A recurrent formula of A_{\infty}-quasi inverses of dg-natural transformations between dg-lifts of derived functors
aa r X i v : . [ m a t h . C T ] N ov A recurrent formula of A ∞ -quasi inverses of dg-naturaltransformations between dg-lifts of derived functors Zhaoting WeiNovember 19, 2019
Abstract
A dg-natural transformation between dg-functors is called an objectwise homotopy equivalence ifits induced morphism on each object admits a homotopy inverse. In general an objectwise homotopyequivalence does not have a dg-inverse but has an A ∞ quasi-inverse. In this note we give a recurrentformula of the A ∞ quasi-inverse. This result is useful in studying the compositions of dg-lifts ofderived functors of schemes. In [Sch18], Schn¨urer constructed Grothendieck six functor formalism of dg-enhancements for ringedspaces over a field k . In more details, for each k -ringed space X we have a dg k -category I ( X ) which isa dg-enhancement of D ( X ) , the derived category of sheaves of O X -modules. Moreover for a morphism f ∶ X → Y of k -ringed space we have dg k -functors f ∗ ∶ I ( Y ) → I ( X ) and f ∗ ∶ I ( X ) → I ( Y ) which are dg-lifts of the derived functos L f ∗ and R f ∗ , respectively. In addition, Schn¨urer showedthat for two composable morphisms f ∶ X → Y and g ∶ Y → Z , we have zig-zags of dg-naturaltransformations which are objectwise homotopy equivalences between ( gf ) ∗ and f ∗ g ∗ and ( gf ) ∗ and f ∗ g ∗ .We call a dg k -natural transformation Φ ∶ F → G an objectwise homotopy equivalence if for anyobject E , the induced morphism Φ E ∶ F (E) → G (E) has a homotopy inverse. This does not mean that we could find a homotopy inverse of Φ because theobjectwise homotopy inverses are not compatible with morphisms as illustrated in the following diagrams F (E ) Φ E ÐÐÐ→ G (E )×××Ö F ( α ) ↻ G ( α ) ×××Ö F (E ) Φ E ÐÐÐ→ G (E ) but F (E ) Φ − E ←ÐÐÐ G (E )×××Ö F ( α ) ↻̸ G ( α ) ×××Ö F (E ) Φ − E ←ÐÐÐ G (E ) . Nevertheless, by [Lyu03, Proposition 7.15] we know that we can extend Φ − to an A ∞ natural trans-formation Ψ ∶ G ⇒ F and Ψ is an A ∞ quasi-inverse of Φ . In this note we give a detailed constructionof the recurrent formula of Ψ as suggested in [Lyu03, Appendix B]. In particular we show that we canconstruct Ψ by compositions of objectwisely chosen homotopies. See Theorem 4.1 below. This formulawill be used in [Wei19]. 1 cknowledgments The author would like to thank Nick Gurski and Olaf Schn¨urer for very helpful discussions. A ∞ -natural transforma-tions In this section we review some concepts around dg-functors, dg-natural transformations, and A ∞ -natural transformations. Remark . We would like to point out that the best way to describe A ∞ -categories/functors /natural-transformations is in the framework of bar constructions and dg-cocategories, see [Lyu03]. In this notewe just take the by-hand definition, which requires minimal amount of preparation but involves morecomplicated notations. Definition 2.1 (dg-categories) . Let k be a commutative ring with unit. A differential graded or dg k -category is a category C whose morphism spaces are cochain complexes of k -modules and whosecompositions of morphisms C( Y, Z ) ⊗ k C( X, Y ) → C( X, Z ) are morphisms of k -cochain complexes. Furthermore, there are obvious associativity and unit axioms. Definition 2.2 (dg-functors) . Let k be a commutative ring with unit and C and D be two dg k -categories.A dg k -functor F ∶ C → D consists of the following data:1. A map F ∶ obj (C) → obj (D) ;2. For any objects X , Y ∈ obj (C) , a closed, degree morphism of complexes of k -modules F ( X, Y ) ∶ C(
X, Y ) → D( F X, F Y ) which is compatible with the composition and the units. Definition 2.3 (dg-natural transformation) . Let k be a commutative ring with unit and F , G ∶ C → D betwo dg k -functors between dg k -categories. A dg k -prenatural transformation Φ ∶ F ⇒ G of degree n consists of a morphism Φ X ∈ D n ( F X, GX ) for each object X such that for any morphism u ∈ C m ( X, Y ) we have Φ Y F u = (− ) mn Gu Φ X . The differential on Φ is defined objectwisely and it is clear that d Φ is a dg k -prenatural transformationof degree n + . We call Φ a dg k -natural transformation if Φ is closed and of degree . Definition 2.4 ( A ∞ -prenatural transformation) . Let k be a commutative ring with unit and F , G ∶ C → D be two dg k -functors between dg k -categories. An A ∞ k -prenatural transformation Φ ∶ F ⇒ G ofdegree n consists of the following data:1. For any object X ∈ obj (C) , a morphism Φ X ∈ D n ( F X, GX ) ;2. For any l ≥ and any objects X , . . . , X l ∈ obj (C) , a morphism Φ lX ,...,X l ∈ Hom n − lk (C( X l − , X l ) ⊗ k . . . ⊗ k C( X , X ) , D( F X , GX l )) efinition 2.5 (Differential of A ∞ -prenatural transformation) . Let k be a commutative ring with unitand F , G ∶ C → D be two dg k -functors between dg k -categories. Let Φ ∶ F ⇒ G be an A ∞ k -prenaturaltransformation of degree n as in Definition 2.4. Then the differential d Φ ∶ F ⇒ G is an A ∞ k -prenaturaltransformation of degree n + whose components are given as follows:1. For any object X ∈ obj (C) , ( d ∞ Φ ) X = d ( Φ X ) ∈ D n + ( F X, GX ) ;2. For any l ≥ and a collection of morphisms u i ∈ C( X i − , X i ) i = , . . . , l , ( d ∞ Φ ) l ( u l ⊗ . . . ⊗ u ) = d ( Φ l ( u l ⊗ . . . ⊗ u )) + (− ) ∣ u l ∣− G ( u l ) Φ l − ( u l − ⊗ . . . ⊗ u )+ (− ) n ∣ u ∣−∣ u ∣− ... −∣ u l ∣+ l − Φ l − ( u l ⊗ . . . ⊗ u ) F ( u )+ l ∑ i = (− ) ∣ u l ∣+ ... +∣ u i + ∣+ l − i + Φ l ( u l ⊗ . . . du i ⊗ . . . u )+ l − ∑ i = (− ) ∣ u l ∣+ ... +∣ u i + ∣+ l − i + Φ l − ( u l ⊗ . . . u i + u i ⊗ . . . u ) (1)We can check that d ∞ ○ d ∞ = on A ∞ k -prenatural transformations. Definition 2.6 ( A ∞ -natural transformation) . Let k be a commutative ring with unit and F , G ∶ C → D be two dg k -functors between dg k -categories. Let Φ ∶ F ⇒ G be an A ∞ k -prenatural transformation.We call Φ an A ∞ k -natural transformation if Φ is of degree and closed under the differential d ∞ inDefinition 2.5.It is clear that a dg k -natural transformation Φ can be considered as an A ∞ k -natural transformationwith Φ l = for all l ≥ . Definition 2.7 (Compositions) . Let k be a commutative ring with unit and F , G , H ∶ C → D be three dg k -functors between dg k -categories. Let Φ ∶ F ⇒ G be a dg k -natural transformation and Ψ ∶ G ⇒ H be an A ∞ k -natural transformation. Then the composition Ψ ○ Φ is defined as follows: For any object X ∈ obj (C) ( Ψ ○ Φ ) X ∶ = Ψ X Φ X ∶ F X → GX → HX and for any u i ∈ C( X i − , X i ) , i = , . . . , l ( Ψ ○ Φ ) l ( u l ⊗ . . . u ) ∶ = Ψ l ( u l ⊗ . . . u ) Φ X We can check that Ψ ○ Φ is an A ∞ k -natural transformation.Similarly, Let Φ ∶ F ⇒ G be an A ∞ k -natural transformation and Ψ ∶ G ⇒ H be a dg k -naturaltransformation. Then the composition Ψ ○ Φ is defined as follows: For any object X ∈ obj (C)( Ψ ○ Φ ) X ∶ = Ψ X Φ X ∶ F X → GX → HX and for any u i ∈ C( X i − , X i ) , i = , . . . , l ( Ψ ○ Φ ) l ( u l ⊗ . . . u ) ∶ = Ψ X l Φ l ( u l ⊗ . . . u ) We can check that Ψ ○ Φ is an A ∞ k -natural transformation. Remark . We can define compositions for general A ∞ k -prenatural transformations. See [Lyu03, Sec-tion 3] or [Sei08, Section I.1(d)]. 3 efinition 2.8 ( A ∞ quasi-inverse) . Let k be a commutative ring with unit and F , G ∶ C → D be twodg k -functors between dg k -categories. Let Φ ∶ F ⇒ G be a dg k -natural transformation. We call an A ∞ k -natural transformation Ψ ∶ G ⇒ F an A ∞ quasi-inverse of Φ if there exists A ∞ k -prenaturaltransformations η ∶ F ⇒ F and ω ∶ G ⇒ G both of degree − such that Ψ ○ Φ − id F = d ∞ η, and Φ ○ Ψ − id G = d ∞ ω. In more details, this means that we have Ψ X Φ X − id F X = dη X , and Φ X Ψ X − id GX = dω X for any X ∈ obj C and for any l ≥ and any u i ∈ C( X i − , X i ) , i = , . . . , l , we have Ψ l ( Φ ( u l ) ⊗ . . . ⊗ Φ ( u )) = d ( η l ( u l ⊗ . . . ⊗ u )) + (− ) ∣ u l ∣− G ( u l ) η l − ( u l − ⊗ . . . ⊗ u )+ (− ) −∣ u ∣− ... −∣ u l ∣+ l − η l − ( u l ⊗ . . . ⊗ u ) F ( u )+ l ∑ i = (− ) ∣ u l ∣+ ... +∣ u i + ∣+ l − i + η l ( u l ⊗ . . . du i ⊗ . . . u )+ l − ∑ i = (− ) ∣ u l ∣+ ... +∣ u i + ∣+ l − i + η l − ( u l ⊗ . . . u i + u i ⊗ . . . u ) (2)and Φ ( Ψ l ( u l ⊗ . . . ⊗ u )) = d ( ω l ( u l ⊗ . . . ⊗ u )) + (− ) ∣ u l ∣− G ( u l ) ω l − ( u l − ⊗ . . . ⊗ u )+ (− ) −∣ u ∣− ... −∣ u l ∣+ l − ω l − ( u l ⊗ . . . ⊗ u ) F ( u )+ l ∑ i = (− ) ∣ u l ∣+ ... +∣ u i + ∣+ l − i + ω l ( u l ⊗ . . . du i ⊗ . . . u )+ l − ∑ i = (− ) ∣ u l ∣+ ... +∣ u i + ∣+ l − i + ω l − ( u l ⊗ . . . u i + u i ⊗ . . . u ) (3) The main reference of this section is [Sch18].
Let k be a field and X be a k -ringed space. Let C ( X ) be the dg k -category of complexes of sheaveson X and I ( X ) its full dg k -subcategory of h-injective complexes of injective sheaves. Let I b ( X ) and I + ( X ) be the full subcategories of I ( X ) consisting of complexes with bounded or bounded belowcohomology sheaves, respectively. See [Spa88] or [KS06, Chapter 14] for an introduction to h-injectivecomplexes.It is clear that I ( X ) is a strongly pretriangulated dg k -category hence its homotopy category [ I ( X )] is a triangulated k -category and the obvious functor [ I ( X )] → D ( X ) is a triangulated equivalence.We could construct an equivalence in the other direction.4 roposition 3.1. [[Sch18, Corollary 2.3]] Let k be a field. Let ( X, O ) be a k -ringed site and let C ( X ) hflat, cwflat denote the full dg k -subcategory of C ( X ) of h-flat and componentwise flat objects. Thenthere exists dg k -functors i ∶ C ( X ) → I ( X ) e ∶ C ( X ) → C ( X ) hflat, cwflat (4) together with dg k -natural transformations ι ∶ id → i ∶ C ( X ) → C ( X ) ǫ ∶ e → id ∶ C ( X ) → C ( X ) (5) whose evaluations ι F ∶ F → i F and ǫ F ∶ e F → F at each object F ∈ C ( X ) are quasi-isomorphisms. It is clear that the induced functor [ i ] ∶ [ C ( X )] → [ I ( X )] sends acyclic objects to zero, hence itfactors to an equivalence ¯ [ i ] ∶ D ( X ) ∼ → [ I ( X )] of triangulated k -categories.Intuitively the dg k -functor i in Proposition 3.1 could be considered as a functorial injective resolu-tion. Remark . The result in Proposition 3.1 is an adaption of general results from enriched model categorytheory in [Rie14], in particular [Rie14, Corollary 13.2.4].
Remark . The assumption that k is a field is essential for Proposition 3.1. Actually if k = Z , then thepair ( i , ι ) in Proposition 3.1 does not exist. See [Sch18, Lemma 4.4] for a counterexample. Definition 3.1. [[Sch18, 2.3.4]] Let k be a field. For a morphism of k -ringed spaces f ∶ X → Y , wedefine the injective pull back dg k -functor f ∗ as f ∗ ∶ = i ○ f ∗ ○ e (6)Similarly we define the injective push forward dg k -functor f ∗ as f ∗ ∶ = i ○ f ∗ (7)where i and e are defined in Proposition 3.1. Remark . Actually in [Sch18] all Grothendieck’s six functors were lifted to dg k -functors. Proposition 3.2. [[Sch18, Proposition 6.5]] Let k be a field. For a morphism of k -ringed spaces f ∶ X → Y , the dg k -functors f ∗ and f ∗ in Definition 3.1 are dg-lifts of the derived pull back and derivedpush forward functors L f ∗ ∶ D ( Y ) → D ( X ) and R f ∗ ∶ D ( X ) → D ( Y ) , respectively. More precisely,the diagrams D ( Y ) L f ∗ ÐÐÐ→ D ( X ) ¯ [ i ] ×××Ö ∼ ¯ [ i ] ×××Ö ∼ [ I ( Y )] [ f ∗ ] ÐÐÐ→ [ I ( X )] nd D ( X ) R f ∗ ÐÐÐ→ D ( Y ) ¯ [ i ] ×××Ö ∼ ¯ [ i ] ×××Ö ∼ [ I ( X )] [ f ∗ ] ÐÐÐ→ [ I ( Y )] commute up to a canonical -isomorphism. By Definition 3.1 it is clear that we do not have ( gf ) ∗ = f ∗ g ∗ . Actually ( gf ) ∗ and f ∗ g ∗ areconnected by a zig-zag of dg natural transformations. To describe this relation more clearly, we introducethe following definitions. Definition 3.2. [[Sch18, 2.1.3]] Let k be a field and X , Y be k -ringed spaces. Let F, G ∶ I ( X ) → I ( Y ) bedg k -functors. A dg k -natural transformation Φ ∶ F ⇒ G is called an objectwise homotopy equivalence if for any object E ∈ obj ( I ( X )) , the morphism Φ E ∶ F E → G E has a homotopic inverse. Proposition 3.3. [[Sch18, Proposition 6.17, Lemma 6.21]] Let k be a field and X f → Y g → Z be mor-phisms of k -ringed spaces. Then there exist zig-zags of objectwise homotopy equivalences T f,g ∶ ( gf ) ∗ ∼ → f ∗ g ∗ T f,g ∶ ( gf ) ∗ ∼ → g ∗ f ∗ T id ∶ id ∗ ∼ → id T id ∶ id ∗ ∼ → id (8) Proof.
We give the relation between ( gf ) ∗ and f ∗ g ∗ to illustrate the idea. We use the dg k -naturaltransformations ι ∶ id → i and ǫ ∶ e → id in Proposition 3.1 and have the following objectwise homotopyequivalences ( gf ) ∗ = i ( gf ) ∗ e ∼ → i f ∗ g ∗ e ǫ ←Ð ∼ i f ∗ e g ∗ e ι Ð→ ∼ i f ∗ ei g ∗ e = f ∗ g ∗ . A ∞ quasi-inverses Definition 4.1.
Let k be a field and X , Y be k -ringed spaces. Let F , G ∶ I ( X ) → I ( Y ) be two dg k -functors and Φ ∶ F → G be a dg k -natural transformation which is an objectwise homotopy equivalence.For each object E ∈ obj ( I ( X )) we can choose Ψ E ∈ I ( Y )( G E , F E ) , h E ∈ I − ( Y )( F E , F E ) , p E ∈ I − ( Y )( G E , G E ) , such that Ψ E Φ E − id F E = dh E , Φ E Ψ E − id G E = dp E . We call such a choice an objectwise homotopy inverse system of Φ .For a objectwise homotopy equivalence Φ , its homotopy invese system always exists.The following theorem is the main result of this note.6 heorem 4.1. Let k be a field and X , Y be k -ringed spaces. Let F , G ∶ I ( X ) → I ( Y ) be two dg k -functors and Φ ∶ F → G be a dg k -natural transformation which is an objectwise homotopy equivalence.Then there exists an A ∞ quasi-inverse of Φ . More precisely, we choose and fix an objectwise homotopyinverse system H of Φ as in Definition 4.1 and there exist an A ∞ k -natural transformation Ψ ∶ G ⇒ F and A ∞ k -prenatural transformations η ∶ F ⇒ F and ω ∶ G ⇒ G of degree − such that Ψ ○ Φ − id F = dη, and Φ ○ Ψ − id G = dω. Moreover, Ψ , η , and ω are defined by compositions of F , G , Φ , and H .Proof. The proof is a refinement of [Lyu03, Proposition 7.15]. We construct Ψ , η , and ω by induction.First we construct the left inverse. Let Ψ E = Ψ E and η E = p E as in Definition 4.1. Now suppose that foran m ≥ we have constructed Ψ i and η i , i = , . . . , m − by compositions of F , G , Φ , and H such thatthe auxiliary A ∞ k -prenatural transformations ̃ Ψ = ( Ψ , Ψ , . . . , Ψ m − , , . . . )̃ η = ( η , η , . . . , η m − , , . . . ) satisfy ( d ∞ ̃ Ψ ) l = , and (̃ Ψ ○ Φ − id F ) l = ( d ∞ ̃ η ) l hold for l = , . . . , m − . Now we introduceWe denote ( d ∞ ̃ Ψ ) m by λ m . For objects E , . . . , E m , λ m can by considered as a degree − m map I ( X )( E m − , E m ) ⊗ . . . ⊗ I ( X )( E , E ) → I ( Y )( G E , F E m ) . For later applications we consider λ m as a degree map λ m ∈ Hom ( I ( X )( E m − , E m )[ ] ⊗ . . . ⊗ I ( X )( E , E )[ ] , I ( Y )( G E , F E m )[ ]) Moreover we denote (̃ Ψ ○ Φ − id F − d ∞ ̃ η ) m by µ m . As before we have µ m ∈ Hom − ( I ( X )( E m − , E m )[ ] ⊗ . . . ⊗ I ( X )( E , E )[ ] , I ( Y )( F E , F E m )[ ]) Lemma 4.2.
In the above notation, λ m and µ m are defined by by compositions of F , G , Φ , and H .Proof. By Definition 2.5 we have λ m ( u m ⊗ . . . ⊗ u ) = ( d ∞ ̃ Ψ ) m ( u m ⊗ . . . ⊗ u ) = ( − ) ∣ u m ∣ − G ( u l ) Ψ m − ( u m − ⊗ . . . ⊗ u ) + ( − ) n ∣ u ∣ − ∣ u ∣ − ... − ∣ u m ∣ + m − Ψ m − ( u m ⊗ . . . ⊗ u ) F ( u ) + m − ∑ i = ( − ) ∣ u m ∣ + ... + ∣ u i + ∣ + m − i + Ψ m − ( u m ⊗ . . . u i + u i ⊗ . . . u ) The claim for λ m is clear by the induction hypothesis. We can prove the claim for µ m in the sameway. Lemma 4.3.
In the above notation, we have dλ m = , and dµ m = λ m Φ E . roof. By the induction hypothesis we have ( d ∞ ̃ Ψ ) l = for l = , . . . , m − . Then d ∞ d ∞ ̃ Ψ = d [( d ∞ ̃ Ψ ) m ] = dλ m . Bu we have d ∞ d ∞ ̃ Ψ = hence dλ m = .Since (̃ Ψ ○ Φ − id F − d ∞ ̃ η ) l = for l = , . . . , m − , it is clear that [ d ∞ (̃ Ψ ○ Φ − id F − d ∞ ̃ η )] m = dµ m On the other hand since Φ and id F are dg k -natural transformations, we have d ∞ Φ = and d ∞ id F = .Therefore d ∞ (̃ Ψ ○ Φ − id F − d ∞ ̃ η ) = ( d ∞ ̃ Ψ ) ○ Φ . Compare the degree m component we have dµ m = λ m Φ E .As suggested by [Lyu03, Appendix B] we let Ψ m = λ m p E − µ m Ψ E (9)and η m = − µ m h E + µ m Ψ E Φ E h E − λ m p E Φ E h E − µ m Ψ E p E Φ E + λ m p E p E p E Φ E . (10)It is clear that d ( Ψ m ) = − λ m , and d ( η m ) = Ψ m Φ E + µ m . (11)Let ̃̃ Ψ = ( Ψ , Ψ , . . . , Ψ m − , Ψ m , , . . . )̃̃ η = ( η , η , . . . , η m − , η m , , . . . ) It is clear by Equation (11) and the induction hypothesis that ( d ∞ ̃̃ Ψ ) l = , and (̃̃ Ψ ○ Φ − id F ) l = ( d ∞ ̃̃ η ) l hold for l = , . . . , m . Then by induction we construct an A ∞ k -natural transformation Ψ ∶ G ⇒ F ofdegree and an A ∞ k -prenatural transformation η ∶ F ⇒ F of degree − such that d ∞ Ψ = , and Ψ ○ Φ − id F = d ∞ η. Notice that Ψ and η are defined by compositions of F , G , Φ , and H .We need to construct the homotopy of the other composition. Actually in the same way we can con-struct an A ∞ k -natural transformation Ψ ′ ∶ G ⇒ F of degree and an A ∞ k -prenatural transformation ω ′ ∶ G ⇒ G of degree − such that d ∞ Ψ ′ = , and Φ ○ Ψ ′ − id G = d ∞ ω ′ where Ψ ′ and ω ′ are also defined by compositions of F , G , Φ , and H . It is clear that Ψ ′ = Ψ + d ∞ ( Ψ ω ′ − η Ψ ′ ) . Therefore let ω ∶ = ω ′ + Φ η Ψ ′ − ΦΨ ω ′ then we have Φ ○ Ψ − id G = d ∞ ω. and ω is also defined by compositions of F , G , Φ , and H .8 emark . The recurrent definition of Ψ and η is given by Equation (9) and (10). Corollary 4.4.
Let k be a field and X f → Y g → Z be morphisms of k -ringed spaces. Then there exist A ∞ k -natural transformations T f,g ∶ ( gf ) ∗ ∼ → f ∗ g ∗ T f,g ∶ ( gf ) ∗ ∼ → g ∗ f ∗ T id ∶ id ∗ ∼ → id T id ∶ id ∗ ∼ → id (12) which only depend on the choice of objectwise homotopy inverse systems as in Definition 4.1.Proof. It is a direct corollary of Proposition 3.3 and Theorem 4.1.
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