PART IV.2. FORMAL MODULI

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1 PART IV.2. FORMAL MODULI Contents Introduction 1 1. Formal moduli problems Formal moduli problems over a prestack Situation over an affine scheme Formal moduli problems under a prestack Formal moduli problems under an affine scheme The pointed case Formal groups 6 2. Groupoids Digression: groupoids and groups over spaces Groupoids in formal geometry Taking the quotient by a formal groupoid Constructing the classifying space of a groupoid Verification of deformation theory 11 References 13 Introduction Date: June 19,

2 2 FORMAL MODULI 1. Formal moduli problems In this section we introduce the notions of formal moduli problem over and under a given prestack. We note the essential difference between our set-up and that of [Lu2], where the case = pt Formal moduli problems over a prestack. Unlike [Lu2], we define the category of formal moduli problems over a given to be a full subcategory in (PreStk laft ) /. The equivalence of this definition and the one in loc. cit. will be established in Sect Let us fix PreStk laft. We define FormMod / (PreStk laft ) / to be the full subcategory of spanned by those, for which the above map is: inf-schematic (see [Book-III.2, Definition 3.1.5] for what this means); a nil-isomorphism (i.e., red red is an isomorphism). We shall refer to FormMod / as the category of formal moduli problems over The following result easily from the definitions: Lemma Let be a map in PreStk laft. for every S < Sch aff aft, the prestack S isomorphism The following will be useful: Lemma The functor is an equivalence. FormMod / Then FormMod / if and only if ) red S is an is an infscheme and red (S lim (Z,x) (( < Sch aff aft ) /) op FormMod /Z 1.2. Situation over an affine scheme. In this subsection we will assume that that = Sch aff aft. We will show that a formal moduli problem over, as a functor (( < Sch aff ft ) / ) op Spc, is completely determined by its value on those Z ( < Sch aff ft ) / for which red Z red is an isomorphism. Note that when = pt, the category (( < Sch aff ft ) / ) op is the same as that of connective finite-dimensional commutative DG algebras over k, whose 0-th cohomology is local. This brings us in contact with the defintion of formal moduli problems in [Lu2] We will prove the following assertion: Proposition (a) Every FormMod /, viewed as a functor (( < Sch aff aft) / ) op Spc, is the left Kan extension of its restriction to the full subactegory (1.1) (( < Sch aff aft) nil-isom to ) op (( < Sch aff aft) / ) op. (b) Let nil-isom be a presheaf on the category ( < Sch aff aft) nil-isom to, satisfying: nil-isom ( red ) = { };

3 FORMAL MODULI 3 For a push-out diagram S 1 S S in ( < Sch aff aft) nil-isom to, where S S has a structure of square-zero extension, the resulting map nil-isom (S 1 S ) nil-isom (S 1 ) nil-isom (S ) S nil-isom (S) is an isomorphism. Then if (PreStk laft ) / denotes the left Kan extension of nil-isom under (1.1), then FormMod /. (c) The assignments ( < Sch aff aft ) and nil-isom to nil-isom are mutually inverse equivalences of categories. Proof. Point (a) follows from [Book-III.2, Corollary 4.3.4]. Point (b) follows from [Book-III.2, Proposition 4.4.5]. Point (c) follows from [Book-III.2, Corollary 4.4.6] The following assertion will be used extensively in [Book-IV.3]: Corollary For FormMod /, the map colim f IndCoh (Z,f) (ω Z ) ω is an isomorphism, where the colimit is taken over the category Proof. By Proposition 1.2.2, the functor is cofinal. Hence, the restriction functor (( < Sch aff ) nil-isom to ) /. (( < Sch aff ) nil-isom to ) / ( < Sch aff ) / IndCoh() lim (Z,f) IndCoh(Z) is an isomorphism, where the limit is taken over the category (( < Sch aff ) nil-isom to ) /. By [Book-III.3, Corollary 4.3.4] and [GL:DG, Lemma 1.3.3], we obtain that the functors : IndCoh(Z) IndCoh() define an equivalence f IndCoh colim IndCoh(Z) IndCoh(), (Z,f) is an equivalence, where the colimit is taken with respect to the (IndCoh, )-direct image functors. In particular, we obtain that as required. ω colim f IndCoh f! (ω ) colim f IndCoh (ω Z ), (Z,f) (Z,f) 1.3. Formal moduli problems under a prestack. In this subsection we consider a prestack PreStk laft-def. We will consider another paradigm for formal moduli problems, by looking at prestack under.

4 4 FORMAL MODULI We define the category FormMod / to be the full subcategory of (PreStk laft ) / spanned by those, for which: PreStk laft-def ; The map realizes as an object of FormMod / Note that when = pt, there is no difference between FormMod / and FormMod / Formal moduli problems under an affine scheme. In this subsection we specialize to the case when = < Sch aff aft. We will show that a formal moduli problem under, viewed as a functor ( < Sch aff aft) op Spc is completely determined by its value on the category of affine schemes Z, equipped with a nil-isomorphism Z Let ( < Sch aff aft) nil-isom from ( < Sch aff aft) / be the full subcategory formed by those f : Z, for which f is a nil-isomorphism. We claim: Proposition (a) For FormMod /, viewed as a functor the map is an isomorphism. ( < Sch aff aft) op Spc, LKE (( < Sch aff aft ) nil-isom from ) op ( < Sch aff aft )op( ( < Sch aff aft ) nil-isom from ) (b) Let nil-isom be a presheaf on ( < Sch aff aft) nil-isom from, satisfying: nil-isom () = { }; For a push-out diagram S 1 S S in ( < Sch aff aft) nil-isom from, where S S has a structure of square-zero extension, the resulting map is an isomorphism. nil-isom (S 1 S ) nil-isom (S 1 ) nil-isom (S ) S nil-isom (S) Then if PreStk laft denotes the left Kan extension of nil-isom under (( < Sch aff aft) nil-isom from ) op ( < Sch aff aft) op, then the canonical map makes into an object of FormMod /. (c) The assignments ( < Sch aff aft ) { } and nil-isom from nil-isom Maps(,) are mutually inverse equivalences of categories.

5 FORMAL MODULI 5 Proof. Applying [Book-III.2, Corollary 4.3.4, Proposition and Corollary 4.4.6], it is enough to evaluate the functor as in the proposition on the subcategory ( < Sch aff aft) { red }. red Sch aff ft The assertion of the proposition follows now from the fact that the forgetful functor admits a left adjoint, given by ( < Sch aff aft) nil-isom from ( < Sch aff aft) { red } red Sch aff ft S S. red As a corollary, we obtain: Corollary For FormMod /, the map colim f IndCoh (Z,f) (ω Z ) ω is an isomorphism, where the colimit is taken over the category (( < Sch aff ) nil-isom from ) /. Proof. Same as that of Corollary The pointed case Note that for PreStk laft-def, we have: Ptd(FormMod / ) (Ptd(FormMod / ) / (Ptd(FormMod / ) /. Combining Propositions and 1.4.2, we obtain: Corollary For < Sch aff we have: (a) Any Ptd(FormMod / ), viewed as a functor (( < Sch aff aft) / ) op Spc, receives an isomorphism from the left Kan extension from its restriction to the category Ptd(( < Sch aff ) nil-isom to ). (b) Let nil-isom be a presheaf on Ptd(( < Sch aff ) nil-isom to ), satisfying: nil-isom () = { }; For a push-out diagram S 1 S S in Ptd(( < Sch aff ) nil-isom to ), where S S has a structure of square-zero extension, the resulting map is an isomorphism. nil-isom (S 1 S ) nil-isom (S 1 ) nil-isom (S ) S nil-isom (S)

6 6 FORMAL MODULI Then if (PreStk laft ) / denotes the left Kan extension of nil-isom under (Ptd(( < Sch aff ) nil-isom to )) op (( < Sch aff aft) / ) op, then the canonical map makes into an object of Ptd(FormMod / ). (c) The assignments Ptd(( < Sch aff ) nil-isom to ) { } and nil-isom Maps(,) are mutually inverse equivalences of categories. Remark The informal meaning of this corollary is that in order to know an object Ptd(FormMod / ) as a prestack, it is is enough to test it on affine schemes S, equipped with a nil-immersion to and a section of this nil-immersion As in the case of Corollary 1.2.4, from Corollary 1.5.2, we obtain: Corollary For Ptd(FormMod / ), the map colim f IndCoh (Z,f) (ω Z ) ω is an isomorphism, where the colimit is taken over the category (Ptd(( < Sch aff ) nil-isom to )) / Formal groups. In this subsection we introduce formal groups over as group-objects in the category of formal moduli problems. In [Book-IV.3] we will show that the category of formal groups identifes with that of Lie algebras in IndCoh(), thereby generalizing the main result in [Lu2] Let Monoid(FormMod / ) be the category of monoid-objects in Ptd(FormMod / ), and let Grp(FormMod / ) Monoid(FormMod / ) be the full subcategory spanned by group-like objects. Lemma The inclusion Grp(FormMod / ) Monoid(FormMod / ) is an equivalence. Proof. We need to show that for H Monoid(FormMod / ) the map is an isomorphism. H H H H, (h 1, h 2 ) (h 1, h 1 h 2 ) This follows from [Book-III.1, Corollary 8.3.6], applied to the Cartesian square id H H h (1,h) H H h (1,h) H (h 1,h 2) (h 1,h 1 h 2) H.

7 FORMAL MODULI We have a naturally defined functor (1.2) Ω : Ptd(FormMod / ) Grp(FormMod / ),. In Sect we will prove: Theorem The functor Ω of (1.2) is an equivalence. In what follows we shall denote by B the functor inverse to Ω. Grp(FormMod / ) Ptd(FormMod / ), 2. Groupoids 2.1. Digression: groupoids and groups over spaces Given a space, recall the category Grpoid() of groupoids acting on (see [Lu0], Sect ). We shall symbolically depict a groupoid via a diagram (2.1) p s R p t, while properly we should be thinking about the entire simplicial object of Spc with 0 = and 1 = R. The category Grpoid() contains an initial object, the identity groupoid, where all R i =. We shall denote it by diag. The category Grpoid() contains also a final object, namely The following assertion will be used repeatedly: Lemma The forgetful functor Grpoid() Spc that sends a groupoid to R commutes with sifted colimits Riven a groupoid R acting on we can consider the quotient space which receives a natural map from : /R := R, = R 0 R /R. Vice versa, given a space under, we construct the groupoid over by R := i.e., R := is the Čech nerve of the above map. It is clear that the two functors are mutually adjoint. We have: Grpoid() Spc / Lemma The above two functors define equivalences between the category Grpoid() and the full subcategory Spc /,surj of Spc / spanned by objects i :, for which the map i is surjective on π 0.,

8 8 FORMAL MODULI Riven a space, consider the category Grp(Spc / ), of group-objects in the category of spaces over. We have: Grp(Spc / ) := Grpoid() / diag, where Grpoid() is the identity groupoid. Consider also the category Ptd(Spc / ), which is the same as the category of of retraction diagrams (2.2) i : : s, s i id. We have a natural functor (2.3) Grp(Spc / ) Ptd(Spc / ), given by G B (G), where B (G) is the relative classifying space over. We also have the adjoint loop functor Ω : Ptd(Spc / ) Grp(Spc / ). Lemma The functors B and Ω define an equivalence between Grp(Spc / ) and the full subcategory Ptd(Spc / ) isom of Ptd(Spc / ), spanned by those objects for which the map i is an isomorphism on π 0. i : : s Note that the condition on an object (i : : s) Ptd(Spc / ) to belong to Ptd(Spc / ) isom Ptd(Spc / ) is equivalent to the map i being a surjection on π Groupoids in formal geometry The definitions of Sect. 2.1 carry over automatically to the algebro-geometric setting. For PreStk laft, we consider the categories (PreStk laft ) /, Ptd((PreStk laft ) / ), Grp((PreStk laft ) / ), and their full subcategories FormMod /, Ptd(FormMod / ), Grp(FormMod / ), formed by objects that are formal as prestacks over. and Grpoid laft (), and FormGrpoid(), Examples. Let be a map between objects of PreStk laft. To it we attach the object of Grpoid laft (), namely, the Čech nerve of this map. Thus, the corresponding R is given by. If the above map is an inf-schematic nil-isomorphism, then R FormGrpoid().

9 FORMAL MODULI Ind-coherent sheaves equivariant with respect to a groupoid. Let PreStk laft ; let R be an object of FormGrpoid(), and let R be the corresponding simplicial object of PreStk laft. We define the category IndCoh() R of ind-coherent sheaves equivariant with respect to a R to be By [Book-III.3, Proposition 3.3.2], we have: Tot(IndCoh(R )). Proposition Let R be the formal groupoid corresponding to a map in PreStk laft, which is an inf-schematic nil-isomorphism. Then the pullback functor defines an equivalence IndCoh() IndCoh() R Taking the quotient by a formal groupoid. In this subsection we state one of the main results of this book Assume that PreStk laft-def. Recall the category FormMod /, see Sect We have a naturally defined functor (2.4) FormMod / FormGrpoid(), namely,, see Sect The main result of this section is the following: Theorem The functor (2.4) is an equivalence An example. Let be a map in PreStk laft-def. Consider the formal completion of along, i.e., := dr dr. Then the map defines an object of FormMod /. Consider the groupoid, (see Sect ) and its formal completion along the diagonal map, It is easy to see that ( ). ( ), where the latter is an object of FormGrpoid() by Sect Thus, the formal completion can be recovered from ( ) by taking the functor inverse to that in Theorem

10 10 FORMAL MODULI Note that Theorem implies Theorem 1.6.4: Proof. To prove Theorem we can assume that Sch aff aft. In particular, we can assume that PreStk laft-def. Then the required assertion follows from Theorem by noting that Grp(FormMod ) = FormGrpoid() / diag, and Ptd(FormMod / ) = (FormMod / ) / Constructing the classifying space of a groupoid. In this subsection we will begin the proof of Theorem For R FormGrpoid() we define an object B (R) PreStk laft as follows: For Z < Sch aff aft, we let Maps(Z, B (R)) be the groupoid consisting of data: {( Z Z) FormMod /Z, Z, a map of groupoids Z Z R}, Z where we require that the diagram Z Z R Z Z be Cartesian, where the vertical arrows are either of the projections We have a tautological map B (R) that sends Z to where the fiber product Z corresponds to p 2 : R. Z := Z R, R is formed using the map p 1 : R, and the map Z R Let us show that the map B (R) makes into an object of FormMod /B (R). Indeed, for a given map Z B (R), the fiber product identifies with Z. The latter observation also implies that Z B (R) R, B (R) so the functor is the right inverse to the functor (2.4). R B (R)

11 FORMAL MODULI We claim that it suffices to show that the functor B (R) PreStk laft-def. Indeed, let us assume this for a moment and conclude the proof of the theorem. For FormMod / we have a tautological map (2.5) B ( ), which becomes an isomorphism after applying the functor (2.4). However, it follows from [Book-III.1, Proposition 8.3.2] that the functor (2.4) is conservative Verification of deformation theory. In this subsection we will prove that the object B (R) PreStk laft constructed in Sect , admits deformation theory Let Z be an object of Z < Sch aff aft, equipped with a map to B (R). We will now construct a certain object of Pro(QCoh(S) ) laft, which we will later identify with the procotangent space to B (R) at our given point Z B (R). Consider the Čech nerve Z of the corresponding map Z Z, and consider the resulting map of simplicial prestacks Z R. Let T (Z /R ) Tot ( Pro(QCoh(Z ) ) fake) be the corresponding relative pro-cotangent complex (see [Book-III.1, Sect. receives a canonically defined map from the pullback of T (Z) ]), which By nil-decent for Pro(QCoh( ) ) fake with respect to Z Z (see [Book-III.3, Corollary 3.3.4]), we obtain that T (Z /R ) gives rise to a canonically defined object, which receives a map from T (Z). Set We will show that the object T (Z/B (R)) Pro(QCoh(Z) ), T (B (R)) Z := ker (T (Z) T (Z/B (R))). T (B (R)) Z Pro(QCoh(Z) ), identifies with the pro-cotangent space of B (R) at the above point Z B (R) We need to show that, given a square-zero extension Z Z, corresponding to γ : T (Z) I[1], I Coh(Z) 0, the groupoid of extensions of the initial map Z B (R) to a map Z B (R), identifies canonically with groupoid of factorizations of γ as T (Z) T (Z/B (R)) Z I[1]. This will show that B (R) admits pro-cotangent spaces that are indeed identified with ones constructed in Sect , and that B (R) is infinitesimally cohesive. The fact that B (R) admits a pro-cotangent complex (i.e., that the formation of pro-cotangent spaces is compatible with pullback) follows from the construction in Sect

12 12 FORMAL MODULI For Z Z as above, by [Book-III.1, Proposition ], the datum of a prestack Z Z, equipped with a Cartesian diagram Z Z Z Z is equivalent to that of a map T (Z) I Z [1] (in the category Pro(QCoh(Z) ) fake ), and a homotopy between the composition T (Z) Z T (Z) I Z [1] and γ Z. Moreover, by [Book-III.1, Proposition ], such Z is automatically an inf-scheme. The same discussion applies to each term of the Čech nerve Z. Furthermore, by [Book-III.1, Proposition ] the datum of a compatible system of maps from the Čech nerve Z to R, extending the initial system Z R, is equivalent to a compatible system of factorizations of the resulting maps as T (Z ) I Z [1] T (Z ) T (Z /R ) I Z [1] Hence, we obtain that the datum of extension of the initial map Z B (R) to a map Z B (R) is equivalent to that of a compatible family of maps and homotopies between and γ Z. T (Z /R ) I Z [1], T (Z) Z T (Z /R ) I Z [1] By nil-descent for Pro(QCoh( ) ) fake with respect to Z Z, the latter datum is equivalent to that of factorizations of γ as T (Z) T (Z/B (R)) Z I[1].

13 FORMAL MODULI 13 References [Fra] J. Francis. [GL:DG] D. Gaitsgory, Notes on Geometric Langlands: Generalities on DG categories, available at gaitsgde/gl/. [GL:Stacks] D. Gaitsgory, Notes on Geometric Langlands: Stacks, available at gaitsgde/gl/. [GL:QCoh] D. Gaitsgory, Notes on Geometric Langlands: Quasi-coherent sheaves on stacks, available at gaitsgde/gl/. [GL:IndCoh] D. Gaitsgory, Ind-coherent sheaves, ariv: [GL:IndSch] D. Gaitsgory and N. Rozenblyum, DG indschemes, ariv: [GL:Crystals] D. Gaitsgory and N. Rozenblyum, Crystals. [Book-II.2] [Book-III.1] [Book-III.2] [Book-III.3] [Book-IV.3] [Lu0] J. Lurie, Higher Topos Theory, Princeton Univ. Press (2009). [Lu1] J. Lurie, DAG-VIII, available at lurie. [Lu2] J. Lurie, DAG-, available at lurie. [Lu?] J. Lurie, 2-Categories, available at lurie.

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