We provide a new and very short proof of the fact that a spherical functor
between certain triangulated categories induces an autoequivalence.Comment: Final version. Accepted for publication in Bull. Lond. Math. So

Meachan, Ciaran

English

arXiv

We provide a new and very short proof of the fact that a spherical functor between certain triangulated categories induces an auto‐equivalence

Enlighten

Bull. London Math. Soc. 0 (2021) 1–7 doi:10.1112/blms.12476A note on spherical functorsCiaran MeachanAbstractWe provide a new and very short proof of the fact that a spherical functor between certaintriangulated categories induces an auto-equivalence.IntroductionLet D(X) denote the bounded derived category of coherent sheaves on a smooth projectivevariety X. If Y is another smooth projective variety, then any object P ∈ D(X × Y ) gives rise toa Fourier–Mukai functor F = ΦP : D(X) → D(Y ), and we refer to the object P as the Fourier–Mukai kernel of F . Similarly, if Z is a third smooth projective variety and Q ∈ D(Y × Z) isany object, then the composition ΦQ ◦ ΦP is induced by the convolution Q ∗ P := π13∗(π∗12P ⊗π∗23Q) ∈ D(X × Z).Now, since such an F = ΦP : D(X) → D(Y ) has a left adjoint L = ΦPL and a right adjointR = ΦPR , we can use the unit and counit of adjunction to define new kernels via the followingtriangles:C → Δ∗OXη−→ PR ∗ P and P ∗ PR ε−→ Δ∗OY → T .The induced functors C = ΦC : D(X) → D(X) and T = ΦT : D(Y ) → D(Y ) are called thecotwist and twist of F , respectively.In this brief note, we give a short and simple proof, relying solely on the structure ofadjunctions, and a classical result found in [5], of the following theorem.Theorem [2.3 and 3.2]. Let F = ΦP : D(X) → D(Y ) be a Fourier–Mukai functor betweenthe bounded derived categories of two smooth projective varieties X and Y , with leftadjoint L = ΦPL and right adjoint R = ΦPR . Suppose that the cotwist C = ΦC is an auto-equivalence of D(X) and PR  C ∗ PL[1] is any isomorphism. Then the canonical map PR →PR ∗ P ∗ PL → C ∗ PL[1] is an isomorphism and the twist T = ΦT is an auto-equivalence ofD(Y ).The observation that spherical functors give rise to interesting auto-equivalences is welldocumented. It is hard to underestimate the importance of the paper [10]. Their ideas werefurther developed in [2, 8] and the foundations were finally completed in [3]. Other notableworks include [1, 6, 9]. Our proof is different from all of these and so we feel it is worthyof mention.Received 14 December 2019; revised 3 December 2020.2020 Mathematics Subject Classification 18G80, 18E30 (primary).This research is supported by the EPSRC Doctoral Prize Research Fellowship Grant no. EP/K503034/1.C2021 The Authors. Bulletin of the London Mathematical Society is copyright CLondon MathematicalSociety. This is an open access article under the terms of the Creative Commons Attribution License, whichpermits use, distribution and reproduction in any medium, provided the original work is properly cited.2 CIARAN MEACHAN1. PreliminariesDefinition 1.1. If F = ΦP : D(X) → D(Y ) is a Fourier–Mukai functor with left adjointL = ΦPL and right adjoint R = ΦPR , then we distinguish the units and counits with subscriptsas follows:Δ∗OXηR−−→ PR ∗ P P ∗ PR εR−−→ Δ∗OY Δ∗OYηL−−→ P ∗ PL PL ∗ P εL−→ Δ∗OX .If an argument only deals with one adjoint pair, then we shall drop the subscripts.Lemma 1.2. Units and counits are exchanged under the adjunction isomorphisms:Hom(Δ∗OX ,PR ∗ P)  Hom(PL ∗ P,Δ∗OX) ; ηR → εLHom(P ∗ PR,Δ∗OY )  Hom(Δ∗OY ,P ∗ PL) ; εR → ηL.Proof. For the first one, recall from [7, Chapter IV] that an adjunction is a bijection whichassigns arrows according to a specific recipe. In particular, we have:where ε is the counit associated to the adjoint pair (PL ∗ P,PR ∗ P). Using naturality ofcounits, we can rewrite this universal arrow as ε  εL ◦ (PL ∗ εR ∗ P). Finally, the compositionof the arrows being εL follows from convolving the triangular identity (εR ∗ P) ◦ (P ∗ ηR)  idPon the left with PL; see [7, Theorem IV.1.1.(ii)]. The second statement follows from a similarargument. Definition 1.3. Let F = ΦP : D(X) → D(Y ) be a Fourier–Mukai functor between thederived categories of two smooth projective varieties X and Y with left adjoint L = ΦPL andright adjoint R = ΦPR . Using the units and counits above, we can define the twist T = ΦT andcotwist C = ΦC of F by the following exact triangles:P ∗ PR εR−−→ Δ∗OY αR−−→ TβR−−→ P ∗ PR[1] and C δR−−→ Δ∗OXηR−−→ PR ∗ PγR−−→ C[1].Similarly, we define the dual twist T ′ = ΦT ′ and dual cotwist C ′ = ΦC′ of F byT ′ δL−→ Δ∗OYηL−−→ P ∗ PLγL−−→ T ′[1] and PL ∗ PεL−→ Δ∗OXαL−−→ C′ βL−−→ PL ∗ P[1].Lemma 1.4. C ′ = ΦC′ and T ′ = ΦT ′ are left adjoint to C = ΦC and T = ΦT , respectively.Proof. This can be found in [6, Remark 2.10]. For a direct argument, first take left adjointsof the triangle C → Δ∗OXηR−−→ PR ∗ P defining the kernel of the cotwist C = ΦC to get an exacttriangle PL ∗ P → Δ∗OX → C′ and then use Lemma 1.2 to see that the Hom(Δ∗OX ,PR ∗ P) Hom(PL ∗ P,Δ∗OX) maps the unit ηR to the counit εL. This shows that C ′ 	 C. Similarly,we can show T ′ 	 T . Lemma 1.5. Let F = ΦP : D(X) → D(Y ) be a Fourier–Mukai functor with right adjointR = ΦPR . Then we have the following natural isomorphisms:T ∗ P[−1] βR∗P[−1]−−−−−−→ P∗PR ∗ PP∗γR−−−→ P ∗ C[1], andPR ∗ T [−1]PR∗βR[−1]−−−−−−−→ PR∗P ∗ PRγR∗PR−−−−→ C ∗ PR[1].A NOTE ON SPHERICAL FUNCTORS 3Similarly, if F = ΦP : D(X) → D(Y ) is a Fourier–Mukai functor with left adjoint L = ΦPL ,then we have the following natural isomorphisms:P ∗ C′[−1] P∗βL[−1]−−−−−−→ P∗PL ∗ PγL∗P−−−→ T ′ ∗ P[1] andC′ ∗ PL[−1]βL∗PL[−1]−−−−−−−→ PL∗P ∗ PLPL∗γL−−−−→ PL ∗ T ′[1].Proof. These identities are standard, and all stem from the triangular identities associatedto the adjoint pairs. For example, [7, Theorem IV.1.1(ii)] tells us that the compositionPRηR∗PR−−−−→ PR ∗ P ∗ PR PR∗εR−−−−→ PR is the identity on PR. This provides a splitting of PR ∗P ∗ PR and allows us to complete the following diagram:using the octahedral axiom to get a functorial isomorphism:PR ∗ T [−1]PR∗βR[−1]−−−−−−−→ PR ∗ P ∗ PRγR∗PR−−−−→ C ∗ PR[1]. The following result is the key technical lemma which will allow us to easily deduce that thetwist associated to a spherical functor is an auto-equivalence.Lemma 1.6. Let F = ΦP : D(X) → D(Y ) be a Fourier–Mukai functor with a right adjointR = ΦPR . If there is any natural isomorphism (not necessarily the unit of adjunction) betweenPR ∗ P and Δ∗OX , then the unit η : Δ∗OX ∼−→ PR ∗ P of adjunction is an isomorphism. Thatis, F = ΦP is fully faithful.Proof. This statement is the dual of [5, Lemma 1.1.1] translated into the setting of Fourier–Mukai functors. 2. Spherical functorsDefinition 2.1. We say that a Fourier–Mukai functor F = ΦP : D(X) → D(Y ) with leftadjoint L = ΦPL and right adjoint R = ΦPR is spherical if the cotwist C = ΦC is an auto-equivalence of D(X) and the canonical map:(γR ∗ PL) ◦ (PR ∗ ηL) : PR → PR ∗ P ∗ PL → C ∗ PL[1],is a functorial isomorphism.Remark 2.2. Proposition 3.2 shows that if C = ΦC is an auto-equivalence, then anyisomorphism PR  C ∗ PL[1] ensures that (γR ∗ PL) ◦ (PR ∗ ηL) is an isomorphism.Theorem 2.3. Let F = ΦP : D(X) → D(Y ) be a Fourier–Mukai functor between thebounded derived categories of two smooth projective varieties X and Y , with left adjointL = ΦPL and right adjoint R = ΦPR .4 CIARAN MEACHAN(i) If the canonical map (γR ∗ PL) ◦ (PR ∗ ηL) : PR → PR ∗ P ∗ PL → C ∗ PL[1] is an iso-morphism, then the unit ηT : Δ∗OY ∼−→ T ∗ T ′ of adjunction is an isomorphism.(ii) If the canonical map (γR ∗ PL) ◦ (PR ∗ ηL) : PR → PR ∗ P ∗ PL → C ∗ PL[1] is an iso-morphism and C = ΦC is an auto-equivalence of D(X), then T = ΦT is an auto-equivalence ofD(Y ).Proof. (i) We use the triangles P ∗ PR → Δ∗OY → T and T ′ → Δ∗OY → P ∗ PL toconstruct a commutative diagram:If we consider the top right square of the previous diagram together with the commutativesquare:consisting of four copies of P ∗ PL and all maps being the identity, then we can use the naturalmap P ∗ ηR ∗ PL : P ∗ PL → P ∗ PR ∗ P ∗ PL to form a commutative diagram:Indeed, the left face is just the triangular identity convolved with PL on the right; the topand bottom faces are clearly commutative and the commutativity of the right face follows fromthe commutativity of the other faces of the cube.Applying the octahedral axiom to the top and right faces of this commutative diagramproduces the following commutative diagram of triangles (figure 1).The canonical map (γR ∗ PL) ◦ (PR ∗ ηL) : PR → PR ∗ P ∗ PL → C ∗ PL[1] is an isomor-phism by assumption. Therefore, convolving the canonical map with P on the left, to get(P ∗ γR ∗ PL) ◦ (P ∗ PR ∗ ηL), must also be an isomorphism. This implies P ∗ Q[1]  0 whichA NOTE ON SPHERICAL FUNCTORS 5Figure 1. Diagram of functors associated to T ∗ T ′[1].in turn provides a natural isomorphism Δ∗OY [1] ∼−→ T ∗ T ′[1]. By Lemma 1.6, this implies thatthe unit ηT : Δ∗OY ∼−→ T ∗ T ′ of adjunction is an isomorphism.(ii) Part (i) proves that T ′ = ΦT ′ : D(Y ) → D(Y ) is fully faithful and so it remains to showthat T ′ = ΦT ′ is an equivalence. By [4, Lemma 1.50], it is enough to show that ker ΦT = 0. Tosee this, suppose that ΦT (E) = 0 for some E ∈ D(Y ). Then, by Lemma 1.5, we have ΦC∗PR(E) ΦPR∗T (E)[−2] = (ΦPR(ΦT (E))[−2] = 0 which implies ΦPR(E) = 0 since the cotwist C = ΦCis an auto-equivalence by assumption. Now, the defining triangle ΦP(ΦPR(E)) → E → ΦT (E)shows that E = 0 and so ker ΦT = 0 as required. Finally, we know that T = ΦT is right adjointto T ′ = ΦT ′ by Lemma 1.4 and so T = ΦT must be an equivalence as well. Corollary 2.4. The left (or right) adjoint of a spherical functor F = ΦP with twist T =ΦT and cotwist C = ΦC is a spherical functor with twist C−1 and cotwist T−1.Proof. This follows from the fact that the units and counits are exchanged under adjunction;see Lemma 1.2. 3. Identifying adjoints by an auto-equivalenceWe work with the same notation that was introduced in Section 1. For details on adjunctions,we refer to [7, § IV.1–4].Lemma 3.1. If C = ΦC is an auto-equivalence of D(X), then the canonical map:PR ∗ PPR∗ηL∗P−−−−−−→ PR ∗ P ∗ PL ∗ PγR∗PL∗P−−−−−−→ C ∗ PL ∗ P[1],is an isomorphism.6 CIARAN MEACHANProof. First, let us observe that we have a commutative diagram:Indeed, the left-hand side is just the triangular identity convolved with PR on the left, and thesquare commutes since the arrows act on separate variables. That is,γR  (C ∗ εL[1]) ◦ (γR ∗ PL ∗ P) ◦ (PR ∗ ηL ∗ P). (3.1)Now, by Lemma 1.2 we know that εL is the left adjunct of ηR. Moreover, in the proof ofLemma 1.4, we observed that the dual cotwist triangle is the left adjoint of the cotwist triangle.Thus, by comparing triangles, we see that αL must be the left adjunct of δR, or equivalently,δR is the right adjunct of αL. That is, we have:(3.2)Combining the right-hand side of (3.2) with (3.1), we can observe that we have a commutativediagram of triangles:where εC : C ∗ C′ → Δ∗OX is an isomorphism since C = ΦC is an auto-equivalence by assump-tion. Since the second and third vertical arrows are isomorphisms, it follows that the firstvertical arrow is also an isomorphism. Proposition 3.2. Suppose the cotwist C = ΦC is an auto-equivalence of D(X). Then anyisomorphism ϕ : PR ∼−→ C ∗ PL[1] implies the canonical map:χ : PRPR∗ηL−−−−→ PR ∗ P ∗ PLγR∗PL−−−−→ C ∗ PL[1],is an isomorphism.Proof. If we consider the triangle PRχ−→ C ∗ PL[1] → Q, then χ is an isomorphism if and onlyif Q  0. To show that Q  0 it is sufficient to prove that the induced Fourier–Mukai functorΦQ : D(Y ) → D(Y ) is zero on a spanning class of D(Y ). To this end, we use the spanningclass Ω = im ΦP ∪ ker ΦPR from [1, § 2.4]. Indeed, convolving the triangle defined by χ withP gives:PR ∗ Pχ∗P−−−→ C ∗ PL ∗ P[1] → Q ∗ P,and Lemma 3.1 tells us that χ ∗ P is an isomorphism. That is, Q ∗ P  0 and so we see thatΦQ is zero on im ΦP . Next, we take an object E ∈ ker ΦPR and evaluate the induced triangleof Fourier–Mukai functors on it to get:ΦPR(E) = 0 → ΦC ◦ ΦPL(E)[1] → ΦQ(E).A NOTE ON SPHERICAL FUNCTORS 7By assumption, we have some isomorphism ϕ : PR ∼−→ C ∗ PL[1] allowing us to conclude thatΦC ◦ ΦPL(E) = 0 and hence ΦQ(E) = 0. This shows that ΦQ is also zero on ker ΦPR and thusQ  0, which completes the proof. Remarks 3.3. The hypotheses of the results in this section are stronger than necessary.Indeed, Lemma 3.1 and Proposition 3.2 only use the weaker statements that ΦC′ is fully faithfuland ker ΦPR = ker ΦPL , respectively.Acknowledgements. I thank Michael Wemyss and Richard Thomas for their encouragement,support and expert advice, as well as Alexander Kuznetsov for helpful discussions. I also thankArend Bayer, Andreas Hochenegger, Andreas Krug and David Ploog for their constructivecomments on a previous version. I am very grateful to the referee for their suggestions on howto improve readability and clarity.References1. N. Addington, ‘New derived symmetries of some hyperkähler varieties’, Algebr. Geom. 3 (2016) 223–260.2. R. Anno, ‘Spherical functors’, Preprint, 2007, arXiv:0711.4409v5.3. R. Anno and T. Logvinenko, ‘Spherical DG-functors’, J. Eur. Math. Soc. 19 (2017) 2577–2656.4. D. Huybrechts, Fourier-Mukai transforms in algebraic geometry, Oxford Mathematical Monographs (TheClarendon Press Oxford University Press, Oxford, 2006).5. P. T. Johnstone, Sketches of an elephant: a topos theory compendium, vol. 1, Oxford Logic Guides 43(The Clarendon Press, Oxford University Press, New York, 2002).6. A. Kuznetsov, ‘Calabi–Yau and fractional Calabi–Yau categories’, J. reine angew. Math., 753 (2019)239–267.7. S. MacLane, Categories for the working mathematician, Graduate Texts in Mathematics 5 (Springer,New York–Berlin, 1971).8. R. Rouquier, ‘Categorification of sl2 and braid groups’, Trends in representation theory of algebras andrelated topics, Contemporary Mathematics 406 (American Mathematical Society, Providence, RI, 2006)137–167.9. E. Segal, ‘All autoequivalences are spherical twists’, Int. Math. Res. Not. (2018) 3137–3154.10. P. Seidel and R. Thomas, ‘Braid group actions on derived categories of coherent sheaves’, Duke Math.J. 108 (2001) 37–108.Ciaran MeachanSchool of Mathematics & MathematicsUniversity of GlasgowUniversity PlaceGlasgow, Scotland G12 8QQUnited Kingdomciaran.meachan@ed.ac.ukThe Bulletin of the London Mathematical Society is wholly owned and managed by the London MathematicalSociety, a not-for-profit Charity registered with the UK Charity Commission. All surplus income from itspublishing programme is used to support mathematicians and mathematics research in the form of researchgrants, conference grants, prizes, initiatives for early career researchers and the promotion of mathematics.

A note on spherical functors

Abstract

Similar works

Full text

Available Versions

Enlighten