PART II.1. INDCOHERENT SHEAVES ON SCHEMES


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1 PART II.1. INDCOHERENT SHEAVES ON SCHEMES Contents Introduction 1 1. Indcoherent sheaves on a scheme Definition of the category tstructure 3 2. The direct image functor Direct image for an individual morphism Upgrading to a functor 6 3. The functor of usual inverse image Inverse image with respect to eventually coconnective morphisms Base change for eventually coconnective morphisms Tensoring up Open embeddings Restriction to an open Zariski descent Proper maps The!pullback Base change for proper maps Pullback compatibilty Closed embeddings Category with support A conservativeness result for proper maps Products Convergence Groupoids and descent The BeckChevalley condition Proper descent 30 References 33 Introduction Date: September 30, 2013; previous: Sept. 5,
2 2 INDCOHERENT SHEAVES ON INFSCHEMES 1. Indcoherent sheaves on a scheme In this section we introduce the category IndCoh() and study its basic properties Definition of the category. In this subsection we define IndCoh() and the functors that connect it to the usual category QCoh() of quasicoherent sheaves For Sch aft we consider the category QCoh() and its full (but not cocomplete) subcategory Coh(), consisting of bounded complexes with coherent cohomologies. We define the category IndCoh() by IndCoh() := Ind(Coh()) By construction, we have a naturally defined functor Ψ : IndCoh() QCoh() obtained by indextension of the tautological inlcusion Coh() QCoh(). We have: Lemma Assume that is a smooth classical scheme. Then Ψ is an equivalence. Proof. It is known by [TT] (see also [Ne]) that for which is a quasiseparated and quasicompact classical scheme, QCoh() Ind(QCoh() perf ). Now, for a regular classical scheme, we have as subcategories of QCoh(). QCoh() perf = Coh(), Remark It is shown in [GL:IndCoh, Proposition 1.5.4] that the assertion of the above lemma is in fact if and only if The monoidal action of QCoh. We note: Proposition There exists a uniquely defined monoidal action of QCoh(), viewed as a monoidal category, on IndCoh(), such that the functor Ψ is compactible with the QCoh() actions. Proof. The action in question is obtained by indextension of the action of the noncocomplete monoidal category QCoh() perf on Coh() We give the following definition: Definition We shall say that Sch aft is eventually coconnective if the structure sheaf O belongs to Coh(). I.e., is eventually coconnective if, Zariski locally, the structure sheaf had nonzero cohomologies in finitely many degrees. We have: Lemma If is eventually coconnective, the functor Ψ denoted Ξ, and this left adjoint is fully faithful. admits a left adjoint, to be
3 INDCOHERENT SHEAVES ON INFSCHEMES 3 Proof. If is eventually coconnective, we have QCoh() perf Coh(), and, using the fact that QCoh() Ind(QCoh() perf ), the functor Ξ indextension of the above embedding. The composition Ψ Ξ is the indextension of the functor QCoh() perf Coh() QCoh(), and it is manifest that its map to Id QCoh() is an isomorphism. is obtained as the Remark In [GL:IndCoh, Proposition 1.5.2] it is shown that Ψ admits a left adjoint if and only if is eventually coconnective tstructure. Some of the most basic operations on the IndCoh category (such as the functor of direct image studied in the next section) are inherited from those on QCoh using the tstructures on both categories. The crucial fact is that the eventually coconnective (a.k.a., bounded below) parts of the two categories are equivalent. The goal of this subsection is to define the tstructure on IndCoh() and establish its basic properties We claim: Proposition The category IndCoh() carries a unique tstructure that satisfies Moreover: (a) The functor Ψ is texact. IndCoh() 0 = {F IndCoh() Ψ (F) QCoh() 0 }. (b) This tstructure is compatible with filtered colimits (i.e., the subcategory IndCoh() 0 is closed under filtered colimits). (c) The induced functor is an equivalence for any n. As a corollary we obtain: Ψ : IndCoh() n QCoh() n Corollary The functor Ψ defines an equivalence IndCoh() + QCoh() +. Proof of Proposition It is clear that the condition of the proposition determines the t structure uniquely. To establish its properties we will use the following general assertion: Lemma Let C 0 be a (noncocomplete) DG category, endowed with a tstructure. Then C := IndCoh(C 0 ) carries a unique tstructure, which is compatible with filtered colimits, and for which the tautological inclusion C 0 C is texact. Moreover: (1) The subcategory C 0 (resp., C 0 ) is compactly generated under filtered colimits by C 0 0 (resp., C 0 0 ). (2) Let D be another DG category endowed with a tstructure which is compatible with filtered colimits, and let F : C D a continuous functor. Then F is texact (resp., left texact, right texact) if and only if F C0 is.
4 4 INDCOHERENT SHEAVES ON INFSCHEMES We apply Lemma 1.2.4(1) to C 0 = Coh() and obtain a (a priori different) tstructure on IndCoh(), which satisfies point (b) of the proposition. It also satisfies point (a) of the proposition, by Lemma 1.2.4(2) applied to D = QCoh() and F = Ψ. To show that this tstructure coincides with the one introduced earlier, it suffices to show that Ψ is conservative when restricted to IndCoh() 0. Hence, it remains to show that the tstructure, given by Lemma 1.2.4, satisfies points (c) of the proposition. To prove point (c), it is sufficient to consider the case of n = 0. Using Lemma 1.2.4(1), the required assertion follows from the next statement: Lemma The category QCoh() 0 is compactly generated under filtered colimits by Coh() Note that Proposition implies that, as long as the functor Ψ is not an equivalence (i.e., is not a classical smooth scheme), the category IndCoh() is not leftcomplete in its tstructure. The latter means that for F IndCoh(), the canonical arrow is not necessarily an isomorphism. lim n τ n () Furthermore, we see that for any, the functor Ψ realizes QCoh() as the left completion of IndCoh() From Proposition we also obtain the following: Corollary The inclusion Coh() IndCoh() c is an equality. Proof. Since the category Coh() compactly generates IndCoh(), the category IndCoh() c is the Karoubicompletion of Coh(). I.e., every object F IndCoh() c can be realized as a direct summand of an object F Coh(). In particular, F IndCoh() +. The object Ψ (F) is a direct summand of Ψ (F ). Hence, Ψ (F), regarded as an object of QCoh(), belongs to Coh(). Let us denote this object of Coh() by F. Thus, we can regard F and F as objects of IndCoh() + such that Ψ (F) Ψ ( F). Applying Proposition 1.2.2(c), we obtain that F F as objects of IndCoh(). The assignment 2. The direct image functor IndCoh() is very functorial. However, all of this functoriality is born from a single cource: the operation of direct image, defined in this section Direct image for an individual morphism. In this subsection we perform the first step in developing the formalism of direct image for IndCoh: we define the corresponding functor for one given morphism between schemes.
5 INDCOHERENT SHEAVES ON INFSCHEMES Let f : be a morphism in Sch aft. We claim: Proposition There exists a uniquely defined functor which is left texact, and makes the diagram f IndCoh : IndCoh() IndCoh( ), IndCoh() f IndCoh Ψ QCoh() f commute. IndCoh( ) Ψ QCoh( ) Proof. By continuity, the functor f IndCoh is the indextension of its retstriction to Coh() IndCoh(). The commutative diagram in the proposition implies that Ψ f IndCoh Coh() = f Coh(), as functors Coh() QCoh( ). Furthermore, f IndCoh Coh() is a functor that takes values in QCoh( ) +. is invertible by Proposition 1.2.2(c). Hence, f IndCoh Coh() is recov Note that Ψ QCoh( ) + ered as (Ψ QCoh( ) +) 1 (f Coh() ) Recall (see [GL:QCoh, Proposition 2.3.6]) that for Sch aft, the monoidal category QCoh() is rigid. Hence, by [GL:DG, Sect. 6], for a morphism f :, the functor f : QCoh() QCoh( ) has a canonical structure of morphism in QCoh( )mod, where QCoh( ) acts on QCoh() via f. Combining the arguments proving Proposition and Proposition 1.1.6, we prove the following: Proposition For a morphism f :, the functor f IndCoh : IndCoh() IndCoh( ) has a unique structure of 1morphism in QCoh( )mod which makes the square IndCoh() f IndCoh Ψ QCoh() f commute in QCoh( )mod. IndCoh( ) Ψ QCoh( )
6 6 INDCOHERENT SHEAVES ON INFSCHEMES 2.2. Upgrading to a functor. We now claim that the assigment upgrades to a functor IndCoh(), f f IndCoh (2.1) Sch aft DGCat cont, to be denoted IndCoh Schaft. Such an extension is not altogether automatic because we live in the world of higher categories. But constructing it will not be very difficult First, we recall the functor see [GL:QCoh,???]. QCoh Sch aft : (Sch aft ) op DGCat cont, Applying [Funct, Sect. 11.2], we obtain a functor QCoh Schaft : Sch aft DGCat cont, obtained from QCoh Sch aft by passage to right adjoints Now, we claim: Proposition There exists a uniquely defined functor equipped with a natural transformation IndCoh Schaft : Sch aft DGCat cont, Ψ Schaft : IndCoh Schaft QCoh Schaft, which at the level of objects and 1morphisms is given by the assignment IndCoh(), f f IndCoh. The rest of this subsection is devoted to the proof of this proposition Consider the following (, 1)categories: DGCat +cont and DGCat t cont : The category DGCat +cont consists if of noncocomplete DG categories C, endowed with a tstructure, such that C = C +. We also require that C 0 contain filtered colimits and that the embedding C 0 C commute with filtered colimits. As 1morphisms we take those exact functors F : C 1 C 2 that are left texact up to a finite shift, and such that F 0 C 1 commutes with filtered colimits. The higher categorical structure is uniquely determined by the requirement that the forgetful functor be 1fully faithful. DGCat +cont DGCat The category DGCat t cont consists of cocomplete DG categories C, endowed with a tstructure, such that C 0 is closed under filtered colimits, and such that C is compactly generated by objects from C +. As 1morphisms we allow those exact functors F : C 1 C 2 that are continuous and left texact up to a finite shift. The higher categorical structure is uniquely determined by the requirement that the forgetful functor be 1fully faithful. DGCat t cont DGCat cont
7 INDCOHERENT SHEAVES ON INFSCHEMES 7 We have a naturally defined functor (2.2) DGCat t cont DGCat +cont, C C +. Lemma The functor (2.2) is 1fully faihful We will use the following general assertion. Let T : D D be a functor between (, 1)categories, which is 1fully faithful. Let I be another (, 1)category, and let (2.3) i F (i), be an assignment, such that the assignment has been extended to a functor F : I D. i T F (i) Lemma Suppose that for every α Maps I (i 1, i 2 ), the point F (α) Maps D (F (i 1 ), F (i 2 )) lies in the connected component corresponding to the image of Maps D (F (i 1 ), F (i 2 )) Maps D (F (i 1 ), F (i 2 )). Then there exists a unique extension of (2.3) to a functor F : I D equipped with an isomorphism T F F. Let now F 1 and F 2 be two assignments as in (2.3), satisfying the assumption of Lemma Let us be given an assignment (2.4) i ψ i Maps D (F 1(i), F 2(i)). Lemma Suppose that the assignment i T (ψ i) Maps D (F 1 (i), F 2 (i)) has been extended to a natural transformation ψ : F 1 F 2. Then there exists a unique extension of (2.4) to a natural transformation ψ : F 1 F 2 equipped with an isomorphism T ψ ψ We are now ready to prove Proposition 2.2.3: Step 1. We start with the functor and consider and the assignment QCoh Schaft : Sch aft DGCat cont, I = Sch aft, D = DGCat cont, D := DGCat t cont, F = QCoh Schaft, Apping Lemma 2.2.7, we obtain a functor ( Sch aft ) (QCoh() DGCat t cont). (2.5) QCoh t Sch aft : Sch aft DGCat t cont. Step 2. Note that Proposition defines a functor and the natural transformation at the level of objects and 1morphisms. IndCoh t Sch aft : Sch aft DGCat t cont, Ψ t Sch aft : IndCoh t Sch aft QCoh t Sch aft
8 8 INDCOHERENT SHEAVES ON INFSCHEMES Since the functor DGCat t cont DGCat cont is 1fully faithful, by Lemmas and 2.2.8, the existence and uniqueness of the pair (IndCoh Schaft, Ψ Schaft ) with a fixed behavior on objects and 1morphisms, is equivalent to that of (IndCoh t Sch aft, Ψ t Sch aft ). Step 3. By Lemma 2.2.5, combined with Lemmas and 2.2.8, we obtain that the existence and uniqueness of the pair (IndCoh t Sch aft, Ψ t Sch aft ), with a fixed behavior on objects and 1 morphisms is equivalent to the existence and uniqueness of the pair obtained by composing with the functor (2.2). The latter, however, is given by (IndCoh + Sch aft, Ψ + Sch aft ), IndCoh + Sch aft := QCoh + Sch aft and Ψ + Sch aft := Id. 3. The functor of usual inverse image We now construct another piece of functoriality in the assignment IndCoh(), namely, the functor of *pullback. Unlike the case of QCoh, its role in the theory is rather limited a more important functor is that of!pullback. However, the *pullback is a necessary step in the construction of the!pullback, which is why we discuss it Inverse image with respect to eventually coconnective morphisms. Unlike the case of QCoh, the functor of *pullback on IndCoh is not defined for all maps of schemes, but only for eventually coconnective ones. In this subsection we give the corresponding construction Let f : be a morphism in Sch aft. Definition We shall say that f is eventually coconnective if the functor sends Coh( ) QCoh( ) to QCoh() +. f : QCoh( ) QCoh() It is easy to see that if f is eventually coconnective, then it sends Coh( ) to Coh(): indeed, for any morphism f, the functor f sends objects of QCoh( ) with coherent cohomologies to objects with a similar property on. In addition, we have the following fact, established in [GL:IndCoh, Lemma 3.4.2]: Lemma The following conditions are equivalent: (a) f is eventually coconnective; (b) f is of finite Tor amplitude, i.e., is left texact up to a finite cohomological shift. Corollary The class of eventually coconnective morphisms is stable under base change.
9 INDCOHERENT SHEAVES ON INFSCHEMES Let f : be eventually coconnective. Indextending the functor we obtain a functor which makes the diagram commute. We have: f : Coh( ) Coh() f IndCoh, : IndCoh( ) IndCoh(), Ψ IndCoh() QCoh() f IndCoh, f IndCoh( ) Ψ QCoh( ) Proposition The functor f IndCoh, is a left adjoint to f IndCoh. Proof. It is sufficient to construct a functorial isomorphism (3.1) Maps IndCoh() (f IndCoh, (F ), F ) Maps IndCoh( ) (F, f IndCoh (F )) for F Coh( ) and F Coh(). However, by construction, the lefthand side in (3.1) is Maps Coh() (f (F ), F )) Maps QCoh() (f (F ), F )), while the righthand side maps isomorphically by the functor Ψ Maps QCoh( ) (F, f (F )). Now, (3.1) follows from the (f, f )adjunction on QCoh. Remark It is shown in [GL:IndCoh], that the functor f IndCoh only if the morphism f is eventually coconnective. to admits a left adjoint if and Note that by [GL:DG, Sect. 6, Adjoint Functors], the functor f IndCoh, carries a canonical structure of morphism in QCoh( )mod. It is easy to see that this is the same structure as obtained by indextending the structure of compatibility with the action of QCoh( ) perf on f : Coh( ) Coh() Let (Sch aft ) eventcoconn Sch aft be the 1full subcategory, where we restrict 1morphisms to maps that are eventually coconnective. By [Funct, Sect. 11.2], combining Propositions and 3.1.6, we obtain: Corollary The assignment IndCoh(), f f IndCoh, canonically extends to a functor, to be denoted IndCoh (Sch aft ) eventcoconn, obtained from by adjunction. ((Sch aft ) eventcoconn ) op DGCat cont, IndCoh Schaft (Schaft ) eventcoconn 3.2. Base change for eventually coconnective morphisms. An important property of the *pullback (and one which is crucial for the construction of the!pullback) is base change. It closely mimics the corresponding phenomenon for QCoh.
10 10 INDCOHERENT SHEAVES ON INFSCHEMES Let 1 g 2 be a Cartesian diagram in Sch aft. f 1 f 2 1 g 2 Suppose that f 2 is eventually coconnective. By Corollary 3.1.4, the morphism f 1 is also eventually coconnective. Then the isomorphism of functors (g ) IndCoh gives rise to a natural transformation. (f 1 ) IndCoh (3.2) (f 2 ) IndCoh, (g ) IndCoh (f 2 ) IndCoh Proposition The map (3.2) is an isomorphism. Proof. It is enough to show that (g ) IndCoh (g ) IndCoh (f 1 ) IndCoh,. (3.3) (f 2 ) IndCoh, (g ) IndCoh (F) (g ) IndCoh (f 1 ) IndCoh, (F) is an isomorphism for F Coh( 1 ). In this case both sides of (3.3) belong to IndCoh( 2 ) +. By Proposition 1.2.2, it is therefore sufficient to show that the map (3.4) Ψ 2 (f 2 ) IndCoh, (g ) IndCoh is an isomorphism. and We have: Ψ 2 (f 2 ) IndCoh, (g ) IndCoh Ψ 2 (g ) IndCoh (f 2 ) Ψ 2 (g ) IndCoh (f 1 ) IndCoh, (f 2 ) (g ) Ψ 1 Ψ 2 (g ) IndCoh (f 1 ) IndCoh, (g ) Ψ 1 (f 1 ) IndCoh, (g ) (f 1 ) Ψ 1. Now, it follows from the construction of the (f IndCoh,, f IndCoh )adjunctions, that the map in (3.4) corresponds to the map obtained from the (f, f )adjunctions. (f 2 ) (g ) Ψ 1 (g ) (f 1 ) Ψ 1, Hence, (3.4) is anisomorphism by base change for QCoh Tensoring up. In this section we study the following question: given a map f :, how closely can we approximate IndCoh() from knowing IndCoh( ) and the QCoh categories on both schemes. In the process we will come across several convergencetype assertions, that are of significant technical importance: some maps that are isomorphisms in QCoh are much less obviously so in the IndCoh context.
11 INDCOHERENT SHEAVES ON INFSCHEMES Let f : be an eventually coconnective map. Regarding the functor f IndCoh, as a map in QCoh( )mod, we obtain a functor (3.5) (Id QCoh() f IndCoh, ) : QCoh() IndCoh( ) IndCoh(). QCoh( ) We claim: Proposition The functor (3.5) is fully faithful. Remark It is shown in [GL:IndCoh, Proposition 4.4.9] that the functor (3.5) is an equivalence when f is smooth. In Proposition 4.1.6, we will prove this in the case when f is an open embedding Note that for a diagram of (, 1)categories T D 1 D 2 F 1 C if the functors F 1 and F 2 admit right adjoints, we have a natural transformation of the resulting endofunctors of C: F R 1 F 1 F R 2 F 2. Furthermore, if T is fully faithful, then the above natural transformation is an isomorphism. From here we obtain that the functor (Id QCoh() f IndCoh, ) gives rise to a map of endofunctors of IndCoh( ): (3.6) (f (O ) ) ((f f ) Id IndCoh( ) ) f IndCoh f IndCoh,, where f (O ) denotes the functor of action of f (O ) QCoh( ) on IndCoh( ). Furthermore, from Proposition we obtain: Corollary The map (3.6) is an isomorphism The rest of the subsection is devoted to the proof of Proposition We note that the lefthand side in (3.5) is compactly generated by objects of the form E F QCoh() F 2 QCoh( ) IndCoh( ), where E QCoh() perf and F Coh( ). Moreover, the functor (Id QCoh() f IndCoh, ) sends these objects to compact objects in IndCoh(). Hence, it is enough to show that for E 1, E2 and F1, F2 (3.7) Maps QCoh() IndCoh( )(E 1 F 1, E 2 F 2 ) QCoh( ) as above, the map Maps IndCoh() (E 1 f IndCoh, (F 1 ), E 2 f IndCoh, (F 2 )) is an isomorphism, where in the righthand side denotes the action of QCoh on IndCoh. We can rewrite the map in (3.7) as (3.8) Maps QCoh() IndCoh( )(O F 1, E F 2 ) QCoh( ) Maps IndCoh() (O f IndCoh, (F 1 ), E f IndCoh, (F 2 )),
12 12 INDCOHERENT SHEAVES ON INFSCHEMES where E E 2 (E1 ). Furthermore, we rewrite the map in (3.8) as Maps IndCoh( ) (F 1, f (E) F 2 ) Maps IndCoh( ) (F 1, f IndCoh (E f IndCoh, (F 2 ))). I.e., we are reduced to showing that the following version of the projection formula: Proposition For an eventally coconnective map, the natural transformation between the functors QCoh() IndCoh( ) IndCoh( ), that sends E QCoh() and F IndCoh() to the map (3.9) f (E ) F f IndCoh (E f IndCoh, (F 2 )), is an isomorphism. Remark Note that there is another kind of projection formula, that encodes the compatibility of f IndCoh with the monoidal action of QCoh( ), and which holds tautologically for any morphism f, see Proposition 2.1.4: For E QCoh( ) and F IndCoh() we have: f IndCoh (f (E ) F ) E f IndCoh (F ) Proof of Proposition It is enough to prove the isomorphism (3.9) holds for E QCoh() perf and F Coh( ). We also note that the map (3.9) becomes an isomorphism after applying the functor Ψ, by the usual projection formula for QCoh. For E QCoh() perf and F Coh( ) we have f IndCoh (E f IndCoh, (F )) IndCoh( ) +. Hence, by Proposition 1.2.2, it suffices to show that in this case f (E ) F IndCoh( ) +. We note that the object f (E ) QCoh( ) b is of bounded Tor dimension. The required fact follows from the next general observation: Lemma For Sch aft and E QCoh() b, whose Tor dimension is bounded on the left by an integer n, the functor E : IndCoh() IndCoh() has a cohomological amplitude bounded on the left by n Proof of Lemma We need to show that the functor E sends IndCoh() 0 to IndCoh() n. By Lemma 1.2.4(1), it is sufficient to show that this functor sends Coh() 0 to IndCoh() n. By cohomological devissage, the latter is equivalent to sending Coh() to IndCoh() n. Let i denote the closed embedding cl =:. The functor i IndCoh induces an equivalence Coh( ) Coh(). So, it is enough to show that for F Coh( ), we have F i IndCoh (F ) IndCoh() n. We have: E i IndCoh (F ) i IndCoh (i (E) F ).
13 Note that the functor i IndCoh the same integer n. INDCOHERENT SHEAVES ON INFSCHEMES 13 is texact (since i is), and i (E) has Tor dimension bounded by This reduces the assertion of the lemma to the case when is classical. Further, by Proposition (which will be proved independently later), the statement is Zariski local, so we can assume that is affine. In the latter case, the assumption on E implies that it can be represented by a complex of flat O modules that lives in the cohomological degrees n. This reduces the assertion further to the case when E is a flat O module in degree 0. In this case we claim that the functor is texact. E : IndCoh() IndCoh() The latter follows from Lazard s lemma: such an E is a filtered colimit of locally free O  modules E, while for each such E, the functor E : IndCoh() IndCoh() is by definition the indextension of the functor and the latter is texact. E : Coh() Coh(), 4. Open embeddings The behavior of direct and inverse image functors for open embeddings is, obviously, an important piece of information about IndCoh Restriction to an open. In this subsection we show that the behavior of IndCoh with respect to open embeddings is exactly the same as that of QCoh Let now j : be an open embedding. We claim: Proposition The functor j IndCoh Proof. We need to show that the counit of the adjunction is an isomorphism. : IndCoh( ) IndCoh() is fully faithful. j IndCoh, j IndCoh Id IndCoh( ) Since the functors in question are continuous, it is enough to check that j IndCoh, j IndCoh (F) F is an isomorphism for F Coh(). However, in this case both j IndCoh, j IndCoh (F) and F belong to IndCoh() +, so by Proposition 1.2.2, it is sufficient to check that is an isomorphism. However, Ψ j IndCoh, j IndCoh Ψ j IndCoh, j IndCoh j IndCoh, Ψ j IndCoh Ψ j IndCoh, j IndCoh Ψ, and it follows from the construction of the (f IndCoh,, f IndCoh )adjunction that the resulting map j IndCoh, j IndCoh Ψ Ψ
14 14 INDCOHERENT SHEAVES ON INFSCHEMES comes from the counit of the (j, j )adjunction. Therefore, it is an isomorphism, as is an isomorphism. j j Id QCoh() The next assertion follows immediately from Lemma 1.2.4: Lemma For an open embedding j, the functor j IndCoh, is texact Finally, let us recall the functor (3.5). We claim: Proposition Assume that f is an open embedding. Then the functor (3.5) is an equivalence. Proof. We already know that the functor in question is fully faithful. Hence, it remains to show that its essential image generates the target category. But this follows from Proposition Zariski descent. In this subsection we will show that IndCoh can be glued locally from a Zariski cover, in a way completely parallel to QCoh Let now f : U be a Zariski cover, i.e., U is the disjoint union of open subschemes of, whose union is all of. Let U be the Čech nerve of f. The functors of (IndCoh, )pullback define a cosimplicial category IndCoh(U ), which is augmented by IndCoh(). We claim: Proposition The functor is an equivalence. IndCoh() Tot(IndCoh(U )) Proof. The usual argument reduces the assertion of the proposition to the following. Let = U 1 U 2 ; U 12 = U 1 U 2. Let j 1 j 2 j 12 j 12,1 j 12,2 U 1, U 2, U 12, U 12 U 1, U 12 U 2 denote the corresponding open embeddings. We need to show that the functor that sends F IndCoh() to the datum of IndCoh() IndCoh(U 1 ) IndCoh(U 1 ) IndCoh(U 12) {j IndCoh, 1 (F 1 ), j IndCoh, 2 (F 2 ), j IndCoh, 12,1 (j IndCoh, 1 (F)) j IndCoh, 12 (F) j IndCoh, 12,2 (j IndCoh, 2 (F))} is an equivalence. We construct a right adjoint functor by sending IndCoh(U 1 ) IndCoh(U 1 ) IndCoh() IndCoh(U 12) {F 1 IndCoh(U 1 ), F 2 IndCoh(U 2 ), F 12 IndCoh(U 12 ), j IndCoh, 12,1 (F 1 ) F 12 j IndCoh, 12,2 (F 2 )}
15 INDCOHERENT SHEAVES ON INFSCHEMES 15 to ( ((j1 ker ) IndCoh (F 1 ) (j 2 ) IndCoh (F 1 ) ) ) (j 12 ) IndCoh (F 12 ), where the maps (j i ) IndCoh (F i ) (j 12 ) IndCoh (F 12 ) are (j i ) IndCoh (F i ) (j i ) IndCoh (j 12,i ) IndCoh (j 12,i ) IndCoh, (F i ) = = (j 12 ) IndCoh (j 12,i ) IndCoh, (F i ) (j 12 ) IndCoh (F 12 ). It is strightforward to see from Propositions and that the composition IndCoh(U 1 ) IndCoh(U 1 ) IndCoh() IndCoh(U 1 ) IndCoh(U 1 ) IndCoh(U 12) IndCoh(U 12) is canonically isomorphic to the identity functor. To prove that the composition IndCoh() IndCoh(U 1 ) IndCoh(U 1 ) IndCoh() IndCoh(U 12) is also isomorphic to the identity functor, it is sufficient to show that for F IndCoh(), the canonical map from it to ( ((j1 (4.1) ker ) IndCoh (j 1 ) IndCoh, (F) (j 2 ) IndCoh (j 2 ) IndCoh, (F) ) ) (j 12 ) IndCoh j IndCoh, 12 (F) is an isomorphism. Since all functors in question are continuous, it is sufficient to do so for F Coh(). In this case, both sides of (4.1) belong to IndCoh() +. So, it is enough to prove that the map in question becomes an isomorphism after applying the functor Ψ. However, in this case we are dealing with the map ( ) Ψ (F) ker ((j 1 ) (j 1 ) (Ψ (F)) (j 2 ) (j 2 ) (Ψ (F))) (j 12 ) j12(ψ (F)), which is known to be an isomorphism We also have Proposition Let f : U be a Zariski cover. Then F IndCoh() belongs to IndCoh() 0 (resp., IndCoh() 0 ) if and only if f IndCoh, (F) does. Proof. The only if direction for both statetements follows from Lemma For the if direction, assuming that f (F) IndCoh() 0, it is sufficient to show that Ψ (F) QCoh() 0, and the assertion follows from the corresponding assertion for QCoh. If f IndCoh, (F) IndCoh() 0, the assertion follows from the construction of the inverse functor Tot(IndCoh(U )) IndCoh().
16 16 INDCOHERENT SHEAVES ON INFSCHEMES 5. Proper maps If until now, the theory of IndCoh has run in parallel to (and was inherited from that of) QCoh, in this section we will come across to the main piece of difference between the two. This piece of difference is the functor of!pullback, studied in this section The!pullback. In this subsection we introduce the functor of!pullback for proper maps. Its extension for arbitrary maps between schemes is the subject of [BookII.2, Sect. 3] Let f : be a map in Sch aft. We recall the following definition: Definition The map f is said to be proper (resp., closed embedding) if the corresponding map cl cl has this property Let f : be a proper map. We claim: Lemma The functor f IndCoh : IndCoh() IndCoh( ) sends Coh() IndCoh() to Coh( ) IndCoh( ). Proof. By the construction of f IndCoh, it is sufficient to show that the functor Coh() QCoh() to Coh( ) QCoh( ). f : QCoh() QCoh( ) First, we note that the assertion holds when f is a closed embedding. In general, by the devissage with respect to the tstructure, it is sufficient to show that for F Coh(), we have f (F) Coh( ). Let i denote the canonical closed embedding cl. The functor i is an equivalence Coh( cl ) Coh(). Hence, F = i (F ) for F Coh( cl ). This reduces the assertion to the case when is classical. We factor the map f : as cl i. Since i is a closed embedding, we have reduced the assertion to the case when is classical as well. In the latter case, the assertion is wellknown The above lemma implies that the functor f IndCoh sends IndCoh() c to IndCoh( ) c. Hence, f IndCoh admits a continuous right adjoint, to be denoted f! : IndCoh() IndCoh( ). Remark The continuity of the functor f! is the raison d etre of the category IndCoh, and its main difference from QCoh By [GL:DG, Sect. 6, Adjoint Functors], we obtain that the functor f! has a natural structure of 1morphism in QCoh( )mod.
17 INDCOHERENT SHEAVES ON INFSCHEMES Note that the functor f IndCoh is right texact, up to a finite shift. Hence, the functor f! is left texact up to a finite shift. In particular, f! maps IndCoh( ) + to IndCoh() +. Let f QCoh,! denote the not necessarily continuous right adjoint to f. It also has the property that it maps QCoh( ) + to QCoh() +. Lemma The diagram IndCoh() + f! IndCoh( ) + Ψ Ψ QCoh() + f QCoh,! QCoh( ) + obtained by passing to right adjoints along the horizontal arrows in IndCoh() + Ψ f IndCoh IndCoh( ) + Ψ commutes. QCoh() + f QCoh( ) +, Proof. Follows from the fact that the vertical arrows are equivalences, by Proposition Remark It is not in general true that the diagram IndCoh() Ψ f! IndCoh( ) Ψ QCoh() f QCoh,! QCoh( ) obtained by passing to right adjoints along the horizontal arrows in IndCoh() Ψ f IndCoh IndCoh( ) Ψ commutes. QCoh() f QCoh( ), For example, take = pt = Spec(k), = Spec(k[t]/t 2 ) and 0 F IndCoh( ) be in the kernel of the functor Ψ. Then Ψ f! (F) 0. Indeed, Ψ is an equivalence, and f! is conservative, see Corollary Let (Sch aft ) proper be a 1full subcategory of Sch aft when we restrict 1morphisms to be proper maps. By [Funct, Sect. 11.2], we obtain: Corollary There exists a canonically defined functor obtained from by passing to right adjoints. IndCoh! (Sch aft ) proper : ((Sch aft ) proper ) op DGCat cont, IndCoh (Schaft ) proper := IndCoh (Schaft ) ((Schaft ) proper) op
18 18 INDCOHERENT SHEAVES ON INFSCHEMES 5.2. Base change for proper maps. A crucial property of the!pullback is base change against the *direct image. We establish it in this subsection Let 1 g 2 f 1 f 2 g 1 2 be a Cartesian diagram in Sch aft, with the vertical maps being proper. The isomorphism of functors (f 2 ) IndCoh gives rise to a natural transformation: (g ) IndCoh (5.1) (g ) IndCoh We will prove: (g ) IndCoh f! 1 f! 2 (g ) IndCoh. Proposition The map (5.1) is an isomorphism. (f 1 ) IndCoh Proof. Since all functors involved are continuous, it is enough to show that the map (g ) IndCoh f 1(F)! f 2! (g ) IndCoh (F) is an isomorphism for F Coh( 1 ). Hence, it is enough to show that (5.1) is an isomorphism when restricted to IndCoh( 1 ) +. By Lemma and Proposition 1.2.2, this reduces the assertion to showing that the natural transformation (5.2) (g ) f QCoh,! 1 f QCoh,! 2 (g ) is an isomorphism for the functors QCoh( 1 ) + f QCoh,! 1 (g ) QCoh(2 ) + f QCoh,! 2 QCoh( 1 ) + (g ) QCoh(2 ) +, where the natural transformation comes from the commutative diagram QCoh( 1 ) + (g ) QCoh(2 ) + (f 1) (f 2) QCoh( 1 ) + by passing to right adjoint along the vertical arrows. We consider the commutative diagram QCoh( 1 ) (g ) QCoh(2 ) + (g ) QCoh(2 ) (f 1) (f 2) QCoh( 1 ) (g ) QCoh(2 ),
19 INDCOHERENT SHEAVES ON INFSCHEMES 19 and the the diagram QCoh( 1 ) f QCoh,! 1 (g ) QCoh(2 ) f QCoh,! 2 (g ) QCoh( 1 ) QCoh(2 ), obtained by passing to right adjoints along the vertical arrows. (Note, however, that the functors involved are no longer continuous). We claim that the resulting natural transformation (5.3) (g ) f QCoh,! 1 f QCoh,! 2 (g ) between the functors QCoh( 1 ) QCoh( 2 ) is an isomorphism. This would imply that (5.2) is an isomorphism by restricting to the eventually coconnective subcategory. To prove that (5.3) is an isomorphism, we note that this map is obtained by passing to right adjoints in the natural transformation (5.4) (g ) (f 2 ) (f 1 ) (g ) as functors in the commutative diagram QCoh( 2 ) QCoh( 1 ) QCoh( 1 ) (g ) QCoh( 2 ) (f 1) (f 2) QCoh( 1 ) (g ) QCoh( 2 ). Now, (5.4) is an isomorphism by the usual base change for QCoh. Hence, (5.3) is an isomorphism as well Pullback compatibilty. The!pullback for arbitrary maps between schemes will be defined in such a way that it is the!pullback for proper morphisms, and the *pullback for open embeddings. Hence, if we want that!pullback to be welldefined, a certain compatibility must take place, when we decompose a morphism in two different ways as a composition of a proper morphism and an open embedding. A basic case of such compatibility is established in this subsection Let 1 g 2 f 1 f 2 g 1 2 be a Cartesian diagram in Sch aft, with the vertical maps being proper, and horizontal maps being eventually coconnective. We start with the base change isomorphism (f 1 ) IndCoh g IndCoh, g IndCoh, (f 2 ) IndCoh
20 20 INDCOHERENT SHEAVES ON INFSCHEMES of Proposition 3.2.2, and by the (f IndCoh, f! )adjunction obtain a map (5.5) g IndCoh, Remark Note that one can get another map (5.6) g IndCoh, f! 2 f! 1 g IndCoh,. f! 2 f! 1 g IndCoh,, namely, via the (g IndCoh,, g IndCoh )adjunction from the isomorphism f! 2 (g ) IndCoh (g ) IndCoh f! 1 of Proposition A diagram chase shows that the map (5.6) is canonically the same as (5.5) We are going to prove: Proposition Suppose that g (and hence g ) are open embeddings. Then the map (5.5) is an isomorphism. Remark It is shown in [GL:IndCoh, Proposition 7.1.6] that the map (5.5) is an isomorphism for any eventually coconnective g. Proof. By Proposition 4.1.2, it suffices to show that the induced map is an isomorphism. (g ) IndCoh Using Proposition 5.2.2, we have (g ) IndCoh g IndCoh, f! 1 g IndCoh, Hence, we need to show that the map (5.7) (g ) IndCoh is an isomorphism. (5.8) Consider the diagram (g ) IndCoh g IndCoh, f! 2 (g ) IndCoh f! 2 (g ) IndCoh f! 2 f! 2 (g ) IndCoh (5.7) g IndCoh, f 2! f 2! (g ) IndCoh f! 2 id f! 2, f! 1 g IndCoh, g IndCoh,. g IndCoh, g IndCoh, where the vertical maps are given by the unit of the (g IndCoh,, g )adjunction. We will prove: Lemma The diagram (5.8) commutes. Let us finish the proof of the proposition modulo this lemma. By Corollary 3.3.2, for F IndCoh( 2 ), we have canonical isomorphisms (g ) IndCoh g IndCoh, f! 2(F) (g ) (O 1 ) f! 2(F) and (g ) IndCoh g IndCoh, (F) (g ) (O 1 ) F, where denotes the action of QCoh on IndCoh.
21 INDCOHERENT SHEAVES ON INFSCHEMES 21 By the compatibilty of the action of QCoh with the!pullback, we have f! 2 (g ) IndCoh g IndCoh, (F) f2 ((g ) (O 1 )) f 2(F)! (g ) (O 1 ) f 2(F).! Hence, the map (5.7), evaluated on F, can be thought of as an endomorphism, to be denoted by φ, of Furthermore, by Lemma 5.3.6, the map (g ) (O 1 ) f! 2(F). f! 2(F) O 2 f! 2(F) (g ) (O 1 ) f! 2(F) is canonically homotopic to the tautological map f! 2(F) (g ) (O 1 ) f! 2(F). φ (g ) (O 1 ) f! 2(F) We claim that this implies that φ is itself canonically homotopic to the identity. Indeed, for any F, F IndCoh( 2 ), the map Maps((g ) (O 1 ) F, (g ) (O 1 ) F ) Maps(F, (g ) (O 1 ) F ) identifies by Corollary with the map Maps((g ) IndCoh (g ) IndCoh, (F ), (g ) IndCoh and the latter is an isomorphism by Proposition (g ) IndCoh, (F )) Maps(F, (g ) IndCoh (g ) IndCoh, (F )), Proof of Lemma We remark that the reason that the proof is not completely straightforward is that the natural map in the isomorphism (f 1 ) IndCoh g IndCoh, g IndCoh, (f 2 ) IndCoh, goes from right to left. So, the actual arrow involved in the construction of the map (5.5) is somewhat implicit. By definition, the map followed by in (5.8) is the composition (g ) IndCoh f 1! (f 1 ) IndCoh (g ) IndCoh g IndCoh, f 2! (g ) IndCoh f 1! g IndCoh, g IndCoh, f 2! (g ) IndCoh f 1! g IndCoh, f! 2 f! 2 (g ) IndCoh where the horizontal arrow is given by the inverse of the isomorphism (5.9) g IndCoh, (f 2 ) IndCoh (f 1 ) IndCoh g IndCoh,. (f 2 ) IndCoh f! 2 g IndCoh,,
22 22 INDCOHERENT SHEAVES ON INFSCHEMES Note that the composed map,, in the above diagram can be rewritten as (g ) IndCoh f! 1 (f 1 ) IndCoh g IndCoh, f 2! f 2! (g ) IndCoh f! 2 (g ) IndCoh (f 1 ) IndCoh g IndCoh, f! 2 (g ) IndCoh g IndCoh, f! 2 (f 2 ) IndCoh f! 2 g IndCoh,, where now the second map is again given by the inverse to (5.9). Hence, we need to establish the commutativity of the diagram (g ) IndCoh f 1! (f 1 ) IndCoh (g ) IndCoh g IndCoh, f 2! f 2! (g ) IndCoh (f 1 ) IndCoh g IndCoh, f 2! f 2! (g ) IndCoh g IndCoh, f 2! f 2! (g ) IndCoh Consider the map (5.10) f 2! f 2! (g ) IndCoh equal to the composition g IndCoh, (f 2 ) IndCoh f! 2 g IndCoh, f! 2 (f 2 ) IndCoh f! 2 g IndCoh,. f 2! f 2! (f 2 ) IndCoh f 2! f 2! (g ) IndCoh g IndCoh, (f 2 ) IndCoh f 2,! where both maps are given by the units of the corresponding adjunctions. Now, diagram chase shows that both of the following diagrams commute: (g ) IndCoh and f 1! (f 1 ) IndCoh (g ) IndCoh g IndCoh, f 2! f 2! (g ) IndCoh (f 1 ) IndCoh g IndCoh, f! 2 g IndCoh, f 2! f 2! (g ) IndCoh g IndCoh, (f 2 ) IndCoh f 2! (5.10) id f! 2 f! 2 (5.10) f! 2 (g ) IndCoh id f! 2 g IndCoh, f 2! f 2! (g ) IndCoh Together, this implies the required commutativity. 6. Closed embeddings (f 2 ) IndCoh f! 2 g IndCoh,. The behavior of IndCoh with respect to closed embedding is better than that of QCoh. The main point of difference is that the direct image functor under a closed embedding for IndCoh preserves compactness, which is not the case for QCoh.
23 INDCOHERENT SHEAVES ON INFSCHEMES Category with support. In this subsection we study the full subcategory of IndCoh() corresponding with objects with support on a given closed subscheme Let be an object of Sch aft, and let i : Z be a closed embedding. Let j : U be the complementary open. We let IndCoh() Z denote the full subcategory of IndCoh() equal to ( ) ker j IndCoh, : IndCoh() IndCoh(U). Note that the embedding admits a right adjoint given by sending F to which by Corollary is the same as We claim: Proposition IndCoh() Z IndCoh(Z) ker(f j IndCoh j IndCoh, (F)), ker(o j (O U )) F. (a) The subcategory IndCoh() Z IndCoh() is compatible with the tstructure (i.e., is preserved by the truncation functors). (b) The subcategory IndCoh() Z IndCoh() is generated by the essential image of the functor i IndCoh : IndCoh(Z) IndCoh(). (c) The functor i! : IndCoh() IndCoh(Z) is conservative, when restricted to IndCoh() Z. (d) The category IndCoh() Z identifies with the indcompletion of ( ) Coh() Z := ker j : Coh() Coh(U). Proof. Point (a) follows from the fact that the functor j IndCoh, is texact. Let us prove point (b). The category IndCoh() is generated by IndCoh() +. The description of the right adjoint to IndCoh() Z IndCoh(Z) implies that the same is true for IndCoh() Z. For every F IndCoh() +, the map colim n τ n (F) F, is an isomorphism, by Proposition and because the corresponding fact is true in QCoh +. Hence, by point (a), every F IndCoh() + Z is a colimit of cohomologically bounded objects from IndCoh() Z. This implies that IndCoh() Z is generated by IndCoh() Z = Coh() Z. Now, it is easy to see that every object F Coh() Z has a filtration F = n F n with F n /F n 1 i (Coh(Z) ). This proves point (b). Point (c) follows from point (b) by adjunction.
24 24 INDCOHERENT SHEAVES ON INFSCHEMES To prove point (d), it suffices to note that the objects from Coh() Z are compact in IndCoh() Z (because they are compact in IndCoh()) and that they generate IndCoh() Z, by point (b) As a corollary we obtain: Corollary Let f : be a nilisomorphism, i.e., a map such that red red is an isomorphism. Then the functor f! is conservative. Equivalently, the essential image of IndCoh( ) under f IndCoh generates IndCoh(). Proof. We can assume that = red, so f is also a closed embedding. In this case the assertion of the corollary follows from Proposition 6.1.3(b) A conservativeness result for proper maps. The main result established in this subsection, Proposition 6.2.2, is of technical significance We shall now use Proposition to prove the following: Proposition Let f : be a proper map, which is surjective at the level of geometric points. Then the functor f! : IndCoh( ) IndCoh() is conservative. The rest of this subsection is devoted to the proof of the proposition By Corollary we can assume that both and are classical and reduced. We argue by Noetherian induction, assuming that the statement is true for all proper closed subschemes of. We need to show that the essential image of IndCoh() under f IndCoh generates IndCoh( ). By Proposition 6.1.3(a), it is sufficient to show that contains an open subscheme such that for := f 1 ( ), the essential image of IndCoh( ) under (f ) IndCoh IndCoh( ). generates Since is classical and reduced, it contains a nonempty open smooth subscheme, which we take to be. By Lemma and Lemma 1.1.9, we are reduced to showing the following: Lemma Let f : be a proper morphism of classical schemes with smooth. Then the essential image of QCoh() under f generates QCoh( ). Proof of Lemma Let C QCoh( ) be the full subcategory generated by the essential image of QCoh() under f. Note that C is a monoidal ideal, since the functor f respects the action of QCoh( ). Consider E := f (O ) C. This is an object of QCoh( ), whose fiber at every geometric point of is nonzero. However, it is easy to see that for any Noetherian classical scheme and E C QCoh( ) with the above properties, we have C = QCoh( ).
25 INDCOHERENT SHEAVES ON INFSCHEMES Products. It is known that for a pair of quasicompact schemes 1 and 2, tensor product defines an equivalence of DG categories QCoh( 1 ) QCoh( 2 ) QCoh( 1 2 ). In this subsection we will establish a similar assertion for IndCoh. This is not altogether tautological; for example the validity of this fact replies on the assumption that the ground field k be perfect Let 1 and 2 be two objects of Sch aft. We claim: Lemma There exists a uniquely defined functor (6.1) : IndCoh( 1 ) IndCoh( 2 ) IndCoh( 1 2 ) that preserves compactness and makes the diagram commute. IndCoh( 1 ) IndCoh( 2 ) IndCoh( 1 2 ) Ψ 1 Ψ 2 Ψ 1 2 QCoh( 1 ) QCoh( 2 ) QCoh( 1 2 ) Proof. The anticlockwise composition sends the compact generators of the category IndCoh( 1 ) IndCoh( 2 ) (i.e., objects of the form F 1 F 2 for F i Coh( i )) to QCoh( 1 2 ) +. Hence, it sends all of (IndCoh( 1 ) IndCoh( 2 )) c to QCoh( 1 2 ) +. The soughtfor functor is the indextension of (IndCoh( 1 ) IndCoh( 2 )) c QCoh( 1 2 ) + Ψ 1 2 IndCoh(1 2 ) +. This functor preserves compactness since the objects belong to Coh( 1 2 ) We now claim: F 1 F 2 QCoh( 1 2 ) +, F i Coh( i ) Proposition (a) The functor (6.1) is fully faithful. (b) If the ground field k is perfect, then (6.1) is an equivalence. Proof. Since the functor (6.1) preserves compactness, for point (a) it is sufficient to show that for F i, F i Coh( i), i = 1, 2, the map Maps IndCoh(1) IndCoh( 2)(F 1 F 2, F 1 F 2 ) Maps IndCoh(1 2)(F 1 F 2, F 1 F 2 ) is an isomorphism. We have a commutative diagram Maps IndCoh(1) IndCoh( 2)(F 1 F 2, F 1 F 2 ) Maps IndCoh(1 2)(F 1 F 2, F 1 F Maps QCoh(1)(F 1, F 1 ) Maps QCoh(2)(F 2, F 2 ) Maps QCoh(1 2)(F 1 F 2, F 1 F 2 ), 2 )
26 26 INDCOHERENT SHEAVES ON INFSCHEMES where the vertical arrows are isomorphisms, by the coherence assumption. Hence, it remains to show that Maps QCoh(1)(F 1, F 1 ) Maps QCoh(2)(F 2, F 2 ) Maps QCoh(1 2)(F 1 F 2, F 1 F 2 ) is an isomorphism. This is not immediate since the objects F i QCoh(S i) are not compact. To circumvent this, we proceed as follows. It is enough to show that ( ) τ n Maps QCoh(1)(F 1, F 1 ) Maps QCoh(2)(F 2, F 2 ) is an isomorphism for any fixed n. τ n ( Maps QCoh(1 2)(F 1 F 2, F 1 F 2 ) Choose α 1 : F 1 F 1 (resp., α 2 : F 2 F 2) with F 1 (resp., F 2) in QCoh( 1 ) perf (resp., QCoh( 2 ) perf ), such that for N 0. Cone(α 1 ) QCoh( 1 ) N and Cone(α 2 ) QCoh( 2 ) N By choosing N large ( enough, we can ensure that ) ( ) τ m Maps QCoh(Si)(F i, F i ) τ m Maps QCoh(Si)( F i, F i ) is an isomorphism for a given integer m, which in turn implies that ( ) τ n Maps QCoh(1)(F 1, F 1 ) Maps QCoh(2)(F 2, F 2 ) and ( ) τ n Maps QCoh(1 2)(F 1 F 2, F 1 F 2 ) are isomorphisms. Hence, it is enough to show that τ n ( Maps QCoh(1)( F 1, F 1 ) Maps QCoh(2)( F 2, F 2 ) ( τ n Maps QCoh(1 2)( F 1 F ) 2, F 1 F 2 ) Maps QCoh(1)( F 1, F 1 ) Maps QCoh(2)( F 2, F 2 ) Maps QCoh(1 2)( F 1 F 2, F 1 F 2 ) is an isomorphism. But this follows from the fact that the fuctor (6.2) QCoh( 1 ) QCoh( 2 ) QCoh( 1 2 ) is an equivalence. This finishes the proof of point (a). To prove point (b), we have to show that the essential image of the functor (6.1) generates IndCoh( 1 2 ). By Corollary we can assume that both 1 and 2 are classical and reduced. We argue by Noetherian induction, assuming that the statement is true for all proper closed subschemes i i. By Proposition 6.1.3(b), it is sufficient to show that 1 and 2 contain nonempty open subschemes i i, for which the statement of the proposition is true. ) )
27 INDCOHERENT SHEAVES ON INFSCHEMES 27 Since i are classical and reduced, we can take i to be a nonempty open smooth subscheme of i. Note that the assumption that k be perfect implies that 1 2 is also smooth. Now, the assertion follows from Lemma and the fact that (6.2) is an equivalence Upgrading to a functor. Consider the category Sch aft as endowed with a symmetric monoidal structure given by Cartesian product. We consider the category DGCat cont also as a symmetric monoidal category with respect to the operation of tensor product. First we recall: Proposition The functor has a natural symmetric monoidal structure. QCoh Sch aft : (Sch aft ) op DGCat cont Proof. Note that the functor QCoh Sch aft upgrades to a functor (Sch aft ) op DGCat SymMon cont = ComAlg(DGCat cont ). We have the following general construction. Let O be a symmetric monoidal category, and let F : I op ComAlg(O) be a functor, where I is a category with finite products. Then F, when viewed as a functor F : I op O, has a natural structure of rightlax symmetric monoidal functor, where the symmetric monoidal structure on I is given by Cartesian product. Informally, for i 1, i 2 I, we have the map F (i 1 ) F (i 2 ) F (π1) F (π2) F (i 1 i 2 ) F (i 1 i 2 ) F (i 1 i 2 ), where π i are the two projections i 1 i 2 i i, and the map F (i 1 i 2 ) F (i 1 i 2 ) F (i 1 i 2 ) is given by the commutative algebra structure on F (i 1 i 2 ). Hence, we obtain that QCoh Sch aft has a canonical rightlax symmetric monoidal structure. The resulting maps QCoh( 1 ) QCoh( 2 ) QCoh( 1 2 ) are given by the operation of external tensor product. The fact that the above rightlax symmetric monoidal structure is strict is the fact that the maps (6.2) are equivalences for i Sch aft by [GL:QCoh, Proposition 1.4.4] Passing to adjoints, by [Funct, Adjunction and Symmetric Monoidal Structures], we obtain that the functor QCoh Schaft : Sch aft DGCat cont also has a natural symmetric monoidal structure. Now, arguing as in Proposition we show: Proposition There exists a unique symmetric monoidal structure on the functor IndCoh Schaft : Sch aft DGCat cont and the functor Ψ Schaft, which at the level of objects is given by Lemma
28 28 INDCOHERENT SHEAVES ON INFSCHEMES 6.4. Convergence. In this subsection we will establish another crucial property of IndCoh that distinguishes it from QCoh. Namely, we will show that for a given scheme, its category IndCoh can be recovered from IndCoh on the ncoconnective truncations of this scheme Let be an object of Sch aft. For each n let i n denote the closed embedding n, and for n 1 n 2, let i n1,n 2 denote the closed embedding n 1 n2. By [GL:DG, Lemma 1.3.3], we have a canonical equivalence colim n IndCoh( n ) lim n IndCoh( n ), where in the lefthand side the transition functors are (i n1,n 2 ) IndCoh, and in the righthand side (i n1,n 2 )!. The functors i! n define a functor (6.3) IndCoh() lim n IndCoh( n ), whose left adjoint (6.4) colim IndCoh( n ) IndCoh() n is given by the compatible family of functors (i n ) IndCoh We are going to establish the following property of the category IndCoh: Proposition The functors (6.3) and (6.4) are mutually inverse equivalences. Proof. First, we note that Corollary shows that the functor (6.3) is conservative. It is clear that the functor (6.4) sends compact objects to compact ones. Hence, it remains to prove the following: for and k N, the map is an isomorphism. colim n We will prove more generally the following: Lemma For the map F 1, F 2 Coh( 0 ) Coh() Maps Coh( n )(F 1, F 2 [k]) Maps Coh() (F 1, F 2 [k]) F 1 (QCoh( n )) 0, F 2 (QCoh( n )) k, (6.5) Maps QCoh( n )(F 1, F 2 ) Maps QCoh() ((i n ) (F 1 ), (i n ) (F 2 )) is an isomorphism for n k.
PART III.3. INDCOHERENT SHEAVES ON INDINFSCHEMES
PART III.3. INDCOHERENT SHEAVES ON INDINFSCHEMES Contents Introduction 1 1. Indcoherent sheaves on indschemes 2 1.1. Basic properties 2 1.2. tstructure 3 1.3. Recovering IndCoh from indproper maps
More informationIndCoh Seminar: Indcoherent sheaves I
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