PART II.2. THE!-PULLBACK AND BASE CHANGE

Transcription

1 PART II.2. THE!-PULLBACK AND BASE CHANGE Contents Introduction 1 1. Factorizations of morphisms of DG schemes Colimits of closed embeddings The closure Transitivity of closure 5 2. IndCoh as a functor from the category of correspondences The category of correspondences Proof of Proposition The functor of!-pullback Definition of the functor Some properties h-descent Extension to prestacks The multiplicative structure and duality IndCoh as a symmetric monoidal functor Duality Convolution monoidal categories and algebras Convolution monoidal categories Convolution algebras The pull-push monad Action on a map 20 References 21 Introduction Date: September 30,

2 2 THE!-PULLBACK AND BASE CHANGE 1. Factorizations of morphisms of DG schemes In this section we will study what happens to the notion of the closure of the image of a morphism between schemes in derived algebraic geometry. The upshot is that there is essentially nothing new as compared to the classical case Colimits of closed embeddings. In this subsection we will show that colimits exist and are well-behaved in the category of closed subschemes of a given ambient scheme Recall that a map X Y in Sch is called a closed embedding if the map is a closed embedding of classical schemes. cl X cl Y Let f : X Y be a morphism in Sch. We let denote the full subcategory of Sch X/ /Y where the map f is a closed embedding. We claim: Sch X/, closed in Y consisting of diagrams X X f Y, Proposition Suppose that f is quasi-compact. (a) The category Sch X/, closed in Y has finite colimits (including the initial object). (b) The formation of colimits commutes with Zariski localization on Y. Proof. Step 1. Assume first that Y is affine, Y = Spec(A). Let (1.1) i (X X i be a finite diagram in Sch X/, closed in Y. f i Y ), Set B := Γ(X, O X ). This is a (not necessarily connective) commutative k-algebra. Set also X i = Spec(B i ). Consider the corresponding diagram (1.2) i (A B i B) in ComAlg A/ /B. Set ( B B) := lim i (B i B), where the limit taken in ComAlg /B. Note that we have a canonical map A B, and (A B B) ComAlg A/ /B maps isomorphically to the limit of (1.2) taken in, ComAlg A/ /B. Set ( ) B := τ 0 ( B ) H 0 ( B ) Im H 0 (A) H 0 ( B ), where the fiber product is taken in the category of connective commutative algebras (i.e., it is τ 0 of the fiber product taken in the category of all commutative algebras).

3 THE!-PULLBACK AND BASE CHANGE 3 We still have the canonical maps A B B, and it is easy to see that for X := Spec(B ), the object is the colimit of (1.1). (X X Y ) Sch X/, closed in Y Step 2. To treat the general case it it suffices to show that the formation of colimits in the affine case commutes with Zariski localization. I.e., we need to show that if Y is affine, Y Y is a basic open, then for X := f 1 ( Y ), X i := (φ i ) 1 ( Y ), X := (f ) 1 ( Y ), then the map colim X i X, i is an isomorphism, where the colimit is taken in Sch. X/, closed Y However, the required isomorphism follows from the description of the colimit in Step 1. (The assumption that f be quasi-compact is used in the commutation of the formation of Γ(X, O X ) with localization along Y.) We note the following property of colimits in the situation of Proposition Let g : Y Ỹ be a closed embedding. Set (X X Y ) = colim i where the colimits are taken in Sch X/, closed in Y Consider the composition and the corresponding object (X X i Y ) and (X X Ỹ ) = colim (X X i Ỹ ), i X X Y Ỹ, (X X Ỹ ) Sch X/, closed Ỹ. It is endowed with a compatible family of maps and Sch X/, closed Ỹ, respectively. (X X i Ỹ ) (X X Ỹ ). Hence, by the universal property of (X X Ỹ ), we obtain a canonically defined map (1.3) X X. We claim: Lemma The map (1.3) is an isomorphism. Proof. We construct the inverse map as follows. We note that by the universal property of (X X Ỹ ), we have a canonical map This produces a compatible family of maps and hence the desired map (X X Ỹ ) (X Y Ỹ ). (X X i Y ) (X X Y ), X X.

4 4 THE!-PULLBACK AND BASE CHANGE In the situation of Proposition let us consider the case of X =. We shall denote the resulting category by Sch closed in Y. Thus, Proposition guarantees the existence and compatibility with Zariski localization of finite colimits in Sch closed in Y. Explicitly, if Y = Spec(A) is affine and i Y i Y is a diagram of closed subschemes, Y i = Spec(A i ), then colim Y i = Y, i where ( Y = Spec(A ), A := τ 0 (Ã ) Im A Ã ), Ã := lim A H 0 (Ã ) i i The closure. In this subsection we will define the notion of closure of the image of a morphism of schemes In what follows, in the situation of Proposition 1.1.3, we shall refer to the initial object in the category Sch X/, closed in Y as the closure of X and Y, and denote it by f(x). Explicitly, if Y = Spec(A) is affine, we have: (1.4) f(x) = Spec(A ), A = τ 0 (Γ(X, O X )) Im ( H 0 (A) H 0 (Γ(X, O X )) ). H 0 (Γ(X,O X )) A particular case of Lemma says: Corollary If f : X Y is a closed embedding, then X f(x) is an isomorphism The following property of the operation of taking the closure will be used in the sequel. Let us be in the situation of Proposition 1.1.3, X = X 1 X 2, where X i X are open and set X 12 = X 1 X 2. Denote f i := f Xi. We have a canonical map (1.5) f 1 (X 1 ) f 2 (X 2 ) f(x), f 12(X 12) where the colimit is taken in Sch closed in Y. Lemma The map (1.5) is an isomorphism. Proof. Follows by reducing to the case when Y is affine, and in the latter case by (1.4) We give the following definition: Definition A map f : X Y is said to be a locally closed embedding, if Y contains an open Y Y, such that f defines a closed embedding X Y. We have: Lemma Suppose that f is a quasi-compact locally closed embedding. Then f defines an open embedding of X into f(x). Proof. Follows by combining Corollary and Proposition 1.1.3(b).

5 THE!-PULLBACK AND BASE CHANGE Transitivity of closure. The basic fact established in this subsection, Proposition 1.3.2, will be of crucial technical importance for the proof of Theorem Consider now a diagram where f and g are quasi-compact. X f Y g Z, Set Y := f(x) and g := g Y. By the universal property of closure, we have a canonical map (1.6) g f(x) g (Y ). We claim: Proposition The map (1.6) is an isomorphism. The rest of this subsection is devoted to the proof of this proposition Step 1. As in the proof of Proposition 1.1.3, the assertion reduces to the case when Z = Spec(A) is affine. Assume first that Y is affine as well, Y = Spec(B). Then we have the following descriptions of the two sides in (1.6). Set C := Γ(X, O X ). We have Y := Spec(B ), where B = τ 0 (C) Im ( H 0 (B) H 0 (C) ), H 0 (C) where here and below the fiber product is taken in the category of connective commutative algebras. Furthermore, g f(x) = Spec(A ), where Finally, g (Y ) = Spec(A ), where Note that A = τ 0 (C) Im ( H 0 (A) H 0 (C) ). H 0 (C) A = B Im ( H 0 (A) H 0 (B ) ). H 0 (B ) H 0 (B ) = Im ( H 0 (B) H 0 (C) ) and Im ( H 0 (A) H 0 (B ) ) = Im(H 0 (A) H 0 (C)). The map (1.6) corresponds to the homomorphism A = B Im(H 0 (A) H 0 (B )) = H 0 (B ) = ( ) τ 0 (C) H 0 (C) Im(H 0 (B) H 0 (C)) Im(H 0 (B) H 0 (C)) Im(H 0 (A) H 0 (C)) which is an isomorphism, as required. τ 0 (C) Im(H 0 (A) H 0 (C)) A, H 0 (C) Step 2. Let Y be arbitrary. Choose an open affine cover Y = i Y i and set X i = f 1 (Y i ). Then the assertion of the proposition follows from Step 1 using Lemma

6 6 THE!-PULLBACK AND BASE CHANGE 2. IndCoh as a functor from the category of correspondences This sectionrealizes one of the main goal of our project, namely, the construction of IndCoh as a functor out of the category of correspondences. It will turn out that IndCoh, equipped with the operation of direct image, and left and right adjoints, corresponding to open embeddings and proper morphisms, respectively will uniquely extend to the sought-for formalism of correspondences The category of correspondences. In this subsection we introduce the category of correspondences on schemes and state our main theorem We consider the category Sch aft equipped with the following classes of morphisms: and consider the corresponding category see [Funct, Sect. 8]. vert = all, horiz = all, adm = proper, (Sch aft ) proper corr:all;all, Our goal in this section is to extend the functor IndCoh Schaft of [Book-II.1, Sect. 2.2] to a functor We shall do so in several stages We start with the functor : Sch aft DGCat cont IndCoh (Schaft ) proper : (Sch aft) proper -Cat corr:all;all corr:all;all DGCat2cont. IndCoh Schaft and consider the class of morphisms : Sch aft DGCat cont open all. By [Book-II.1, Proposition 3.2.2], the functor IndCoh Schaft, viewed as a functor Sch aft ( DGCat 2 cont -Cat ) 2 -op, satisfies the left base-change condition. Applying [Funct, Theorem ], we extend IndCoh Schaft to a functor IndCoh (Schaft ) open : (Sch aft) open corr:all;open corr:all;open ( DGCat 2 cont -Cat ) 2 -op. We restrict the latter functor to (Sch aft ) corr:all;open (Sch aft ) open corr:all;open, and denote the resulting functor by IndCoh (Schaft ) corr:all;open, viewed as a functor The main result of this section reads: (Sch aft ) corr:all;open DGCat cont. Theorem There exists a unque extension of the functor IndCoh (Schaft ) corr:all;open functor IndCoh (Schaft ) proper : (Sch aft) proper -Cat corr:all;all corr:all;all DGCat2cont. to a

7 THE!-PULLBACK AND BASE CHANGE Proof of Theorem We start with the following three classes of morphisms vert = all, horiz = all, et = open, adm = proper. We note that the class open proper is that of embeddings of a connected component. This implies that the condition of [Funct, Sect ] holds. The fact that IndCoh (Schaft ) corr:all;open Schaft = IndCoh Schaft Schaft satisfies the left base change condition with respect to the class of proper maps is the content of [Book-II.1, Proposition 5.2.1]. Finally, the fact that the condition of [Funct, Sect ] holds is the content of [Book-II.1, Proposition 5.3.4]. Hence, in order to deduce Theorem from [Funct, Theorem ], it remains to verify that the condition of [Funct, Sect ]. I.e. we need to prove the following: Proposition For a morphism f : X Y in Sch aft, the category Factor(f) of factorizations of f as (2.1) X j Z g Y, where j is an open embedding, and g is proper, is contractible Proof of Proposition Modulo the classical Nagata theorem, the proof will be a simple manipulation with the notion of closure, developed in the previous section Step 1. First we show that Factor(f) is non-empty. By Nagata s theorem, we can factor the morphism red X red Y as red X Z red red Y, where red X Z red is an open embedding and Z red red Y is proper. We define an object of Factor(f) by setting Z := X Z red, red X where we use [Book-III.1, Corollary 1.4.4(a)] for the existence and the properties of push-out in this situation Step 2. Let Factor(f) dense Factor(f) be the full subcategory consisting of those objects for which the map is an isomorphism. We claim that the tautological embedding X j Z g Y, j(x) Z Factor(f) dense Factor(f) admits a right adjoint that sends a given object (2.1) to X j(x) Y.

8 8 THE!-PULLBACK AND BASE CHANGE Indeed, the fact that the map X j(x) is an open embedding follows from Proposition 1.1.3(b). The fact that the above operation indeed produces a right adoint follows from Proposition Hence, it suffices to show that the category Factor(f) dense is contractible. Since it is nonempty (by Step 1), it suffices to show that it contains products Step 3. Given two objects (X Z 1 Y ) and (X Z 2 Y ) of Factor(f) dense consider W := Z 1 Z 2, Y and let h denote the resulting map X W. Set Z := h(x). We claim that the map X Z is an open embedding. Indeed, consider the open subscheme of W W equal to X Y X. By Proposition 1.1.3(b), Z := Z W is the closure of the map X/Y : X X X. Y However, X X/Y (X) is an isomorphism by Corollary Step 4. Finally, we claim that the resulting object X Z Y is the product of X Z 1 Y and X Z 2 Y in Factor(f) dense. Indeed, let X Z Y be another object of Factor(f) dense, endowed with maps to X Z 1 Y and to X Z 2 Y. Let i denote the resulting morphism We have a canonical map in Factor(f) Z Z 1 Z 2 = W. Y (X Z Y ) (X i(z ) Y ). However, from Proposition we obtain that the natural map Z i(z ) is an isomorphism. This gives rise to the desired map (X Z Y ) (X Z Y ). 3. The functor of!-pullback Having defined IndCoh as a functor out of the category of correspondences, restricting to horizontal morphisms, we in particular obtain the functor of!-pullback, which is now defined on all morphisms. In this section we study the basic properties of this functor.

9 THE!-PULLBACK AND BASE CHANGE Definition of the functor. In this subsection we summarize the basic properties of the!-pullback that follow formally from Theorem We let IndCoh! Sch aft denote the restriction of the functor IndCoh (Schaft ) proper to corr:all;all (Sch aft ) op (Sch aft ) proper corr:all;all. In particular, for a morphism f : X Y, we let f! : IndCoh(Y ) IndCoh(X) the resulting morphism. The functor IndCoh! Sch aft is essentially defined by the following two properties: The restriction IndCoh! Sch aft ((Schaft ) proper) op identifies with IndCoh! (Sch aft ) proper. The restriction IndCoh! Sch aft ((Schaft ) open) op identifies with IndCoh (Sch aft ) open. In the above formula, IndCoh (Sch aft ) open := IndCoh (Sch aft ) event-coconn ((Schaft ) op, open) see [Book-II.1, Corollary ], where the functor is introduced. IndCoh (Sch aft ) event-coconn : ((Sch aft ) event-coconn ) op DGCat cont In what follows we shall denote by ω X IndCoh(X) the canonical object equal to where p X : X pt Base change. Let be a Cartesian diagram in Sch aft. Theorem ], we obtain: X 1 p! X(k), g X X2 f 1 f 2 g Y Y 1 Y2 As the main corollary of Theorem 2.1.4, using [Funct, Corollary There exists a canonical isomorphism of functors (3.1) g! Y (f 2 ) IndCoh (f 1 ) IndCoh g! X, compatible with compositions of vertical and horizontal morphisms in the natural sense. Furthermore: (a) Suppose that g Y (and hence g X ) is proper. Then the morphism in (3.1) comes by the, g! )-adjunction from the isomorphism (g IndCoh (f 2 ) IndCoh (g X ) IndCoh (g Y ) IndCoh (f 1 ) IndCoh. (b) Suppose that f 2 (and hence f 1 ) is proper. Then the morphism in (3.1) comes by the, f! )-adjunction from the isomorphism (f IndCoh f! 1 g! Y g! X f! 2. (c) Suppose that g Y (and hence g X ) is an open embedding. Then the morphism in (3.1) comes by the (g!, g IndCoh )-adjunction from the isomorphism (f 2 ) IndCoh (g X ) IndCoh (g Y ) IndCoh (f 1 ) IndCoh.

10 10 THE!-PULLBACK AND BASE CHANGE (d) Suppose that f 2 (and hence f 1 ) is an open embedding. Then the morphism in (3.1) comes by the (f!, f IndCoh )-adjunction from the isomorphism f! 1 g! Y g! X f! 2. Remark The real content of Theorem is that there exists a uniquely defined family of functors f!, that satisfies the properties listed in Corollary and those of Sect Some properties Let IndCoh! Sch denote the restriction aff aft We claim: IndCoh! Sch aft (Sch aff aft )op. Lemma The functor IndCoh! Sch aft is an isomorphism. RKE (Sch aff aft )op (Sch aft ) op(indcoh! Sch aff aft) IndCoh! Sch aft Proof. Follows from the fact that IndCoh! Sch aft satisfies Zariski descent ([Book-II.1, Proposition 4.2.2]), combined with the fact that affine schemes form a basis for the Zariski topology Convergence. Let IndCoh! < Sch ft and IndCoh! < Sch aff ft denote the restrictions of IndCoh! Sch aft to the corresponding subcategories. We claim: Lemma The functors and are isomorphisms. IndCoh! Sch aft RKE ( < Sch ft ) op (Sch aft ) op(indcoh! < Sch ft ) IndCoh! Sch aff aft RKE ( < Sch aff ft )op (Sch aff aft )op(indcoh! < Sch ) aff Proof. Both statements are equivalent to the assertion that for X Sch aft, the functor is an equivalence. IndCoh(X) lim n IndCoh( n X) aft The latter assertion is the content of [Book-II.1, Proposition 6.4.3] h-descent. We will now use proper descent for IndCoh to show that it in fact has h- descent.

11 THE!-PULLBACK AND BASE CHANGE Let C be a category with Cartesian products, and let α be an isomorphism class of 1-morphisms, closed under base change. We define the Grothendieck topology generated by α to be the minimal Grothendieck topology that contains all morphisms from α and has the following 2-out-of-3 property: If X f Y g Z are maps in C such that f and g f are coverings, then so is g. The following is well-known: Lemma Let F be a presheaf on C that satisfies descent with respect to morphisms from the class α. Then F is a sheaf with respect to the Grothendieck topology generated by α We recall that the h-topology on Sch aft is the one generated by the class of proper surjective maps and Zariski covers. From Lemma 3.3.2, combined with [Book-II.1, Propositions and 7.2.2] we obtain: Corollary The functor satisfies h-descent We have: IndCoh! Sch aft : (Sch aft ) op DGCat op Lemma Any fppf covering is an h-covering. Proof. Let f : X Y be an fppf covering. Consider the Cartesian square cl Y X X Y cl Y Y. It suffices to show that cl Y X cl Y is an h-covering. By flatness, cl Y X is classical. Y Hence, we are reduced to an assertion at the classical level, in which case it is well-known. Hence, combining, we obtain: Corollary The functor satisfies fppf-descent. IndCoh! Sch aft : (Sch aft ) op DGCat op 3.4. Extension to prestacks. The functor of!-pullback for arbitrary morphisms of schemes allows to define the category IndCoh on arbitrary prestacks (locally almost of finite type) We consider the category PreStk laft and define the functor IndCoh! PreStk laft : (PreStk laft ) op DGCat cont as the right Kan extension of IndCoh! Sch along the Yoneda functor aff aft (Sch aff aft) op (PreStk laft ) op. According to Lemmas and 3.2.4, we can equivalently define IndCoh! PreStk laft as the right Kan extension of IndCoh! Sch aft, IndCoh! < Sch ft or IndCoh! < Sch aff ft Y

12 12 THE!-PULLBACK AND BASE CHANGE from the corresponding subcategories. For X PreStk laft we let IndCoh(X) denote the value of IndCoh! PreStk laft on it. For a morphism f : X 1 X 2 we let denote the corresponding functor. f! : IndCoh(X 2 ) IndCoh(X 1 ) For X PreStk laft, we let ω X where p X : X pt. IndCoh(X) denote the canonical object equal to p! X (k), We now consider the category where sch & proper (PreStk laft ) corr:sch & qc;all, sch & qc and sch & proper signify the classes of schematic and quasi-compact (resp., schematic and proper) morphisms between prestacks. We claim: Theorem There exists a uniquely defined functor IndCoh sch & proper (PreStklaft ) corr:sch & qc;all equipped with isomorphisms and sch & proper -Cat : (PreStk laft ) corr:sch & qc;all DGCat2cont, IndCoh sch & proper (PreStklaft ) corr:sch & qc;all (PreStklaft ) op IndCoh! PreStk laft IndCoh sch & proper (PreStklaft ) (Schaft ) proper IndCoh corr:sch & qc;all corr:all;all (Sch aft ) proper, corr:all;all where the latter two isomorphisms are compatible in a natural sense. Proof. We consider the functor and define (Sch aft ) proper corr:all;all (PreStk laft) sch & proper corr:sch & qc;all, IndCoh sch & proper (PreStklaft ) corr:sch & qc;all to be the right Kan extension of IndCoh (infschlaft ) proper corr:all;all what this means). (see [Funct, 1-categorical RKE]) for To prove the theorem, it suffices to show that for X PreStk laft, the natural map is an equivalence. IndCoh sch & proper (PreStklaft ) (X) IndCoh! PreStk corr:sch & qc;all laft (X) However, the latter is given by [Funct, Horizontal Extension].

13 THE!-PULLBACK AND BASE CHANGE The actual content of Theorem can be summarized as follows: First, for any schematic morphism f : X Y we have a well-defined functor f IndCoh : IndCoh(X) IndCoh(Y). Furthermore, if f : X Y is schematic and proper, the functor f IndCoh is the left adjoint of f!. When Y is a scheme (and hence X is one as well), the above functor f IndCoh defined in this case, and similarly for the (f IndCoh, f! )-adjunction. f IndCoh (3.2) Second, let X 1 g X X2 f 1 f 2 Y 1 g Y Y2 is the usual be a Cartesian diagram in PreStk laft, where the vertical maps are schematic. Then we have a canonical isomorphism of functors (3.3) g! Y (f 2 ) IndCoh (f 1 ) IndCoh g! X, compatible with compositions. Furthermore, if the vertical (resp., horizontal) morphisms are proper (resp., schematic and proper), the map in (3.3) comes by adjunction in a way similar to Corollary 3.1.4(a) (resp., Corollary 3.1.4(b)) In [Book-III.3, Proposition 5.3.6], we will show that for a morphism f : X Y, which is an open embedding, the functor f IndCoh is the right adjoint of f!. Furthermore, if in the Cartesian diagram (3.2) the vertical (resp., horizontal) morphisms are open embeddings, the map in (3.3) comes by adjunction in a way similar to Corollary 3.1.4(c) (resp., Corollary 3.1.4(d)) For future use, we note that the statement and proof of [Book-II.1, Proposition 7.2.2] for groupoid objects in (PreStk laft ) sch & proper. 4. The multiplicative structure and duality In this section we will show that the functor IndCoh, when viewed as a functorout of the category of correspondences, and equipped with a natural symmetric monoidal structure, encodes Serre duality IndCoh as a symmetric monoidal functor. In this subsection we show that the functor IndCoh possesses a natural symmetric monoidal structure We recall that by [Book-II.1, Proposition 6.3.6], the functor IndCoh Schaft carries a natural symmetric monoidal structure. : Sch aft DGCat cont Applying [Funct, Multiplicative structure on correspondences], we obtain: Theorem The functor IndCoh (Schaft ) proper : (Sch aft) proper -Cat corr:all;all corr:all;all DGCat2cont carries a canonical symmetric monoidal structure that extends one on IndCoh Schaft.

14 14 THE!-PULLBACK AND BASE CHANGE In particular, we obtain that the functors and IndCoh (Schaft ) corr:all;all : (Sch aft ) corr:all;all DGCat cont IndCoh! Sch aft : (Sch aft ) op DGCat cont both carry natural symmetric monoidal structures Note that the symmetric monoidal structure on IndCoh! Sch aft automatically upgrades the functor IndCoh! Sch aft to a functor (4.1) (Sch aft ) op DGCat SymMon cont, due to the fact that the identity functor on (Sch aft ) op naturally lifts to a functor via the diagonal map. (Sch aft ) op ComAlg ((Sch aft ) op ) Explicitly, the monoidal operation on IndCoh(X) is given by IndCoh(X) IndCoh(X) IndCoh(X X)! X IndCoh(X). We shall denote the above monoidal operation by! : F 1, F 2 IndCoh(X) F 1! F 2 IndCoh(X). The unit for this symmetric monoidal structure is given by ω X IndCoh(X) Applying the functor of right Kan extension along (Sch aft ) op (PreStk laft ) op of the functor (4.1), we obtain that the functor (4.2) IndCoh! PreStk laft : (PreStk laft ) op DGCat cont naturally upgrades to a functor (4.3) IndCoh! PreStk laft : (PreStk laft ) op DGCat SymMon cont. As in [Book-II.1, Proposition 6.3.6], the datum of the functor (4.3) is equivalent to that of right-lax symmetric monoidal structure on the functor (4.2) The above right-lax symmetric monoidal structure on IndCoh! PreStk laft can be enhanced: Indeed, applying [Funct, Horizontal extension monoidal], we obtain that the functor IndCoh sch & proper (PreStklaft ) corr:sch & qc;all carries a canonical right-lax symmetric monoidal structure. sch & proper -Cat : (PreStk laft ) corr:sch & qc;all DGCat2cont 4.2. Duality. In this subsection we will formally deduce Serre duality for schemes from the symmetric monoidal structure on IndCoh.

15 THE!-PULLBACK AND BASE CHANGE Let O be a symmetric monoidal category, and let O dualizable O be the full subcategory spanned by dualizable objects. This subcategory carries a canonical symmetric monoidal antiinvolution ( O dualizable ) op dualization O dualizable, given by the passage to the dual object. o o. Recall now that if F : O 1 O 2 is a symmetric monoidal functor between symmetric monoidal categories, then it maps and the following diagram commutes O dualizable 1 O dualizable 2, (O dualizable 1 ) op F op dualization O dualizable 1 (O dualizable F 2 ) op dualization O dualizable Recall now that by [Funct, Anti-involution on Corr] the category (Sch aft ) corr:all;all carries a canonical anti-involution ϖ, which is the identity on objects, and at the level of 1-morphisms is maps a 1-morphism f X 12 X 1 g to X 2 X 12 f g X 2 X 1. Moreover, by [Funct, Anti-involution on Corr], we have: Theorem The inclusion ((Sch aft ) corr:all;all ) dualizable (Sch aft ) corr:all;all is an isomorphism. The anti-involution ϖ identifies canonically with the dualization functor ( ((Sch aft ) corr:all;all ) dualizable) op ((Schaft ) corr:all;all ) dualizable Combining Theorem with Theorem we obtain: Theorem We have the following commutative diagram of functors (IndCoh (Schaft )corr:all;all ) op ( ((Sch aft ) corr:all;all ) op ϖ ) op DGCat dualizable cont dualization (Sch aft ) corr:all;all IndCoh (Schaft ) corr:all;all DGCat dualizable cont.

16 16 THE!-PULLBACK AND BASE CHANGE Let us explain the concrete meaning of this theorem. It says that for X Sch aft there is a canonical equivalence : IndCoh(X) IndCoh(X), D Serre X and for a map f : X Y an isomorphism where (f! ) is viewed as a functor (f! ) f IndCoh, IndCoh(X) (DSerre X ) 1 IndCoh(X) (f! ) IndCoh(Y ) DSerre Y IndCoh(Y ) Let us write down explicitly the unit and co-unit for the identification D Serre X : The co-unit, denoted ɛ X is given by IndCoh(X) IndCoh(X) IndCoh(X X)! X IndCoh(X) (p X ) IndCoh IndCoh(pt) = Vect, where p X : X pt. The unit, denoted µ X is given by Vect = IndCoh(pt) p! X IndCoh(X) ( X ) IndCoh IndCoh(X X) IndCoh(X) IndCoh(X). Remark One does not need to rely on Theorems and in order to show that the maps µ X and ɛ X, defined above, give rise to an identification Indeed, the fact that the composition IndCoh(X) Id IndCoh(X) µ X IndCoh(X) IndCoh(X). IndCoh(X) IndCoh(X) IndCoh(X) ɛ X Id IndCoh(X) is isomorphic to the identity functor follows by base change from the diagram IndCoh(X) X X X id X p X X X X id X X X X p X id X X, X id X X X X and similarly for the other composition. A similar diagram chase implies the isomorphism (f! ) f IndCoh. Remark Let us also note that one does not need the (difficult) Theorem either in order to construct the pairing ɛ X : Indeed, both functors involved in ɛ X, namely,! X and (p X) IndCoh are elementary. If one believes that the functor ɛ X defined in the above way is the co-unit of a duality (which is a property, and not an extra structure), then one recover the object ω X IndCoh(X). Namely, ω X := (p X id X ) IndCoh (µ X (k)).

17 THE!-PULLBACK AND BASE CHANGE Relation to the usual Serre duality. By passage to compact objects, the equivalence gives rise to an equivalence D Serre X D Serre X : IndCoh(X) IndCoh(X) : (Coh(X)) op Coh(X). is the usual Serre duality anti- It is shown in [GL:IndCoh, Proposition 8.3.5] that D Serre X equivalence of Coh(X), given by internal Hom into ω X. 5. Convolution monoidal categories and algebras In this section we will apply the the formalism of IndCoh as a functor out of the category of correspondences to carry out the following constructions and its generalizations: Let R X be a groupoid-object in the category of schemes. Then the category IndCoh(R) has a natural monoidal structure, given by convolution, and as such it acts on the category IndCoh(X). The contents of this section were suggested to us by S. Raskin Convolution monoidal categories. In this sections we will show that IndCoh of a groupoid-object (and, more generally, category-object) carries a natural monoidal structure. We note that this construction only uses the underlying 1-category of the category of correspondences Let X be a category-object in PreStk laft. I.e., this is a simplicial object in PreStk laft, such that for any S < Sch aff, the object is a complete Segal space. Maps(S, X ) Spc op We denote X := X 0 and R := X 1. Let us assume that the target map p t : R X and the composition map R R R t,x,s are schematic and quasi-compact Note that by [Funct, Algebras in corr], the object R acquires a natural structure of algebra in the (symmetric) monoidal category and as such it acts on X. Applying the (symmetric) monoidal functor (PreStk laft ) corr:sch & qc;all, IndCoh (PreStklaft ) corr:sch & qc;all : (PreStk laft ) corr:sch & qc;all DGCat cont, we obtain that the DG category IndCoh(R) acquires a structure of monoidal DG category, and IndCoh(X) acquires a structure of IndCoh(R). Unwinding the definitions, we obtain that the binary operation on IndCoh(R) is given by the convolution product, i.e., pull-push along the diagram R R R R X R.

18 18 THE!-PULLBACK AND BASE CHANGE Consider a particular case when X = X Sch aft, and X n = X n. So R = X X. We obtain that IndCoh(X X) acquires a structure of monoidal category, and as such it acts on IndCoh(X). I.e., we obtain a monoidal functor (5.1) IndCoh(X X) Funct cont (IndCoh(X), IndCoh(X)). It is easy to see that as a functor of plain categories, (5.1) identifies with IndCoh(X X) IndCoh(X) IndCoh(X) (DSerre X ) 1 Id IndCoh(X) IndCoh(X) Funct cont (IndCoh(X), IndCoh(X)). In particular, the functor (5.1) is an equivalence of monoidal categories We note that the above discussion goes through as is with IndCoh replaced by QCoh. We note, however, that for the subsequent discussion that involves a 2-categorical structure, if one considers QCoh instead of IndCoh, the direction of the 2-morphisms must be reversed. In particular, what are algebras and monads for IndCoh become co-algebras and co-monads for QCoh Convolution algebras. In this subsection we will show how certain maps between categoryobjects give rise to lax monoidal functors between the corresponding convolution categories Let now f : X 1 X 2 be a map between category-objects in PreStk laft. Assume that the map is schematic and proper. R 1 X 1 R 1 R 1 R 2 (R 2 X 2 R 2 ) For example, let X 1 (resp., X 1) be the Čech nerve of a map X 1 Y 1 (resp., X 2 Y 2 ). Let us be given a commutative (but not necessarily Cartesian) diagram X 1 X 2 Y 1 Y 2. Then the above condintion is satisfied if the map is schematic and proper. X 1 Y 1 Y 2 X 2

19 THE!-PULLBACK AND BASE CHANGE In the above situation, by [Funct, Maps of algs in corr], the map R 1 R 2 defines a right-lax map of algebra objects R 2 R 1 (note the direction of the arrow!) in given by the diagram sch & proper (PreStk laft ) corr:sch & qc;all, R 1 R 2 id R 1. Applying the (symmetric) monoidal functor IndCoh (PreStklaft ) corr:sch & qc;all : (PreStk laft ) corr:sch & qc;all DGCat cont, we obtain a right-lax monoidal functor In particular, the image of the unit f! : IndCoh(R 2 ) IndCoh(R 1 ). f! (unit R2 ) IndCoh (ω X2 ) IndCoh(R 1 ) acquires a structure of algebra in the monoidal category IndCoh(R 1 ) The pull-push monad. In this subsection we will show that for a category-object with a proper multiplication, its dualizing complex has a natural algebra structure in the convolution category Let us consider a particular case when X 2 = pt. Denote X 1 =: X, so our condition is that the composition map R R R X be proper. Let M R denote the monad acting on X (where X is viewed as an object of the 2-category sch & proper (PreStk laft ) corr:sch & qc;all ), obtained from the action on R on X and the right-lax monoidal map pt R By Sect , ω R IndCoh(R) acquires a structure of algebra. Hence, the action of ω R defines a monad on X. It is easy to see that this monad, viewed as a plain endo-functor, identifies with (p t ) IndCoh (p s )!. By definition, the above monad on IndCoh(X) can also be viewed as one obtained by applying the functor IndCoh (PreStklaft ) corr:sch & qc;all to the monad M R Assume now that X is a groupoid object of PreStk laft, with the maps p s, p t : R X being proper. In then according to Sect , the endo-functor (p t ) IndCoh (p s )! acqures a (a priori different) structure of monad. We claim, however that the above two ways of introducing a structure of monad on (p t ) IndCoh (p s )! coincide. Indeed, this follows from Sect for Y := X.

20 20 THE!-PULLBACK AND BASE CHANGE 5.4. Action on a map. In this subsection we will show that the pull-push monad from the previous subsection is canonically isomorphic to one from [Book-II.1, Proposition 7.2.2(a)] We retain the assumptions of Sect Let Y be an object of PreStk laft, and let us assume that X is a category-object over Y. Let g denote the map X Y. Assume now that the map p t : R X is schematic and proper. In this case, by [Funct, Modules of algs in corr], the monad M R, introduced above acts on the 1-morphism sch & proper (Y X) (PreStk laft ) corr:sch & qc;all (not the direction of the arrow!), given by the diagram id X X. g Y Applying the functor IndCoh (PreStklaft ) corr:sch & qc;all, we obtain that the functor upgrades to a functor g! : IndCoh(Y) IndCoh(X) (5.2) (g! ) enh : IndCoh(Y) ((p t ) IndCoh (p s )! )-mod(indcoh(x)) Assume now that g : X Y is schematic and proper, and that X equals the Čech nerve of g. In this case, the functor (5.2) defines a map of monads (5.3) ((p t ) IndCoh (p s )! ) (g! g IndCoh ), and it is easy to see that at the level of plain endo-functors, the map (5.3) is given by base change along p t X X X Y p x g g X Y. In particular, the map (5.3) is an isomorphism of monads.

21 THE!-PULLBACK AND BASE CHANGE 21 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, arxiv: [GL:IndSch] D. Gaitsgory and N. Rozenblyum, DG indschemes, arxiv: [GL:Crystals] D. Gaitsgory and N. Rozenblyum, Crystals. [Book-II.1] [Book-III.1] [Book-III.2] [Book-III.3] [Funct] [InfSch] [Lu0] J. Lurie, Higher Topos Theory, Princeton Univ. Press (2009). [Lu1] J. Lurie, DAG-VIII, available at lurie. [Lu?] J. Lurie, 2-Categories, available at lurie.