAlgdg

C[W01] can be identied CAlgCofE1-algebras

overC: see Proposition A.7.1.4.11). Every dierential graded Lie algebragadmits a canonical map g !0, so that its Chevalley-Eilenberg complex is equipped with an augmentationC(g)!C(0)'C.

We may therefore reneto a functor Liedg

C[W1]op!CAlgaug

Ctaking values in the1-category CAlgaug

CofaugmentedE1-algebras overC. We will see that this functor admits a left adjointD: CAlgaug C! Lie dg

C[W1]op(Theorem 2.3.1). The functor: N(Liedg

C)!Moduli appearing in the statement of Theorem

0.0.13 can then be dened by the formula

(g)(R) = MapLiedg

C[W1](D(R);g):

In more abstract terms, the relationship between commutative algebras and Lie algebras suggested by Theorem 0.0.13 is an avatar ofKoszul duality. More specically, Theorem 0.0.13 re ects the fact that the

commutative operad is Koszul dual to the Lie operad (see [29]). This indicates that should be many other

versions of Theorem 0.0.13, where we replace commutative and Lie algebras by algebras over some other

pair of Koszul dual operads. For example, the Koszul self-duality of theEn-operads (see [17]) suggests an

analogue of Theorem 0.0.13 in the setting of \noncommutative" derived algebraic geometry, which we also

prove (see Theorems 3.0.4 and 4.0.8).

Let us now outline the contents of this paper. Inx1, we will introduce the general notion of adeformation

theory: a functor of1-categoriesD: op! satisfying a suitable list of axioms (see Denitions 1.3.1 and

1.3.9). We will then prove an abstract version of Theorem 0.0.13: every deformation theoryDdetermines

an equivalence 'Moduli, where Moduliis a suitably dened1-category of formal moduli problems 6

(Theorem 1.3.12). This result is not very dicult in itself: it can be regarded as a distillation of the purely

formal ingredients needed for the proof of results like Theorem 0.0.13. In practice, the hard part is to

construct the functorDand to prove that it satises the axioms of Denitions 1.3.1 and 1.3.9. We will give

a detailed treatment of three special cases: (a) Inx2, we treat the case where is the1-category CAlgaug kof augmentedE1-algebras over a eldk of characteristic zero, and use Theorem 1.3.12 to prove a version of Theorem 0.0.13 (Theorem 2.0.2). (b) Inx3, we treat the case where is the1-category Algaug kof augmentedE1-algebras over a eldk (of arbitrary characteristic), and use Theorem 1.3.12 to prove a noncommutative analogue of Theorem

0.0.13 (Theorem 3.0.4).

(c) Inx4, we treat the case where is the1-category Alg(n);aug kof augmentedEn-algebras over a eldk (again of arbitrary characteristic), and use Theorem 1.3.12 to prove a more general noncommutative analogue of Theorem 0.0.13 (Theorem 4.0.8). In each case, the relevant deformation functorDis given by some variant of Koszul duality, and our main result gives an algebraic model for the1-category of formal moduli problems Moduli. Inx5, we

will use these results to study some concrete examples of formal moduli problems which arise naturally in

quotesdbs_dbs43.pdfusesText_43

[PDF] histoire de la géométrie algébrique

[PDF] tv orange cote d'ivoire apk

[PDF] méthode naturelle pour planification familiale

[PDF] telecharger formulaire cession de ligne orange

[PDF] lettre de changement de titulaire de la ligne téléphonique

[PDF] telecharger formulaire cession de ligne mobile orange

[PDF] régulation des naissances svt

[PDF] changement de titulaire orange

[PDF] demande de cession de contrat d'abonnement orange business

[PDF] lettre resiliation orange mobile suite deces

[PDF] contrat de cession ligne fixe orange

[PDF] délai cession de ligne orange

[PDF] en quoi les politiques conjoncturelles peuvent elles lutter contre le chomage

[PDF] comment les politiques conjoncturelles peuvent-elles agir sur les fluctuations économiques

[PDF] les politiques conjoncturelles sont elles efficaces