PART III.3. INDCOHERENT SHEAVES ON INDINFSCHEMES


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1 PART III.3. INDCOHERENT SHEAVES ON INDINFSCHEMES Contents Introduction 1 1. Indcoherent sheaves on indschemes Basic properties tstructure Recovering IndCoh from indproper maps Direct image for IndCoh on indschemes 5 2. Proper base change for indschemes st version nd version Groupoids and descent IndCoh on (ind)infschemes Nil base change Basic properties Indnilproper descent tstructure for indinfschemes The direct image functor for indinfschemes Recovering from nilclosed embeddings Recovering from nilisomomorphisms Constructing the direct image functor Proof of Theorem Base change Extending the formalism of correspondences to infschemes Setup for extension Enlarging 2morphisms to indnilproper maps Extending to prestacks Open embeddings Selfduality and multiplicative structure of IndCoh on indinfschemes The multiplcative structure Duality Convolution categories and algebras 33 References 35 Introduction Date: June 19,
2 2 INDCOHERENT SHEAVES ON INDINFSCHEMES 1. Indcoherent sheaves on indschemes In order to develop the theory of IndCoh on indinfschemes, we first need to do this for indschemes. The latter theory follows rather easily from one on schemes. In this section we will mainly review the results from [GL:IndSch], Sect Basic properties. In this subsection we will express the category IndCoh on an indscheme X from that on schemes equipped with a closed embedding into X Let IndCoh! indsch laft denote the restriction of the functor IndCoh! PreStk laft to the full subcategory (indsch laft ) op (PreStk laft ) op. In particular, for X indsch laft we have a welldefined category IndCoh(X) Suppose X has been written as (1.1) X conv X X colim a A X a, where X a Sch aft with the maps i a,b : X a X b being closed embeddings. In this case we have: Proposition Under the above circumstances,!restriction defines an equivalence IndCoh(X) lim a A op IndCoh(Xa ), where for a b, the corresponding functor IndCoh(X b ) IndCoh(X a ) is i! a,b. Proof. This follows from the convergence property of the functor IndCoh! Sch, see [BookII.2, aff aft Lemma and Sect ]. Remark The reason we exhibit an indscheme X as conv (colim X a) rather than just as X a A a is that the former presentation comes up in practice more often: many indschemes colim a A are given in this form. The fact that the resulting prestack is indeed an indscheme (i.e., can be written as a colimit of schemes under closed embeddings) is [BookIII.2, Corollary 1.4.4] and is somewhat nontrivial Combining the above proposition with [GL:DG, Lemma 1.3.3], we obtain: Corollary For X written as in (1.1), we have IndCoh(X) colim a A IndCoh(Xa ), where for a b, the corresponding functor IndCoh(X a ) IndCoh(X b ) is (i a,b ). Corollary For X indsch laft and a closed embedding i : X X, the functor left adjoint to i!, is welldefined. i IndCoh : IndCoh(X) IndCoh(X), For X indsch laft, let Coh(X) denote the full subcategory of IndCoh(X) spanned by objects i IndCoh (F), i : X X is a closed embedding and F Coh(X). From Corollary we obtain:
3 INDCOHERENT SHEAVES ON INDINFSCHEMES 3 Corollary For an indscheme X, the category IndCoh(X) is compactly generated and IndCoh(X) c = Coh(X). Proof. Follows from [DrGa, Corollary and Lemma ] Here is another convenient fact about the category IndCoh(X), where X indsch laft. Let X i X i X be closed embeddings. We would like to calculate the composition (i )! (i ) IndCoh : IndCoh(X ) IndCoh(X ). Let A denote the category (Sch aft ) closed in X, so that X and X correspond to indices a and a, respectively. Let i a and i a denote the corresponding closed embeddings, i.e., the maps i and i, respectively. Let B be any category cofinal in A a a / := A a / A a /. A For b B, let X i a,b i a,b = X a Xb Xa = X denote the corresponding maps. The next assertion results from [GL:DG, Sect ]: Lemma Under the above circumstances, we have a canonican isomorphism (i )! (i ) IndCoh colim (i a,b)! (i a,b) IndCoh. b B 1.2. tstructure. In this subsection we will study the naturally defined tstructure on IndCoh of an indscheme For X indsch laft we introduce a tstructure on the category IndCoh(X) as follows: An object F IndCoh(X) belongs to IndCoh(X) 0 if and only if for every closed embedding i : X X, where X Sch aft, we have i! (F) IndCoh(X) 0. By construction, this tstructure is compatible with filtered colimits, which by definition means that IndCoh(X) 0 is preserved by filtered colimits We can describe this tstructure and the category IndCoh(X) 0 more explicitly. Write where X a ( cl Sch aft ) closed in X. cl X colim a A X a, For each a, let i a denote the corresponding map (automatically, a closed embedding) X a X. By Corollary 1.1.7, we have a pair of adjoint functors (i a ) IndCoh : IndCoh(X a ) IndCoh(X) : i! a. Lemma Under the above circumstances we have: (a) An object F IndCoh(X) belongs to IndCoh 0 if and only if for every a, the object i! a(f) IndCoh(X a ) belongs to IndCoh(X a ) 0. (b) The category ( IndCoh(X) 0 is generated under colimits by the essential images of the functors (i a ) IndCoh Coh(Xa ) 0).
4 4 INDCOHERENT SHEAVES ON INDINFSCHEMES Proof. It is easy to see that for a quasicompact DG scheme X, the category IndCoh(X) 0 is generated under colimits by Coh( cl X) 0. In particular, by adjunction, an object F IndCoh(X) is coconnective if and only if its restriction to cl X is coconnective. Hence, in the definition of IndCoh(X) 0, instead of all closed embeddings X X, it suffices to use only those with X a classical scheme. Note that the category A is cofinal in ( cl Sch aft ) closed in X. This implies point (a) of the lemma. Point (b) follows formally from point (a) Suppose i : X X is a closed embedding of a scheme into an indscheme. By Corollary 1.1.7, we have a welldefined functor which is the right adjoint to i!. However, we claim: i IndCoh Lemma The functor i IndCoh : IndCoh(X) IndCoh(X), Since i! is left texact, the functor i IndCoh is texact. is right texact. Proof. We need to show that for F IndCoh(X) 0, and a closed embedding i : X X, we have (i )! i IndCoh (F) IndCoh(X ) 0. This follows from Lemma : in the notations of loc.cit., each of the functors (i a,b) IndCoh is texact (because i a,b is a closed embedding), each of the functors (i a,b)! is left texact (because i a,b is a closed embedding), and the category B is filtered. Corollary The subcategory is preserved by the truncation functors. Proof. Follows from Lemma Coh(X) = IndCoh(X) c Corollary The tstructure on IndCoh(X) is obtained from the tstructure on Coh(X) by the procedure of [BookII.1, Lemma 1.2.4] Recovering IndCoh from indproper maps. The contents of this subsection are rather formal: we show that the functor IndCoh on indschemes can be recovered from the corresponding functor on schemes, where we restrict 1morphisms to be proper, or even closed embeddings. This is not surprising, given the definition of indschemes Recall what it means for a map in PreStk to be indproper (resp., indfinite, indnilclosed embedding, indclosed embedding), see [BookIII.2, Definitions and ] Consider the corresponding 1full subactegories (indsch laft ) indclosed (indsch laft ) indnilclosed (indsch laft ) indfinite (indsch laft ) indproper and the corresponding categories (Sch aft ) closed (Sch aft ) nilclosed (Sch aft ) finite (Sch aft ) proper. Consider the corresponding fully faithful embeddings (Sch aft ) closed (indsch laft ) indclosed, (Sch aft ) nilclosed (indsch laft ) indnilclosed, (Sch aft ) finite (indsch laft ) indfinite
5 INDCOHERENT SHEAVES ON INDINFSCHEMES 5 and (indsch laft ) proper (indsch laft ) indproper. Let IndCoh! (Sch aft ) proper denote the functor IndCoh! Sch aft ((Schaft ) proper) op : ((Sch aft) proper ) op DGCat cont, and similarly, for proper replaced by nilclosed, finite or closed. Let IndCoh! (indsch laft ) indproper denote the functor IndCoh! indsch laft ((indschlaft ) indproper ) op : ((indsch laft) indproper ) op DGCat cont, and similarly, for proper replaced by finite, nilclosed or closed We claim: Proposition The naturally defined functors ( ) IndCoh! (indsch laft ) indproper RKE ((Schaft ) proper) op ((indsch laft ) indproper ) op IndCoh! (Sch aft ) proper, ( ) IndCoh! (indsch laft ) indfinite RKE ((Schaft ) finite ) op ((indsch laft ) indfinite ) op IndCoh! (Sch aft ) finite, and IndCoh! (indsch laft ) indnilclosed RKE ((Schaft ) nilclosed ) op ((indsch laft ) indnilclosed ) op ( IndCoh! (Sch aft ) nilclosed ) IndCoh! (indsch laft ) indclosed RKE ((Schaft ) closed ) op ((indsch laft ) indclosed ) op ( IndCoh! (Sch aft ) closed ) are isomorphisms. Proof. By [BookIII.2, Corollary 1.7.5(b)], for X indsch laft, the functor (Sch aft ) closed in X (Sch aft ) closed (indsch laft ) indclosed ((indsch laft ) indclosed ) /X (Sch aft ) /X is cofinal. This implies that the functor IndCoh! (indsch laft ) indclosed RKE ((Schaft ) closed ) op ((indsch laft ) indclosed ) op ( IndCoh! (Sch aft ) closed ) is an isomorphism. The other two isomorphisms follow from the fact that the functors (Sch aft ) closed (indsch laft ) indclosed ((indsch laft ) indclosed ) /X are fully faithful. (Sch aft ) nilclosed (indsch laft ) indnilclosed ((indsch laft ) indnilclosed ) /X (Sch aft ) finite (indsch laft ) indfinite ((indsch laft ) indfinite ) /X (Sch aft ) proper (indsch laft ) indproper ((indsch laft ) indproper ) /X 1.4. Direct image for IndCoh on indschemes. In this subsection we show how to construct the functor of direct image on IndCoh for morphisms between indschemes.
6 6 INDCOHERENT SHEAVES ON INDINFSCHEMES We now consider the functor IndCoh Schaft : Sch aft DGCat cont, where for a morphism f : X 1 X 2 in Sch aft, the functor is f IndCoh, see [BookII.1, Sect. 2.2]. Recall the notation IndCoh(X 1 ) IndCoh(X 2 ) IndCoh (Schaft ) proper = IndCoh Schaft (Schaft ) proper : (Sch aft ) proper DGCat cont, and let consider also the corresponding functors IndCoh (Schaft ) finite : (Sch aft ) finite DGCat cont, and Denote and let IndCoh (Schaft ) nilclosed : (Sch aft ) nilclosed DGCat cont IndCoh (Schaft ) closed : (Sch aft ) closed DGCat cont. IndCoh indschlaft := LKE (Schaft ) (indsch laft )(IndCoh (Schaft )), and IndCoh (indschlaft ) indproper, IndCoh (indschlaft ) indnilclosed, IndCoh (indschlaft ) indfinite IndCoh (indschlaft ) indclosed denote its restriction to the corresponding 1full subcategories. The same proof as in the case of Proposition gives: Proposition The natural maps LKE (Schaft ) proper (indsch laft ) indproper (IndCoh (Schaft ) proper ) IndCoh (indschlaft ) indproper, LKE (Schaft ) finite (indsch laft ) indfinite (IndCoh (Schaft ) finite ) IndCoh (indschlaft ) indfinite, LKE (Schaft ) nilclosed (indsch laft ) indnilclosed (IndCoh (Schaft ) nilclosed ) IndCoh (indschlaft ) indnilclosed and LKE (Schaft ) closed (indsch laft ) indclosed (IndCoh (Schaft ) closed ) IndCoh (indschlaft ) indclosed are isomorphisms Recall from [Funct, Sect. 11.2] the notion of two functors obtained from each other by passing to adjoints. The following is a particular case of [Funct, Corollary ]: Let F : C 1 C 2 be a functor between categories. Let Φ 1 : C 1 DGCat cont be a functor such that for every c 1 c 1, the corresponding functor Φ 1 (c 1) Φ 1 (c 1) admits a right adjoint. Let Ψ 1 : C op 1 DGCat cont be the resulting functor given by taking the right adjoints. Let Φ 2 and Ψ 2 be the left (resp., right) Kan extension of Φ 1 (resp., Ψ 1 ) along F (resp., F op ). We have: Lemma Under the above circumstances, the functor Ψ 2 is obtained from Φ 2 by taking right adjoints.
7 INDCOHERENT SHEAVES ON INDINFSCHEMES We apply Lemma in the following situation: and F to be the natural embedding. We take C 1 := (Sch aft ) proper, C 2 := (indsch laft ) indproper, Φ 1 := IndCoh (Schaft ) proper and Ψ 1 := IndCoh! (Sch aft ) proper. These two functors are obtained from one another by passage to adjoints, by the definiton of the functor IndCoh! (Sch aft ) proper, see [BookII.1, Corollary ]. Corollary The functor is obtained from the functor by passing to right adjoints. IndCoh! (indsch laft ) indproper : ((indsch laft ) indproper ) op DGCat cont IndCoh (indschlaft ) indproper : (indsch laft ) indproper DGCat cont Remark Let us informally explain the concrete meaning of this corollary. Namely, it says that for any morphism f : X 1 X 2 between objects of indsch laft, there is a welldefined functor f IndCoh : IndCoh(X 1 ) IndCoh(X 2 ). Moreover, when f is indproper, this functor is the left adjoint of the functor f! We can now make the following observation pertaining to the behavior of the tstructure with respect to direct images: Lemma Let f : X 1 X 2 be a map of indschemes. Then the functor f IndCoh texact. Furthermore, if f is indaffine, then it is texact. is left Proof. Let F IndCoh(X 1 ) 0. We wish to show that f IndCoh (F) IndCoh(X 2 ) 0. By Corollary 1.2.7, we can assume that F = (i 1 ) IndCoh (F 1 ) for F 1 IndCoh(X 1 ) 0 where is a closed embedding of a scheme. Now, let i 1 : X 1 X 1 X 1 g X2 i 2 X2 be a factorization of f i 1, where i 2 is a closed embedding of a scheme. Thus, it suffices to show that the functor f IndCoh (i 1 ) IndCoh (i 2 ) IndCoh g IndCoh is left texact. However, (i 2 ) IndCoh since g is a map between schemes. is texact by Lemma 1.2.5, while g IndCoh (F 1 ) is left texact, Now, suppose that f is indaffine. In this case, we wish to show that f IndCoh is also right t exact. Let F IndCoh(X 1 ) 0. We can assume that F = (i 1 ) IndCoh (F 1 ) for F 1 IndCoh(X 1 ) 0 where i 1 : X 1 X 1 is a closed embedding. In the above notations, it suffices to show that is texact. f IndCoh (i 1 ) IndCoh (i 2 ) IndCoh g IndCoh By Lemma 1.2.5, (i 2 ) IndCoh is texact. Hence, it suffices to show that g IndCoh However, g is an affine map between schemes, and the assertion follows. is texact.
8 8 INDCOHERENT SHEAVES ON INDINFSCHEMES 2. Proper base change for indschemes Base change for IndCoh is a crucial property needed for its definition as a functor out of the category of correspondences. In this section we make two (necessary) preparatory steps, establsihing base change for morphisms between indschemes st version Recall the notion of indschematic map in PreStk, see [BookIII.2, Defintion 1.6.5(a)]. Let X 1 f g 1 X1 f X g 2 2 X2, be a Cartesian diagram in PreStk laft with f being indschematic and indproper. We claim: Proposition The functors f! and (f )! admit left adjoints, to be denoted f IndCoh (f ) IndCoh, respectively. The natural transformation (2.1) (f ) IndCoh arrising by adjunction from is an isomorphism. g! 1 g! 2 f IndCoh, g! 1 f! (f )! g! 2, The rest of this subsection is devoted to the proof of this proposition We begin by making the following observation. Let G : C 2 C 1 be a functor between  categories. Let A be a category of indices, and suppose we are given an Afamily of commutative diagrams and C a 1 G a i a 1 C 1 G C a i a 2 2 C 2. Assume that for each a A, the functor G a admits a left adjoint F a. Furthermore, assume that for each map a a in A, the diagram C a 1 i a,a 1 C a 1 F a F a C a 2 i a,a 2 C a 2, which a priori commutes up to a natural transformation, actually commutes. Finally, assume that the functors are equivalences. Under the above circumstances we have: C 1 lim a A C a 1 and C 2 lim a A C a 2
9 INDCOHERENT SHEAVES ON INDINFSCHEMES 9 Lemma The functor G admits a left adjoint, and for every a A, the diagram C a 1 F a i a 1 C 1 F C a 2 i a 2 C 2, which a priori commutes up to a natural transformation, commutes Applying Lemma 2.1.4, we obtain that the assertion of Proposition reduces to the case when X 2 = X 2 Sch aft and X 2 = X 2 Sch aft. In this case X 1, X 1 indsch laft. Write X 1 colim X a, a A where X a Sch aft and i a : X a X 1 are closed embeddings. Set X a := X 2 X a. X 2 We have: X 1 colim a A X a, Let i a denote the correspoding closed embedding X a X 1, and let g a denote the map X a X a. Note that the maps f i a : X a X 2 and f i a : X a X 2 are proper, by assumption By Corollary 1.1.6, we have: Id IndCoh(X1) colim (i a) IndCoh a A Hence, we can rewrite the lefthand side in (2.1) as and the righthand side as (i a )! and Id IndCoh(X 1 ) colim a A (i a) IndCoh (i a)!. colim (f ) IndCoh (i a) IndCoh (i a)! g 1,! a A colim a A g! 2 f IndCoh (i a ) IndCoh (i a )!. It follows from the construction that the map in (2.1) is given by a compatible system of maps for each a A (f ) IndCoh (i a) IndCoh (i a)! g! 1 (f i a) IndCoh (f i a) IndCoh where the arrow (f i a) IndCoh is base change for the Cartesian square (g 1 i a)! (i a g a )! (f i a) IndCoh g! 2 (f i a ) IndCoh g! a i! a i! a g! 2 f IndCoh g! a g! 2 (f i a ) IndCoh (i a ) IndCoh (i a )!, f i a X a g a Xa f i a X 2 g 2 X2.
10 10 INDCOHERENT SHEAVES ON INDINFSCHEMES Hence, the required isomorphism follows from proper base change in the case of schemes, see [BookII.2, Proposition 3.1.4(b)] nd version Let now X 1 f g 1 X1 f X g 2 2 X2 be a Cartesian diagram in indsch laft, and assume now that the map g 2, and hence g 1, are indproper. From the isomorphism (g 2 ) IndCoh (f ) IndCoh by adjunction, we obtain a natural transformation: (2.2) (f ) IndCoh We claim: f IndCoh g! 1 g! 2 f IndCoh. Proposition The map (2.2) is an isomorphism. (g 1 ) IndCoh, The rest of this subsection is devoted to the proof of the proposition By Corollary 1.1.6, we need to show that (2.2) becomes an isomorphism after precomposing both sides with the functor (i 1 ) IndCoh for a closed embedding X 1 X 1 with X 1 Sch aft. This allows to assume that X 1 = X 1 Sch aft. By [BookIII.2, Corollary 1.7.5(b)], we can factor the map X 1 X 2 as a composition X 1 X 1 X 2, where X 1 Sch aft and X 1 X 2 is a closed embedding. Hence, we can consider separately the following two cases: (I) The case when X 1 = X 1 Sch aft and X 1 X 2 is a closed embedding. (II) The case when X 1 = X 1 Sch aft and X 2 = X 2 Sch aft We note that that in Case (I), the map f is indproper. Moreover, the map (2.2) equals the map (2.1). Hence, it is an isomorphism by Proposition The proof of Case (II) essentially repeats the argument in Sects We include it for completeness. Write X 2 colim a A X 2,a, where the maps i 2,a : X 2,a X 2 are closed embeddings. In this case where X 1 colim a A X 1,a, X 1,a X 2,a X 2 X 1.
11 INDCOHERENT SHEAVES ON INDINFSCHEMES 11 For every a denote by f a the corresponding map X 1,a X 2,a, and by i 1,a : X 1,a X 1 the corresponding closed embedding. Note that the compositions are proper, by assumption By Corollary 1.1.6, we have: Id IndCoh(X 2 ) colim (i 2,a) IndCoh a A g 2 i 2,a : X 2,a X 2 and g 1 i 1,a : X 1,a X 1 Hence, we can rewrite the lefthand side in (2.2) as and the righthand side as (i 2,a )! and Id IndCoh(X 1 ) colim (i 1,a) IndCoh (i 1,a )!. a A colim (f ) IndCoh (i 1,a ) IndCoh (i 1,a )! g 1,! a A colim (i 2,a) IndCoh (i 2,a )! g 2! f IndCoh. a A It follows from the construction that the map in (2.2) is given by a compatible system of maps for each a A: (f ) IndCoh where the map (i 1,a ) IndCoh (i 2,a ) IndCoh (i 1,a )! g! 1 (i 2,a ) IndCoh (f a ) IndCoh (f a ) IndCoh is base change for the Cartesian square (f a ) IndCoh (g 1 i 1,a )! (i 2,a ) IndCoh (i 1,a )! g! 1 (g 2 i 2,a )! f IndCoh (i 2,a ) IndCoh (g 1 i 1,a )! (g 2 i 2,a )! f IndCoh (i 2,a )! g! 2 f IndCoh, X 1,a f a g 1 i 1,a X1 f X 2,a g 2 i 2,a X2. Hence, the required isomorphism follows from proper base change for schemes, see [BookII.1, Proposition 5.2.1] Groupoids and descent Let X be a groupoid simplicial object in PreStk laft, see [Lu0], Definition Denote by (2.3) p s, p t : X 1 X 0 the corresponding maps. We form a cosimplicial category IndCoh(X )! using the!pullback functors, and consider its totalization Tot(IndCoh(X )! ). Consider the functor of evaluation on 0simplices: ev 0 : Tot(IndCoh(X )) IndCoh(X 0 ).
12 12 INDCOHERENT SHEAVES ON INDINFSCHEMES We claim: Proposition Assume that the maps p s, p t in (2.3) are indschematic and indproper. (a) The functor ev 0 admits a left adjoint. The resulting monad on IndCoh(X 0 ), viewed as a plain endofunctor, is canonically isomorpic to (p t ) IndCoh (p s )!. The adjoint pair is monadic. IndCoh(X 0 ) Tot(IndCoh(X )! ) (b) Suppose that X is the Čech nerve of a map f : X0 Y, where f is indschematic and indproper. Assume also that f is surjective at the level of kpoints. Then the resulting map is an equivalence. IndCoh(Y) Tot(IndCoh(X )! ) Proof. Follows by repeating the argument of [BookII.1, Proposition 7.2.2]. 3. IndCoh on (ind)infschemes In this section be begin the development of the theory of IndCoh on indinfschemes. We will essentially bootstrap it from IndCoh on indschemes, using nil base change Nil base change. As was just mentioned, nil base change is a crucial property of the category IndCoh. Its proof relies on the structural results on infschemes from [BookIII.2, Sect. 4] Recall the notion of infschematic map in PreStk, see [BookIII.2, Definition 3.1.6]. We are going to show: Proposition Let f : X 1 X 2 be a map in PreStk laft, and assume that f is an infschematic nilisomomorphism. (a) The functor f! : IndCoh(X 2 ) IndCoh(X 1 ) admits a left adjoint, to be denoted f IndCoh. (b) The functor f! is conservative. (c) For a Cartesian daigram X 1 f g 1 X1 f the natural transformation arrising by adjunction from is an isomorphism. X 2 (f ) IndCoh g 2 X2, g! 1 g! 2 f IndCoh, g! 1 f! (f )! g! 2,
13 INDCOHERENT SHEAVES ON INDINFSCHEMES Proof of Proposition Using Lemma 2.1.4, we reduce the assertion to the case when X 2 = X 2 Sch aff aft, for points (a) and (b), and further to the case when X 2 = X 2 Sch aff aft for point (c). In this case X 1 has the property that red X 1 = X 1 red Sch aff ft. By [BookIII.2, Corollary 4.3.4], we can write where A is the category X 1 colim X 1,a, a A (Sch aff aft) /X1 ( red Sch aff aft ) /X 1 {X 1 }, and the colimit is taken in the category PreStk laft. In particular, for every a, the resulting map X 1,a X 2 is a nilisomomorphism, and in particular proper. Moreover, for every morphism a a, the corresponding map is also a nilisomomorphism and is proper. We have: i a,a : X 1,a X 1,a IndCoh(X 1 ) lim a A op IndCoh(X 1,a). The fact that f! is conservative follows from the fact that each (f i a )! is conservative. Using [GL:DG, Lemma 1.3.3], we can therefore rewrite (3.1) IndCoh(X 1 ) colim IndCoh(X 1,a), a A where the colimit is taken with respect to the functors (i a,a )IndCoh : IndCoh(X 1,a ) IndCoh(X 1,a ). Now, the left adjoint to f is given by the compatible collection of functors (f i a ) IndCoh : IndCoh(X 1,a ) IndCoh(X 2 ). Thus, it remains to establish the base change property. repeating the argument in Sects However, the latter follows by 3.2. Basic properties. We will now use nil base change to establish some basic properties of the category IndCoh on an indinfscheme First, as a corollary of Proposition we obtain: Corollary Let X be an object of indinfsch laft. Then the category IndCoh(X) is compactly generated. Proof. Consider the canonical map i : red X X. The category IndCoh( red X) is compactly generated by Corollary Now, Proposition implies that the essential image of i IndCoh (IndCoh( red X) c ) compactly generates IndCoh(X).
14 14 INDCOHERENT SHEAVES ON INDINFSCHEMES Let X 1 f g 1 X1 f X g 2 2 X2, be a Cartesian diagram in PreStk laft with f being indinfschematic and indnilproper (see [BookIII.2, Definition ] for what this means). We claim: Proposition The functors f! and (f )! admit left adjoints, to be denoted f IndCoh (f ) IndCoh, respectively. The natural transformation (3.2) (f ) IndCoh arrising by adjunction from is an isomorphism. g! 1 g! 2 f IndCoh, g! 1 f! (f )! g! 2, Proof. By Lemma 2.1.4, the assertion of the proposition reduces to the case when X 2 and (resp., both X 2 and X 2) belong to Sch aff aft. Denote these objects by X 2 and X 2, respectively. In this case X 1 (resp., both X 1 and X 1) belong to indinfsch laft. Let X 0 be any object of indsch laft endowed with a nilisomomorphism to X 1 ; e.g., X 0 = red X 1. Set X 0 := X 2 X 0, X 2 and consider the diagram X g 0 0 X0 i i and X 1 f g 1 X1 f in which both squares are Cartesian. X 2 g 2 X2, By Proposition 3.1.2(b), it is sufficient to prove the assertion for the top square and for the outer square. Now, the assertion for the outer square is given by Proposition 2.1.2, and for the top square by Proposition 3.1.2(c) Indnilproper descent By the same logic as in Proposition we obtain: Proposition Let X be a groupoid simplicial object in PreStk laft where the maps are indinfschematic and indnilproper. (a) The functor p s, p t : X 1 X 0 ev 0 : IndCoh(X ) IndCoh(X 0 )
15 INDCOHERENT SHEAVES ON INDINFSCHEMES 15 admits a left adjoint. The resulting monad on IndCoh(X 0 ), viewed as a plain endofunctor, is canonically isomorpic to (p t ) IndCoh (p s )!. The adjoint pair IndCoh(X 0 ) Tot(IndCoh(X )! ) is monadic. (b) Suppose that X is the Čech nerve of a map f : X0 Y, where f is indinfschematic and indnilproper. Assume also that f is surjective at the level of kpoints. Then the resulting map is an equivalence. IndCoh(Y) Tot(IndCoh(X )! ) For X PreStk laft recall the category Pro(QCoh(X) ) fake, see [BookIII.1, Sect ]. From Proposition and [BookIII.1, Corollary 4.3.7] we obtain: Corollary Let X Y be a map in PreStk laft, which is indinfschematic, infproper and surjective at the level of kpoints. Then the pullback functor Pro(QCoh(Y) ) fake Tot ( Pro(QCoh(X /Y) ) fake) is an equivalence Descent for maps. Let f : Y X be an indnilschematic and indnilproper map between objects of PreStk laft, which is surjective at the level of kpoints. We claim: Proposition For Z PreStk laftdef, the natural map is an isomorphism. Tot(Maps(Y, Z)) Maps(X, Z) Proof. The statement automatically reduces to the case when X = X < Sch aff aft. Furthermore, by [BookIII.2, Corollary 4.1.5], we can assume that Y = Y < Sch aft. The assertion of the proposition is evident if X is reduced: in this case the simplicial object Y is split. Hence, by [BookIII.1, Proposition 5.4.2], by induction, it suffices to show that if the assertion holds for a given X and we have a squarezero extension X X by means of F Coh(X) 0, then the assertion holds also for Y (Sch aff aft) nilisom to X. Set Y := Y X X. By [BookIII.1, Proposition 5.3.2], the map Y Y has a structure of simplicial object in the category of squarezero extensions. By the induction hypothesis, it is enough to show that for a given map z : X Z, the map (3.3) Maps(X, Z) {z} Tot(Maps(Y, Z)) {z} Maps(X,Z) Tot(Maps(Y,Z)) is an isomorphism. Since Z admits deformation theory, the lefthand side in (3.3) is canonically isomorphic to the groupoid of nullhomotopies of the composition T z (Z) T (X) F. We rewrite the righthand side in (3.3) as the totalization of the simplicial space Maps(Y, Z) {f}. Maps(Y,Z)
16 16 INDCOHERENT SHEAVES ON INDINFSCHEMES The above simplicial groupoid identifies with that of nullhomotopies of the composition where f denotes the map Y X. T z f (Z) T (Y ) f (F), Now, the desired property follows the the descent property of Pro(QCoh( ) ) fake, see Corollary above tstructure for indinfschemes. The category IndCoh on an indinfscheme also possesses a tstructure. However, it has less favorable properties than in the case of indschemes Let X be an indinfscheme. We define a tstructure on the category IndCoh(X) by declaring that an object F IndCoh(X) belongs to IndCoh(X) 0 if and only if i! (F) belongs to IndCoh( red X) 0, where is the canonical map. i : red X X Equivalently, we let IndCoh(X) 0 be generated under colimits by the essential image of IndCoh( red X) 0 under i IndCoh. It is easy to see that if f is a nilclosed map, then functor f! is left texact Assume for a moment that X is actually an indscheme. We claim that the tstructure defined above, when we view X as a mere indinfscheme, coincides with one for X considered as an indscheme of Sect This follows from the next lemma: Lemma Let f : X 1 X 2 be a nilisomomorphism of indschemes. Then for F IndCoh(X 2 ) we have: F IndCoh(X 2 ) 0 f! (F) F IndCoh(X 1 ) 0. Proof. The implication is tautological. For the implication, by the definition of the t structure on IndCoh(X 2 ), we can assume that X 2 = X 2 Sch aft and X 1 = X 1 Sch aft That, is f is a nilisomomorphism of schemes X 1 X 2. By the definition of the tstructure on IndCoh(X 2 ) and adjunction, it suffices to show that Coh(X 2 ) 0 is generated by the essential image of Coh(X 1 ) 0 underf IndCoh, which is obvious. Corollary Let X be an object of indinfsch laft. (a) For F IndCoh(X), we have F IndCoh(X) 0 if and only if for every nilclosed map f : X X with X red Sch ft we have f! (F) IndCoh(X) 0. (b) The category IndCoh(X) 0 is generated under colimits by the essential images of the categories IndCoh(X) 0 for f : X X with X red Sch ft and f nilclosed.
17 INDCOHERENT SHEAVES ON INDINFSCHEMES As we have seen above, if f is a nilclosed map X 1 X 2 of indinfschemes, then the functor f! is left texact. By adjunction, this implies that the functor f IndCoh is right texact. Lemma Let f : X 1 X 2 is be a nilclosed and indschematic map between indinfschemes. Then the functor f IndCoh is texact. Proof. We only have to prove that f IndCoh is left texact. By Proposition 3.1.2(c), the assertion reduces to the case when X 2 = indsch laft. In the latter case, X 1 is an indscheme, and the assertion follows from the fact that the functor f IndCoh for a map of indschemes is left texact, by Lemma Remark It is easy to see that the assertion of the lemma is false without the assumption that f be indschematic. 4. The direct image functor for indinfschemes In this section we construct the direct image functor on IndCoh for maps between indinfschemes. The idea is that one can bootstrap it from the case of maps that are nilclosed embeddings, while for the latter the soughtfor procedure is obtained as left/right Kan extension from the case of schemes Recovering from nilclosed embeddings. In this subsection we show that if we take IndCoh on the category of schemes, with morphisms restricted to nilclosed maps, then its right Kan extension to indinfschemes recovers the usual IndCoh Consider the fully faithful embeddings Denote Sch aff aft Sch aft indinfsch laft PreStk laft. IndCoh! indinfsch laft := IndCoh! PreStk laft (indinfschlaft ) op. Since IndCoh! Sch aft RKE (Sch aff aft )op (Sch aft ) op(indcoh! Sch ) aff aft is an isomorphism, the map (4.1) IndCoh! indinfsch laft RKE (Schaft ) op (indinfsch laft ) op(indcoh! Sch aft ) is an isomorphism Let (indinfsch laft ) nilclosed indinfsch laft denote the 1full subcategory, where we restrict 1morphisms to be nilclosed. Denote IndCoh! (indinfsch laft ) nilclosed := IndCoh! indinfsch laft ((indinfschlaft ) nilclosed ) op. From the isomorphism (4.1), we obtain a canonically defined map (4.2) IndCoh! (indinfsch laft ) nilclosed We will prove: RKE ((Schaft ) nilclosed ) op ((indinfsch laft ) nilclosed ) op(indcoh! (Sch aft ) nilclosed ). Proposition The map (4.2) is an isomorphism.
18 18 INDCOHERENT SHEAVES ON INDINFSCHEMES Proof. We need to show that for X indinfsch laft, the functor is an equivalence. IndCoh(X) lim IndCoh(Z) op Z ((Sch aft ) nilclosed in X ) However, this follows from [BookIII.2, Corollary 4.1.4], since the functor IndCoh takes colimits in PreStk laft to limits Recovering from nilisomomorphisms. The material in this subsection is not needed for the sequel and is included for the sake of completeness Let (indinfsch laft ) nilisom indinfsch laft and (indsch laft ) nilisom indsch laft denote the 1full subcategories, where we restrict 1morphisms to be nilisomomorphisms. Denote also and IndCoh! (indinfsch laft ) nilisom := IndCoh! indinfsch laft ((indinfschlaft ) nilisom ) op IndCoh! (indsch laft ) nilisom := IndCoh! indsch laft ((indschlaft ) nilisom ) op From Proposition we deduce: Corollary The natural map IndCoh! (indinfsch laft ) nilisom RKE ((indschlaft ) nilisom ) op ((indinfsch laft ) nilisom ) op(indcoh! (Sch aft ) nilisom ) is an isomorphism. Proof. By Proposition and Proposition 4.1.3, the map IndCoh! (indinfsch laft ) nilclosed is an isomorphism. RKE ((indschlaft ) nilclosed ) op ((indinfsch laft ) nilisom ) op(indcoh! (Sch aft ) nilclosed ) Hence, it remains to show that for X indinfsch laft, the restriction map lim IndCoh(Y) lim IndCoh(Y) Y ((indsch laft ) nilclosed in X ) op Y ((indsch laft ) nilisom to X ) op is an isomorphism. We claim that the map (indsch laft ) nilisom to X (indsch laft ) nilclosed in X is cofinal. Indeed, it admits a left adjoint, given by sending an object (Y X) (indsch laft ) nilclosed in X Y red X X, red Y where the pushout is taken in the category PreStk laft Constructing the direct image functor. In this subsection we finally construct the direct image functor. The crucial assertion is Theorem 4.3.2, which says that this functor is the right one.
19 INDCOHERENT SHEAVES ON INDINFSCHEMES Consider again the functor IndCoh Schaft : Sch aft DGCat cont, where for a morphism f : X 1 X 2, the functor IndCoh(X 1 ) IndCoh(X 2 ) is f IndCoh. Recall the notation: IndCoh (Schaft ) nilclosed := IndCoh Schaft (Schaft ) nilclosed. Denote and denote IndCoh indinfschaft := LKE Schaft indinfsch aft (IndCoh Schaft ), IndCoh (indinfschaft ) nilclosed := IndCoh indinfschaft (indinfschaft ) nilclosed. We have a canonical map (4.3) LKE (Schaft ) nilclosed (indinfsch aft ) nilclosed (IndCoh (Schaft ) nilclosed ) IndCoh (indinfschaft ) nilclosed. We claim: Theorem The map (4.3) is an isomorphism Note that by combining Lemma and Theorem 4.3.2, we obtain: Corollary The functors IndCoh (indinfschaft ) nilclosed and IndCoh! (indinfsch aft ) nilclosed are obtained from one another by passing to adjoints. The upshot of Theorem and Corollary is that for a morphism f : X 1 X 2 we have a welldefined functor f IndCoh : IndCoh(X 1 ) IndCoh(X 2 ), and when f is nilclosed, the above functor is the left adjoint to f!. In what follows, for X 1 = X and X 2 = pt, we shall also us the notation Γ(X, ) IndCoh for the functor (p X ) IndCoh, where p X : X pt is the projection. Remark In the sequel, we will show that the assertion of Corollary can be strengthened by replacing the class of nilclosed morphisms by that of indnilproper ones Proof of Theorem Step 1. We need to show that the functor is an equivalence. colim IndCoh(Z) colim IndCoh(Y ) Z (Sch aft ) nilclosed in X Y (Sch aft ) /X First, the convergence property of the IndCoh functor allows to repace Sch aft by < Sch aft. Thus, we need to show that the functor is an equivalence. colim IndCoh(Z) colim IndCoh(Y ) Z ( < Sch aft ) nilclosed in X Y ( < Sch aft ) /X
20 20 INDCOHERENT SHEAVES ON INDINFSCHEMES Step 0. Consider the commutative diagram (4.4) colim IndCoh(Y ) colim Y ( < Sch aft ) /X colim IndCoh(Z) Z ( < Sch aft ) nilclosed in X S ( < Sch aff aft ) /X where C is the category of S Z X, with S < Sch aff aft and (Z X) ( < Sch aft ) nilclosed in X. IndCoh(S) colim IndCoh(S), (S Z X) C We will show that the horizontal arrows and the right vertical arrow in this diagram are equivalences. This will prove that the left vertical arrow is also an equivalence Step 1. Consider the functor C ( < Sch aff aft) /X, appearing in the right vertical arrow in (4.4). We claim that it is cofinal, which would prove that the right vertical arrow in (4.4) is an equivalence. We note that the functor C ( < Sch aff aft) /X is a Cartesian fibration. Hence, the fact that it is cofinal is equivalent to the fact that it has contractible fibers. The fiber over a given object S X is the category of factorizations S Z X, (Z X) ( < Sch aft ) nilclosed in X. This category is contractible by [BookIII.2, Proposition 4.1.3] Step 2. Consider the functor C ( < Sch aft ) nilclosed in X, appearing in the bottom horizontal arrow in (4.4). It is a cocartesian fibration. Hence, in order to show that this arrow in the diagram is an equivalence, it suffices to show that for a given Z ( < Sch aft ) nilclosed in X, the functor (4.5) colim IndCoh(S) IndCoh(Z) S ( < Sch aff aft ) /Z is an equivalence. We have the following assertion, proved below: Proposition The functor IndCoh Schaft, regarded as a presheaf on Sch aft with values in (DGCat cont ) op satisfies Zariski descent. Now, the fact that (4.5) is an equivalence (for any Z < Sch aft ) follows from [GL:IndCoh, Proposition 6.4.3], applied to < Sch aff aft < Sch aft Step 3. To treat the top horizontal arrow in (4.4), we consider the category D of and functor S Y Y, S < Sch aff aft, Y < Sch aft, (4.6) colim IndCoh(S) colim IndCoh(S). (S Y X) D S ( < Sch aff aft ) /X We note that the functor (4.6) is an equivalence, because the corresponding forgetful functor D ( < Sch aff aft) /X is cofinal (it is a Cartesian fibration with contractible fibers).
21 INDCOHERENT SHEAVES ON INDINFSCHEMES 21 Hence, it is remains to show that the composition of (4.6) with the top horizontal arrow in (4.4), i.e., colim IndCoh(S) colim IndCoh(S) (S Y X) D Y ( < Sch aft ) /X is an equivalence. The above functor corresponds to the cocartesian fibration D ( < Sch aft ) /X. Hence, it suffices to show that for a fixed Y ( < Sch aft ) /X, the functor is an equivalence. colim IndCoh(S) IndCoh(Y ) S ( < Sch aff aft ) /Y However, this follows as in Step 2 from Proposition Proof of Proposition The assertion of the proposition is equivalent to the following. Let X Sch aft be covered by two opens U 1 and U 2. Denote Then the claim is that the diagram j 1 j 2 j 12 U 1 X, U 2 X, U 1 U 2 X, U 1 U 2 j 12,1 U 1, U 1 U 2 j 12,1 U 2. (j 12,1) IndCoh IndCoh(U 1 U 2 ) IndCoh(U 1 ) (j 12,2) IndCoh IndCoh(U 1 ) is a pushout square in DGCat cont. (j 2) IndCoh (j 1) IndCoh IndCoh(X) That is to say, given C DGCat and a triple of continuous functors F 1 : IndCoh(U 1 ) C, F 2 : IndCoh(U 2 ) C, F 12 : IndCoh(U 1 U 2 ) C endowed with isomorphisms F 1 (j 12,1 ) IndCoh F 12 F 2 (j 12,2 ) IndCoh, we need to show that this data comes from a uniquely defined functor F : IndCoh(X) C. The soughtfor functor F is recovered as follows: for F IndCoh(X), we have F (F) = F 1 (j! 1(F)) F 12(j! 12 (F)) F 2 (j! 2(F)), where the maps F i (j! i (F)) F 12(j! 12(F)) are given by F i (j! i(f)) F i ((j 12,i ) IndCoh j! 12,i j! i(f)) = F i ((j 12,i ) IndCoh j! 12(F) F 12 (j! 12(F)) Base change. As in the case of indschemes, there are two types of base change isomorphism for nilclosed infschematic maps.
22 22 INDCOHERENT SHEAVES ON INDINFSCHEMES One has to do with a Cartesian diagram X 1 f g 1 X1 f X 2 g 2 X2, be a Cartesian diagram in PreStk laft with f being infschematic and nilclosed. We claim that Proposition The natural transformation (4.7) (f ) IndCoh arrising by adjunction from is an isomorphism. g! 1 g! 2 f IndCoh, g! 1 f! (f )! g! 2, Proof. This is a particular case of Proposition Let now X 1 f g 1 X1 f X g 2 2 X2, be a Cartesian diagram of objects of indinfsch laft with g 2 being nilclosed. From the isomorphism (g 2 ) IndCoh (f ) IndCoh by adjunction, we obtain a natural transformation: (4.8) (f ) IndCoh We claim: f IndCoh g! 1 g! 2 f IndCoh. Proposition The map (4.8) is an isomorphism. Proof. Let X 0 := red X 1. Set and consider the diagram X 0 := X 1 X 1 X 0, X 0 g 0 X0 (g 1 ) IndCoh, X 1 f g 1 X1 f in which both squares are Cartesian. X 2 g 2 X2, By Proposition 3.1.2(b), it suffices to show that the outer square and the top square each satisfy base change. Now, base change for the top square is given by Proposition 3.1.2(c). This reduces the assertion of the proposition to the case when X 1 is a classical reduced indscheme. In this case the map f factors as X 1 X 3/2 X 2,
23 INDCOHERENT SHEAVES ON INDINFSCHEMES 23 where X 3/2 = red X 2. Set X 3/2 := X 2 X 2 X 3/2, and consider the diagram X 1 g 0 X1 in which both squares are Cartesian. X 3/2 f X 2 g 3 X3/2 f g 2 X2, It is sufficient to show that the two inner squares each satisfy base change. Now, for the top square, this is given by Proposition For the bottom square, this is given by Proposition 3.1.2(c). Remark Let f : X 1 X 2 be a indnilproper map between indinfschemes. On the one hand, we have a welldefined functor f IndCoh : IndCoh(X 1 ) IndCoh(X 2 ). On the other hand, by Proposition 3.2.4, we have the functor left adjoint to f!. (f! ) L : IndCoh(X 1 ) IndCoh(X 2 ), We will ultimately show that there is a canonical isomorphism f IndCoh (f! ) L, see Sect Extending the formalism of correspondences to infschemes In this section we will take the formalism of IndCoh to what (in our opinion) is its ultimate domain of definition: the category of correspondences, where the objects are all prestacks (locally almost of finite type), pullbacks are taken with respect to any maps, pushforwards are taken with respect to indinfschematic maps, and adjunctions hold for indnilproper maps Setup for extension. As a first step, we will consider the category of correspondences, where the objects are indinfschemes, pullbacks and pushforwards are taken with respect to any maps, and adjunctions are for nilclosed maps. We will construct the required functor by a 2categorical left Kan extension procedure from [Funct, Theorem ] We consider the category indinfsch laft with the following three classes of morphisms Let vert = all, horiz = all, adm = nilclosed. (indinfsch laft ) nilclosed be the resulting (, 2)category of correspondences.
24 24 INDCOHERENT SHEAVES ON INDINFSCHEMES Consider also the category and the functor (Sch aft ) nilclosed, (IndCoh Schaft ) nilclosed : (Sch aft ) nilclosed DGCat 2 Cat cont, obtained from the functor (IndCoh Schaft ) proper of [BookII.2, Theorem 2.1.4] by restriction along (Sch aft ) nilclosed (Sch aft ) proper. We wish to extend the functor (IndCoh Schaft ) nilclosed to a functor (IndCoh indinfschlaft ) nilclosed : (indinfsch laft ) nilclosed DGCat 2 Cat cont, along the tautological functor Sch laft indinfsch laft. We refer the reader to [Funct, Sect ], where sufficient conditions for the existence and canonicity of such an extension are formulated. In the present context, conditions (1), (2) and (3) amount to to Proposition (for property (1)), Theorem (for property (2)), and Propositions and (for property (3)) Applying [Funct, Theorem ], we obtain: Theorem There exists a canonically defined functor whose restrictions to identify with IndCoh (indinfschlaft ) : (indinfsch laft) nilclosed nilclosed DGCat 2 cont Cat, indinfsch op laft and indinfsch laft IndCoh! indinfsch laft and IndCoh indinfschlaft, respectively, and such that the restriction of IndCoh (indinfschlaft ) nilclosed (Sch aft ) nilclosed (indinfsch laft ) nilclosed identifies canonically with IndCoh (Schaft ) nilclosed Let us explain the concrete content of the above theorem. Let X 1 f g 1 X1 f X g 2 2 X2 be a Cartesian diagram of objects of indinfsch laft. Theorem says that we have a canonical isomorphism (5.1) g! 2 (f X ) IndCoh (f Y ) IndCoh g! 1. under Moreover, this isomorphism coincides with the base change isomorphism of (4.7) (resp., (4.8)), when f X (resp., g 2 ) is nilclosed. Moreover, if the indinfschemes in the above diagram are schemes, then the isomorphism (5.1) equals one defined a priori in this case by [BookII.2, Corollary 3.1.4].
25 INDCOHERENT SHEAVES ON INDINFSCHEMES 25 Remark In Sect. 5.2 we will show that the isomorphism (5.1) comes from an adjunction one of the maps f or g 2 is indnilproper (but not necessarily nilclosed) Enlarging 2morphisms to indnilproper maps. In this subsection we will extend the functor from the previous subsection, where we include adjunctions for indnilproper maps Consider now the (, 2)category (indinfsch laft ) indnilproper, where we enlarge the class of 2morphisms to that of indnilproper maps. Consider the 2fully faithful functor We are going to prove: (indinfsch laft ) nilclosed (indinfsch laft ) indnilproper. Theorem There exists a unique extension of the functor functor (indinfsch laft ) nilclosed to a functor IndCoh (indinfschlaft ) indnilproper There exists a unique isomorphism : (indinfsch laft ) indnilproper DGCat 2 Cat cont. IndCoh (indinfschlaft ) indnilproper (Schlaft ) proper IndCoh (Sch laft ) proper, compatible with the isomorphism IndCoh (indinfschlaft ) nilclosed (Sch laft ) IndCoh nilclosed (Sch laft ). nilclosed By [Funct, Theorem ], from Theorem 5.2.2, we obtain: Corollary (a) The functors IndCoh (indinfschlaft ) indnilproper := IndCoh indinfschlaft (indinfschlaft ) indnilproper and IndCoh! (indinfsch laft ) indnilproper := IndCoh! indinfsch laft ((indinfschlaft ) indnilproper ) op are obtained from each other by passing to adjoints. (b) Let X 1 f g 1 X1 f X g 2 2 X2, be a Cartesian diagram of objects of indinfsch laft with g 2 being indnilproper. Then the natural transformation (f ) IndCoh g 1! g 2! f IndCoh, obtained by adjunction from the isomorphism, is an isomorphism. (g 2 ) IndCoh (f ) IndCoh f IndCoh (g 1 ) IndCoh, The rest of this subsection is devoted to the proof of Theorem
26 26 INDCOHERENT SHEAVES ON INDINFSCHEMES The case of indschemes. Consider the category indsch laft with the following three classes of morphisms vert = all, horiz = all, adm = indnilproper. We claim that we have the following result: Theorem There exists a canonically defined functor IndCoh (indschlaft ) indnilproper : (indsch laft ) indnilproper DGCat 2 Cat cont, whose restrictions to identify with indsch op laft and indsch laft IndCoh! indsch laft and IndCoh indschlaft, respectively, and such that the restriction of IndCoh (indschlaft ) indnilproper under (Sch aft ) proper (indsch laft) indnilproper identifies canonically with IndCoh (Schaft ). Furthermore, there is a canonical isomorphism nilclosed between the restrictions IndCoh (indschlaft ) indnilproper (indschlaft ) nilclosed IndCoh (indinfsch laft ) indnilproper (indschlaft ). nilclosed Proof. This follows from [Funct, Theorem ] applied to the functor Sch aft indsch laft, using Proposition 1.3.4, Proposition 1.4.2, and Propositions and The last assertion of the theorem follows from the canonicity of the construction in [Funct, Theorem ]. Remark The difference between the case of indschemes and that of infschemes that it is not true (at least, not obviously so) that propertes (1) and (2) of [Funct, Sect ] are satisfied for the functors and IndCoh! (Sch laft ) proper : ((Sch laft ) proper ) op DGCat cont IndCoh (Schlaft ) proper : (Sch laft ) proper DGCat cont with respect to the functor while this is true for Sch laft indinfsch laft, Sch laft indsch laft.
27 INDCOHERENT SHEAVES ON INDINFSCHEMES We are going to deduce Theorem from Theorem by applying [Funct, Theorem ]. Thus, we need to check that the inclusion satisfies the condition of [Funct, Sect ]. That is, we consider a infproper morphism nilclosed indnilproper f : X 1 X 2 of objects of indinfsch laft, and the Cartesian square: The diagonal map p 2 X 1 X 1 X1 X 2 p 1 f X 1 f X 2. X1/X 2 : X 1 X 1 X 2 X 1 is nilclosed. Hence, from the (( X1/X 2 ) IndCoh, ( X1/X 2 )! adjunction, we obtain a natural transformation ( X1/X 2 ) IndCoh ( X1/X 2 )! Id IndCoh(X1 X 1). X 2 By composing, the latter natural transformation gives rise to (5.2) Id IndCoh(X1) (id X1 ) IndCoh (p 1 ) IndCoh (id X1 )! ( X1/X 2 ) IndCoh ( X1/X 2 )! p! 2 (p 1 ) IndCoh p! 2 f! f IndCoh. We need to show that the natural transformation (5.2) is the unit of an adjunction. I.e., that for F 1 IndCoh(X 1 ) and F 2 IndCoh(X 2 ), the map (5.3) Maps(f IndCoh is an isomorphism. (F 1 ), F 2 ) Maps(f! f IndCoh (F 1 ), f! (F 2 )) Maps((p 1 ) IndCoh p! 2(F 1 ), f! (F 2 )) Maps(F 1, f! (F 2 )) We note that by Theorem the map (5.3) is the unit for the (f IndCoh, f! ) adjunction, when f is nilclosed Note that the natural transformation (5.2) is defined for any map f which is nilseparated, i.e., one for which X1/X 2 is nilclosed. Let g : X 0 X 1 be another nilseparated map between objects of indinfsch laft. Diagram chase implies: Lemma For F 0 IndCoh(X 0 ) and F 2 IndCoh(X 2 ), the diagram Maps(g IndCoh (F 0 ), f! (F 2 )) Maps(F 0, g! f! (F 2 )) id commutes. Maps(f IndCoh g IndCoh (F 0 ), F 2 ) Maps(F 0, g! f! (F 2 ))
28 28 INDCOHERENT SHEAVES ON INDINFSCHEMES Let us take X 0 := red X 1 and g to be the canonical embedding. By Proposition 3.1.2, it is sufficient to show that (5.3) is an isomorphism for F 1 of the form g IndCoh (F 0 ) for F 0 IndCoh(X 0 ). Using Lemma , and the fact that the map Maps(g IndCoh (F 0 ), f! (F 2 )) Maps(F 0, g! f! (F 2 )) is an isomorphism in this case, since g is nilclosed, we obtain that it is sufficient to show that (5.3) is an isomorphism, when the initial map f is replaced by f g. I.e., in proving that (5.3) is an isomorphism, we can assume that X 1 is a reduced indscheme Let us now factor f as X 1 X 3/2 X 2, where X 3/2 := red X 2. Applying Lemma again, we obtain that it is enough to show that (5.3) is an isomorphism for f replaced by X 1 X 3/2 and X 3/2 X 2 separately. For the map X 3/2 X 2, the assertion follows from the fact that the map in question is nilclosed Hence, we are further reduced to the case when f is a indnilproper map between indschemes. However, in this case, the required isomorphism follows from Theorem 5.2.6: it follows by [Funct, Theorem ] from the existence of the functor IndCoh (indschlaft ) indnilproper : (indsch laft ) indnilproper whose restriction to (indsch laft ) nilclosed is isomorphic to IndCoh (infschlaft ) nilclosed (indsch laft ) nilclosed. DGCat 2 Cat cont, 5.3. Extending to prestacks. In this subsection we will finally extend the formalism to the category of correspondences that has all laft prestacks as objects Consider now the category PreStk laft, and the three classes of morphisms indinfsch, all, indinfsch & indnilproper, where indinfsch stands for the class of indinfschematic morphisms, and indnilproper for the class of morphisms that are indnilproper. Consider the tautological embedding indinfsch laft PreStk laft. It satisfies the conditions of [Funct, Horizontal Extension], with respect to the classes (all, all, indnilproper) (indinfsch, all, indinfsch & indnilproper). Consider now the functor IndCoh (indinfschlaft ) indnilproper and the corresponding functor Clearly, the map : (indinfsch laft ) indnilproper IndCoh! indinfsch laft : (indinfsch laft ) op DGCat cont. DGCat 2 Cat cont, IndCoh! PreStk laft RKE (indinfschlaft ) op (PreStk laft ) op(indcoh! indinfsch laft )
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