Flag Hilbert schemes, colored projectors and Khovanov-Rozansky homology
FFLAG HILBERT SCHEMES, COLORED PROJECTORS ANDKHOVANOV-ROZANSKY HOMOLOGY
EUGENE GORSKY, ANDREI NEGUT , , AND JACOB RASMUSSENA BSTRACT . We construct a categorification of the maximal commutative subalgebra of the type A Hecke algebra. Specifically, we propose a monoidal functor from the (symmetric) monoidalcategory of coherent sheaves on the flag Hilbert scheme to the (non-symmetric) monoidal cat-egory of Soergel bimodules. The adjoint of this functor allows one to match the Hochschildhomology of any braid with the Euler characteristic of a sheaf on the flag Hilbert scheme. Thecategorified Jones-Wenzl projectors studied by Abel, Elias and Hogancamp are idempotents inthe category of Soergel bimodules, and they correspond to the renormalized Koszul complexesof the torus fixed points on the flag Hilbert scheme. As a consequence, we conjecture that theendomorphism algebras of the categorified projectors correspond to the dg algebras of functionson affine charts of the flag Hilbert schemes. We define a family of differentials d N on thesedg algebras and conjecture that their homology matches that of the gl N projectors, generalizingearlier conjectures of the first and third authors with Oblomkov and Shende.
1. I
NTRODUCTION
HHH categorifying the HOMFLY-PT polynomial [41]. We havelearned a lot about the structure of this invariant in the intervening time, but there is muchthat remains mysterious. In [27], the third author conjectured a relation between
HHH of the ( n, n + 1) torus knot and the q, t -Catalan numbers studied by Haiman and Garsia [26, 34].A key feature of this conjecture is that it relates HHH( T ( n, n + 1)) to the cohomology of aparticular sheaf on the Hilbert scheme of n points in C . This idea was developed further in[32], and later in [30], which identified the sheaves which should correspond to arbitrary torusknots T ( m, n ) . This paper grew out of our attempts to understand whether HHH of any closed n -strand braid in the solid torus can be described as the cohomology of some element of thederived category of coherent sheaves on the Hilbert scheme.We conjecture that this is indeed the case (Conjecture 1.1 below). More importantly, weintroduce a mechanism which we hope can be used to prove it. Two ideas play an impor-tant role in our construction. The first (already present in [30]) is that one should use the flagHilbert scheme rather than the usual Hilbert scheme. The second is the notion of categoricaldiagonalization introduced by Elias and Hogancamp in [23]. In Theorem 1.6, we give a geo-metric characterization of categorical diagonalization in terms of the bounded derived categoryof sheaves on projective spaces. Using this formulation, we show that Conjecture 1.1 wouldfollow from some very specific facts about the Rouquier complex of certain braids. Finally, asan application of our ideas, we describe how the homology of colored Jones-Wenzl projectorsis related to the local rings at fixed points of the natural torus action on the flag Hilbert scheme.1.2. Recall the Hecke algebra H n of type A n , whose objects can be perceived as isotopyclasses of braids on n strands modulo the relation: (cid:16) σ k − q (cid:17) (cid:16) σ k + q − (cid:17) = 0 a r X i v : . [ m a t h . G T ] A ug EUGENE GORSKY, ANDREI NEGUT , , AND JACOB RASMUSSEN where σ k denotes a single crossing between the k and ( k + 1) –th strands. The product inthe Hecke algebra corresponds to stacking braids on top of each other, from which the non-commutativity of H n is manifest. Ocneanu constructed a collection of linear maps:(1.1) χ : ∞ (cid:71) n =0 H n → C ( a, q ) which is uniquely determined by the fact that ∀ σ, σ (cid:48) ∈ H n we have χ ( σσ (cid:48) ) = χ ( σ (cid:48) σ ) , and:(1.2) χ ( i ( σ )) = χ ( σ ) · − aq − q − , χ ( i ( σ ) σ n ) = χ ( σ ) , χ ( i ( σ ) σ − n ) = χ ( σ ) · a where i ( σ ) ∈ H n +1 is the braid obtained by adding a single free strand to the right of σ . Jones([38], [25]) showed that the map (1.1) is an invariant of the closure σ of the braid:(1.3) HOMFLY–PT ( σ ) = χ ( σ ) which in fact coincides with the well-known HOMFLY-PT knot invariant. The map χ factorsthrough a maximal commutative subalgebra C n :(1.4) C n ι ∗ (cid:44) → H n ι ∗ −→ C n by which we mean that χ : H n ι ∗ −→ C n (cid:82) −→ C ( a, q ) for some linear map (cid:82) that will be explained later. As a vector space, the commutative algebra C n is spanned by the Jones-Wenzl projectors to irreducible subrepresentations of the regularrepresentation of H n . As such, dim C n equals the number of standard Young tableaux of size n , while dim H n = n ! . Alternatively, one can describe C n in terms of the twists :(1.5) FT k = ( σ · · · σ k − ) k for all k ∈ { , ..., n } . Note that FT = , while FT n is central in the braid group. The factthat FT , ..., FT n generate a maximal commutative algebra (precisely our C n ) is well-known.1.3. The Hecke algebra admits a well-known categorification, namely the monoidal category: (SBim n , ⊗ R ) (cid:32) K (SBim n ) = H n of certain bimodules over R = C [ x , ..., x n ] called Soergel bimodules (see [53],[52]). Thiscategory admits three gradings: • the internal grading given by considering graded bimodules with respect to deg x i =1 . We write q for the variable that keeps track of this grading. • the homological grading that arises from chain complexes in the homotopy category K b (SBim n ) . We write s for the variable that keeps track of this grading. • the Hochschild grading that appears when considering D b (SBim n ) , namely the clo-sure of SBim n in D b ( R –mod– R ) . We write a for the corresponding variable.Khovanov ([39]) used the above structure to construct the functor:(1.6) HHH : K b ( D b (SBim n )) −→ triply graded vector spacessuch that: the Poincar´e polynomial of HHH( σ ) = ∞ (cid:88) i,j,k =0 q i s j a k · dim HHH( σ ) i,j,k only depends on σ and specializes to (1.3) when we substitute s (cid:55)→ − and a (cid:55)→ − a . One of themain goals of this paper is to construct a geometric version of the functor (1.6), by categorifyingthe maximal commutative subalgebra C n and the maps of (1.4). The natural place to look is the LAG HILBERT SCHEMES, COLORED PROJECTORS AND KHOVANOV-ROZANSKY HOMOLOGY 3 category of coherent sheaves on an algebraic space. In our case, the appropriate choice will bethe flag Hilbert scheme
FHilb n ( C ) which parametrizes full flags of ideals: I n ⊂ ... ⊂ I ⊂ I = C [ x, y ] such that each successive inclusion has colength and is supported on the line { y = 0 } . Forevery k ∈ { , ..., n } , there is a tautological rank k vector bundle:(1.7) T k on FHilb n ( C ) , T k | I n ⊂ ... ⊂ I ⊂ I = C [ x, y ] /I k which is naturally equivariant with respect to the action: C ∗ × C ∗ (cid:121) FHilb n ( C ) with equivariant parameters q and t that is induced by the standard action C ∗ × C ∗ (cid:121) C × C . These parameters are related to thegradings on the category of Soergel bimodules via:(1.8) s = −√ qt In Subsection 2.7 we will introduce a certain dg version of the flag Hilbert scheme, denoted by
FHilb dg n ( C ) , which is rigorously speaking a sheaf of dg algebras over FHilb n ( C ) . Our mainconjecture is the following: Conjecture 1.1.
There exists a pair of adjoint functors which preserve the q and t gradings: (1.9) K b (SBim n ) ι ∗ − (cid:0) == (cid:1) − ι ∗ D b (cid:0) Coh C ∗ × C ∗ (cid:0) FHilb dg n ( C ) (cid:1)(cid:1) where ι ∗ is monoidal and fully faithful. Furthermore, we have: (1.10) FT k ι ∗ − (cid:0) == (cid:1) − ι ∗ (det T k ) ⊗ O FHilb dg n ( C ) for all k ∈ { , ..., n } . Moreover, the map HHH of (1.6) factors as: (1.11) HHH : K b (SBim n ) ι ∗ −→ D b (cid:0) Coh C ∗ × C ∗ (cid:0) FHilb dg n ( C ) (cid:1)(cid:1) (cid:82) −→ (cid:82) refers to the derived push-forward map in equivariant cohomology.Remark . To account for the a grading in (1.9) and (1.11), we conjecture that one can liftthe setup of Conjecture 1.1 to functors:(1.12) K b ( D b (SBim n )) (cid:101) ι ∗ − (cid:0) == (cid:1) − (cid:101) ι ∗ D b (cid:16) Coh C ∗ × C ∗ (cid:16) Tot
FHilb dg n ( C ) T n [1] (cid:17)(cid:17) which preserve the q, t and a gradings, defined by:(1.13) (cid:101) ι ∗ ( σ ) = ι ∗ ( σ ) ⊗ ∧ • T ∨ n where a keeps track of the exterior degree in the right hand side. With this in mind, we notethat the target of the map (cid:82) from (1.11) can be lifted to quadruply graded vector spaces, sincewe may separate the derived category grading on FHilb dg n ( C ) from the exterior grading a . EUGENE GORSKY, ANDREI NEGUT , , AND JACOB RASMUSSEN D b (Coh C ∗ × C ∗ (FHilb dg n ( C ))) and the functors ι ∗ , ι ∗ categorify (1.4), one of the main applications of Conjecture 1.1 is a geometric incarnation ofKhovanov’s Hochschild homology functor. Indeed, since SBim n is a categorification of theHecke algebra, to any braid σ one may associate a homonymous object σ ∈ K b (SBim n ) (seeSection 3 for an overview). Therefore, we have:(1.14) HHH( σ ) = (cid:90) FHilb dg n ( C ) B ( σ ) ⊗ ∧ • T ∨ n where B ( σ ) := ι ∗ ( σ ) is the sheaf on the dg scheme FHilb dg n ( C ) that our construction associates to the braid σ . Wetensor with ∧ • T ∨ n as in Remark 1.2 in order to pick up the a grading on HHH( σ ) (if we had nottaken this tensor product, we would recover HHH( σ ) | a =0 ). While it is difficult to describe at themoment the sheaves B ( σ ) for arbitrary braids σ , properties (1.10) and the projection formula(4.5) imply that: B (cid:32) n (cid:89) k =1 FT a k k (cid:33) = n (cid:79) k =1 (det T k ) ⊗ a k Therefore, (1.14) immediately implies the following Corollary for all products of twists:
Corollary 1.3.
For all ( a , . . . , a n ) ∈ Z n , let us consider the twist braid σ = (cid:81) k FT a k k .Assuming Conjecture 1.1, the HOMFLY-PT homology of the closure of σ is given by: (1.15) HHH ( σ ) = (cid:90) FHilb dg n ( C ) n (cid:79) k =1 (det T k ) ⊗ a k (cid:79) ∧ • T ∨ n where the integral denotes the derived equivariant pushforward to a point. When the a i are sufficiently positive, we expect that the higher cohomology of the sheafappearing in the the right-hand side of (1.15) should vanish. If this is the case, the right-handside of (1.15) can be computed using the Thomason localization formula as in [30] to give:(1.16) HHH( σ ) = (1 − q ) − n (cid:88) T n (cid:89) i =1 z a i + ... + a n i (1 + az − i )1 − z − i (cid:89) ≤ i The composition: K b (SBim n ) ι ∗ −→ D b (cid:0) Coh C ∗ × C ∗ (cid:0) FHilb dg n ( C ) (cid:1)(cid:1) ν ∗ −→ D b (Coh C ∗ × C ∗ (Hilb n )) associates to a braid σ a complex of sheaves:(1.18) F ( σ ) = ν ∗ ( B ( σ )) We may tensor this complex with ∧ • T ∨ n as in Remark 1.2 if we also wish to encode the a grading. This is the object we conjecture gives rise to the geometrization of (1.1). Conjecture 1.4. The objects F ( σ ) satisfy the following properties: (1.19) F ( σσ (cid:48) ) ∼ = F ( σ (cid:48) σ ) for all braids σ and σ (cid:48) on n strands, and: (1.20) F ( i ( σ )) = α ( F ( σ )) where: (1.21) α : D b (Coh C ∗ × C ∗ (Hilb n )) −→ D b (Coh C ∗ × C ∗ (Hilb n +1 )) denotes the simple correspondence of Nakajima and Grojnowski (as in Subsection 3.10). For any braid σ , the Euler characteristic of F ( σ ) at t = q coincides with χ ( σ ) of (1.1). Remark . While the present paper was being written, Oblomkov and Rozansky ([45]) inde-pendently gave an alternative construction of objects very similar to B ( σ ) and F ( σ ) , althoughin a very different presentation. Specifically, their construction associates to any braid an objectin the category of matrix factorizations, which descends to an object on the commuting vari-ety. The authors then show that the corresponding object is actually supported on the Hilbertscheme. We strongly suspect that their objects coincide with ours, and hope that the connectionwill be elucidated in the near future.1.6. We show that Conjecture 1.1 would follow from certain computations in the Soergelcategory, which we believe may be proved using the techniques developed in an upcomingpaper of Elias and Hogancamp (see [22] for a special case). In the present paper, we developthe geometric machinery necessary to prove such results. Specifically, we outline a strategyfor constructing the functors ι ∗ , ι ∗ with equation (1.10) in mind. The starting point for us isto reinterpret geometrically a concept introduced by Elias and Hogancamp under the nameof categorical diagonalization ([23]). Suppose that C is a graded monoidal category withmonoidal unit , and F is an object in the homotopy category K b ( C ) . Elias and Hogancampcall F diagonalizable if there exist grading shifts λ , ..., λ n and morphisms: α i : λ i · → F, i = 0 , . . . , n satisfying certain conditions (see Definitions 7.6 and 7.7). Under these conditions, it is provedin [23] that there exist objects P i ∈ K ( C ) (a certain completion, whose relation with the origi-nal category K b ( C ) is analogous to the relation between the categories of left unbounded chaincomplexes and bounded chain complexes) such that tensoring Id P i with α i yields an isomor-phism:(1.22) λ i · P i ∼ = F ⊗ P i , i = 0 , . . . , n It is natural to call the P i eigenobjects of F and the λ i the eigenvalues of F . The maps α i arecalled the eigenmaps for F , and they are a particular feature of the categorical setting. Undermild assumptions on C and F , we show the following: EUGENE GORSKY, ANDREI NEGUT , , AND JACOB RASMUSSEN Theorem 1.6. An object F ∈ C is diagonalizable in the sense of [23] if and only if there is apair of adjoint functors: K b ( C ) ι ∗ − (cid:0) == (cid:1) − ι ∗ D b (Coh( P nA )) , where A = End C ( ) . If the category C is graded and the maps α i preserve the grading, then ι ∗ and ι ∗ can be lifted to the equivariant derived category: K b ( C ) ι ∗ − (cid:0) == (cid:1) − ι ∗ D b (Coh T ( P nA )) , where T is a torus acting on P n with weights prescribed by the eigenvalues of F . Furthermore, the following result of Elias-Hogancamp provides one of the first proved factsabout our conjectural connection between SBim n and FHilb dg n ( C ) . Theorem 1.7 ([23]) . The full twist FT n is diagonalizable in SBim n , and its eigenvalues agreewith the equivariant weights of det T n at fixed points. The flag Hilbert scheme is more complicated than a projective space, but it turns out to bepresented by a tower of projective fibrations. More precisely, the fibers of the natural projection: FHilb n ( C ) → FHilb n − ( C ) × C , ( I n ⊂ ... ⊂ I ) (cid:55)→ ( I n − ⊂ ... ⊂ I ) × supp ( I n − /I n ) are projective spaces. They are rather badly behaved, but we will show in Section 2.7 that thecorresponding map on the level of our dg schemes: π n : FHilb dg n ( C ) → FHilb dg n − ( C ) × C is the projectivization of a two-step complex of vector bundles. The strategy we propose is touse a relative version of Theorem 1.6 (developed in Section 4) in order to construct a commu-tative tower of functors:(1.23)Here I n : SBim n − ⊗ C [ x n ] → SBim n denotes the natural full embedding of categories, while Tr n : SBim n → SBim n − ⊗ C [ x n ] is the partial trace map of [36] (see Subsection 3.5 fordetails, as well as an overview of the construction of its derived version). We prove that theexistence of the horizontal functors in (1.23) is equivalent to the computation of Tr n ( FT ⊗ kn ) for all integers k (see 3.9 below), together with certain compatibility conditions that must bechecked. Assuming these computations, we show how Conjecture 1.1 follows. LAG HILBERT SCHEMES, COLORED PROJECTORS AND KHOVANOV-ROZANSKY HOMOLOGY 7 FT k in the Soergel category. The easiest of these conjecturesinvolves the objects L k := FT k ⊗ FT − k − ∈ K b (SBim n ) for all k ∈ { , ..., n } : Conjecture 1.8. There exist objects T n , ..., T ∈ K b (SBim n ) and morphisms T n → T n − → ... → T , which satisfy: (1.24) L k ∼ = [ T k → T k − ] for all k ∈ { , ..., n } . Furthermore, there exist two commuting morphisms: X : qT k → T k Y : s q T k → T k which commute: [ X, Y ] = 0 and are compatible with the isomorphisms (1.24) . Moreover, X | L k is multiplication by the element x k ∈ R and Y | L k = 0 . Various matrix elements of products of X and Y can be used to construct morphisms betweenvarious L k . See Conjecture 3.9 for more conjectures of similar kind.1.8. An important role in the geometry of flag Hilbert schemes is played by torus fixed points: FHilb n ( C ) C ∗ × C ∗ = { I T } T is a standard Young tableau of size n While the flag Hilbert scheme is badly behaved, the dg scheme FHilb dg n ( C ) is by definitiona local complete intersection. As such, the skyscraper sheaves at the torus fixed points arequasi-idempotents in the derived category of coherent sheaves on FHilb dg n ( C ) : O I T ⊗ O I T ∼ = O I T ⊗ ∧ • (cid:0) Tan I T (cid:0) FHilb dg n ( C ) (cid:1)(cid:1) where Tan denotes the tangent bundle (which makes sense for a local complete intersection asa complex of vector bundles). Inspired by the constructions of Elias–Hogancamp ([23]), wemake sense of the objects: P T “ = ” (cid:34) O I T ∧ • Tan I T (cid:0) FHilb dg n ( C ) (cid:1) (cid:35) ∈ a certain extension of Coh C ∗ × C ∗ (cid:0) FHilb dg n ( C ) (cid:1) and conjecture that the functor ι ∗ sends this object to the categorified Jones–Wenzl projector:(1.25) ι ∗ ( P T ) = P T These projectors are among the main actors of [23], where the authors construct them induc-tively as eigenobjects for the full twists FT n following the categorical diagonalization proce-dure described in (1.22). In the present paper, we exhibit an affine covering of the flag Hilbertscheme: FHilb n ( C ) = (cid:91) T ˚FHilb T ( C ) If we restrict the structure sheaf O FHilb dg n ( C ) to these open pieces, we obtain dg algebras: A T ( C ) = Γ (cid:16) ˚FHilb T ( C ) , O FHilb dg n ( C ) (cid:17) We expect that these dg algebras coincide with the endomorphism algebras of the categori-fied Hecke algebra idempotent indexed by the standard Young tableau T , as in the followingconjecture. EUGENE GORSKY, ANDREI NEGUT , , AND JACOB RASMUSSEN Conjecture 1.9. The endomorphism algebra of the categorified Jones-Wenzl projector P T isisomorphic as an algebra to: (1.26) End( P T ) = A T ( C ) ⊗ (cid:16) ∧ • T ∨ n | ˚FHilb T ( C ) (cid:17) Note that T ∨ n is a trivial rank n vector bundle on the affine chart ˚FHilb T ( C ) , and so theexterior power that appears in (1.26) is free on n odd generators, whose equivariant weightsmatch the inverse q, t –weights of the boxes in the Young tableau T . Following recent results ofAbel and Hogancamp [1, 36], we prove (1.26) in the two extremal cases, corresponding to thesymmetric and anti–symmetric projectors: Theorem 1.10. If T = ( n ) or (1 , . . . , then the endomorphis algebra of the resulting projectoris isomorphic to the right hand side of (1.26) . Explicitly: (1.27) End( P ( n ) ) (cid:39) C [ x , . . . , x n , y i,j ] i>j y i,j ( x i − x j ) − ( y i − ,j − y i,j +1 ) ⊗ ∧ • ( ξ , . . . , ξ n ) where deg x i = q , deg y i,j = tq j − i and deg ξ i = aq − i , while: (1.28) End( P (1 ,..., ) (cid:39) C [ u , . . . , u n ] ⊗ ∧ • ( ξ , . . . , ξ n ) where deg u i = qt − i and deg ξ i = at − i . As further evidence for Conjecture 1.9, we prove that it holds at the decategorified level. Theorem 1.11. For all standard Young tableaux T , the Euler characteristic of the algebra: ˚ A T ( C ) ⊗ (cid:16) ∧ • T ∨ n | ˚FHilb T ( C ) (cid:17) equals the Markov trace of the Hecke idempotent p λ , where λ is the partition associated to T . reduced HOMFLY-PThomology. Indeed, it is proven in [49] that the HOMFLY-PT homology of any braid is a freemodule over the homology of the unknot, which is isomorphic to a free algebra in one even andone odd variable. Let us explain how these variables arise from the geometry. First, define the reduced flag Hilbert scheme FHilb n ( C ) as the subscheme in FHilb n ( C ) cut out by the equation Tr( X ) = x + . . . + x n = 0 . It is not hard to see that there is an isomorphism:(1.29) r : FHilb n ( C ) → FHilb n ( C ) × C We will denote two components of this isomorphism by r and r . As a result, the homologyof any sheaf on FHilb n ( C ) is a free module over the polynomial ring in one (even) variable.To identify the odd variable, remark that T n has a nowhere vanishing section given by thepolynomial ∈ C [ x, y ] . It is not hard to see that this section splits, so we may write: T n (cid:39) O ⊕ T n = ⇒ T ∨ n (cid:39) O ⊕ T ∨ n = ⇒ ∧ • T ∨ n (cid:39) ∧ • ( ξ ) ⊗ ∧ • T ∨ n To sum up, we get the following corollary analogous to Corollary 1.3: Corollary 1.12. Assuming Conjecture 1.1, the reduced HOMFLY-PT homology of any object σ ∈ K b (SBim n ) is: HHH red ( σ ) ∼ = (cid:90) FHilb dg n ( C ) ( r ◦ ι ) ∗ ( σ ) ⊗ ∧ • T ∨ n . LAG HILBERT SCHEMES, COLORED PROJECTORS AND KHOVANOV-ROZANSKY HOMOLOGY 9 gl N Khovanov-Rozansky ho-mology [40, 41] for all N . Recall that in [49] the third author constructed a spectral sequencefrom the HOMFLY-PT homology to the gl N homology of any knot. For any pair of nonnegativeintegers N, M , there is an equivariant section: s N,M ∈ Γ (FHilb n ( C ) , T n ) , s N,M | I n ⊂ ... ⊂ I = x N y M ∈ C [ x, y ] I n = T n | I n ⊂ ... ⊂ I Conjecture 1.13. For all braids σ , the gl N spectral sequence on the homology of σ is inducedby the contraction of: ∧ • T ∨ n on FHilb dg n ( C ) with the section s N, , which induces a differential on the vector space (1.14) .Remark . A similar conjecture can be stated for the reduced gl N homology. However, themap (1.29) does not commute with the differential, and hence the unreduced homology is nolonger a free module over the homology of the unknot.We are hopeful that the contraction with more general s N,M may correspond to an (as yetundefined) knot homology theory associated to the Lie superalgebra gl N | M (see some conjec-tural properties in [28]). In particular, the differential induced by s , = xy should give rise to aknot homology theory associated to gl | . Recent work of Ellis, Petkova and V´ertesi [24] showsthat the tangle Floer homology of [48] gives a sort of categorification of the gl | Reshitikhin-Turaev invariant. In the spirit of the above conjecture, contraction with s , may give rise to adifferential on HHH whose homology is knot Floer homology, as conjectured in [21].In an earlier joint work with A. Oblomkov and V. Shende ([32]), the first and the third authorsgave a precise conjectural description of the stable gl N homology of ( n, ∞ ) torus knots, whichis known ([15, 36, 50, 51]) to be isomorphic to the gl N homology of the categorified projector P (1 ,..., . Conjecture 1.15 ([32]) . The spectral sequence from HOMFLY-PT homology (given by (1.28) )to the gl N homology of P (1 ,..., degenerates after the first nontrivial differential d N , which isgiven by the equation: (1.30) d N (cid:32) n (cid:88) k =1 z k − ξ k (cid:33) = (cid:32) n (cid:88) k =1 z k − u k (cid:33) N mod z n , d N ( u i ) = 0 . This conjecture has been extensively verified against computer-generated data for N = 2 and (see [29, 31]). We prove that Conjecture 1.15 immediately follows from Conjecture 1.13.1.11. This paper is naturally divided into two parts. The first part (Sections 2, 3, 4) presentsthe non-equivariant picture, which relates the global geometry of the flag Hilbert scheme withthe Soergel category. Sections 5 and 6 present examples of many of our constructions for n = 2 and n = 3 , respectively. The second part of the paper (Sections 7, 8, 9) is an equivariantrefinement of the previous framework, which relates the local geometry of the flag Hilbertscheme with categorical idempotents in the Soergel category. More specifically: • In Section 2, we define flag Hilbert schemes and the associated dg schemes, and werealize them as towers of projective bundles. • In Section 3, we recall the necessary facts about the Hecke algebra and the Soergelcategory, and formulate the main conjectures. • In Section 4, we develop a framework of monoidal categories over dg schemes, whichencapsulates the existence of adjoint functors as in (1.9), with all the desired properties.We show what computations one needs to make in order to prove Conjecture 1.1. , , AND JACOB RASMUSSEN • In Section 5, we present examples for n = 2 . • In Section 6, we present examples for n = 3 . • In Section 7, we show how the categorical setup of Section 4 can be enhanced to theequivariant setting. Inspired by the constructions of Elias–Hogancamp, we categorifythe equivariant localization formula on projective space. • In Section 8, we work out local equations for flag Hilbert schemes, and connect thestructure sheaves of torus fixed points with the categorical projectors of [23]. • In Section 9, we discuss differentials and Conjecture 1.13. • In Section 10, we collect certain foundational facts about dg categories and dg schemes.A CKNOWLEDGMENTS The authors would like to thank Michael Abel, Ben Elias and Matt Hogancamp for explain-ing to us their results [1, 23, 36], and Mikhail Gorsky, Daniel Halpern-Leistner, Mikhail Kho-vanov, Ivan Losev, Davesh Maulik, Michael McBreen, Hugh Morton, Alexei Oblomkov, An-drei Okounkov, Claudiu Raicu, Sam Raskin, Raphael Rouquier, Lev Rozansky, Peter Samuel-son, Peng Shan and Monica Vazirani for very useful discussions. Special thanks to Ian Gro-jnowski for explaining Example 2.5 to us. The work of E. G. was partially supported by theNSF grant DMS-1559338 and the Hellman fellowship. The work of A. N. was partially sup-ported by the NSF grant DMS-1600375. The work of E.G. in sections 3, 5 and 6 was supportedby the grant 16-11-10018 of the Russian Science Foundation.2. T HE FLAG H ILBERT SCHEME Definition. Let us recall the usual Hilbert scheme of n points on C : Hilb n = { ideal I ⊂ C [ x, y ] , dim C C [ x, y ] /I = n } There is a tautological bundle of rank n on the Hilbert scheme given by: T n | I = C [ x, y ] /I Similarly, one can define the flag Hilbert scheme FHilb n ( C ) of n points on C [16, 54] asthe moduli space of complete flags of ideals:(2.1) FHilb n ( C ) = { I n ⊂ ... ⊂ I ⊂ I = C [ x, y ] , dim I k − /I k = 1 , ∀ k } Clearly, FHilb n ( C ) can be thought of as the closed subscheme of Hilb n × ... × Hilb × Hilb cut out by the inclusions I k ⊂ I k − for all k . We will not pursue this description, and insteadwork with an alternative one given in the next Subsection. Meanwhile, let us point out severalgeneral features of the flag Hilbert scheme (2.1). We may pull T n back to FHilb n ( C ) , wherewe have a full flag of tautological bundles:of ranks n, ..., . For any k ∈ { , ..., n } , the fibers of T k over flags I n ⊂ ... ⊂ I are preciselythe quotients C [ x, y ] /I k . We define the tautological line bundles as the successive kernels:(2.2) L k = Ker ( T k (cid:16) T k − ) LAG HILBERT SCHEMES, COLORED PROJECTORS AND KHOVANOV-ROZANSKY HOMOLOGY 11 Moreover, there is a morphism:(2.3) ρ : FHilb n ( C ) −→ C n = C n × C n ( I n ⊂ ... ⊂ I ) (cid:55)→ ( x , . . . , x n , y , . . . , y n ) where ( x k , y k ) = supp I k − /I k . We may consider the various fibers of this map: FHilb n ( C ) = ρ − ( C n × { } ) , FHilb n ( point ) = ρ − ( { } × { } ) These will be the moduli spaces of flags of sheaves set-theoretically supported on the line { y = 0 } and at the point (0 , , respectively. The vector bundles T k and L k are defined asbefore. As a rule, we will write: FHilb n for any of FHilb n ( C ) , FHilb n ( C ) or FHilb n (point) when we will make general statements that apply to all our flag Hilbert schemes. Example . It is well-known that Hilb is the blow-up of the diagonal inside ( C × C ) /S .It should be no surprise that:(2.4) FHilb ( C ) = Bl ∆ (cid:0) C × C (cid:1) = Proj (cid:18) C [ x , x , y , y , z, w ]( x − x ) w − ( y − y ) z (cid:19) where the variables x i , y i sit in degree 0, while z, w sit in degree 1 with respect to the Proj.Setting y = y = 0 , respectively x = x = y = y = 0 , we obtain:(2.5) FHilb ( C ) = P × A ∪ A × A = Proj (cid:18) C [ x , x , z, w ]( x − x ) w (cid:19) (2.6) FHilb (point) = P = Proj ( C [ z, w ]) The matrix presentation. Throughout this section, we fix the Lie groups: G = GL n , B = invertible lower triangular n × n matricesand the flag variety Fl = G/B . We will also consider the Lie algebras: g = n × n matrices , b = lower triangular n × n matricesWe will also write n ⊂ b for the nilpotent subgroup of strictly lower triangular matrices, and V for the n dimensional vector space on which all the above matrix groups and algebras act. Proposition 2.2. (ADHM construction, [43] ) The Hilbert scheme of n points is given by: (2.7) Hilb n = µ − (0) cyc /G where the “moment map” is given by: (2.8) µ : g × g × V −→ g , µ ( X, Y, v ) = [ X, Y ] and the superscript cyc stands for the open subset of cyclic triples ( X, Y, v ) , i.e. those for which V is generated by the vectors { X a Y b v } a,b ≥ . Finally, the quotient by G is explicitly given by: g · ( X, Y, v ) = (cid:0) gXg − , gY g − , gv (cid:1) ∀ g ∈ G Remark . The reader accustomed to the construction of symplectic varieties via Hamiltonianreduction will recognize that two of the Lie algebras in (2.8) are usually replaced with theirduals. Here we tacitly assume the identification of g with its dual given by the trace pairing. , , AND JACOB RASMUSSEN Passing between the ideal description of the Hilbert scheme and the ADHM picture is easy: I (cid:32) { V = C [ x, y ] /I, X, Y = multiplication by x, y, and v = 1 mod I } ( X, Y, v ) (cid:32) I = { f ∈ C [ x, y ] such that f ( X, Y ) · v = 0 } To mimic (2.7) for the flag Hilbert scheme, one needs to replace the vector space V by a full flagof vector spaces. Then the maps X, Y must preserve these vector spaces, and so are requiredto lie in the Borel subspace b . In other words, we have:(2.9) FHilb n ( C ) = ¯ µ − (0) cyc /B where: ¯ µ : b × b × V −→ n , ¯ µ ( X, Y, v ) = [ X, Y ] However, using (2.9) as the definition of flag Hilbert schemes leads us into trouble, since there isno general reason why quotients modulo Borel subgroups are good. To remedy this problem, letus consider the following alternative definition of flag Hilbert schemes, built on the observationthat one can let the Borel subgroup vary. Definition 2.4. Consider the following space, inspired by the Grothendieck resolution: z = (cid:110) ( X, Y, v, b ) ∈ g × g × V × Fl , X, Y ∈ b (cid:111) where we identify the flag variety with the set of Borel subalgebras of g . Consider the map:(2.10) ν : z −→ Adj n , ( X, Y, v, b ) (cid:55)→ [ X, Y ] where the target Adj n is the affine bundle over the flag variety with fibers given by the nilpotentradicals n . It is G –equivariant with respect to the adjoint action, hence the notation. Define:(2.11) FHilb n ( C ) = ν − (0) cyc /G where the G action is: g · ( X, Y, v, b ) = (cid:0) gXg − , gY g − , gv, Ad g ( b ) (cid:1) ∀ g ∈ G and the superscript cyc still refers to the open subset of cyclic triples.While mostly a matter of presentation, the definition (2.11) has several advantages. Firstly,note that the map ν : FHilb n ( C ) → Hilb n is simply given by forgetting the flag b . Secondly,the set of quadruples ( X, Y, v, b ) which are cyclic is precisely the set of stable points with re-spect to the action of G on the trivial line bundle on z (endowed with the determinant character).Then geometric invariant theory implies that (2.11) is a geometric quotient.2.3. DG schemes. Because the quotient in (2.7) is taken in the sense of GIT, the Hilbertscheme is a quasi-projective variety. But let us neglect its interesting structure as a topologi-cal space, and describe its ring of functions locally. By definition, the locus of cyclic triples ( g × g × V ) cyc is an open subset of affine space, and the moment map (2.8) gives rise to a sectionof the trivial g bundle: µ ∈ Γ (cid:0) O ( g × g × V ) cyc ⊗ g (cid:1) over ( g × g × V ) cyc . We may write down the Koszul complex corresponding to this section: ( ∧ • g , µ ) := (cid:104) O ( g × g × V ) cyc ⊗ ∧ dim G g µ −→ ... µ −→ O ( g × g × V ) cyc ⊗ g µ −→ O ( g × g × V ) cyc (cid:105) LAG HILBERT SCHEMES, COLORED PROJECTORS AND KHOVANOV-ROZANSKY HOMOLOGY 13 Since the Hilbert scheme is smooth, this complex is exact except at the rightmost cohomologygroup, where it is isomorphic to O µ − (0) cyc . Moreover, since all the maps are G –equivariant, wemay write locally: O Hilb n q . i . s . ∼ = (cid:0) ∧ • adj g , µ (cid:1) = (cid:104) ∧ dim G adj g µ −→ ... µ −→ O ( g × g × V ) cyc /G (cid:105) where adj g denotes the vector bundle on ( g × g × V ) cyc /G , obtained by descending the trivialvector bundle g on g × g × V , endowed with the G –action by conjugation. One may write downthe analogous Koszul complex for the map ν of (2.10), but observe that:(2.12) O FHilb n ( C ) is not q . i . s . ∼ = ( ∧ • adj n , ν ) := (cid:104) ∧ dim N adj n ν −→ ... ν −→ O ( g × g × V × Fl ) cyc /G (cid:105) (recall that adj n denotes the vector bundle on ( g × g × V × Fl ) cyc /G , obtained by descendingthe vector bundle Adj n on Fl, endowed with the G –action by conjugation). The fact that theKoszul complex (2.12) is not exact anymore boils down to the fact that FHilb n ( C ) is not alocal complete intersection, and so we choose to work instead with the dg scheme:(2.13) O FHilb dg n ( C ) := ( ∧ • adj n , ν ) Note that we think of the left hand side as a sheaf of dg algebras, given precisely by the complexin (2.12) supported on the smooth scheme ( g × g × V × Fl ) cyc /G , which is nothing but a flagvariety bundle over the smooth scheme ( g × g × V ) cyc /G . This will allow us to ignore thesubtleties of the topology of dg schemes.2.4. Explicit matrices. Although the definition of z and FHilb n ( C ) is given by allowing theBorel subgroup to vary, to keep the presentation explicit we will henceforth fix it to be B = B .Therefore, points of the flag Hilbert scheme will be triples:(2.14) X = x ∗ x ∗ ∗ ... ∗ ∗ ∗ x n , Y = y ∗ y ∗ ∗ ... ∗ ∗ ∗ y n , v = such that [ X, Y ] = 0 , and the vectors { X a Y b v } a,b ≥ generate the space V . This latter conditionimplies that the first entry of v must be non-zero, so we may use the B = B action to fix v asin equation (2.14). Therefore, we will abuse notation and re-write (2.11) as:(2.15) FHilb n ( C ) = (cid:110) ( X, Y, v ) , X, Y lower triangular , [ X, Y ] = 0 , v cyclic (cid:111) /B In this language, the map: FHilb n ( C ) ρ −→ C n is given by taking the joint eigenvalues of the matrices X and Y . Therefore, we conclude that:(2.16) FHilb n ( C ) = (cid:110) ( X, Y, v ) as in (2.15) , Y strictly lower triangular (cid:111) (2.17) FHilb n (point) = (cid:110) ( X, Y, v ) as in (2.15) , X, Y strictly lower triangular (cid:111) We may use the descriptions (2.15)–(2.17) to obtain the following estimates of the dimensionsof flag Hilbert schemes: dim FHilb n ( C ) ≥ dim ( affine space of ( X, Y, v )) − equations [ X, Y ] = 0) − dim B = (2.18) = n + 2 n − n ( n − − n ( n + 1)2 = 2 n =: exp dim FHilb n ( C ) , , AND JACOB RASMUSSEN The right hand side stands for “expected (or virtual) dimension”. Similarly, we have:(2.19) dim FHilb n ( C ) ≥ n =: exp dim FHilb n ( C ) (2.20) dim FHilb n (point) ≥ n − n (point) The reason why the expected dimension in (2.20) is n − rather than 0 is that when X and Y are both strictly lower triangular matrices, the commutator [ X, Y ] = 0 is not only strictlylower triangular, but has the first sub-diagonal equal to zero by default. Therefore, the firstsub-diagonal entries are n − equations that need not be placed on FHilb n (point) . Example . If the inequalities in (2.18)–(2.20) were equalities, then we would conclude thatflag Hilbert schemes were local complete intersections. However, this is not the case. We givean example of how the bound in (2.20) can fail, which we learned from Ian Grojnowski. Let n = 10 , and consider the affine space of matrices X, Y which are lower triangular, and havezero blocks of sizes , , and on the diagonal:(2.21) X, Y = ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ The dimension of the affine space consisting of triples ( X, Y, v ) equals 35 + 35 + 10 = 80 .Since the commutator [ X, Y ] = 0 must have the × , × and × blocks under the diagonalequal to zero by default, the number of equations we need to impose is only . Taking intoaccount the fact that the Borel subgroup has dimension , we conclude that: dim FHilb (point) ≥ − − 55 = 10 > (point) We may translate this example in terms of flags of ideals inside C [ x, y ] . Let d = 4 , n = (cid:0) d +12 (cid:1) ,and m ⊂ C [ x, y ] be the maximal ideal of the origin, and let us consider the locus of flags: L = ( I ⊃ I ⊃ . . . ⊃ I n ) ⊂ FHilb n (point) such that:(2.22) I ( k +12 ) = m k , k = 0 , . . . , d. By the defining property of the maximal ideal m , for each k ∈ { , ..., d − } the flag of ideals: m k ⊃ I ( k +12 ) +1 ⊃ . . . I ( k +22 ) − ⊃ m k +1 can be chosen as an arbitrary complete flag of vector subspaces in m k / m k +1 (cid:39) C k +1 . Since thedimension of the corresponding flag variety is (cid:0) k +12 (cid:1) , we conclude that: dim L = d − (cid:88) k =0 (cid:18) k + 12 (cid:19) = (cid:18) d + 13 (cid:19) (cid:29) n − n (point) LAG HILBERT SCHEMES, COLORED PROJECTORS AND KHOVANOV-ROZANSKY HOMOLOGY 15 as d becomes large (although the inequality is strict as soon as d ≥ ). This construction alsoshows that the stratum L is non-empty, since there always exist flags of ideals with the property(2.22), something which was not immediately apparent from the matrix construction (2.21).2.5. Projective tower construction. Let us consider the action:(2.23) C ∗ × C ∗ (cid:121) FHilb n which scales the matrices X, Y independently. We denote the basic characters of this action by q and t , so the C ∗ × C ∗ action is explicitly given by: ( z , z ) · ( X, Y ) = ( q ( z ) X, t ( z ) Y ) , ∀ ( z , z ) ∈ C ∗ × C ∗ In the matrix presentation, the tautological bundle T n on FHilb n has fibers consisting simplyof the vector spaces V on which the matrices X, Y act. The fact that flag Hilbert schemesare defined as B –quotients means that this vector bundle need not be trivial. Therefore, thematrices X, Y : V → V give rise to endomorphisms of the tautological bundle on the whole of FHilb n , which we will denote by the same letters: q T n X −→ T n , t T n Y −→ T n In the formulas above, one must twist the tautological bundle by the torus characters q, t inorder for the endomorphisms X, Y to be C ∗ × C ∗ equivariant. Since a point of the flag Hilbertscheme entails the choice of a fixed flag of V , there is a full flag of tautological vector bundles: T n (cid:16) T n − (cid:16) ... (cid:16) T on FHilb n . Flag Hilbert schemes are easier to work with than usual Hilbert schemes becausethey can be built inductively. Specifically, we have the maps:(2.24)for any ∗ ∈ { C , C , point } . When ∗ = C we set y n +1 = 0 and when ∗ = point we furtherset x n +1 = y n +1 = 0 . What makes (2.24) manageable is that it is a projective bundle , so weconclude that flag Hilbert schemes are projective towers. Specifically, consider the complexes:(2.25)for any ∗ ∈ { C , C , point } , with the maps defined by:(2.26) Ψ( w ) = (cid:16) − ( Y − y n +1 ) w, ( X − x n +1 ) w, (cid:17) (2.27) Φ( w , w , f ) = ( X − x n +1 ) w + ( Y − y n +1 ) w + f v Here, x n +1 , y n +1 are the coordinates on the second factor of FHilb n ( C ) × C , which arespecialized to y n +1 = 0 (resp. x n +1 = y n +1 = 0 ) when ∗ = C (resp. ∗ = point ). When ∗ = point , the leftmost bundle in the complex (2.25) is T n − . This implicitly uses the fact that , , AND JACOB RASMUSSEN the maps X, Y : T n → T n become nilpotent, hence they factor through T n (cid:16) T n − . In the nextSubsection, we will prove the following inductive description of flag Hilbert schemes ([44]): Theorem 2.6. The maps π of (2.24) can be written as projectivizations: (2.28) FHilb n +1 = P (cid:0) H ( E n ) ∨ (cid:1) := Proj FHilb n (cid:0) S • (cid:0) H ( E n ) (cid:1)(cid:1) This holds for each of the three variants ∗ ∈ { C , C , point } of flag Hilbert schemes. The linebundle L n +1 on the left hand side coincides with the tautological sheaf O (1) on the right.Example . Example 2.1 shows that the space FHilb can be obtained as Proj of an explicitalgebra. Let us obtain the same result using Theorem 2.6. Since T = O , we have: E ( C ) = (cid:104) qt O ( − y + y ,x − x , −−−−−−−−−−→ q O ⊕ t O ⊕ O ( x − x ,y − y , −−−−−−−−−→ O (cid:105) (cid:39)(cid:39) (cid:104) qt O ( − y + y ,x − x ) −−−−−−−−−→ q O ⊕ t O (cid:105) ⇒ S • (cid:0) E ( C ) (cid:1) = C [ x , x , y , y , z, w ]( x − x ) w − ( y − y ) z precisely as in (2.4). Here, z and w are the two basis vectors of q O ⊕ t O . If we set y = y = 0 in the above computation, we obtain the case ∗ = C of (2.5). Finally, we have: E (point) = (cid:104) q O ⊕ t O ⊕ O (0 , , −−−→ O (cid:105) (cid:39) [ q O ⊕ t O ] ⇒ S • ( E (point)) = C [ z, w ] as expected from (2.6). Example . Let us study Theorem 2.6 in the case when n = 2 and ∗ = point , in which case: FHilb (point) = P with respect to which we have T = O and T = O ⊕ O (1) . With this in mind, the complex(2.25) is explicitly given by: E (point) = (cid:104) qt O Ψ −→ q O ⊕ t O ⊕ O ⊕ q O (1) ⊕ t O (1) Φ −→ O ⊕ O (1) (cid:105) and the maps are given by: Ψ = − z z , Φ = (cid:18) z z (cid:19) It is clear from the above that the map Φ is surjective, which is a general phenomenon thatfollows from the cyclicity of triples ( X, Y, v ) . Therefore, we have: E (point) q.i.s. ∼ = (cid:104) qt O (0 , − z ,z ) −−−−−→ qt O ( − ⊕ q O (1) ⊕ t O (1) (cid:105) q.i.s. ∼ = qt O ( − ⊕ O (2) Therefore, Theorem 2.6 implies that:(2.29) FHilb (point) = P P (cid:18) O (1) qt ⊕ O ( − (cid:19) which is a Hirzebruch surface. It is also the resolution of the singular cubic cone, which isnothing but the subvariety of the Hilbert scheme consisting of ideals supported at the origin. LAG HILBERT SCHEMES, COLORED PROJECTORS AND KHOVANOV-ROZANSKY HOMOLOGY 17 Proving Theorem 2.6. Without loss of generality, we will treat the case ∗ = C . We willproceed by induction by n , by studying the fibers of the map (2.24):(2.30)Recall that points of FHilb n ( C ) are triples ( X, Y, v ) consisting of two commuting lower tri-angular matrices (for simplicity, we fix the flag of vector spaces), together with a cyclic vector.Over such a triple, fibers of π are completely determined by extending X, Y, v by a bottom row: ¯ X = (cid:18) X w x n +1 (cid:19) , ¯ Y = (cid:18) Y w y n +1 (cid:19) , ¯ v = (cid:18) vf (cid:19) where w , w ∈ T ∨ n and f ∈ O . The triple ( w , w , f ) must satisfy the following properties: • The closed condition [ ¯ X, ¯ Y ] = 0 is equivalent to:(2.31) w · ( Y − y n +1 ) = w · ( X − x n +1 ) • ( w , w , f ) is only defined up to conjugation by: V (cid:111) C ∗ = Ker ( B n +1 (cid:16) B n ) = (cid:18) Id w c (cid:19) ( w, c ) ∈ V (cid:111) C ∗ In other words, we do not consider the action of the group of n × n lower triangularmatrices B n because it has already been trivialized locally on FHilb n ( C ) . In formulas:(2.32) ( w , w , f ) ∼ ( cw + w · ( X − x n +1 ) , cw + w · ( Y − y n +1 ) , cf + w · v ) • Since we already know that ( X, Y, v ) is cyclic, the extra condition that ( ¯ X, ¯ Y , ¯ v ) becyclic is equivalent to the fact that:(2.33) C n +1 is generated by (cid:110) ¯ v, Im ( ¯ X − x n +1 ) , Im ( ¯ Y − y n +1 ) (cid:111) This fails precisely when there exists a linear functional λ : C n → C such that: λ ( v ) = f, λ (( X − x n +1 ) w ) = w · w, λ (( Y − y n +1 ) w ) = w · w for all w ∈ V . This is equivalent to ( w , w , f ) ∼ (0 , , with respect to (2.32). Proof. of Theorem 2.6: The three bullets above establish the fact that the triple ( w , w , f ) thatdetermines points in the fibers of FHilb n +1 ( C ) → FHilb n ( C ) × C is a non-zero element in:(2.34) H (cid:18) T ∨ n qt Ψ ∨ ←− T ∨ n q ⊕ T ∨ n t ⊕ O Φ ∨ ←− T ∨ n (cid:19) modulo rescaling. Note that (2.34) is the dual of (2.25), which completes the proof. (cid:3) Remark . Note that the map Φ of (2.25) is surjective, according to the equivalent description(2.33) of a point being cyclic. This implies that E n ( ∗ ) is quasi-isomorphic to a complex:(2.35) E n ( ∗ ) q . i . s . ∼ = (cid:104) qt T n − δ ∗ point Ψ −→ Ker Φ (cid:105) of vector bundles on FHilb n ( ∗ ) × ∗ , which lie in degrees − and . , , AND JACOB RASMUSSEN The dg scheme. We will now give an alternative definition of the dg scheme (2.13), andwe leave it as an exercise to the interested reader to show that the two descriptions are equivalent(we will only use the definition in this Subsection for the remainder of this paper). The ideais to note that the map Ψ of the complex (2.35) fails to be injective on many fibers, and thiswill lead to the flag Hilbert scheme misbehaving. To remedy this issue, we replace the middlecohomology sheaf H ( E n ) in (2.25) by the entire complex E n (we tacitly suppress the symbol ∗ ∈ { C , C , point } since the construction applies equally well to all three choices). Proposition 2.10. There exist dg schemes FHilb dg n endowed with flags of objects: T n → T n − → ... → T ∈ D b (Coh(FHilb dg n )) together with maps q T n X → T n , t T n Y → T n that respect the above flag, and O v → T n such that: (2.36) FHilb dg n +1 = P FHilb dg n ( E ∨ n ) := Proj FHilb dg n ( S • E n ) where the complex E n is defined by formula (2.25) . See Subsection 10.4 for the definition of theProj construction of a two-step complex of vector bundles (according to (2.35) ).Proof. Let us write L n +1 = O (1) for the tautological line bundle on the projectivization (2.36),and π : FHilb n +1 → FHilb n for the natural map. Take the defining map of projective bundles:Taut ∈ Hom( π ∗ E n , O (1)) and compose it with the natural map T n [ − i → E n . We obtain an object: i ∗ ( Taut ) =: T n +1 ∈ Hom ( π ∗ T n [ − , L n +1 ) Composing the map i with q T n ⊕ t T n ⊕ O ( X,Y,v ) −−−−→ T n yields 0, hence: ( X, Y, v ) ∗ ( T n +1 ) ∈ Hom ( π ∗ ( q T n ⊕ t T n ⊕ O )[ − , L n +1 ) equals 0 as well. This precisely gives rise to a splitting:and the dotted map is the desired extension of the arrows X, Y, v from T n to T n +1 . Note thatwe may write the above diagram as an equality in the derived category of FHilb dg n +1 :(2.37) (cid:104) qt L n +1 ( − y,x ) −−−→ q L n +1 ⊕ t L n +1 ( x,y ) −−→ L n +1 (cid:105) ∼ = (cid:104) E n +1 −→ (cid:94) π ∗ ( E n ) (cid:105) where we have underlined the –th terms of both complexes. In the above equation, we write x and y for the operators of multiplication by x n − x n +1 and y n − y n +1 , respectively, and:(2.38) (cid:94) π ∗ ( E n ) denotes π ∗ ( E n ) with the variables ( x n , x n +1 ) and ( y n , y n +1 ) switched (cid:3) One can run the proof of Proposition 2.10 with E n replaced by H E n . We leave it as an exerciseto the interested reader to show that one would obtain the schemes FHilb n of Theorem 2.6. LAG HILBERT SCHEMES, COLORED PROJECTORS AND KHOVANOV-ROZANSKY HOMOLOGY 19 Serre duality. As explained in Subsection 10.4 of the Appendix, we may embed the dgscheme FHilb dg n +1 into an actual projective bundle:(2.39)where we implicitly use the description (2.35) of the complex of vector bundles E n . This allowsus to compute the push-forward π ∗ of sheaves by factoring them through the diagram (2.39). Proposition 2.11. Let π : FHilb dg n +1 ( C ) → FHilb dg n ( C ) × C be the projection. Then: (2.40) π ∗ ( A ) ∨ ∼ = π ∗ ( A ∨ ⊗ L − n +1 ) for any A ∈ D b (Coh(FHilb dg n +1 ( C ))) . The functor π ∗ is derived, and ∨ denotes Serre dualityon the dg scheme FHilb dg n ( C ) , which is defined inductively by Proposition 2.11. This is a direct application of Proposition 10.9 in the Appendix, together with the fact thatthe determinant of the complex E n ( C ) of (2.25) is trivial. Applying formula (2.40) to A = O gives us the following formulas for all k ≥ :(2.41) π ∗ ( L − − kn +1 ) = π ∗ ( L kn +1 ) ∨ = S k E ∨ n concentrated in degree 0 Remark . The analogue of (2.40) when C is replaced by C holds exactly as stated. Mean-while, when C is replaced by point we must replace formula (2.40) by the following equation:(2.42) (cid:101) π ∗ ( A ) ∨ ∼ = (cid:101) π ∗ (cid:18) A ∨ ⊗ qt L n L n +1 (cid:19) [ − where (cid:101) π : FHilb dg n +1 → FHilb dg n is the standard projection.3. T HE H ECKE ALGEBRA AND S OERGEL CATEGORY The Hecke algebra. Recall that the Hecke algebra of type A n has n − generators: H n = C ( q ) (cid:104) σ , ..., σ n − (cid:105) modulo relations:(3.1) (cid:16) σ i − q (cid:17) (cid:16) σ i + q − (cid:17) = 0 ∀ i ∈ { , . . . , n − } (3.2) σ i σ i +1 σ i = σ i +1 σ i σ i +1 ∀ i ∈ { , . . . , n − } (3.3) σ i σ j = σ j σ i ∀ | i − j | > . The algebra H n is a q -deformation of the group algebra of the symmetric group C [ S n ] . Theirreducible representations V λ of H n at generic parameter q are labeled by partitions of n , or,equivalently, by Young diagrams of size n . The multiplicity of V λ in the regular representa-tion is equal to its dimension, which is itself equal to the number of standard Young tableaux(henceforth abbreviated SYT) of shape λ . Therefore, the regular representation of H n splitsinto a direct sum of irreducible representations labeled by standard tableaux. For each such , , AND JACOB RASMUSSEN tableau T , let P T denote the projector onto the irreducible summand in H n labeled by T . Byconstruction, these projectors have the following properties:(3.4) P T P T (cid:48) = δ TT (cid:48) P T , (cid:88) T P T = 1 . The projectors P T can be written very explicitly in terms of the generators σ i , see [4, 33] fordetails. They satisfy the following branching rule :(3.5) i ( P T ) = (cid:88) (cid:3) P T + (cid:3) , where i : H n → H n +1 is the natural inclusion and the summation in the right hand side is overall possible SYT obtained from T by adding a single box labeled by n + 1 .The renormalized Markov trace χ : H n → C ( a, q ) satisfies the relations:(3.6) χ ( σσ (cid:48) ) = χ ( σ (cid:48) σ ) , χ ( i ( σ )) = χ ( σ ) · − aq − q − , χ ( i ( σ ) σ n ) = χ ( σ ) . There is a natural pairing (cid:104)· , ·(cid:105) : H n × H n → C ( a, q ) given by (cid:104) σ, τ (cid:105) = χ ( στ † ) , where σ † is the “Hermitian conjugate” of σ (this is the C -antilinear map determined by the relations q † = q − , σ † i = σ − i , and ( στ ) † = τ † σ † ). With respect to this pairing, the adjoint of theinclusion i : H n → H n +1 is the partial Markov trace : Tr : H n +1 → H n ⊗ C [ a ] . It follows easily from the definitions that for all σ ∈ H n , we have χ ( σ ) = Tr n ( σ ) . The Markov trace of a projector P T only depends on the underlying Young diagram λ of theSYT T , and is equal to the λ -colored HOMFLY-PT polynomial of the unknot. Specifically, wehave the following result: Proposition 3.1. (e.g. [3] ) The Markov trace of P T equals: Tr n ( P T ) = (cid:89) (cid:3) ∈ λ − aq c ( (cid:3) ) − q h ( (cid:3) ) , where c ( (cid:3) ) and h ( (cid:3) ) respectively denote the content and the hook length of a square (cid:3) in λ . The braid group. The Hecke algebra is a quotient of the braid group on n strands, whichis defined by removing relation (3.1). Specifically, the braid group is generated by σ ± , ..., σ ± n − modulo relations (3.2) and (3.3). By definition, the full twist on n strands is the braid: FT n = ( σ · · · σ n − ) n . The full twist is known to be central in the braid group, and hence its image is central in theHecke algebra. If we interpret the generator σ i as a single crossing between the strands i and i + 1 , then the full twist corresponds to the pure braid where each strand wraps around all theother ones (see Figure 1). We may also define the partial twists: FT , ..., FT n − where FT k is the braid which consists of the full twist on the leftmost k strands, with therightmost n − k strands simply vertical lines. We will also work with the generalized Jucys-Murphy elements (the name is due to the fact that their images in H n deform the well-knownJucys-Murphy elements in C [ S n ] ): L k = FT − k − · FT k LAG HILBERT SCHEMES, COLORED PROJECTORS AND KHOVANOV-ROZANSKY HOMOLOGY 21 F IGURE 1. The full twist FT which are easily seen to be given by the formula: L k = σ k − ...σ σ σ ...σ k − . Either the braids { FT k } k =1 ,...,n or the braids { L k } k =1 ,...,n generate a certain commutative sub-F IGURE 2. The braid L algebra of the braid group, and hence also of the Hecke algebra, which we will denote by: C n ⊂ H n . It is well-known that the projectors P T lie in this subalgebra for all SYTx T . Proposition 3.2. (e.g. [4, Theorem 5.5] ) The projectors are eigenvectors for twists with thefollowing eigenvalues: (3.7) FT k · P T = q c ( (cid:3) )+ ... + c ( (cid:3) k ) · P T = ⇒ (3.8) = ⇒ L k · P T = q c ( (cid:3) k ) · P T where (cid:3) k denotes the box labeled by k in the standard Young tableau T . In fact, equations (3.5) and (3.7) allow one to inductively construct the elements P T , asfollows: given P T for a standard Young tableau T of size n , all projectors P T + (cid:3) are eigenvectorsfor the full twist FT n +1 with different eigenvalues, and hence can be uniquely reconstructedas the projections of i ( P T ) onto the corresponding eigenspaces. This is precisely the viewpointthat is categorified in [23], and which inspired Section 7 of the present paper.3.3. Soergel bimodules. The category of Soergel bimodules, which we will denote SBim n , isa categorification of the Hecke algebra. We will consider R = C [ x , ..., x n ] and study graded R − bimodules, where deg x i = 1 . We will write qM for the graded module M with the gradingshifted by 1. Among the most important such R − bimodules are the elementary Bott-Samelson bimodules:(3.9) B i = q − R ⊗ R i,i +1 R , , AND JACOB RASMUSSEN for any simple transposition s i = ( i, i + 1) , where we write R i,i +1 for those polynomials whichare invariant under s i . In other words, R i,i +1 consists of polynomials which are symmetric in x i and x i +1 , and therefore R has rank 2 over R i,i +1 . Therefore, B i has rank 2 as an R − module. Definition 3.3. The category SBim n is the Karoubian envelope of the smallest full subcategoryof R –mod– R that contains the Bott-Samuelson modules B i and is closed under ⊗ R and gradingshifts. Objects of SBim n will be called Soergel bimodules .The category SBim n is monoidal with respect to the operation of tensoring bimodules over R . Clearly, the unit object is := R , viewed as a bimodule over itself. Note that SBim n isneither abelian, nor symmetric. Let: B i,i +1 = q − R ⊗ R i,i +1 ,i +2 R where R i,i +1 ,i +2 denotes the set of polynomials which are symmetric in x i , x i +1 , x i +2 . Thenone can check the following identities [39, 53]:(3.10) B i (cid:39) q B i ⊕ q − B i , B i B j (cid:39) B j B j for | i − j | > , (3.11) B i B i +1 B i (cid:39) B i ⊕ B i,i +1 ⇒ B i B i +1 B i ⊕ B i +1 (cid:39) B i +1 B i B i +1 ⊕ B i . It was shown in [53] that the split graded Grothendieck group of SBim n is generated by theclasses of B i and is isomorphic to H n . Indeed, one can identify [ B i ] = σ i + q − and show that(3.10)–(3.11) imply (3.1)–(3.3).3.4. From Rouquier complexes to Khovanov-Rozansky homology. Since σ i = [ B i ] − q − ,it is clear that σ i does not correspond to any Soergel bimodule. However, Rouquier showed that σ i can be realized in the homotopy category of complexes: K b (SBim n ) where we use the variable s to keep track of homological degree. Explicitly, objects in thehomotopy category of complexes will be denoted by: (cid:104) s k M k → ... → s k (cid:48) M k (cid:48) (cid:105) for some k ≤ k (cid:48) ∈ Z . The variable s may seem redundant when writing down chain complexes,but we keep track of it for two reasons: first of all, it will give rise to the equivariant parameter t of Section 2 via (1.8). Second of all, we think of the object: [ M → sM (cid:48) ] ∈ K b (SBim n ) as the cone of a morphism between the objects M and sM (cid:48) , and thus the power of s makes thehomological degrees of our formulas manifest. Recall the Bott-Samuelson bimodules (3.9) andconsider the Rouquier complexes:(3.12) σ i := (cid:20) B i ⊗ (cid:55)→ −−−−→ sRq (cid:21) , σ − i := (cid:34) q Rs (cid:55)→ x i ⊗ − ⊗ x i +1 −−−−−−−−−−→ B i (cid:35) They satisfy the following equations [39, 52] (which can be deduced from (3.10) and (3.11)): σ i ⊗ σ − i ∼ = σ − i ⊗ σ i ∼ = ,σ i ⊗ σ j ∼ = σ j ⊗ σ i for | i − j | > ,σ i ⊗ σ i +1 ⊗ σ i ∼ = σ i +1 ⊗ σ i ⊗ σ i +1 , and hence categorify the braid group. To any braid σ = (cid:81) n − i =1 σ a i i (where α i ∈ {− , , } ) onecan associate a complex of bimodules obtained by tensoring together the various complexes LAG HILBERT SCHEMES, COLORED PROJECTORS AND KHOVANOV-ROZANSKY HOMOLOGY 23 (3.12). We abuse notation and denote the resulting complex also by σ . Khovanov [39] definedthe HOMFLY-PT homology of a braid σ as:(3.13) HHH( σ ) := RHom K b (SBim n ) ( , σ ) . The right hand side is a triply graded vector space, endowed with the internal grading q , the ho-mological grading s of the complexes (3.12) and their coproducts, and the Hochschild grading a given by taking the RHom . The appropriate derived category formalism can be found in [36].With respect to these three gradings, Khovanov proved that (3.13) is a topological invariant ofthe closure of σ , after a certain renormalization. Corollary 3.4. Let σ, σ (cid:48) be any two braids. Then: HHH( σσ (cid:48) ) = RHom K b (SBim n ) ( , σ ⊗ σ (cid:48) ) and HHH( σ (cid:48) σ ) = RHom K b (SBim n ) ( , σ (cid:48) ⊗ σ ) are isomorphic as R -modules, up to a twist by w σ . The above formula follows from Corollary 4.19, which applies to all invertible objects in amonoidal category. Proposition 3.5. The Soergel bimodule B i is self biadjoint, for all i . The Rouquier complex σ for a braid σ is biadjoint to σ − . The second statement of the above Proposition also follows from Corollary 4.18 below,which is quite general, and actually implies the following stronger result: Corollary 3.6. For any A, A (cid:48) ∈ SBim n and any braid σ there are canonical isomorphisms: RHom K b (SBim n ) ( A ⊗ σ, A (cid:48) ⊗ σ ) ∼ = RHom K b (SBim n ) ( A, A (cid:48) ) ∼ = RHom K b (SBim n ) ( σ ⊗ A, σ ⊗ A (cid:48) ) . The trace functor. We will henceforth write R n = C [ x , ..., x n ] to avoid confusion as towhich number n we are considering. For an extra variable x n +1 , we consider the category:(3.14) SBim n [ x n +1 ] of Soergel bimodules which are equipped with an additional endomorphism denoted by x n +1 that commutes with the action of R n . In other words, SBim n [ x n +1 ] is the Karoubian envelopeof the smallest full subcategory of R n +1 –mod– R n +1 that contains the modules B , . . . B n − and is closed under ⊗ R n +1 and grading shifts. It is easy to see that the functors: SBim n [ x n +1 ] − (cid:0) = (cid:1) − SBim n that forget the action of x n +1 , respectively tensor with C [ x n +1 ] , are adjoint with respect to eachother. We will now recall the functors I and Tr defined in [36], upgraded to the level of thecategory (3.14). At the level of additive categories, these functors are quite simple: I : SBim n [ x n +1 ] −→ SBim n +1 is the full embedding. Meanwhile: Tr : SBim n +1 −→ SBim n [ x n +1 ] , M (cid:55)→ Ker (cid:16) M x n +1 ⊗ − ⊗ x n +1 −−−−−−−−−−→ M (cid:17) As shown in [36], these functors can be upgraded to the derived categories: D b (SBim n [ x n +1 ]) Tr − (cid:0) == (cid:1) − I D b (SBim n +1 ) where the trace functor now encodes the full operation of multiplication by x n +1 ⊗ − ⊗ x n +1 ,instead of simply the kernel: Tr( M ) = (cid:104) M x n +1 ⊗ − ⊗ x n +1 −−−−−−−−−−→ M (cid:105) . , , AND JACOB RASMUSSEN Remark . When working in the upgraded category (3.14) rather than SBim n , one must becareful with Markov invariance, i.e. the statement ([39]) that for M ∈ SBim n +1 one has: Tr( M ⊗ σ ± n ) (cid:39) M ∈ SBim n In the upgraded category, this equation becomes:(3.15) Tr( M ⊗ σ ± n ) (cid:39) (cid:104) M ⊗ C [ x n +1 ] x n ⊗ − ⊗ x n +1 −−−−−−−−→ M ⊗ C [ x n +1 ] (cid:105) ∈ SBim n [ x n +1 ] The proof is straightforward and we leave it to the reader. Remark that in the category SBim n the complex (3.15) is quasi-isomorphic to M , but this is no longer true in SBim n [ x n +1 ] . M (cid:39) M x n ⊗ − ⊗ x n +1 −−−−−−−−→ M F IGURE 3. Markov move in SBim n [ x n +1 ] The main conjectures. For the remainder of this Section, we will write FHilb dg n =FHilb dg n ( C ) and E n = E n ( C ) , in the notation of Section 2. Our main Conjecture can be re-stated more precisely as follows: Conjecture 1.1. There exists a pair of adjoint functors: (3.16) K b (SBim n ) ι ∗ − (cid:0) == (cid:1) − ι ∗ D b (cid:0) Coh C ∗ × C ∗ (cid:0) FHilb dg n (cid:1)(cid:1) where ι ∗ is monoidal and fully faithful. Moreover, we have: (3.17) ι ∗ ( ι ∗ N ⊗ M ⊗ ι ∗ N ) ∼ = N ⊗ ι ∗ ( M ) ⊗ N for all N , N ∈ D b (cid:0) Coh C ∗ × C ∗ (cid:0) FHilb dg n (cid:1)(cid:1) and M ∈ K b (SBim n ) . In addition: (3.18) ι ∗ = O and L k = ι ∗ ( L k ) (3.17) = ⇒ (3.19) (3.17) = ⇒ ι ∗ L k = L k ∀ k ∈ { , ..., n } , where O is the structure sheaf of FHilb dg n and L k is the line bundle (2.2) . Finally, the followingdiagrams of functors commute (we write ι = ι ( n ) to keep track of n ): (3.20) LAG HILBERT SCHEMES, COLORED PROJECTORS AND KHOVANOV-ROZANSKY HOMOLOGY 25 (3.21) where the map π : FHilb dg n +1 → FHilb dg n × C is the particular case of (2.24) for ∗ = C . In broad strokes, the functor ι ∗ is given by sending each object M ∈ K b (SBim n ) to:(3.22) ι ∗ M = (cid:77) a ,...,a n ∈ N Hom K b (SBim n ) (cid:32) , M n (cid:79) k =1 L a k k (cid:33) which is naturally a module for the N n –graded dg algebra:(3.23) A = (cid:77) a ,...,a n ∈ N Hom K b (SBim n ) (cid:32) , n (cid:79) k =1 L a k k (cid:33) This algebra is commutative and ι ∗ M gives rise to a coherent sheaf on (Spec A ) / ( C ∗ ) n . Ourconjecture entails the fact that this sheaf is actually supported on the n –fold iterated projec-tivization Proj A (cid:44) → (Spec A ) / ( C ∗ ) n , and that in fact:(3.24) Proj A = FHilb dg n To upgrade to the setting of Remark 1.2, we must replace the Hom spaces by RHom in (3.22)and (3.23). We expect that this can be dealt with as in the following conjecture. Conjecture 3.8. Given the setup of Conjecture 1.1 we consider the object: T n = ι ∗ ( T n ) ∈ K b (SBim n ) Then we claim that for any object M ∈ K b (SBim n ) , we have an isomorphism: (3.25) RHom K b (SBim n ) ( , M ) ∼ = Hom K b (SBim n ) ( , M ⊗ ∧ • T ∨ n ) which is functorial with respect to the action of the algebra A (cid:121) M of (3.23) . Assuming Conjecture 3.8, one may ask if there is a sheaf on the flag Hilbert scheme whichis defined by replacing Hom with RHom in (3.22). By (3.25) and (3.17), this sheaf would be: ι ∗ ( M ⊗ ι ∗ ( ∧ • T ∨ n )) = ι ∗ M ⊗ ∧ • T ∨ n This sheaf should naturally be thought to live on Tot FHilb dg n ( T n [1]) = Spec FHilb dg n ( ∧ • T ∨ n ) , as inRemark 1.2. The entire picture presented in this Subsection will be explained in more detail inSection 4, when we develop the formalism of categories over schemes in general. Proof of Corollary 1.3. The fact that ι ∗ is a monoidal functor, together with (3.18), imply that: σ := n (cid:89) k =1 FT a k k = n (cid:89) k =1 ι ∗ (det T k ) ⊗ a k = ι ∗ (cid:32)(cid:79) k (det T k ) ⊗ a k (cid:33) . Corollary 4.18 below implies that: HHH( σ ) := RHom K b (SBim n ) ( , σ ) = RHom K b (SBim n ) ( σ − , ) , , AND JACOB RASMUSSEN while (3.25) implies that: HHH( σ ) = Hom K b (SBim n ) ( σ − ⊗∧ • T n , ) = Hom K b (SBim n ) (cid:34) ι ∗ (cid:32)(cid:79) k (det T k ) − a k ⊗ ∧ • T n (cid:33) , (cid:35) The adjunction of ι ∗ and ι ∗ , together with the conjectured fact that ι ∗ = O , imply that: HHH( σ ) = RHom FHilb dg n (cid:32)(cid:79) k (det T k ) − a k ⊗ ∧ • T n , O (cid:33) Dualizing the RHom produces the desired result. (cid:3) E n ∼ = π ∗ ( L n +1 ) ∈ D b (cid:0) Coh C ∗ × C ∗ (cid:0) FHilb dg n (cid:1)(cid:1) Define the following object:(3.26) E n := Tr( L n +1 ) ∈ K b (SBim n ) Conjecture 1.1 implies that:(3.27) ι ∗ ( E n ) = ι ∗ (Tr( L n +1 )) = π ∗ ( ι ∗ ( L n +1 )) = π ∗ ( L n +1 )) ∼ = E n . Conjecture 3.9. The following topological facts hold for all n ≥ . (a) E n is an explicit complex in terms of I ( E n − ) and L n , as in (3.31) below. (b) The following equation holds in K b (SBim n [ x n +1 ]) : (3.28) S k E n ∼ = Tr( L kn +1 ) ∀ k ≥ . (c) The Koszul complex (3.29) (cid:104) ... η −→ I ( ∧ E n ) ⊗ L − n +1 η −→ I ( E n ) ⊗ L − n +1 η −→ R (cid:105) is acyclic, where I ( E n ) η → L n +1 denotes the adjoint map to (3.26) . Statement (a) implies that E n lies in the monoidal subcategory of K b (SBim n ) generated by L , ..., L n . Since this subcategory is symmetric and Karoubian, the objects S k E n and ∧ k E n thatappear in (b) and (c) are well-defined: as in [20], they are simply the projections of E ⊗ kn definedby the symmetric and antisymmetric projectors in the symmetric group S k , respectively. Thefollowing result is proved in Section 4.8, and will show how to reduce our main Conjecture 1.1to the topological computations of Conjecture 3.9 (a)–(c). Theorem 3.10. Conjecture 3.9 implies Conjecture 1.1. E n as an explicit braid. The object E n = Tr( L n +1 ) ∈ K b (SBim n [ x n +1 ]) has a simpletopological meaning, represented below.F IGURE 4. The braid L and its partial trace E . LAG HILBERT SCHEMES, COLORED PROJECTORS AND KHOVANOV-ROZANSKY HOMOLOGY 27 − q = (1 − q ) F IGURE 5. Skein relationThe relation between the tangle E n and the complex E n is expected to categorify the classicalformula for E n (e.g. [42]) in the skein algebra. Specifically, skein relations are topologicalequalities between knots which only differ near a crossing:In K b (SBim n ) such equalities must be replaced with exact sequences. For example, considerthe skein relation applied to the bottom right crossing of the braid L n +1 . If one closes the − q = (1 − q ) F IGURE 6. Skein relation for L n +1 last strand in Figure 6 and applies a Markov move, one gets the following formula in theGrothendieck group of SBim n (which is isomorphic to the Hecke algebra):(3.30) (cid:104) E n (cid:105) − (cid:104) I ( E n − ) (cid:105) = (1 − q ) (cid:104) L n (cid:105) In the category K b (SBim n [ x n +1 ]) , the above equality is lifted to an exact sequence:(3.31) (cid:104) qtL n (0 ,x n − x n +1 ) −−−−−−−→ qL n ⊕ tL n ( x n − x n +1 , −−−−−−−→ L n (cid:105) ∼ = (cid:104) E n −→ (cid:94) I ( E n − ) (cid:105) where t = s /q and (cid:94) I ( E n − ) refers to the same braid as I ( E n − ) , but with the variables onthe last two strands switched (compare with (2.37)). This is a crucial feature of the category SBim n [ x n +1 ] , where the variables x n and x n +1 play different roles. Also note that (3.31) con-sists of 4 copies of L n instead of the two of (3.30), due to the modified Markov move (3.15).3.9. Geometric Markov invariance. In the category of Soergel bimodules, equation (3.15)governs the behavior of objects under Markov moves:(3.32) α (cid:32) i ( α ) , α (cid:32) i ( α ) · σ n , α (cid:32) i ( α ) · σ − n where i is the operation of adding an extra strand to a braid α on n strands. We will now studyhow the complexes of sheaves B ( α ) = ι ∗ ( α ) ∈ D b (Coh C ∗ × C ∗ (FHilb dg n ( C )) behave under thesame moves. Throughout this Subsection, we write FHilb dg n = FHilb dg n ( C ) and: π : FHilb dg n +1 → FHilb dg n × C for the standard projection. The following Corollary is an easy consequence of Conjecture 3.9,as we will show in Subsection 4.8. Corollary 3.11. For any braid α on n strands, we have: (3.33) B ( i ( α )) = π ∗ ( B ( α )) . , , AND JACOB RASMUSSEN To tackle the second and third Markov moves of (3.32), we consider the dg subscheme:(3.34) Z n ⊂ FHilb dg n +1 O Z n := (cid:20) . . . y n,n +1 −−−−→ q t L n L n +1 x n − x n +1 −−−−−→ qt L n L n +1 y n,n +1 −−−−→ q O x n − x n +1 −−−−−→ O (cid:21) , where y n,n +1 denotes the last subdiagonal entry of the matrix Y of (2.14), regarded as an endo-morphism t L n → L n +1 on FHilb dg n +1 . The fact that O Z n is a complex follows from: X, Y ] n,n +1 = x n y n,n +1 − y n,n +1 x n +1 Conjecture 3.12. For any braid α on n strands, we have: (3.35) B ( i ( α ) · σ n ) = π ∗ ( B ( α )) ⊗ O Z n . Corollary 3.13. Conjecture 3.12 implies that for any braid α on n strands: (3.36) B ( i ( α ) · σ − n ) = π ∗ ( B ( α )) ⊗ O Z n ⊗ L n L n +1 . Proof. Note the following the equation in the braid group: L n +1 = σ n · L n · σ n ⇒ σ − n = L − n +1 · σ n · L n ⇒ i ( α ) · σ − n = i ( α ) · L − n +1 · σ n · L n = L − n +1 · i ( α ) · σ n · L n . since L n +1 commutes with the image of i . Applying B ( − ) to the above equation implies: B ( i ( α ) · σ − n ) = ι ∗ ( i ( α ) ⊗ σ − n ) = ι ∗ ( L − n +1 ⊗ i ( α ) ⊗ σ n ⊗ L n )) As in Conjecture 1.1, we have L k = ι ∗ ( L k ) for all k , and therefore (3.17) implies (3.36). (cid:3) Equations (3.33)–(3.36) are compatible with the stabilization invariance of HHH at the levelof equivariant Euler characteristic. Proposition 3.14. For any braid α on n strands, we have: (3.37) χ (cid:0) B ( i ( α )) ⊗ ∧ • T ∨ n +1 (cid:1) = 1 − a − q χ ( B ( α ) ⊗ ∧ • T ∨ n ) Assuming Conjecture 3.12, we further have: (3.38) χ (cid:0) B ( i ( α ) · σ n ) ⊗ ∧ • T ∨ n +1 (cid:1) = χ ( B ( α ) ⊗ ∧ • T ∨ n ) (3.39) χ (cid:0) B ( i ( α ) · σ − n ) ⊗ ∧ • T ∨ n +1 (cid:1) = aqt χ ( B ( α ) ⊗ ∧ • T ∨ n ) Proof. We replace the sheaves in (3.37)–(3.39) by their K –theory classes and write: [ T n +1 ] = π ∗ ([ T n ]) + [ L n +1 ] and:(3.40) [ O Z n ] = (1 − q ) (cid:18) − qt [ L n ][ L n +1 ] (cid:19) − Since (cid:82) is just pushforward to a point, it can be decomposed along the projection map π :FHilb dg n +1 → FHilb dg n × C . In other words, for all sheaves A one has: (cid:90) FHilb dg n +1 A = (cid:90) FHilb dg n × C π ∗ A LAG HILBERT SCHEMES, COLORED PROJECTORS AND KHOVANOV-ROZANSKY HOMOLOGY 29 We will apply this equality for the K –theory class: [ A ] = [ B ( i ( α ))] · ∧ • [ T ∨ n +1 ] = π ∗ ([ B ( α )] · ∧ • [ T ∨ n ]) · (cid:18) − a [ L n +1 ] (cid:19) where in the second equality we have used (3.33). Then we may prove (3.37) by noting that: χ (cid:0) [ B ( i ( α ))] · ∧ • [ T ∨ n +1 ] (cid:1) = χ (cid:18) π ∗ (cid:20) π ∗ ([ B ( α )] · ∧ • [ T ∨ n ]) · (cid:18) − a [ L n +1 ] (cid:19)(cid:21)(cid:19) = (3.41) = χ (cid:18) [ B ( α )] · ∧ • [ T ∨ n ] · π ∗ (cid:18) − a [ L n +1 ] (cid:19)(cid:19) = (1 − a ) χ ([ B ( α )] · ∧ • [ T ∨ n ]) (the additional factor of − q in the right hand side of (3.37) comes from integrating over C ).To establish the last equality in (3.41), we note that it holds at the categorified level:(3.42) π ∗ (cid:16) O FHilb dg n +1 (cid:17) = O FHilb dg n × C = π ∗ (cid:16) O FHilb dg n +1 ⊗ L − n +1 (cid:17) where the first equality is a consequence of the fact that π is the projectivization of E ∨ n , and thesecond equality follows from the first and (2.40) for A = O . Similarly, if we assume formula(3.35) (which would also imply (3.36), according to Corollary 3.13), then relations (3.38) and(3.39) follow from:(3.43) π ∗ ( O Z n ) = (cid:104) q O x n − x n +1 −−−−−→ O (cid:105) , π ∗ (cid:18) O Z n ⊗ L n +1 (cid:19) = 0 (3.44) π ∗ (cid:18) O Z n ⊗ L n +1 (cid:19) = (cid:20) t L n x n − x n +1 −−−−−→ qt L n (cid:21) [1] We will only prove these equalities at the level of K –theory, by using (3.40). Indeed, since themap π is P E ∨ n , the push-forwards of the powers of L n +1 = O (1) are encoded by:(3.45) π ∗ (cid:18) δ (cid:18) L n +1 z (cid:19)(cid:19) = S ∗ z ∼∞ [ E n ] − S ∗ z ∼ [ E n ] == ∧ ∗ z ∼∞ [ qt T n ] ∧ ∗ z ∼∞ [ T n ](1 − z − ) ∧ ∗ z ∼∞ [ q T n ] ∧ ∗ z ∼∞ [ t T n ] − ∧ ∗ z ∼ [ qt T n ] ∧ ∗ z ∼ [ T n ](1 − z − ) ∧ ∗ z ∼ [ q T n ] ∧ ∗ z ∼ [ t T n ] where the δ function is δ ( z ) = (cid:80) ∞ k = −∞ z k . In the right hand side, we write: S ∗ z [ V ] = ∞ (cid:88) k =0 ( − z ) − k · S k V , ∧ ∗ z [ V ] = ∞ (cid:88) k =0 ( − z ) − k · ∧ k V and the notations S ∗ z ∼ , ∧ ∗ z ∼ and S ∗ z ∼∞ , ∧ ∗ z ∼∞ refer to expanding the rational functions S ∗ z , ∧ ∗ z in the domains z ∼ and z ∼ ∞ , respectively. Applying (3.40), we obtain: π ∗ (cid:18) [ O Z n ] · δ (cid:18) L n +1 z (cid:19)(cid:19) = π ∗ (cid:32) − q − qt [ L n ][ L n +1 ] · δ (cid:18) L n +1 z (cid:19)(cid:33) = 1 − q − qt [ L n ] z · π ∗ (cid:18) δ (cid:18) L n +1 z (cid:19)(cid:19) and we can compute the right hand side using (3.45). To obtain (3.43) and (3.44), we mustextract the coefficients of z , z , z in the right hand side of the above equality, and it is easy tosee that one obtains − q , and q − qt [ L n ] , respectively. (cid:3) , , AND JACOB RASMUSSEN Correspondences. Formula (3.33) can be expressed in terms of the complexes of sheaves: F ( σ ) = ν ∗ ( B ( σ )) ∈ D b (Coh C ∗ × C ∗ (Hilb n )) of (1.18), where ν : FHilb dg n → Hilb n is the map (1.17). Specifically, we have the spaces:where Hilb n,n +1 = { I ∈ Hilb n , I (cid:48) ∈ Hilb n +1 , I ⊃ I (cid:48) with quotient supported on { y = 0 }} arethe correspondences studied by Nakajima and Grojnowski to describe the cohomology groupsof Hilbert schemes. At the categorified level, their construction gives rise to a functor: D b (Coh C ∗ × C ∗ (Hilb n )) α −→ D b (Coh C ∗ × C ∗ (Hilb n +1 )) , α = p ∗ p ∗ To establish (1.20), note that F ( i ( σ )) equals: ν n +1 ∗ ( B ( i ( σ ))) = p ∗ ( r ∗ ( B ( i ( σ )))) = p ∗ ( r ∗ ( q ∗ ( B ( σ )))) = p ∗ ( p ∗ ( ν n ∗ ( B ( σ )))) = α ( F ( σ )) where the second equality follows from (3.33), and the third equality follows from the fact thatthe rhombus is cartesian. This latter fact may seem obvious at the level of closed points, butscheme-theoretically it only holds because we have replaced the badly behaved scheme FHilb n with the nicely behaved dg scheme FHilb dg n .3.11. Mirror braids. In this section, we will relate the operation of mirroring braids (i.e.looking at them from behind) with Verdier duality on the category of coherent sheaves on FHilb n . Proposition 3.15. For any F ∈ D b Coh(FHilb dg n ) one has: (cid:90) FHilb dg n F ⊗ ∧ • T ∨ n ∼ = (cid:20)(cid:90) FHilb dg n F ∨ ⊗ ∧ • T n (cid:21) ∨ where the a -grading in the right hand side is reversed from i to n − i . The Proposition is obvious, since it’s just stating that a proper push-forward commutes withVerdier duality. It is natural to conjecture, therefore, that mirroring the braid σ simply corre-sponds to dualizing the complex of sheaves B ( σ ) on FHilb dg n : Conjecture 3.16. For any braid σ , we have: B ( σ ∨ ) = B ( σ ) ∨ , where β ∨ denotes the mirror of β . The following example shows that the computation of a dual sheaf can be nontrivial. Example . As we will see in Section 5 (and also from Section 3.9), the braid σ ∈ SBim corresponds to the structure sheaf of FHilb (point) × C ⊂ FHilb ( C ) , while σ − ∈ SBim corresponds to O ( − on FHilb (point) × C . The fact that the objects B ( σ ) = O FHilb (point) × C and B ( σ − ) = O FHilb (point) × C ( − LAG HILBERT SCHEMES, COLORED PROJECTORS AND KHOVANOV-ROZANSKY HOMOLOGY 31 are dual to each other follows from the fact that the exact sequence: O FHilb (point) × C ←− O FHilb ( C ) x − x ←−−− O FHilb ( C ) w ←− O ( − FHilb (point) × C is self-dual.3.12. Some remarks on support. We now explore what the endpoints of a braid σ say aboutthe sheaf B σ on FHilb dg n . For any braid σ , let w σ ∈ S n denote the underlying permutation. Proposition 3.18. (e.g. [36, Proposition 2.16] ) For any braid σ and for all i ∈ { , ..., n } , theleft action of x i on the complex σ ∈ K b (SBim n ) is homotopic to the right action of x w σ ( i ) . In short, we will say that the left action R (cid:121) σ is homotopic to the right action σ (cid:120) w ( σ ) R , twisted by the permutation w σ . As a consequence, we obtain the following result: Corollary 3.19. The R –module RHom K b (SBim n ) ( , σ ) is supported on the subspace: (cid:8) x i = x w σ ( i ) , i = 1 , . . . , n (cid:9) ⊂ C n . Our construction of Conjecture 1.1 is predicated on the expectation that: Hom K b (SBim n ) ( , σ ) = R Γ(FHilb dg n , B ( σ )) and that moreover B ( σ ) can be reconstructed from the spaces Hom K b (SBim n ) ( , σ · (cid:81) ni =1 L a i i ) forall sequences of large enough natural numbers ( a , ..., a n ) . These Hom spaces in the category SBim n are very hard to compute, and all we can say at this stage is that Corollary 3.19 stillapplies to them. Therefore, we obtain the following: Corollary 3.20. The complex B ( σ ) = ι ∗ ( σ ) is supported on the subvariety: FHilb dg w := ρ − (cid:0)(cid:8) x i = x w σ ( i ) , i = 1 , . . . , n (cid:9)(cid:1) ⊂ FHilb dg n = FHilb dg n ( C ) where ρ : FHilb dg n ( C ) → C n is the map that records the eigenvalues ( x , ..., x n ) , akin to (2.3) . Corollary 3.21. Suppose that the closure of σ is connected. Then B ( σ ) is supported on ρ − ( { x = . . . = x n } ) = FHilb n (point) × C . Remark . Following Section 1.9, one can prove that if the closure of σ is connected, thenthe sheaf B ( σ ) fibers trivially over C , i.e.: B ( σ ) = B ( σ ) (cid:2) O C for some sheaf B ( σ ) ∈ D b Coh(FHilb n (point)) . Since FHilb n (point) is projective, the coho-mology of this sheaf is expected to be finite-dimensional. Moreover, our conjectures imply thefact that this cohomology matches the reduced Khovanov-Rozansky homology of α .In general, FHilb dg w may be quite complicated. However, for certain permutations w = w σ we can describe it explicitly. The baby case is when w = ( j, j + 1) is a transposition. Definition 3.23. Define the dg subscheme Z j ⊂ FHilb dg n by the following equation:(3.46) O Z j := (cid:20) . . . −→ q t L j L j +1 y j,j +1 −−−→ q t L j L j +1 x j − x j +1 −−−−−→ qt L j L j +1 y j,j +1 −−−→ q O x j − x j +1 −−−−−→ O (cid:21) . Here y j,j +1 : t L j → L j +1 is the map of line bundles induced by the homonymous coefficientof the matrix Y in (2.14), and the fact that y j,j +1 ( x j − x j +1 ) = 0 follows from [ X, Y ] = 0 . , , AND JACOB RASMUSSEN Remark . Formula (3.46) implies the following exact sequence:(3.47) (cid:104) q O x j − x j +1 −−−−−→ O (cid:105) ∼ = (cid:20) O Z j Id −→ qt L j L j +1 ⊗ O Z j [2] (cid:21) Our motivation for defining Z j is the fact that:(3.48) O FHilb dg ( j,j +1) = O Z j for all j ∈ { , ..., n − } . The following proposition follows directly by iterating (3.48). Proposition 3.25. Suppose that w has cycle structure: (1 , ..., k )( k + 1 , ..., k ) , ..., ( k r + 1 , ..., n ) for some sequence < k < . . . < k r < n . Then the dg structure sheaf of FHilb dg w has thefollowing periodic resolution by locally free sheaves on FHilb dg n : (3.49) O FHilb dg w ∼ = (cid:79) j / ∈{ k ,...,k r } (cid:20) . . . −→ q t L j L j +1 x j − x j +1 −−−−−→ qt L j L j +1 y j,j +1 −−−→ q O x j − x j +1 −−−−−→ O (cid:21) . Conjecture 3.26. Suppose that α = (cid:81) ri =0 ( σ k i +1 · · · σ k i +1 − ) is a subword of the Coxeter word σ · · · σ n − , for any sequence < k < . . . < k r < n as in Proposition 3.25. Then: B ( α ) = O FHilb dg w . Example . For α = 1 , the conjecture simply reads B ( α ) = O FHilb dg n , as prescribed byConjecture 1.1. For α = σ · · · σ n − , the conjecture reads B ( α ) = O FHilb dg n (point) × C .Conjecture 3.26 gives a full description of B ( α ) for all braids α on two strands (see Section5 for the explicit construction in this case). Moreover, it completely describes B ( α ) for thebraids α = 1 , s , s , s s on 3 strands, multiplied by arbitrary powers of the twists FT , FT .Building upon this, the following conjecture supersedes the main conjecture of [30], and itserves as one of the motivating examples of the present work: Conjecture 3.28. For gcd ( m, n ) = 1 , consider the torus braid α n,m = ( σ · · · σ n − ) m . Then (3.50) B ( α m,n ) = (cid:32) n (cid:79) i =1 L(cid:98) imn (cid:99) − (cid:98) ( i − mn (cid:99) i (cid:33) ⊗ O FHilb dg n (point) × C See Sections 5 and 6 for detailed computations for two and three-strand torus braids. Remark . It was proved in [30] that the equivariant Euler characteristic of the right handside of (3.50) is equal to the “refined Chern-Simons invariant” defined by Aganagic-Shakirov[2] and Cherednik [17]. One can therefore consider Conjecture 3.28 as a categorification of theconjectures in [2, 17] relating the Poincar´e polynomial of Khovanov-Rozansky homology tothese “refined invariants”. 4. C ATEGORIES AND SCHEMES Motivation: maps to projective space. We start by recalling certain classical construc-tions in algebraic geometry which will guide all subsequent generalizations. Let X be a pro-jective algebraic variety and let L be a line bundle (i.e. a rank one locally free sheaf) over X .One says that L is generated by global sections if the map of sheaves: O X ⊗ Γ( X, L ) → L LAG HILBERT SCHEMES, COLORED PROJECTORS AND KHOVANOV-ROZANSKY HOMOLOGY 33 is surjective. If we choose a basis s , ..., s n of the vector space Γ( X, L ) , this comes down torequiring that any local section of L is a linear combination of the sections s , ..., s n . Moreover,the above datum gives rise to a map:(4.1) X ι → P n , x (cid:55)→ [ s ( x ) : ... : s n ( x )] Global generation implies the fact that the sections s , ..., s n cannot all vanish simultaneously.Moreover, while s i are sections of the line bundle L , their ratios are well-defined functions on X . To this end, we may define the open subset: X i = { s i ( x ) (cid:54) = 0 } ⊂ X where the ratios s j /s i are well–defined. Hence the map (4.1) restricts to a map: X i → U i = { z i (cid:54) = 0 } ⊂ P n If we let O (1) denote the tautological line bundle on P n , then we have: ι ∗ ( O ( k )) = L ⊗ k , ∀ k ∈ Z The functor ι ∗ is monoidal, and is the left adjoint of the direct image functor:(4.2) Coh( X ) ι ∗ − (cid:0) == (cid:1) − ι ∗ Coh( P n ) In the remainder of this section, we present a generalization of this construction, where the roleof the map ι : X → P n is replaced by an abstract categorical setup inspired by (4.2). Remark . By deriving the functors in question, we may write (4.2) at the level of derivedcategories. Then the sections can be thought of as complexes: (cid:104) O X s i → L (cid:105) ∈ D b (Coh( X )) which are supported on { X \ X i } = { s i = 0 } . The product of these complexes:(4.3) n (cid:79) i =0 (cid:104) O X s i → L (cid:105) is therefore supported on the set where all s i vanish simultaneously, which by assumption isthe empty set. Therefore, (4.3) is quasi-isomorphic to 0, and hence it vanishes in D b (Coh( X )) .Put differently, the vanishing of (4.3) is forced upon us by the vanishing of the Koszul complex: n (cid:79) i =0 (cid:104) O P n z i → O P n (1) (cid:105) q . i . s . ∼ = 0 ∈ D b (Coh( P n )) and the fact that the derived version of the functor ι ∗ in (4.2) is monoidal. Remark . Projective space can be defined more scheme-theoretically as: P n = Proj (cid:32) ∞ (cid:77) k =0 S k C n +1 (cid:33) Then the map (4.1) is given by the map C n +1 → Γ( X, L ) induced by the choice of the sections s , ..., s n , and in fact global generation translates into: X = Proj (cid:32) ∞ (cid:77) k =0 Γ( X, L ⊗ k ) (cid:33) . , , AND JACOB RASMUSSEN Notations for categories. In this subsection, we would like to collect all homologicalalgebra notations, definitions and assumptions which will be frequently used below. Let C bean additive unital monoidal category with tensor product ⊗ and direct sum ⊕ . The monoidalstructure is not necessary symmetric. We will denote the unit object of C by C , or if the cate-gory is clear from context. The endomorphism algebra End( ) is always commutative, and weassume that it is Noetherian. For any object A ∈ C , the morphism space Hom( , A ) is a moduleover End( ) , and we assume that it is finitely generated. We assume that all morphism spacesare positively graded. We denote by K b ( C ) the homotopy category of bounded complexes ofobjects in C and by K − ( C ) the homotopy category of bounded above complexes. Unless statedotherwise, we will work with bounded above complexes and abbreviate K − ( C ) to K ( C ) .We will consider two types of “semi-infinite completions” of the category C . The first typeis the homotopy category K − ( C ) of bounded above complexes of objects in C (which is well-known to also be a monoidal category). The other type is the category of certain infinite sumsof objects in C , as in the following definition. Definition 4.3. Assume that C is graded, and the grading shift is denoted by A (cid:55)→ A (1) . Wedefine its graded completion C ↑ as follows. The objects are given by countable direct sums:Ob ( C ↑ ) = (cid:40) N (cid:77) i = −∞ A i ( i ) for some N ∈ Z (cid:41) and the morphisms φ : ⊕ A i ( i ) → ⊕ B j ( j ) are collections of arrows { φ ij : A i ( i ) → B j ( j ) } forall i, j , such that for each i there are only finitely many j such that φ ij (cid:54) = 0 .One can check that C ↑ and K − ( C ↑ ) inherit the tensor product from C . Note that K − ( C ↑ ) isendowed with both the grading (1) and the homological degree [1] .Note that the category C may have multiple gradings, and the notion of completion dependson a specific choice of grading among these. For example, if C is graded by Z r , this accountsto choosing a one-dimensional direction in Z r . To clarify homological algebra over C ↑ , wepresent some examples. Example . Let C be the category of graded finitely generated C [ x ] -modules. Consider thefollowing two-term complex in K − ( C ↑ ) : F IGURE LAG HILBERT SCHEMES, COLORED PROJECTORS AND KHOVANOV-ROZANSKY HOMOLOGY 35 We can introduce an auxilary variable y of degree ( − and rewrite the complex as following: C [ x, y ] xy −−−→ C [ x, y ] . At first glance, one could think that since all horizontal arrows in Figure 7 are isomorphisms,the complex is contractible. However, this is not the case, since a homotopy would be: C [ x, y ] H ←− C [ x, y ] such that H (1 + xy ) = (1 + xy ) H = 1 A natural choice for H would be: H ( x, y ) = 11 + xy = 1 − xy + x y − x y + . . . , but this is not a valid morphism in C ↑ since there would be non-zero arrows from the top-mostcopy of C [ x ] to all infinitely many copies below it. Remark . One can check that the homology of the complex in Figure 7 is isomorphic to C [ x, y ] / (1 + xy ) = C [ x, x − ] .4.3. Categories over schemes. In this section, we will develop a general setup relating a cat-egory C with a scheme X , with the goal of reducing Conjecture 1.1 to Conjecture 3.9. Thoughwe will not always say this explicitly, X should be thought of as a dg scheme. Definition 4.6. A morphism from the category C to the scheme X , written as: C ι −→ X consists of a pair of functors:(4.4) C ι ∗ − (cid:0) == (cid:1) − ι ∗ Coh( X ) such that: • ι ∗ is a monoidal functor • ι ∗ is the right adjoint of ι ∗ • the following projection formula holds:(4.5) ι ∗ ( ι ∗ M ⊗ C ⊗ ι ∗ M ) = M ⊗ ι ∗ ( C ) ⊗ M for all M , M ∈ Coh( X ) and C ∈ C .The above definition is modeled on the situation when C = Coh( Y ) for a scheme Y , inwhich case the functors ι ∗ and ι ∗ play the roles of direct and inverse image functors associatedto a map of schemes ι : Y → X . Definition 4.7. We call the map C ι −→ X birational if:(4.6) ι ∗ = O X This terminology, albeit imprecise, is motivated by the important case when C = Coh( Y ) where Y is endowed with a proper birational map to X . Proposition 4.8. Suppose that C ι −→ X is birational. Then ι ∗ is fully faithful, and moreover: (4.7) Hom C ( , ι ∗ M ) = Γ( X, M ) for all M ∈ Coh( X ) . , , AND JACOB RASMUSSEN Proof. The adjunction implies that: Hom C ( ι ∗ M (cid:48) , ι ∗ M ) = Hom X ( M (cid:48) , ι ∗ ι ∗ M ) = Hom X ( M (cid:48) , M ) where the last equality follows from (4.5) and (4.6). When M (cid:48) = O X we obtain precisely (4.7). (cid:3) Most of the time we will consider a derived version of this construction. Definition 4.9. A derived morphism from the category C to the scheme X , written as: C ι −→ X is a pair of mutually adjoint functors:(4.8) K ( C ) ι ∗ − (cid:0) == (cid:1) − ι ∗ D (Coh( X )) All other properties and requirements remain unchanged.4.4. The affine case. Let C be an additive monoidal category. Suppose we are given a Noe-therian commutative ring A and a ring homomorphism(4.9) A f −→ End C ( ) satisfying ( C ( , C ) is finitely generated over A for any object C of C . Then there is a derived morphism:(4.10) C ι −→ Spec A. The functors K ( C ) ι ∗ − (cid:0) == (cid:1) − ι ∗ D ( A –mod ) are defined as follows. There is a functor i : C → A –mod given by:(4.11) ι ∗ ( C ) = Hom C ( , C ) . This extends in the obvious way to a functor i : K ( C ) → K ( A –mod ) , and ι ∗ is defined to bethe composition of i ∗ with the natural inclusion K ( A –mod ) → D ( A –mod ) .In the other direction, let F A –mod be the category of finitely generated free A modules. Theinclusion K ( F A –mod ) → D ( A –mod ) is an equivalence of categories, so we may as well givea functor ι ∗ : K ( F A –mod ) → K ( C ) . We define ι ∗ by setting ι ∗ ( A ) = and ι ∗ ( a ) = f ( a ) for a ∈ A = Hom( A, A ) . This extends to K ( F A –mod ) in the obvious way. If M is an object of D ( A –mod ) , we write ι ∗ ( M ) = M ⊗ A .Let us check that the functors ι ∗ and ι ∗ are adjoint, or equivalently, that(4.12) Hom K ( C ) ( M ⊗ A , C ) = Hom D ( A –mod ) ( M, Hom C ( , C )) for all M ∈ D ( A –mod ) and C ∈ K ( C ) . If C ∈ C , the right-hand side is by definition Ext A ( M, Hom C (1 , C )) . The statement that it is equal to the left-hand side reduces to the wellknown fact that to compute Ext of two modules, it is enough to take a free resolution of one ofthem. Properties (4.5) and (4.6) also follow directly from the definitions. Example . Let Y be an algebraic variety, and C = Coh( Y ) . The unit in Y is given by thestructure sheaf O Y , and indeed C is a category over Spec End C ( ) = Spec Γ( Y, O Y ) . Thisstructure is precisely equivalent with the global section map: ι : Y → Spec Γ( Y, O Y ) LAG HILBERT SCHEMES, COLORED PROJECTORS AND KHOVANOV-ROZANSKY HOMOLOGY 37 More generally, a ring homomorphism A f → Γ( Y, O Y ) corresponds to a map Spec Γ( Y, O Y ) → Spec A , and one can use the composed map from Y to Spec A to define ι ∗ and ι ∗ .4.5. The projective case. In the previous Subsection, we showed that any category can berealized over the spectrum of the endomorphism ring of its unit. We may upgrade this con-struction if we are given an invertible object F ∈ K ( C ) , i.e. one which is endowed withisomorphisms:(4.13) F ⊗ F − ∼ = F − ⊗ F ∼ = Assumption 4.11. We assume that the graded algebra: (4.14) Hom K ( C ) ( , F • ) := ∞ (cid:77) k =0 Hom K ( C ) ( , F k ) is commutative.Remark . Recall that C was a graded category, so for every k the space Hom K ( C ) ( , F k ) isgraded. The algebra Hom K ( C ) ( , F • ) has an extra grading which equals k on Hom K ( C ) ( , F k ) .In this setting, there exists a tautological derived map:(4.15) C ι −→ (Spec R ) / C ∗ for any Noetherian graded commutative ring R and graded ring homomorphism:(4.16) R f −→ Hom K ( C ) ( , F • ) The functors (4.4) are explicitly given by:(4.17) ι ∗ ( C ) = Hom K ( C ) ( , F • ⊗ C ) (4.18) ι ∗ ( M ) = (cid:32) M ⊗ R ∞ (cid:77) k = −∞ F k (cid:33) for all graded R − modules M and all C ∈ K ( C ) . The Hom space in (4.17) is an R − modulevia (4.16). It is straightforward to show that the analogue of (4.12) holds, and that the abovedatum makes C into a category over the stack (Spec R ) / C ∗ :(4.19) K ( C ) ι ∗ − (cid:0) == (cid:1) − ι ∗ D ( R –grmod ) Note that one needs the analogue of condition ( on the category C to ensure that the abovefunctors are well-defined (in particular, that the right hand side of (4.17) is a finitely generated R -module). But given this, the map ι is birational if and only if the map f of (4.16) is anisomorphism. Example . Let us consider the case where R = A [ z , ..., z n ] , for a ring A equipped with ahomomorphism A → End C ( ) . Then the datum of the homomorphism (4.16) boils down togiving n + 1 morphisms:(4.20) z i (cid:32) (cid:110) α i −→ F (cid:111) i =0 ,...,n , , AND JACOB RASMUSSEN This makes C into a category over the stack: C ι −→ A n +1 A / C ∗ . The natural question is when does ι factor through projective space:which amounts to factoring (4.19) through functors:(4.21) K ( C ) ι (cid:48)∗ − (cid:0) === (cid:1) − ι (cid:48)∗ D (Coh( P nA )) It is clear that ι (cid:48)∗ and ι (cid:48)∗ must be given by the same formulas as in (4.17)–(4.18), but one needsto impose a certain relation. Because the zero section of A n +1 A / C ∗ is removed when definingprojective space, the structure sheaf of the zero section becomes quasi-isomorphic to 0. Sincethis structure sheaf can be expressed via the following Koszul complex: (cid:104) ... −→ O ( − ⊕ ( n +12 ) −→ O ( − ⊕ ( n +11 ) ( z ,...,z n ) −−−−−→ O (cid:105) = n (cid:79) i =0 (cid:104) O ( − z i −→ O (cid:105) we conclude that the functors (4.21) are well-defined only if:(4.22) (cid:104) α −→ F (cid:105) ⊗ ... ⊗ (cid:104) α n −→ F (cid:105) h.e. ∼ = 0 ∈ K ( C ) . It is not hard to see that this condition is also sufficient, by invoking Beilinson’s description[9, 10] of the derived category of projective space as equivalent to the homotopy category ofcomplexes of finite direct sums of free A [ x , ..., x n ] –modules with degree shifts ∈ { , ..., n } . Remark . If F = L is a line bundle in C = Coh( X ) , then α i are nothing but sections of L .By Remark 4.1, equation (4.22) is equivalent to the fact that α i generate L , and indeed this is anecessary and sufficient condition for the existence of X → P n , as we saw in Subsection 4.1.4.6. The relative case. The situation of Example 4.13 captures a very interesting problem,namely when can we factor a map from a category to a scheme through another scheme:(4.23)More precisely, ι (cid:48) should satisfy the equations ι ∗ = ι (cid:48)∗ ◦ π ∗ and ι ∗ = π ∗ ◦ ι (cid:48)∗ and all the functorsshould be derived from now on. The situation we will study in this paper is when: Y = P V ∨ := Proj X ( S • V ) where V is a coherent sheaf on X of projective dimension 0 or 1. Let us first study the case ofprojective dimension zero, so assume that V is a vector bundle. Proposition 4.15. Suppose that Y = P V ∨ and that the map ι in (4.23) is constructed. Thedatum of the extension ι (cid:48) is equivalent to an invertible object F ∈ C together with an arrow: (4.24) ι ∗ V α −→ F LAG HILBERT SCHEMES, COLORED PROJECTORS AND KHOVANOV-ROZANSKY HOMOLOGY 39 in C . This gives C the structure of a category over Y if and only if: (4.25) (cid:104) ... α −→ ι ∗ (cid:0) ∧ k V (cid:1) ⊗ F − k α −→ ... (cid:105) h.e. ∼ = 0 ∈ K b ( C ) The map ι (cid:48) is birational if and only if ι satisfies: (4.26) S k V ∼ = ι ∗ ( F k ) ∀ k ≥ Proof. All notations O or O ( k ) will refer to invertible sheaves on P V ∨ . If ι (cid:48) exists and hasall the expected properties, then set F = ι (cid:48)∗ ( O (1)) . In this case, the map (4.24) is simply ι (cid:48)∗ applied to the tautological morphism: π ∗ V −→ O (1) on Y . The fact that the complex (4.25) is quasi-isomorphic to 0 follows by applying ι (cid:48)∗ to theKoszul complex of Y . The birationality of ι (cid:48) implies that ι (cid:48)∗ ∼ = O , from which the projectionformula implies ι (cid:48)∗ ( F k ) ∼ = O ( k ) . Applying π ∗ to this relation implies precisely (4.26). Con-versely, suppose that we are given a morphism (4.24) which satisfies (4.25), and let us constructthe map ι (cid:48) that makes the diagram (4.23) commute. Note that (4.24) gives us an arrow: ι ∗ (cid:0) V ⊗ k (cid:1) −→ F k for all k ≥ . Because F is invertible, this arrow factors through:(4.27) ι ∗ (cid:0) S k V (cid:1) −→ F k for all k ≥ (since F is invertible, so is F k , and hence has no nontrivial endomorphisms; thisimplies that the anti-symmetric projector is zero, hence S k F = F k ) . This allows us to define: ι (cid:48)∗ ( M ) = (cid:32) π ∗ ( M ) (cid:79) S • V ∞ (cid:77) k = −∞ F k (cid:33) A priori, this only determines the functor ι (cid:48)∗ on the level of the homotopy category of coherentsheaves on P V ∨ . To check that it descends to a functor on the derived category, we must showthat ι (cid:48)∗ takes quasi-isomorphic complexes to isomorphic complexes. The fact that this statementis true for the Koszul complex is precisely the assumption (4.25). The fact that this is sufficientis due to Theorem 2.10 of [35] (see also [6]), which asserts that: D b (Coh( P V ∨ )) ∼ = homotopy category of complexes of (cid:32) rank V− (cid:77) i =0 E i ( i ) (cid:33) E , E ,... ∈ D b (Coh( X )) As for the right adjoint functor, we set: ι (cid:48)∗ ( C ) = ι ∗ (cid:32) ∞ (cid:77) k =0 F k ⊗ C (cid:33) as a graded O X − module. To realize the right hand side as a sheaf on Y , we need to endow itwith an action of S ∗ V , namely with an associative homomorphism of graded algebras: S ∗ V ⊗ O X ι ∗ (cid:32) ∞ (cid:77) k =0 F k ⊗ C (cid:33) −→ ι ∗ (cid:32) ∞ (cid:77) k =0 F k ⊗ C (cid:33) The above morphism is obtained via adjunction and (4.27). (cid:3) , , AND JACOB RASMUSSEN Projective dimension one. For the setting of this paper, we will need a version of Propo-sition 4.15 when the vector bundle V is replaced by the quotient: −→ W ψ −→ V −→ Q −→ where W is another vector bundle. More precisely, we are interested in the case when: Y (cid:44) → P V ∨ is the (derived) zero locus of the section:(4.28) s : π ∗ ( W ) ψ −→ π ∗ ( V ) −→ O (1) where π is the map in the following diagram:(4.29)To simplify the geometry, we make the following very important assumption:(4.30) the ideal of Y j (cid:44) → P V ∨ is generated by a regular sequence in Im s which entails that the embedding ψ cuts out Y as a complete intersection in P V ∨ . One could dowithout this assumption, but that would require one to replace Y with the dg scheme determinedby the exterior power of the section s . In other words, we must require the following quasi-isomorphism in the derived category of P V ∨ :(4.31) O Y ∼ = (cid:104) ... s −→ ∧ k π ∗ ( W ) ⊗ O ( − k ) s −→ ... s −→ O (cid:105) In order to construct the lift ι (cid:48)(cid:48) in (4.29), we must first construct the arrow ι (cid:48) , and for this weinvoke Proposition 4.15. Then the following Proposition says precisely when the arrow ι (cid:48) thusdefined factors through Y . Proposition 4.16. Suppose that Y j (cid:44) → P V ∨ as in (4.29) and that the map ι is constructed. Thedatum of the extension ι (cid:48)(cid:48) is equivalent to an invertible object F ∈ C together with an arrow: (4.32) ι ∗ Q β −→ F in C . This gives C the structure of a category over Y if and only if: (4.33) (cid:104) ... β −→ ι ∗ (cid:0) ∧ k Q (cid:1) ⊗ F − k β −→ ... (cid:105) h.e. ∼ = 0 The map ι (cid:48)(cid:48) is birational if and only if ι ∗ gives rise to an isomorphism: (4.34) S k Q ∼ = ι ∗ ( F k ) ∀ k ≥ Note that if we interpret Y as a dg scheme whose structure sheaf is the dg algebra in the righthand side of (4.31), we must replace Q in (4.32), (4.33), (4.34) with the two term complex [ W → V ] . Making sense of the symmetric and exterior powers of such a complex is ratherstraightforward homological algebra, which we relegate to the Appendix. LAG HILBERT SCHEMES, COLORED PROJECTORS AND KHOVANOV-ROZANSKY HOMOLOGY 41 Proof. As we have seen in Proposition 4.15, the existence of a monoidal functor: ι (cid:48)∗ : D b (Coh( P V ∨ )) → K b ( C ) implies the datum of an invertible object F ∈ C (the image of O (1) ) together with an arrow ι ∗ V → F in C (the image of the tautological morphism). The question is when does the functor ι (cid:48)∗ factor through: D b (Coh( P V ∨ )) j ∗ −→ D b (Coh( Y )) M (cid:55)→ M ⊗ O P V∨ O Y = (cid:104) ... s −→ ∧ k π ∗ ( W ) ⊗ M ( − k ) s −→ ... s −→ M (cid:105) where in the last equality we have used the assumption (4.30). In particular, we have:This implies that the functor ι (cid:48)(cid:48)∗ must take the composition π ∗ ( W ) ψ (cid:44) → π ∗ ( V ) → O (1) to zero,and hence the map α of (4.24) must factor through a map β as in (4.32). Sending the Koszulcomplex of β though the functor j ∗ gives rise to the Koszul complex of α , which must be sentto 0 by (4.25). Therefore, we conclude that the existence of the extension ι (cid:48)(cid:48)∗ requires (4.33).Finally, recall that being birational is equivalent to ι (cid:48)(cid:48)∗ ∼ = O Y . The projection formula impliesthat ι (cid:48)(cid:48)∗ ( F • ) ∼ = O Y ( k ) , and applying j ∗ to this isomorphism yields: ι (cid:48)∗ ( F • ) ∼ = (cid:104) ... s −→ ∧ k π ∗ ( W ) ⊗ O ( • − k ) s −→ ... (cid:105) Applying π ∗ to the above isomorphism implies:(4.35) ι ∗ ( F • ) ∼ = (cid:104) ∧ W ⊗ S V (cid:105) • where the differential in the right hand side of (4.35) is given by the map ψ : W → V . As inExample 10.3, the right hand side is a resolution of S • Q , hence we obtain (4.34). (cid:3) Deducing Conjecture 1.1 from Conjecture 3.9. The categorical setup presented in thissection allows one to deduce the main conjecture from Conjecture 3.9 (a)–(c). We will proceedby induction on n , so let assume that the functors (3.16) are well-defined for some fixed n . Ourtask is to construct functors: K b (SBim n +1 ) ι n +1 ∗ − (cid:0) ==== (cid:1) − ι ∗ n +1 D b (Coh(FHilb dg n +1 )) given the functors: K b (SBim n )[ x n +1 ] ι n ∗ − (cid:0) === (cid:1) − ι ∗ n D b (Coh(FHilb dg n × C )) obtained from the inductive hypothesis and tensoring with the extra variable x n +1 . We definethe composed functors: ι ∗ : K b (SBim n +1 ) Tr − (cid:0) == (cid:1) − I SBim n [ x n +1 ] ι n ∗ − (cid:0) === (cid:1) − ι n ∗ D b (Coh(FHilb dg n × C )) : ι ∗ According to Proposition 2.10, we have FHilb dg n +1 = P E ∨ n , where E n is the complex on FHilb dg n × C from (2.25). Relation (2.35) states that this complex has projective dimension 1, and we can , , AND JACOB RASMUSSEN therefore apply Proposition 4.16. To do so, we must exhibit an invertible object F ∈ SBim n +1 and a morphism: ι ∗ E n β −→ F in K b (SBim n ) . We will choose F = L n +1 and take the morphism β to be the adjoint of (3.28): E n = ι ∗ ( L n +1 ) = ι n ∗ (Tr( L n +1 )) The full statement of (3.28) allows one to prove that S k ( E n ) = ι ∗ ( L kn +1 ) , which establishes thefact that SBim n +1 is birational over FHilb n +1 by (4.34). To complete the proof of Conjecture1.1 one needs to also check that (4.33) holds, which is part (c) of Conjecture 3.9.4.9. Invertible objects in monoidal categories. We summarize several important propertiesof invertible objects (4.13) in arbitrary monoidal categories. The proofs are straightforward,and left as exercises to the interested reader. Proposition 4.17. For any invertible object F ∈ C and two arbitrary objects C, C (cid:48) ∈ C , thereexist canonical isomorphisms: Hom C ( F ⊗ C, F ⊗ C (cid:48) ) ∼ = Hom C ( C, C (cid:48) ) ∼ = Hom C ( C ⊗ F, C (cid:48) ⊗ F ) Corollary 4.18. Tensoring with an invertible object and with its inverse yield biadjoint func-tors, that is, we have canonical isomorphisms: Hom C ( C, F ⊗ C (cid:48) ) ∼ = Hom C ( F − ⊗ C, C (cid:48) ) Hom C ( C, C (cid:48) ⊗ F ) ∼ = Hom C ( C ⊗ F − , C (cid:48) ) Corollary 4.19. For any invertible F ∈ C and any object C ∈ C , we have: Hom C ( , F ⊗ C ) ∼ = Hom C (1 , C ⊗ F ) 5. E XAMPLE : THE CASE OF TWO STRANDS The geometry of FHilb . In this section, we will always write FHilb = FHilb ( C ) . Inthis section we construct explicitly the functors ι ∗ and ι ∗ between the category of sheaves on FHilb and the category of Soergel bimodules SBim . We have the matrix presentation: FHilb = (cid:26) X = (cid:18) x z x (cid:19) , Y = (cid:18) w (cid:19) , [ X, Y ] = 0 , v = (cid:18) (cid:19) cyclic (cid:27) conjugation by g = (cid:18) c (cid:19) Note that in the presentation above, we fixed the vector v (and this fixes the first column of theconjugating matrix) to eliminate some coordinates. Unwinding the above gives us:(5.1) FHilb = { ( x , x , z, w ) , ( x − x ) w = 0 , z, w not both zero } ( x , x , z, w ) ∼ ( x , x , cz, cw ) = Proj( A ) where x , x , z, w have degrees , , , in the graded algebra:(5.2) A = C [ x , x , z, w ]( x − x ) w Recall the complex (2.25):(5.3) E = (cid:20) qt O (0 ,x − x , −−−−−−→ q O ⊕ t O ⊕ O ( x − x , , T −−−−−−−→ O (cid:21) LAG HILBERT SCHEMES, COLORED PROJECTORS AND KHOVANOV-ROZANSKY HOMOLOGY 43 on FHilb ( C ) × C = C , from which it is clear the the leftmost map is injective and therightmost map is surjective on all fibers. Therefore, we have H ( E ) ∼ = E and hence: FHilb ∼ = FHilb dg Moreover, letting z and w be coordinates on the first two summands of the middle space of(5.3), we observe that H ( E ) = ( O z ⊕ O w ) / ( x − x ) w , which matches the algebra (5.2).The irreducible components of the flag Hilbert scheme are:(5.4) FHilb = Z ∪ Z where:(5.5) Z = { x (cid:54) = x } = { b = 0 } = C with coordinates ( x , x ) = Proj( A/wA ) (5.6) Z = { x = x } = C × P with coordinates ( x, [ z : w ]) = Proj( A/ ( x − x ) A ) The intersection of these two irreducible components is: Z ∩ Z = C × [1 : 0] = C × { I (2) } while the other torus fixed point I (1 , satisfies: Z (cid:54)(cid:51) I (1 , ∈ Z , I (1 , = (0 , [0 : 1]) •• I (2) I (1 , Z Z F IGURE 8. Flag Hilbert scheme of two points5.2. Cohomology of sheaves on FHilb . On the projectivization (5.1), the line bundles ofimportance for us are L ∼ = O and L ∼ = O (1) , where the latter denotes the Serre twistingsheaf. Note that:(5.7) T ∼ = O ⊕ O (1) We will now compute the cohomology groups of certain line bundles on FHilb . To simplifyour computations by removing a factor of C , we will work with the reduced version all theschemes and dg schemes in question (see Subsection 1.9). Specifically, this means:(5.8) FHilb = Proj( A ) where A = C [ x, z, w ] xw where we set x + x = 0 and x − x = x . The irreducible components of this variety are: Z = C and Z = P = FHilb (point) Note that T = O (1) . The following cohomology computations are well-known: H i ( Z , O ( k )) = q k − q · δ i, , , AND JACOB RASMUSSEN because Z = C , while: H i ( Z , O ( k )) = t k + . . . + q k if i = 0 and k ≥ qt ) − ( t k +2 + . . . + q k +2 ) if i = 1 and k ≤ − otherwisebecause Z = P with equivariant weights q and t . Consider the short exact sequence: (cid:47) (cid:47) q O Z x (cid:47) (cid:47) O FHilb (cid:47) (cid:47) O Z (cid:47) (cid:47) which is induced by (5.4). Because the cohomology of sheaves on Z is concentrated in degree0, we have the following equality of ( q, t ) –equivariant vector spaces: H i (FHilb , O ( k )) = qH i ( Z , O ( k )) + H i ( Z , O ( k )) = (5.9) = t k + . . . + q k + q k +1 − q if i = 0 and k ≥ q k +1 − q if i = 0 and k < qt ) − ( t k +2 + . . . + q k +2 ) if i = 1 and k ≤ − otherwiseThe analogous equalities for the non-reduced version FHilb are obtained by dividing the righthand sides of (5.9) by − q .5.3. Soergel bimodules for n = 2 . The category of Soergel bimodules is generated by twoobjects: R = C [ x , x ] and B = R ⊗ R (12) R . With our grading conventions, we have:(5.10) B = B ⊗ R B ∼ = q B ⊕ q − B In the reduced category, we can set x + x = 0 and x − x = x , and write R = C [ x ] and: B = R ⊗ R s R = C [ x ] ⊗ C [ x ] C [ x ] . This object also satisfies property (5.10), and moreover: Hom( , B ) (cid:39) Ext ( , B ) = R are rank 1 modules over R . In terms of grading, note that Ext differs from Hom by a shift bythe equivariant weight a − q − , which is an incarnation of the wedge product in (1.13). Thus:(5.11) RHom • ( , B ) = ∧ • (cid:18) ξqa (cid:19) ⊗ R for a formal variable ξ . The object in the Soergel category which corresponds to a singlepositive crossing σ is the following complex: σ = (cid:20) B ⊗ (cid:55)→ −−−−→ sRq (cid:21) The powers of s mark homological degree, and so they are always consecutive integers in acomplex. We mainly use them to pinpoint the 0–th term of a complex, and to compare withformulas from geometry. Similarly, the object in the Soergel category which corresponds to asingle negative crossing σ − is: σ − = (cid:34) q Rs (cid:55)→ x ⊗ ⊗ x −−−−−−−→ B (cid:35) LAG HILBERT SCHEMES, COLORED PROJECTORS AND KHOVANOV-ROZANSKY HOMOLOGY 45 Let us write FT = FT for the image of the full twist in the reduced Soergel category, andnote that FT = σ . Therefore, formula (5.10) allows us to write: FT = (cid:20) q B → sBq → s Rq (cid:21) Recall that the connection between the parameters s and t is given by s = qt . The twoeigenmaps that span the space Hom( , FT ) are described in the following diagram:(5.12)where z = (1 (cid:55)→ x ⊗ ⊗ x ) and w = Id. As we will see in more examples in the nextSubsection, it is no coincidence that the only maps from R into non-negative powers of the fulltwist have integer q, t –weights: this is called the “parity miracle” by [23]. Proposition 5.1. We have the following relation in the category SBim : h.e. ∼ = (cid:104) ... α −→ q FT − ⊕ q FT − α −→ FT − ⊕ q FT − α −→ FT − ⊕ FT − z,w ) −−−→ (cid:105) where the maps α and α are given by: (5.13) α = (cid:18) w − z x (cid:19) and α = (cid:18) x z w (cid:19) Proof. Remark that this complex is filtered by complexes:(5.14) (cid:104) FT − k − w, − z ) −−−−→ FT − k − ⊕ FT − k − z,w ) −−−→ FT − k (cid:105) == FT − k − ⊗ Cone (cid:104) w −→ FT (cid:105) ⊗ Cone (cid:104) z −→ FT (cid:105) , so it is sufficient to prove that Cone( z ) ⊗ Cone( w ) (cid:39) Cone( w ) ⊗ Cone( z ) (cid:39) (indeed, thiswould imply that the complexes in the left hand side of (5.14) are contractible). Since: Cone (cid:104) w −→ FT (cid:105) = [ B −→ B ] , and: B ⊗ Cone( z ) (cid:39) B ⊗ [ R → B → B → R ] (cid:39) [ B → B ⊕ B → B ⊕ B → B ] (cid:39) , we conclude that Cone( w ) ⊗ Cone( z ) (cid:39) . The case of Cone( z ) ⊗ Cone( w ) is analogous. (cid:3) Proj construction. The purpose of this Subsection is to construct the functors:(5.15) D b (Coh C ∗ × C ∗ (FHilb )) ι ∗ − (cid:0) == (cid:1) − ι ∗ K b (SBim ) and prove Conjecture 1.1 for n = 2 . To keep our notation simple, we will perform the compu-tation for the reduced versions of the above categories. As was shown in Section 4, in order toconstruct ι ∗ one needs to prove the following isomorphism of graded algebras:(5.16) ∞ (cid:77) k =0 Hom (cid:16) , FT k (cid:17) ∼ = ∞ (cid:77) k =0 Hom (cid:0) FHilb , O ( k ) (cid:1) , , AND JACOB RASMUSSEN To compute the left hand side of (5.16), recall from [39] that we have the following identity in SBim for all k > : FT k (cid:39) q k − B → q k − sB → · · · → s k − Bq k − → s k − Bq k − → s k − Bq k − (cid:124) (cid:123)(cid:122) (cid:125) k → s k Rq k where the maps alternate between x ⊗ − ⊗ x and x ⊗ ⊗ x . Since s = −√ qt , we have: Hom( , FT k ) (cid:39) q k R → q k − t R x −→ · · · → qt k − R x −→ qt k − R → t k − R (cid:124) (cid:123)(cid:122) (cid:125) k x −→ t k R (5.17) ∼ = z k C [ x ] k (cid:77) i =1 w i z k − i C [ x ] x = A k One can think of z , w as formal variables of degrees q , t , but they actually correspond to themaps of (5.12) under the required isomorphism (5.16). This establishes (5.16) as an isomor-phism of C [ x ] –modules. We claim that this isomorphism also preserves the algebra structures,and therefore the functor ι ∗ is well-defined. By construction: ι ∗ ( FT k ) = O ( k ) for all k ≥ . As for the functor ι ∗ of (5.15), we require: ι ∗ ( O ( k )) := FT k and: ι ∗ (cid:16) q O z −→ O (1) (cid:17) and ι ∗ (cid:16) q O w −→ O (1) (cid:17) = the maps (5.12)However, note that this assignment simply defines a functor: Coh (cid:0) Spec A/ C ∗ (cid:1) ι ∗ −→ SBim since A is the homogeneous coordinate ring of FHilb . We wish to show that this functorfactors through D b (Coh(Proj A )) . To do so, we must prove that the object:(5.18) q . i . s . ∼ = A = A ( z, w ) on FHilb goes to −−−→ ι ∗ ( A ) h.e. ∼ = 0 in SBim To compute the image of A under ι ∗ , we need to resolve this object in terms of free A modules.The standard choice is the Koszul resolution, which is infinite because FHilb is singular: q . i . s . ∼ = (cid:104) ... α −→ qA ( − ⊕ qA ( − α −→ A ( − ⊕ qA ( − α −→ A ( − ⊕ A ( − ( z,w ) −−−→ A (cid:105) where the maps alternate between those of (5.13). Then (5.18) follows from Proposition 5.1. Remark . By analogy with (5.17), we have: Ext ( , FT k ) ∼ = q k − R → q k − t R x −→ · · · → qt k − R x −→ t k − R → t k − R (cid:124) (cid:123)(cid:122) (cid:125) k → t k R ∼ = LAG HILBERT SCHEMES, COLORED PROJECTORS AND KHOVANOV-ROZANSKY HOMOLOGY 47 (5.19) ∼ = z k − C [ x ] k − (cid:77) i =1 w i z k − − i C [ x ] x = A k − and therefore: RHom (cid:16) , FT k (cid:17) ∼ = Hom (cid:16) , FT k − (cid:17) This is precisely (3.25) for M = FT k and T = ι ∗ ( T ) = ι ∗ ( O (1)) = FT .5.5. Sheaves for two-strand braids. To construct the sheaf ι ∗ ( M ) for any object M ∈ SBim ,one needs to consider the module Hom( , M ⊗ FT • ) over the graded algebra A = Hom( , FT • ) .In the previous subsection, we have studied the case M = FT k for positive integers k , and wefound that ι ∗ ( M ) = O ( k ) . The computation for negative k is more interesting: FT − k ∼ = t − k R → q − t − k B → q − t − k B → q − t − k B → · · · → q − k t − B → q − k B (cid:124) (cid:123)(cid:122) (cid:125) k for any k ≥ , where the maps alternate between x ⊗ ⊗ x and x ⊗ − ⊗ x . Therefore, we have: Hom( , FT − k ) ∼ = t − k R −→ t − k R −→ t − k R x −→ q − t − k R −→ . . . x −→ q − k t − R −→ q − k R (cid:124) (cid:123)(cid:122) (cid:125) k (5.20) = ⇒ Hom( , FT − k ) ∼ = t H (cid:0) FHilb , O ( − k ) (cid:1) according to (5.9). The case of general a follows by analogy with the previous subsection, sowe conclude the following formula that extends (5.16) to negative integers:(5.21) RHom • SBim ( , FT − k ) ∼ = R Γ (cid:0) FHilb , O ( − k ) ⊗ ∧ • O ( − (cid:1) Remark . Let us observe the fact that the derived functors in the two sides of the aboveequation are very different. In the left hand side, we have the derived Hochschild homologyfunctor, whose degree is measured by a . In the right hand side, we have derived direct imageof sheaves, whose degree is measured by t , and the a grading comes from ∧ • O ( − .To complete the discussion for n = 2 , let us compute B ( σ ) := ι ∗ ( σ ) where σ denotes asingle positive crossing. Together with the projection formula (4.5), this implies that: B ( σ k +1 ) := ι ∗ ( σ k +1 ) = ι ∗ ( σ ⊗ FT k ) = ι ∗ ( σ ) ⊗ O ( k ) = B ( σ ) ⊗ O ( k ) for all integers k . In fact, we have: Hom( , σ k +1 ) ∼ = q k + R x −→ · · · −→ q t k − R x −→ qt k − R −→ qt k R (cid:124) (cid:123)(cid:122) (cid:125) k +1 x −→ t k + R and therefore: (cid:77) k ≥ Hom( , σ ⊗ FT k ) = (cid:77) k ≥ Hom( , σ k +1 ) = t / C [ x, z, w ] x , , AND JACOB RASMUSSEN where recall that z and w are the maps of (5.12). We conclude that B ( σ ) is the structuresheaf of the subscheme { x = 0 } ⊂ FHilb , which is nothing but the connected component Z = FHilb (point) ∼ = P of (5.6). The periodic resolution (3.49) takes the form: B ( σ ) ∼ = O P q . i . s . ∼ = (cid:104) ... w −→ q t O ( − x → qt O ( − w −→ q O ( − x → O (cid:105) where O denotes the structure sheaf of FHilb . In the non-reduced category, one needs toreplace x by x − x everywhere. Finally, let us compute ι ∗ ( B ) , where recall that B = R ⊗ R s R .Since z ⊗ Id B is an isomorphism between B and FT ⊗ B , we have: (cid:77) k ≥ Hom( , B · FT k ) = (cid:77) k ≥ z k Hom( , B ) = C [ x , x , z ] . Therefore ι ∗ ( B ) is the structure sheaf of the irreducible component Z ⊂ FHilb cut out by theequation w = 0 (see (5.5)), which is isomorphic to C with coordinates x and x .6. E XAMPLE : THE CASE OF THREE STRANDS The geometry of FHilb . We will now study the variety FHilb = FHilb ( C ) and formu-late a precise conjecture about the sheaf ι ∗ ( figure eight knot ) . Recall the matrix presentation: FHilb = X = x a x α α x , Y = b β β , [ X, Y ] = 0 , (6.1) v = cyclic (cid:46) conjugation by g = c t d Note that in the presentation above, we fixed the vector v (and this fixes the first column of theconjugating matrix) to eliminate certain coordinates. Note that the map FHilb → FHilb isgiven by only retaining the top × corners of the matrices in question. If one is given theeigenvalues x , x , x and the point [ a : b ] ∈ P , then the datum one needs to construct a pointin FHilb is the vector:(6.2) ( α , α , β , β ) ∈ T ∨ ⊕ T ∨ To ensure that the equation [ X, Y ] = 0 is satisfied, we need to ensure that: ( x − x ) β = α b − β a and ( x − x ) β = 0 Moreover, the fact that we quotient out by conjugation matrices implies that we must identify: ( α , α , β , β ) ∼ ( α + ta, α + t ( x − x ) , β + tb, β ) and ( α , α , β , β ) ∼ d ( α , α , β , β ) for t ∈ C and d ∈ C ∗ . Unwinding these facts, one sees that the datum (6.2) correspondsto a vector in H ( E ∨ ) , where E is the complex in (2.25) when ∗ = C . It is elementary toprove that E and E ∨ are quasi-isomorphic to their zero-th cohomology, so we conclude that FHilb = FHilb dg . The irreducible components of the flag Hilbert scheme FHilb are: FHilb ( C ) = Z ∪ Z ∪ Z ∪ Z ∪ Z where Z , . . . , Z are determined by which eigenvalues x , x , x are equal to each other: Z = { x (cid:54) = x (cid:54) = x (cid:54) = x } , Z = { x = x = x } Z = { x = x (cid:54) = x } , Z = { x = x (cid:54) = x } , Z = { x = x (cid:54) = x } LAG HILBERT SCHEMES, COLORED PROJECTORS AND KHOVANOV-ROZANSKY HOMOLOGY 49 On Z , because the eigenvalues are generically distinct, the commutation relation [ X, Y ] = 0 forces Y = 0 . Then the cyclicity of the vector v implies a, α (cid:54) = 0 , and so conjugation by g allows one to set a = α = 1 and e = 0 . We conclude that:(6.3) Z = C As for Z , note that one can always subtract a constant matrix from X without changing any ofthe other properties of (6.1). By (2.29), we see that:(6.4) Z = FHilb (point) × C = P P (cid:18) O (1) qt ⊕ O ( − (cid:19) × C Torus braids. In this section, we compare our conjectures to the ones of [32, 46] forthree-strand torus braids. The remainder of this Section provides explicit computations thatfollow from Conjectures 1.1 and 3.26. Proposition 6.1. The sheaves on FHilb associated to torus braids on 3 strands are: (6.5) ι ∗ ( σ σ ) k = ι ∗ ( σ σ ) k = L m L m k = 3 m, L m L m ⊗ O Z k = 3 m + 1 , L m +12 L m ⊗ O Z k = 3 m + 2 . Here m (and hence k ) is allowed to be either positive or negative.Proof. Clearly, ( σ σ ) = ( σ σ ) = FT = ι ∗ ( L L ) , so in virtue of the projection formula(3.17) it is sufficient to consider the cases k = 0 , , . For k = 0 , Conjecture 1.1 states that ι ∗ ( ) = O FHilb , which is precisely the content of (6.5). For k = 1 , Conjecture 3.26 implies ι ∗ ( σ σ ) = O Z . Furthermore, for all a, b ∈ N one has:(6.6) Tr( σ σ L a L b ) = Tr( σ σ L a L b ) , since σ commutes with both L and L , and the trace map enjows the property Tr( σσ (cid:48) ) =Tr( σ (cid:48) σ ) . By virtue of the definition (3.22) of the sheaves associated to the braids σ σ and σ σ ,formula (6.6) implies that ι ∗ ( σ σ ) = ι ∗ ( σ σ ) . The case k = 2 of (6.5) follows analogously,because: ( σ σ ) = L σ σ , ( σ σ ) = σ σ L . (cid:3) To compute the Khovanov-Rozansky homology of torus braids, one needs to compute thehomology of the resulting line bundles either on FHilb , or on Z = FHilb (point) × C . Forsimplicity, we will consider only the latter case, which corresponds to knots: Proposition 6.2. The following equations hold: H i (FHilb (point) , L a L b ) = (6.7) = H i ( P , O ( a ) ⊗ S b ( O (2) ⊕ qt O ( − if b ≥ , if b = − ,H i +1 (cid:16) P , O ( a − qt ⊗ S − b − (cid:16) O ( − ⊕ O (1) qt (cid:17)(cid:17) if b ≤ − . Proof. Let π : FHilb (point) → FHilb (point) = P be the natural projection. By (6.4) wehave FHilb (point) = Proj (cid:0) S ∗ P ( O (2) ⊕ qt O ( − (cid:1) . The following properties hold: R i π ∗ (cid:0) L b (cid:1) = S b ( O (2) ⊕ qt O ( − if i = 0 and b ≥ , O ( − qt ⊗ S − b − (cid:16) O ( − ⊕ O (1) qt (cid:17) if i = 1 and b ≤ − , otherwise , , AND JACOB RASMUSSEN Indeed, the second formula follows from the first and Serre duality. This completes the proof. (cid:3) Corollary 6.3. Putting together (6.5) , (6.7) and the well-known formula for the cohomology ofline bundles on P , we have the following formulas for all m ≥ . (6.8) HHH (cid:0) ( σ σ ) m +1 (cid:1) = H ∗ (FHilb (point) , L m L m ) == H (cid:32) P , m (cid:77) i =0 ( qt ) i O (3 m − i ) (cid:33) = m (cid:88) i =0 3 m − i (cid:88) j =0 q i + j t m − i − j (6.9) HHH (cid:0) ( σ σ ) m +2 (cid:1) = H ∗ (FHilb (point) , L m +12 L m ) == H (cid:32) P , m (cid:77) i =0 ( qt ) i O (3 m − i + 1) (cid:33) = m (cid:88) i =0 3 m − i +1 (cid:88) j =0 q i + j t m − i − j +1 This agrees with the a = 0 part of the Khovanov-Rozansky homology of (3 , m + 1) and of (3 , m + 2) torus knots, conjectured in [32, Section 5.2]. To recover the full a dependence, weneed to twist the right hand sides of (6.8) and (6.9) by the exterior power: ∧ • T ∨ = ∧ • ( L “ ⊕ ” L ⊕ O ) ∨ where the symbol “ ⊕ ” refers to the fact that T is a non-trivial extension of L ⊕ O by L . Notethat all of our computations can be easily extended to “twisted torus knots” in the sense of [14],which are presented by the braids ( σ σ ) k ⊗ ι ∗ ( L a ) . We leave the corresponding computationto the interested reader.6.3. The longest word. Let us describe the sheaf for the positive lift σ σ σ of the longestword in S . Remark that the following equation holds for all a and b :(6.10) Tr( σ σ σ L a L b ) = Tr( σ σ σ L a L b ) , since σ commutes both with L and L and the trace satisfies Tr( σσ (cid:48) ) = Tr( σ (cid:48) σ ) . By Corol-lary 3.4, these traces are isomorphic up to a twist by a permutation (1 2) . In particular, theleft hand side of (6.10) is supported on { x = x } , while the right hand side is supported on { x = x } . Furthermore, ι ∗ ( σ σ σ ) = L ⊗ ι ∗ ( σ ) = L ⊗ O FHilb(2 ∼ . Note that in the notations of Section 6, FHilb(2 ∼ 3) = Z ∪ Z . There is a natural involution j on FHilb which exchanges x and x in Z , acts trivially on Z and Z and permutes Z and Z . We arrive at the following conjecture: Conjecture 6.4. One has ι ∗ ( σ σ σ ) = j ∗ ( L ⊗ O Z ∪ Z ) . The figure eight knot. In this section we describe a sheaf for the braid β = σ σ − σ σ − representing the figure eight knot. There is a skein exact sequence relating B β with the follow-ing objects in SBim : σ σ σ σ − = σ σ , σ σ σ − = L σ − . More precisely, there is an exact sequence:(6.11) ←− σ σ ←− Cone (cid:104) L σ − x − x ←−−− L σ − (cid:105) ←− B β ←− . LAG HILBERT SCHEMES, COLORED PROJECTORS AND KHOVANOV-ROZANSKY HOMOLOGY 51 Proposition 6.5. The following identity holds: ι ∗ Cone (cid:104) σ − x − x ←−−− σ − (cid:105) (cid:39) (cid:2) L L − ⊕ qt L − (cid:3) Z . Proof. By (3.49) one has: O FHilb(1 ∼ (cid:39) [ O FHilb ( C ) x − x ←−−− q O FHilb ( C ) y ←− qt L − | FHilb(1 ∼ ] (note that this is also a skein exact sequence for σ − , , σ ) and ι ∗ ( σ − ) = L L − ⊗ O FHilb(2 ∼ . Since O FHilb(2 ∼ ⊗ O FHilb(1 ∼ = O Z , one has an exact sequence: ← L L − | Z ←− ι ∗ Cone (cid:104) σ − x − x ←−−− σ − (cid:105) ←− qt L − | Z ←− . It remains to notice that Ext Z ( L L − , L − ) = H ∗ ( Z , L − ) = H ∗ ( P , O ( − . (cid:3) Proposition 6.6. Consider the braid β = σ σ − σ σ − representing the figure eight knot. Then,assuming Conjecture 1.1 and 3.26, one has ι ∗ ( β ) = O P ⊕ qt L L − . Proof. By (6.11) and Proposition 6.5 one has: ←− O Z α ←− (cid:2) L L − ⊕ qt L L − (cid:3) Z ←− qt ( ι ∗ B β ) ←− . Let us compute the map α . Remark that: Hom Z ( L L − , O ) = H ( Z , L − L ) = H ( P , O (1) ⊕ qt O ( − , Hom Z ( L L − , O ) = H ( Z , L − L ) = H ( P , O ⊕ qt O ( − . Therefore α is the unique degree 1 map L L − → O and vanishes on L L − , so ι ∗ B β (cid:39) L L − ⊕ q − t − Cone[ O α ←− L L − ] (cid:39) O P ⊕ L L − . (cid:3) Using this result, we can compute the reduced homology of β · FT a FT b by computing thehomology of each summand individually. Since FHilb (point) is a blowup of the punctualHilbert scheme of 3 points, and P is the exceptional divisor, the tautological bundle is trivialon P : T ⊗ O P (cid:39) ( q + t ) O P . Similarly, FT ⊗ O P (cid:39) qt O P . We get the following equation:(6.12) (cid:90) FHilb (point) O P ⊗ FT a FT b ⊗ ∧ • T ∨ = (1 + aq − )(1 + at − )( qt ) b (cid:90) P O ( a ) . Equations (6.12) and (6.7) can be used to compute the homology of β · L a L b for all a and b .In particular: H ∗ (FHilb (point) , L L − ) = H ∗ (FHilb (point) , L − ) = 0 ,H ∗ (FHilb (point) , L − ) = H ∗ +1 ( P , O ( − ,H ∗ (FHilb (point) , L L − ) = H ∗ +1 ( P , O ) = C [1] , so (cid:90) FHilb (point) L L − ⊗ ∧ • T ∨ = a C [1] , , , AND JACOB RASMUSSEN and HHH( β ) = (1 + aq − )(1 + at − ) + a √ qt. One can compare this with [21, Table 5.7].7. C ATEGORICAL IDEMPOTENTS AND EQUIVARIANT LOCALIZATION Categories over equivariant schemes. We will now enhance the setup of Section 4 toschemes endowed with a torus action T (cid:121) X . Definition 7.1. A T –equivariant category C is one which the Hom spaces are representationsof T . If the category is monoidal, we require the tensor product to preserve the T action. Definition 7.2. Given a T –equivariant category C , we will say that a map ι : C → X is T –equivariant if the defining functors: C ι ∗ − (cid:0) == (cid:1) − ι ∗ Coh T ( X ) preserve the action of T on all Hom spaces. The derived version is defined analogously. Example . Suppose that X = Spec A with A being a T –graded ring. Recall from Subsection4.4 that realizing C as a category over X amounts to giving a ring homomorphism: A f −→ End C ( ) It is easy to see that C → X is T –equivariant if and only if f is T –equivariant. Example . Going one step further, suppose A is a T –graded ring. Define: X = P nA where the n + 1 coordinate directions of the projective spaces have T –equivariant characters λ , ..., λ n . As in Example 4.13, the map C ι −→ X is the same datum as a ring homomorphism: A f −→ End C ( ) together with an object F ∈ K b ( C ) and n + 1 arrows: (cid:104) λ · α −→ F (cid:105) , ..., (cid:104) λ n · α n −→ F (cid:105) whose tensor product is homotopic to 0. Then ι is T –equivariant if the homomorphism f is T –equivariant, and moreover the arrows α i , i ∈ { , ..., n } are all homogeneous with respect tothe structure of T –modules of the vector spaces Hom K b ( C ) ( λ i · , F ) . Example . Finally, let us treat the relative case of Subsection 4.6. Suppose we have a T –equivariant map: C ι −→ X and we wish to upgrade it to a T –equivariant map: C ι (cid:48) −→ P V ∨ where V is a T –equivariant vector bundle on X . As we saw in Subsection 4.6, the existence ofthe map ι (cid:48) is equivalent to the choice of an object F ∈ C together with an arrow: ι ∗ V α −→ F in C , whose Koszul complex is quasi-isomorphic to 0. It is easy to see that the map ι (cid:48) is T –equivariant if and only if the map α is T –equivariant. The same picture applies when V isreplaced by a coherent sheaf Q of homological dimension 1, as in Subsection 4.7. LAG HILBERT SCHEMES, COLORED PROJECTORS AND KHOVANOV-ROZANSKY HOMOLOGY 53 Categorical diagonalization. In [23], Elias and Hogancamp developed a theory of cat-egorical diagonalization, which we will now recall. Assume we are given an equivariantmonoidal category T (cid:121) C , which can be taken to be triangulated or dg. Definition 7.6. ([23]) Fix an object F ∈ K b ( C ) . An arrow:(7.1) λ · α −→ F is called an eigenmap of F , and the grading shift λ ∈ T ∨ is called an eigenvalue of F . Definition 7.7. ([23]) An object F ∈ K b ( C ) is called diagonalizable if it has a collection ofeigenvalues λ , ..., λ n ∈ T ∨ and eigenmaps: (cid:110) λ i · α i → F (cid:111) i ∈{ ,...,n } such that ⊗ ni =0 Cone( α i ) (cid:39) .The intuition behind the above terminology comes about by considering the Grothendieckgroup [ C ] , which is an algebra because the category C is monoidal. Multiplication by the classof the object [ F ] induces an operator on [ C ] , and the datum of Definition 7.7 amounts to:(7.2) n (cid:89) i =0 ([ F ] − λ i ) = 0 In other words, the condition that the product of the cones of the eigenmaps is 0 amounts torequiring the operator ∗ (cid:32) ∗ · [ F ] to solve its characteristic polynomial. In Lemma 7.8, weestablish the fact that categorical diagonalization is universally represented by the category: D = D b (Coh T ( P nA )) where A is any commutative ring and T (cid:121) P nA acts via:(7.3) t · [ z : ... : z n ] (cid:55)→ (cid:20) z λ ( t ) : ... : z n λ n ( t ) (cid:21) where λ , ..., λ n ∈ T ∨ . An immediate generalization of Example 4.13 yields the following: Lemma 7.8. The datum of a diagonalizable object F ∈ C as in Definition 7.7 is equivalent tothe existence of a T –equivariant map: ι : C → P nA such that F = ι ∗ ( O (1)) , where A = End C ( ) . Eigenobjects. In Definition 7.6 we have recalled the categorical version of eigenvalues.In [23], the authors complete the picture by categorifying eigenvectors: Definition 7.9. If for some P ∈ C the arrow:(7.4) α ⊗ Id P : λ · P ∼ = −→ F ⊗ P is an isomorphism, then we call P an eigenobject for the datum of Definition 7.6.In the decategorified world, the eigenvectors of the operator of multiplication by [ F ] of (7.2)can be computed explicitly, essentially by the Lagrange interpolation formula:(7.5) [ P i ] := (cid:89) ≤ j (cid:54) = i ≤ n λ j − [ F ] λ j − λ i , , AND JACOB RASMUSSEN The reason why we divide by λ j − λ i is to ensure that the elements [ P i ] are idempotents.However, this comes at the cost of enlarging the algebra to account for such denominators.One of the main constructions in [23] is the categorify formula (7.5) in a way which keepstrack of the eigenmaps.The main difficulty, which we will shortly address, is how to lift the denominators of (7.5)from the Grothendieck group to the category C . The idea spelled out in [23] is that in (7.5) oneshould expand: λ j − [ F ] λ j − λ i = (cid:18) − [ F ] λ j (cid:19) (cid:18) λ i λ j + λ i λ j + ... (cid:19) if j < i and: λ j − [ F ] λ j − λ i = (cid:18) [ F ] λ i − λ j λ i (cid:19) (cid:18) λ j λ i + λ j λ i + ... (cid:19) if j > i . To understand the above as an expansion of geometric series, we assume that thereexists a distinguished subtorus C ∗ ⊂ T which we will be called homological , such that:(7.6) λ | C ∗ > ... > λ n | C ∗ To categorify these geometric series, [23] replace the category C by its homological comple-tion C ↑ , as in Section 4.2. Theorem 7.10. ( [23] ) Let F be a diagonalizable object, with eigenmaps α i and eigenvalues λ i satisfying (7.6) . Then there exist a collection of eigenobjects P i as in (7.4) , explicitly given by: (7.7) The objects should be added ⊕ along columns, with differentials according to the arrows. Thecollection { P , ..., P n } yields a semi-orthogonal decomposition of C : (7.8) ∼ = (cid:104) P ⊕ ... ⊕ P n , a certain differential (cid:105) and Hom C ( P i , P j ) = 0 if i > j . Furthermore, P i ⊗ P j (cid:39) for i (cid:54) = j and P i ⊗ P i (cid:39) P i . The main application of [23] is when C = K b (SBim n ) is replaced by C = K − (SBim n ) , andthe homological C ∗ action is by homological degree of chain complexes. We may generalizethis particular case to the following setup.7.4. The geometric realization over a fixed base. As we saw in Lemma 7.8, any categoricaldiagonalization in a category C comes from a T –equivariant map: C → P nA i.e. K b ( C ) ι ∗ − (cid:0) == (cid:1) − ι ∗ D LAG HILBERT SCHEMES, COLORED PROJECTORS AND KHOVANOV-ROZANSKY HOMOLOGY 55 where D = D b (Coh T ( P nA )) , and the action T (cid:121) P nA is given in (7.3). The above functorsextend to functors on the homological completions: K ( C ↑ ) ι ∗ − (cid:0) == (cid:1) − ι ∗ D ↑ which are given by the same formulas, but allow infinite direct sums of objects in decreasinghomological degree. Therefore, we have: P i = ι ∗ ( P i ) where P i ∈ D ↑ are given by formula (7.7) with F replaced by O (1) and α i replaced by multi-plication with the homogeneous coordinate z i :(7.9)The rows in the above diagram make up for the expansion of the geometric series ( λ j − λ i ) − .Meanwhile, observe that the top row is precisely;(7.10) top row of P i = (cid:79) ji (cid:104) λ j λ − i z j −→ O (1) λ − i (cid:105) q . i . s . ∼ = O p i (cid:89) j
Consider a locally closed subset S i = { z = . . . = z i − = 0 , z i (cid:54) = 0 } ⊂ P nA .Then P i is quasi-isomorphic to the pushforward of S • ( ν ∨ S i ) , where ν S i is the normal bundle to S i .Remark . The ordering of coordinates in the definition of S i agrees with the ordering ofeigenvalues of O (1) (that is, the weights of the torus action) on P n . It is easy to see that thestrata S i agree with the cells in the Białynicki-Birula decomposition [11, 12] of P n with respectto this torus action. Similar decompositions of equivariant derived categories with respect toBiałynicki-Birula strata were studied in [35], and we plan to study the relation between thecategorical diagonalization framework and [35] in the future work. Proof. To simplify the notations, we will consider the case n = 1 and omit all the grading shifts(which can be easily reconstructed since all maps are homogeneous). The construction (7.9)yields two different infinite complexes built from the sections z , z : O → O (1) . The first hasa form:Here y is a formal variable corresponding to the shift of the complex down by one unit. It canbe made less formal by considering the projection π : P n × A → P n , so that P = p ∗ (cid:104) O z + yz −−−−→ O (1) (cid:105) = p ∗ O { z + yz =0 } . The projection p identifies the closed subset { z + yz = 0 } ⊂ P n × A with the open subset S = { z (cid:54) = 0 } ⊂ P n , so P = O S . The second complex is more interesting. It has the form:It is supported on S = P \ S = { z = 0 } where the stalk of O is isomorphic to C [ z z ] andthe stalk of O S is isomorphic to C [ z z , z z ] , so the quotient is isomorphic to z z · C [ z z ] . (cid:3) LAG HILBERT SCHEMES, COLORED PROJECTORS AND KHOVANOV-ROZANSKY HOMOLOGY 57 Remark . Note thatOne can use similar arguments to formally match this complex with O { z (cid:54) =0 } ⊗ O ( − 1) = O { z (cid:54) =0 } . However, P ∨ does not belong to the category D ↑ since the gradings of its summandsare unbounded. Corollary 7.14. The endomorphism ring of P i is isomorphic to the local ring of P nA at a fixedpoint p i .Proof. We follow the proof of Proposition 7.11. Indeed, End( P ) = H ( S , O S ) = C [ z z ] . Onthe other hand, End( P ) = End (cid:20) C (cid:20) z z (cid:21) → C (cid:20) z z , z z (cid:21)(cid:21) = C (cid:20) z z (cid:21) . One could also argue that End( P ) = End( P ∨ ) = End( O { z (cid:54) =0 } ) = C (cid:20) z z (cid:21) . The proof for general n is analogous. (cid:3) Remark . Proposition 7.11 shows that the endomorphism rings of the projectors can beinterpreted as the rings of functions on certain open charts. This point of view will be importantin the next section where we define some open charts on the flag Hilbert scheme and computethe rings of functions on them (up to a certain completion). By Conjecture 1.1 and the precedingdiscussion these rings match the homology of the categorified Jones-Wenzl projectors. Remark . The equivariant localization formula makes sense when D = D b (Coh T ( X )) forany local complete intersection X acted on by a torus T :(7.12) [ O X ] = (cid:88) p ∈ X T [ O p ] ∧ • (cid:0) Tan ∨ p X (cid:1) As we have seen, when X = P n the above setup encodes categorical diagonalization as inDefinition 7.7 and 7.9. It would be very interesting to determine which problems in “categoricallinear algebra” are encoded by formula (7.12) for more general schemes X .7.5. The relative case. For the remainder of this Section, we will generalize the objects (7.9)from P nA to projective bundles P V ∨ on an arbitrary base scheme X , as in Example 7.5. Weassume that both X and V are acted on by a torus T , and that we have a decomposition:(7.13) O X ∼ = (cid:34) (cid:77) x ∈ X T P x , a certain differential (cid:35) ∈ D b (Coh T ( X )) , , AND JACOB RASMUSSEN where the indexing set goes over the fixed points of X . We assume that the above is semi-orthogonal, in the sense that Hom( P x , P y ) = 0 whenever x > y with respect to some totalorder. We wish to upgrade the decomposition (7.13) to the projective bundle P V ∨ . Proposition 7.17. Let n +1 = rank V . There exist objects P xi for all i ∈ { , ..., n } and x ∈ X T ,such that we have a semi-orthogonal decomposition: (7.14) O P V ∨ ∼ = (cid:34) ≤ i ≤ n (cid:77) x ∈ X T P ix , a certain differential (cid:35) ∈ D b (Coh T ( P V ∨ )) whenever the homological subtorus C ∗ ⊂ T acts with distinct weights in the fibers V| x for all x ∈ X T . We have Hom( P ix , P jy ) = 0 if x > y or if x = y and i < j . The object P ix is precisely of the form (7.9) if one replaces O with P x , and λ , ..., λ n areprecisely the weights of the torus T in the fiber V| x .8. L OCAL CHARTS AND FIXED POINTS OF FHilb n Affine charts for Hilbert schemes. Recall the action of C ∗ × C ∗ on Hilbert schemesgiven by rescaling the X and Y matrices. The fixed points of this action on the Hilbert schemeare well-known. They are given by monomial ideals, which are indexed by partitions of n : Hilb C ∗ × C ∗ n = { I λ } λ (cid:96) n , I λ = ( x λ , x λ y, ... ) ⊂ C [ x, y ] Haiman described a set of affine charts on the Hilbert scheme, each of which is C ∗ × C ∗ invariantand contains a single fixed point:(8.1) Hilb n = (cid:91) λ (cid:96) n ˚Hilb λ where:(8.2) ˚Hilb λ = (cid:110) I such that { x a y b } ( a,b ) ∈ λ is a basis of C [ x, y ] /I (cid:111) Here and throughout this paper, we identify a partition with its Young diagram, which is the setof × boxes in the first quadrant of the plane with coordinates ( a, b ) ∈ N × N , b < λ a :For example, the above Young diagram corresponds to the partition λ = (4 , , . It would bevery nice to have a clear description of the algebra of functions on each affine chart (8.1), butthis is not at all easy. Instead, Haiman’s construction gives us a set of generators: { f , ..., f n } ∈ m λ / m λ where m λ ∈ C [ ˚Hilb λ ] denotes the maximal ideal of the fixed point I λ . LAG HILBERT SCHEMES, COLORED PROJECTORS AND KHOVANOV-ROZANSKY HOMOLOGY 59 Affine charts for flag Hilbert schemes. The situation is somewhat better in the case offlag Hilbert schemes FHilb n ( ∗ ) for any ∗ ∈ { C , C , point } , where one has affine coverings:(8.3) FHilb n ( ∗ ) = (cid:91) T (cid:96) n ˚FHilb T ( ∗ ) indexed by standard Young tableaux T of size n . Recall that a standard Young tableau is anumbering of the boxes of a Young diagram of size n with the numbers , ..., n such that thenumbers increase as we go up and right in the diagram. A covering (8.3) is called good if allthe charts are C ∗ × C ∗ equivariant and it respects passage from n + 1 to n :where the chart corresponding to any T (cid:48) maps to the chart corresponding to T = T (cid:48) \ (cid:3) n +1 .Here, (cid:3) n +1 denotes the box labeled n + 1 in T , which must necessarily be an outer corner of T and an inner corner of T (cid:48) . Restricting the sheaf of dg algebras FHilb dg n ( ∗ ) to the affine charts(8.3) gives rise to dg algebras:(8.4) ˚ A dg T ( ∗ ) = C (cid:104) ˚FHilb dg T ( ∗ ) (cid:105) Conjecture 8.1. There exists a good covering whose coordinate rings (8.4) satisfy: (8.5) ˚ A T ∪ (cid:3) ( ∗ ) = ˚ A T ( ∗ )[ ∗ , f (cid:3) , f (cid:3) , ... ]( r (cid:4) , r (cid:4) , ... ) where (cid:3) , (cid:3) , ... denote the inner corners of T different from (cid:3) , and (cid:4) , (cid:4) , ... denote the outercorners of T (except for the outer corner labeled n in the case ∗ = point ). The generatorsdenoted by ∗ stand for the affine coordinates { x n +1 , y n +1 } , { x n +1 } , ∅ when ∗ = C , C , point . We do not know how to define the generators f k and the relations r k , but we know how topredict their characters with respect to the C ∗ × C ∗ action. Specifically, for a box (cid:3) = ( a, b ) ina Young diagram, we define its weight as:(8.6) z (cid:3) = q a t b When (cid:4) is the box labeled by i in a Young tableau T , we will write z (cid:4) = z i for brevity. Thenwe expect that the generators and relations of (8.5) have equivariant weights(8.7) weight f (cid:4) = z (cid:4) z (cid:3) , weight r (cid:4) = z (cid:4) z (cid:3) where (cid:3) is the corner that is being added in (8.5). In the remainder of this Section, we willestablish a weaker version of Conjecture 8.1, by constructing affine C ∗ × C ∗ invariant open setsthat contain the fixed points of FHilb n ( ∗ ) , but are not required to cover it.8.3. Defining the charts. In this Section, we will define affine charts on the flag Hilbertscheme which only satisfy Conjecture 8.1 on the local rings around the fixed points. FHilb n will henceforth refer to either of FHilb n ( ∗ ) for ∗ ∈ { C , C , point } . , , AND JACOB RASMUSSEN Definition 8.2. For any point ( X, Y, v ) ∈ FHilb n and standard Young tableau T , consider thefollowing algorithm to construct a basis e = v, e , ..., e n of C n . Suppose e , ..., e k − have beenconstructed and the k -th box looks as in the following picture: i i (cid:48) k Define the vector e k ∈ Ker ( C n (cid:16) C k − ) by the following formula if i > i (cid:48) :(8.8) Xe i = e k + k − (cid:88) j = i x ji e j where x ji are coefficients, and by the following formula if i < i (cid:48) :(8.9) Y e i (cid:48) = e k + k − (cid:88) j = i (cid:48) y ji (cid:48) e j where y ji (cid:48) are coefficients. If the process terminates after having constructed e n , in a way suchthat e , ..., e k form a basis of the quotient C n (cid:16) C k for all k , then we set: ( X, Y, v ) ∈ FHilb T In either (8.8) or (8.9), it is clear that the vector e k is unique, since the coefficients x ji or y ji are uniquely determined by the fact that e k vanishes in the quotient C n (cid:16) C k − . The fact thatsuch an e k exists at each step, and that the resulting collection of vectors forms a basis, is anopen condition and therefore: FHilb T ⊂ FHilb n thus defined is an open subscheme. It is also an affine subscheme, simply because the basis e , ..., e n is unique. We could therefore define FHilb T alternatively as the affine space of matri-ces X, Y of the form prescribed by (8.8) and (8.9) in a fixed basis. It is also clear that the locus FHilb T is C ∗ × C ∗ invariant and that the only fixed point it contains is: I T = (cid:40) C n = n (cid:77) i =1 C · e i with X · e i = e i → , Y · e i = e i ↑ , v = e (cid:41) In the above formula, for any box i ∈ T we write i → and i ↑ for the boxes immediately rightand above (cid:3) , respectively. If there is no box to the right or up of (cid:3) , we naturally set e i → or e i ↑ equal to 0. The fact that the open sets of Definition 8.2 cover the whole of FHilb n follows fromthe following principle:(8.10) any open torus invariant property which holdsnear the fixed points of FHilb n holds everywhere This is because the set of points which do not enjoy said property is closed, torus invariant andcontains no fixed points: any such set must be empty. One must be careful here, because theargument is a priori only true for projective varieties, such as FHilb n (point) . However, it alsoapplies to FHilb n ( C ) and FHilb n ( C ) because the torus C ∗ × C ∗ contracts the affine directions C and C to the origin. LAG HILBERT SCHEMES, COLORED PROJECTORS AND KHOVANOV-ROZANSKY HOMOLOGY 61 The special coefficients. Note that the coefficients x ii and y ii in (8.8) and (8.9) are pre-cisely the eigenvalues of the matrices ( X, Y, v ) ∈ FHilb n . If we are in the case ∗ = C or ∗ = point , then we must set y ii = 0 or x ii = y ii = 0 in (8.8) and (8.9), respectively. Definition 8.3. The coefficients x ji and y ji which appear in (8.8) and (8.9) will be called specialcoefficients . We also apply this terminology to the case when k is an outer corner of the Youngdiagram of T , but in that case (8.8) and (8.9) hold with e k = 0 .Note that the number of special coefficients corresponding to a standard Young tableau T is:(8.11) n (cid:88) i =1 of inner corners of the Young diagram consisting of the boxes labeled , ..., i ) Conjecture 8.1 would suggest that the special coefficients generate the dg ring of functions ˚ A T subject to a number of:(8.12) n (cid:88) i =1 of outer corners of the Young diagram consisting of the boxes labeled , ..., i ) However, this is not true, because this would entail that all coefficients x ji and y ji could be writ-ten as polynomials in the special coefficients. We partially salvage this in the next Subsection,when we will show that the previous sentence holds if we replace the word “polynomials” by“rational functions”. In other words, some open subset of FHilb T can be described by (8.11)generators and (8.12) relations. Example . When T = ( n ) and ∗ = C , only relations (8.8) come into play: Xe i = e i +1 + x i e i unless i = 1 , in which case we have: Y e = n (cid:88) j =2 y j e j Therefore, the special coefficients are { x i , y j } ≤ i ≤ n ≤ j ≤ n . The number of these coefficients is n − ,and it matches (8.11) minus 1, where the minus one stems from the fact that y = 0 for ∗ = C .The non-special coefficients are the y ji with i > , but they can be inferred from the specialones via the commutation relation [ X, Y ] = 0 , which in the case at hand reads:(8.13) y ji ( x i − x j ) = y ji +1 − y j − i for all i < j . Note that (8.13) is precisely (1.28). We make the convention that y ji = 0 for j ≤ i . After solving for y ji in terms of { x i , y j } , we obtain the inductive formulas for any δ > : y i + δi = y δ +11 + i − (cid:88) s =1 y i − s + δ +1 i − s ( x i − s − x i − s + δ +1 ) The above relation also holds when i + δ = n + 1 , in which case the left hand side is 0. Wetherefore obtain a relation among the special coefficients { x i , y j } for all δ > . There are n − such relations, and their number matches (8.12) minus 1, where the minus one stems from thefact that y = 0 for ∗ = C . , , AND JACOB RASMUSSEN Example . When T = (1 , ..., and ∗ = C , only relations (8.9) come into play: Y e i = e i +1 unless i = 1 , in which case we have: Xe = n (cid:88) j =1 x j e j Therefore, the special coefficients are x j for all j . Note that the commutation relation [ X, Y ] =0 implies that: x j − i = x ji +1 ∀ i < j and therefore we conclude that x ji = u j − i +1 for some variables u , ..., u n . Compare with (1.27).8.5. Explicit local coordinates. In this section, we will use the special coefficients to describethe neighborhood of the fixed point I T for any standard Young tableau T :(8.14) ˚FHilb T := (FHilb n ) localized at T and the dg local ring ˚ A dg T = C [ ˚FHilb dg T ] . In fact, we will actually describe an open subschemeof FHilb T given by the non-vanishing of certain torus invariant functions. The resulting opensubschemes also form a cover of FHilb n because of the principle (8.10), so we abuse notationand use (8.14) both for the local neighborhood and for the open subscheme ˚FHilb T ⊂ FHilb T . Proposition 8.6. For any standard Young tableau T (cid:96) n , the complex E n of (2.25) is: (8.15) E n | ˚FHilb T q . i . s . ∼ = (cid:34) (cid:4) outer (cid:77) corner of T O · e (cid:4) ψ −→ (cid:3) inner (cid:77) corner of T O · f (cid:3) (cid:35) Theorem 2.6 describes the map π : FHilb n +1 → FHilb n as the projectivization of H ( E n ) .Locally, this map takes the form: π − (cid:16) ˚FHilb T (cid:17) = (cid:3) inner (cid:91) corner of T ˚FHilb T ∪ (cid:3) where ˚FHilb T ∪ (cid:3) ⊂ P H (cid:0) E ∨ n | ˚FHilb T (cid:1) is the affine chart of (8.15) given by f (cid:3) = 1 . We conclude (8.5) , where the generators are f (cid:3) (cid:48) for inner corners (cid:3) (cid:48) (cid:54) = (cid:3) and the relations are r (cid:4) = ψ ( e (cid:4) ) .Proof. From each box in T , draw two lines of unit length, one going up and one to the right: 134 25 68 7 The lines are of two types: thick or dotted, and black or red. The color of a line is determinedby whether the line points to a box in T or outside of T . The shape of a line is determined LAG HILBERT SCHEMES, COLORED PROJECTORS AND KHOVANOV-ROZANSKY HOMOLOGY 63 by the following rule: If i > i (cid:48) where i (cid:48) is the label of the box to the southeast (respectivelynorthwest) of i , then we make the horizontal (respectively vertical) line starting at i thick. Allthe boxes below and to the left of the diagram are thought to have label for the purpose ofthis rule, and all the boxes above and to the right of the diagram are thought to have label ∞ .By definition:(8.16) E n = (cid:104) qt T n Ψ −→ q T n ⊕ t T n ⊕ O Φ −→ T n (cid:105) When we restrict the complex to the affine chart ˚FHilb T , we observe that the tautologicalbundles are already trivialized by the basis e , ..., e n of Definition 8.2: T n | ˚FHilb T = O · e ⊕ ... ⊕ O · e n Therefore, the middle term of (8.16) has a basis which we will denote by e , ..., e n , e (cid:48) , ..., e (cid:48) n , .We claim that the projection that forgets some of these basis vectors induces an isomorphism:(8.17) Ker Φ | ˚FHilb T ∼ = red or dotted horizontal (cid:77) lines from box i O · e i red or dotted vertical (cid:77) lines from box i O · e (cid:48) i In other words, we claim that if one specifies rescaled basis vectors c i e i and d i e (cid:48) i correspond-ing to the red and dotted edges, then there exist unique rescaled basis vectors γ i e i and δ i e (cid:48) i corresponding to the black thick edges, and a function f , such that: ( X − x n +1 ) (cid:32) (cid:88) i dotted c i e i + (cid:88) i thick γ i e i (cid:33) + ( Y − y n +1 ) (cid:32) (cid:88) i dotted d i e i + (cid:88) i thick δ i e i (cid:33) + f e = 0 Any box k has a unique black thick line going to the left or down. Assume without loss ofgenerality that the black thick line from k leads one step left to the box i . Then (8.8) impliesthat equating the coefficient of e k in the left hand side to 0 yields the equation: γ i ∈ (cid:88) j ( c j or d j ) · coefficients + (cid:88) j ( γ j or δ j ) · ˚ m T This system of equations can be solved in the localization ˚ A T , since its determinant is in m T .Therefore, we conclude that in the local chart ˚FHilb T , we have:(8.18) E n | ˚FHilb T q . i . s . ∼ = (cid:34) n (cid:77) i =1 O · e i Ψ −→ red or dotted horizontal (cid:77) lines from box i O · e i red or dotted vertical (cid:77) lines from box i O · e (cid:48) i (cid:35) The Proposition will be proved once we show that projecting the two terms in the above com-plex to a certain subset of factors induces a quasi-isomorphism. Specifically, in the domain of Ψ we consider the subspace spanned by basis vectors e (cid:4) corresponding to outer corners (cid:4) , andin the codomain of Ψ we project onto one basis vector f (cid:3) corresponding to each inner corner.The rule is that f (cid:3) = e i or e (cid:48) i (cid:48) , depending on whether the number i to the left of (cid:3) is bigger orsmaller than the number i (cid:48) below (cid:3) . In other words, the only e i or e (cid:48) i we will consider in thecodomain of Ψ are the ones corresponding to thick red lines:(8.19) E n | ˚FHilb T q . i . s . ∼ = (cid:34) (cid:4) outer (cid:77) corner of T O · e (cid:4) ψ −→ (cid:3) inner (cid:77) corner of T O · ( e (cid:3) ← or e (cid:48) (cid:3) ↓ ) (cid:35) , , AND JACOB RASMUSSEN In plain English, the claim is that for any e i where i is not an outer corner of T , quotienting thecodomain of (8.18) by the vector:(8.20) red or dotted horizontal (cid:88) lines between boxes ¯ jk ( x ji − δ ij x n +1 ) e j + red or dotted vertical (cid:88) lines between boxes ¯ jk ( y ji − δ ij y n +1 ) e (cid:48) j will allow us to solve for one of the e j , e (cid:48) j . The only basis vectors which remain unsolved forshould be the ones that appear in the codomain of (8.19). For example, suppose we are tryingto solve for e j , where j corresponds to a dotted horizontal edge ¯ jk . Then with the notation inthe following picture: ij j (cid:48) k let us observe that (8.8)–(8.9) imply that y ji = x j (cid:48) i = y kj (cid:48) ∈ m T . Then the vector (8.20) liesin e j + ˚ m T , and e j can therefore be solved for in the localization ˚ A T . (cid:3) Examples. In this subsection, we use the local geometry of the flag Hilbert scheme todescribe the homology of categorified projectors on two and three strands. Example . For the S projector, we have Y = (cid:18) (cid:19) . The commutation relation implies that X is of the form X = (cid:18) u u u (cid:19) . One has deg( u ) = q , deg( u ) = q/t , so the Poincar´e series equals P ( ˚ A T ) = 1(1 − q )(1 − q/t ) . Example . For the Λ projector, we have X = (cid:18) x x (cid:19) , Y = (cid:18) y (cid:19) , and the commutation relation ( x − x ) y = 0 . One has deg( x ) = deg( x ) = q , deg( y ) = t/q , so the Poincar´e series equals P ( ˚ A T ) = 1 − t (1 − q ) (1 − t/q ) = 1(1 − q ) + t/q (1 − q )(1 − t/q ) . Example . For the S projector, we have X = x x x x x x , Y = , LAG HILBERT SCHEMES, COLORED PROJECTORS AND KHOVANOV-ROZANSKY HOMOLOGY 65 One has deg( x ) = q , deg( x ) = q/t , deg x = q/t , so the Poincar´e series equals P ( ˚ A T ) = 1(1 − q )(1 − q/t )(1 − q/t ) . More generally, for the S n projector, we have P ( ˚ A T ) = n (cid:89) i =1 (1 − qt − i ) − . Example . For the Λ projector, we have X = x x 00 1 x , Y = y y y , and the commutation relations ( x − x ) y = ( x − x ) y = 0 and y − y = ( x − x ) y . Note that one can eliminate y using the last equation. One has deg( x ) = deg( x ) =deg( x ) = q , deg( y ) = deg( y ) = t/q , deg( y ) = t/q , so the Poincar´e series equals P ( ˚ A T ) = (1 − t ) (1 − q ) (1 − q/t )(1 − q/t ) . More generally, for the Λ n projector, we have ˚ A T = C [ x , . . . , x n , y i,j ] i>j y i,j ( x i − x j ) − ( y i − ,j − y i,j +1 ) and P ( ˚ A T ) = (1 − t ) n − (1 − q ) n n − (cid:89) i =1 (1 − qt − i ) − . (which can also be seen directly from Proposition 8.12.) Example . For the hook-shaped projector with ( z , z , z ) = (1 , t, q ) , we have X = x x x x x , Y = y , with commutation relations x = x y , ( x − x ) y = 0 , In this case deg( x ) = deg( x ) = q , deg( x ) = q/t , deg( x ) = t , deg( y ) = t /q , so thePoincar´e series equals P ( ˚ A T ) = (1 − t )(1 − q ) (1 − q/t )(1 − t /q ) . For the other hook-shaped projector we have ( z , z , z ) = (1 , q, t ) , so X = x x x x , Y = y y , with commutation relations ( x − x ) y = ( x − x ) y = 0 and x − x + y = x y . , , AND JACOB RASMUSSEN In this case deg( x ) = deg( x ) = deg( x ) = q , deg( x ) = q /t , deg( y ) = t/q , deg( y ) = q , so the Poincar´e series equals P ( ˚ A T ) = (1 − t )(1 − q )(1 − q ) (1 − t/q )(1 − q /t ) . Poincare series. In general, Proposition 8.6 can be used to show Proposition 8.12. For any standard Young tableau of size n , the bigraded Poincar´e series ofgraded algebras ˚ A T ( ∗ ) are given by the following formulas: (8.21) P ( ˚ A T ( C )) = (1 − q ) − n (1 − t ) − n n (cid:89) i =1 − z − i (cid:89) ≤ i If we pass to the decategorified setting by substituting t = q − , we see that the Poincar´eseries depends only on the Young diagram of T : Corollary 8.13. P ( ˚ A T ( C )) | t = q − = 1 (cid:81) (cid:3) ∈ λ (1 − q h ( (cid:3) ) ) . Proof. If we let ζ q,q − ( x ) = ζ ( x ) | t = q − , then clearly ζ q,q − ( x ) = ζ q,q − ( x − ) . It follows that thefunction (cid:81) i 9. D IFFERENTIALS AND gl N HOMOLOGY Spectral sequence for gl N homology. By [49], for each N there exists a spectral se-quence starting at the HOMFLY-PT homology and converging to sl N homology of a givenknot. More precisely, for a given braid σ one can construct a complex of Soergel bimodulesas described in Subsection 3.4. The Hochschild homology of this complex coincides with theHOMFLY-PT homology of the closure of σ . Given a polynomial p ∈ C [ x ] , we can constructan additional differential d − which acts on Soergel bimodules, as we now describe. , , AND JACOB RASMUSSEN Recall that the simple Soergel bimodule can be written as B i = R ⊗ R i,i +1 R . Denote u j = x j ⊗ , v j = 1 ⊗ x j for all j , and U i,i +1 := C [ u , . . . , u n , v , . . . , v n ]( u i + u i +1 − v i − v i +1 , u j − v j , j / ∈ { i, i + 1 } ) , then B i ∼ = (cid:104) U i,i +1 ( v i − u i )( v i − u i +1 ) −−−−−−−−−−→ U i,i +1 (cid:105) . Given a polynomial p ∈ C [ x ] , consider the difference W i,i +1 := p ( u i )+ p ( u i +1 ) − p ( v i ) − p ( v i +1 ) = p ( u i )+ p ( u i +1 ) − p ( v i ) − p ( u i + u i +1 − v i ) ∈ U i,i +1 . Remark that W i,i +1 is divisible by ( v i − u i )( v i − u i +1 ) : indeed, W i,i +1 vanishes if v i = u i or v i = u i +1 . Let p i,i +1 = W i,i +1 / ( v i − u i )( v i − u i +1 ) . We use p i,i +1 to define an additionaldifferential (denoted by d − in [49]) which acts backwards :(9.1) B ( p ) i := (cid:20) U i,i +1 ( v i − u i )( v i − u i +1 ) − (cid:0) =========== (cid:1) − d − := p i,i +1 U i,i +1 (cid:21) . Note that the total complex ( B ( p ) i , d + + d − ) is not a chain complex but a matrix factorizationwith potential W i,i +1 .It is proved in [49] that this additional differential d − can be naturally extended to Bott-Samuelson bimodules (tensor products of B i ), and to Rouquier complexes. One can also prove[7] that d − can be correctly defined on general Soergel bimodules as well. For p (cid:48) ( x ) = x N , thisdifferential is usually denoted by d N , and the homology of the total differential is isomorphicto gl N Khovanov-Rozansky homology [40]. The desired spectral sequence is then induced by d N on HHH( σ ) .In the present section, we wish to present a more geometric viewpoint of this construction.Given N , we define the so-called sl N dg category (SBim n , d N ) , where the objects are Soergelbimodules equipped with the “internal differential” d N . This is a subcategory of the categoryof matrix factorizations with potential x N . There is a monoidal functor: K b (SBim n ) → (cid:0) K b (SBim n ) , d N (cid:1) which is given by endowing complexes of Soergel bimodules with the differential d N .9.2. Sections and schemes. On the geometric side, we have a remarkable family of dg schemesclosely related to FHilb dg n = FHilb dg n ( C ) . Namely, let s be an arbitrary section of the tautolog-ical bundle T n . It defines a contraction map:(9.2) d s : ∧ • T ∨ n → ∧ •− T ∨ n Recall the construction (1.13): (cid:101) ι ∗ ( σ ) = ι ∗ ( σ ) ⊗ ∧ • T ∨ n which is naturally a sheaf of dg modules on Tot FHilb dg n T n [1] . If we endow the exterior powerwith the differential (9.2), we obtain: ( (cid:101) ι ∗ ( σ ) , d s ) which is naturally a sheaf of dg modules on the dg scheme: Tot FHilb dg n ( T n [1] , s ) := the sheaf of dg algebras ( ∧ • T ∨ n , d s ) on FHilb dg n . To construct sections s of the tautological bundle T n , recall that its fibers are given by: T n | I n ⊂ ... ⊂ C [ x,y ] = C [ x, y ] /I n . LAG HILBERT SCHEMES, COLORED PROJECTORS AND KHOVANOV-ROZANSKY HOMOLOGY 69 Therefore every polynomial f ∈ C [ x, y ] defines a section s f ∈ Γ(FHilb n , T n ) for all n , andthese sections are all compatible with each other:The morphism FHilb n π −→ FHilb n − × C therefore induces a map: Tot FHilb dg n ( T n [1] , s f ) π f −→ Tot FHilb dg n − × C ( T n − [1] , s f ) and so one has a commutative diagram of maps of dg schemes:where the vertical maps are simply induced by the map of dg algebras ∧ • T ∨ n → ( ∧ • T ∨ n , d s ) .Note that the dg scheme Tot FHilb dg n ( T n [1] , s f ) is C ∗ × C ∗ equivariant if and only if f is anequivariant section of T n . It is not hard to see that the only such equivariant sections are f ( x, y ) = x N y M for some ( N, M ) ∈ N × N . We denote the corresponding section by s N | M . Remark . In [32, Section 7], the differentials were parametrized by copies of the definingrepresentation of S n in the rational Cherednik algebra, which can be considered as a non-commutative deformation of C [ x , . . . , x n , y , . . . , y n ] . One can check that such a copy nat-urally corresponds to a section of T n , in particular, f ∈ C [ x, y ] corresponds to the subspace Span( f ( x i , y i )) ≤ i ≤ n .9.3. The commutative tower. We conjecture that the differential d N in the Soergel categoryis closely related to the section f = x N of the tautological bundle on the flag Hilbert scheme.More precisely, we propose the following: Conjecture 9.2. There is a map ι N : (SBim n , d N ) → ( Z n ( C ) , s N ) in the sense of Definition4.6. The corresponding functors fit into the commutative diagram: (9.3) Furthermore, there is a tower of commuting squares connected with π N , Tr , I akin to (1.23) .Remark . We expect that the general differential on SBim n corresponding to the polynomial p ( x ) in the right hand side, corresponds to replacing s N by s p ( x ) in the left hand side. , , AND JACOB RASMUSSEN The conjecture is true for n = 1 . Indeed, FHilb = FHilb dg = C , so: Tot FHilb dg ( T n [1] , s N ) = S • C [ x ] (cid:16) C [ x ] x N −→ C [ x ] (cid:17) ∼ = Spec C [ x ] / ( x N ) . The Soergel category SBim has a unique C [ x ] bimodule, namely = C [ x, y ] / ( x − y ) , and thecorresponding object in the dg category (SBim , d N ) is given by: = (cid:20) C [ x, y ] ( W ( x ) − W ( y )) / ( x − y ) − (cid:0) ============= (cid:1) − x − y C [ x, y ] (cid:21) where W ( x ) = x N +1 N +1 . One can eliminate y and rewrite the above = (cid:20) C [ x ] W (cid:48) ( x )= x N −−−−−−→ C [ x ] (cid:21) from where it is clear that the categories Tot FHilb dg ( T n [1] , s N ) and (SBim , d N ) are equivalent.9.4. Differentials in affine charts. Recall the affine charts FHilb T ⊂ FHilb n defined in Sub-section 8.3. In each of these, the vector space C n is endowed with a preferred basis e , ..., e n ,which more abstractly means that the tautological bundle is trivialized: T n | FHilb T ∼ = O · e ⊕ ... ⊕ O · e n The basis vectors are indexed by boxes (cid:3) in the Young diagram of T , and the torus C ∗ × C ∗ acts on the basis vector e (cid:3) by the character z (cid:3) = q a t b for any box (cid:3) = ( a, b ) . We concludethat: ∧ • T ∨ n | FHilb T ∼ = ∧ ( ξ , . . . , ξ n ) where the equivariant weights of the symbols ξ (cid:3) are given by z − (cid:3) = q − a t − b . Recall from Sub-section 9.2 that to any polynomial f ∈ C [ x, y ] , we may associate a section of the tautologicalbundle given by:(9.4) s f | ( X,Y,v ) = f ( X, Y ) v ∈ T n | ( X,Y,v ) We may dualize the above section to obtain s f : T ∨ n → O , and in local coordinates this takesthe form:(9.5) s f ( ξ i ) = [ f ( X, Y ) v ] i = f ( X, Y ) i The local rings of the dg scheme Tot FHilb dg n ( T n [1] , s f ) is then given by the Koszul complexassociated with the first column of the matrix f ( X, Y ) . Lemma 9.4. Suppose that f = x N y M and the diagram of T contains the box with coordinates ( N, M ) . Then the dg algebra ∧ FHilb dg n ( T ∨ n , s f ) is contractible in the local chart ˚FHilb T .Proof. Suppose that (cid:3) = ( N, M ) in T . Using (8.8)–(8.9), one can prove that ( X N Y M )( v ) ∈ e (cid:3) + m T , where m T is the maximal ideal in the local ring ˚ A T = C [ ˚FHilb T ] . Therefore, s f ( ξ (cid:3) ) =1 is invertible in (9.5), and this implies that the Koszul complex of s f is contractible. (cid:3) Corollary 9.5. Suppose that the diagram of T has more than N columns. Then the homologyof the categorified projector P T with respect to d N vanishes.Remark . In [7, Theorem 4] it is proved that ( B w , d N ) ∼ = 0 , if the Robinson-Shenstedtableau of w has more than N columns. One can prove that Soergel bimodules B w with thisproperty generate a tensor subcategory of SBim n , and all categorified projectors P T belongto this subcategory, provided that T has more than N columns. Therefore ( P T , d N ) ∼ = 0 inagreement with Corollary 9.5. LAG HILBERT SCHEMES, COLORED PROJECTORS AND KHOVANOV-ROZANSKY HOMOLOGY 71 For T = (1 , . . . , , the differential corresponding to x N can be written very explicitly. Proposition 9.7. In the chart ˚FHilb (1 ,..., the differential d N is given by the equation (9.6) d N ( ξ + zξ + . . . + z n − ξ n − ) = ( u + zu + . . . + z n − u n ) N mod z n , where u , . . . , u n are local coordinates and z is a formal parameter.Proof. Indeed, in the chart ˚FHilb (1 ,..., one has X = u + Bu + . . . + B n − u n , where B is the n × n Jordan block. Clearly, B n = 0 and the first column of X N contains first n coefficients ofthe polynomial ( u + zu + . . . + z n − u n ) N . (cid:3) As a corollary, we get the following result. Proposition 9.8. Assuming Conjecture 9.2, the sl N homology of the n -th symmetric categori-fied Jones-Wenzl projector is isomorphic to the Koszul homology of the differential (9.6) . This description of d N indeed agrees with the ones in [29, 31, 32], and the homology is quiteinvolved. Indeed, its Poincar´e series for n → ∞ deforms the character of the (2 , N + 1) minimal model for the Virasoro algebra. Extensive computer experiments [29, 31] support thisconjecture for N = 2 and N = 3 . See also [37] for recent developments for N = 2 .The homology of all projectors on two and three strands with respect to d N were describedin [29]. One can check that they agree with the general framework of this paper.10. A PPENDIX Dg algebras. A vector space V will be called dg (short for “differential graded”) if itcomes endowed with a grading: V = (cid:77) n ∈ Z V i and a differential d : V • → V • +1 such that d = 0 . A vector v ∈ V is called homogeneous if v ∈ V i for some integer i . If this is the case, then we will write deg v = i . Definition 10.1. A dg algebra A • is a dg vector space concentrated in non-positive degrees( A n = 0 for n > ), which is endowed with a multiplication that preserves the grading: A i · A j ⊂ A i + j ∀ i, j ∈ N and the differential via the graded Leibniz rule:(10.1) d ( a · a (cid:48) ) = ( da ) · a (cid:48) + ( − deg a a · ( da (cid:48) ) ∀ a, a (cid:48) ∈ A We impose the usual axioms on the dg algebra A • , such as associativity and unit ∈ A .All the dg algebras in this paper will be commutative, in the sense that:(10.2) a · a (cid:48) = ( − (deg a )(deg a (cid:48) ) a (cid:48) · a ∀ a, a (cid:48) ∈ A • We will write H ( A ) for the 0–th cohomology of A • , which is a usual commutative algebra.All the dg algebras studied in this paper will be finitely generated over H ( A ) . Definition 10.2. A dg module M • for a dg algebra A • is a dg vector space M • with a map: A • ⊗ M • −→ M • which is associative, preserves the grading, and satisfies the graded Leibniz rule (i.e. (10.1)with a (cid:48) replaced by m ). Note that all the cohomologies H i ( M • ) are modules for H ( A • ) . , , AND JACOB RASMUSSEN When the grading will not be particularly crucial, we may simplify notation by writing A = A • and M = M • . We will only studied the derived category A –modules: A –Mod = (cid:110) dg modules M (cid:120) A (cid:111) / quasi–isomorphismWhen the dg algebra A is finitely generated over H ( A ) , we will call an object of A –Mod finitely presented if all its cohomologies have this property over H ( A ) . Then we write: A –mod ⊂ A –Modfor the full subcategory of finitely presented modules. The category of dg modules behavesmuch like that of usual modules, but with certain particular features. First of all is the existenceof the grading shift: M • [1] = M • +1 Given two A –modules M and M (cid:48) , one can define the space of degree preserving homomor-phisms between them as Hom A ( M, M (cid:48) ) . But it is more naturally to consider instead:(10.3) Hom • A ( M, M (cid:48) ) = (cid:77) n ∈ Z Hom A ( M, M (cid:48) [ n ]) which is actually a dg vector space with respect to: d ( f ) = d ◦ f − ( − n f ◦ d ∀ f : M → M (cid:48) [ n ] The spaces (10.3) make A –Mod and A –mod into dg categories , which just means a categorywhose Hom spaces are dg vector spaces. We may inquire about the ordinary categories:(10.4) H ( A –Mod ) and H ( A –mod ) whose Hom spaces are, by definition, the 0–th cohomologies of (10.3). Because the zero–cyclesof (10.3) are degree and differential preserving maps f : M → M (cid:48) , while the zero–boundariesare homotopies between such maps, we conclude that (10.4) is nothing but the homotopy cate-gory of A –modules. So the dg category A –mod supersedes the homotopy category.10.2. Symmetric and exterior algebras. There will be two main examples of dg algebras,both associated to a vector space V . The first is the symmetric algebra :(10.5) SV = ∞ (cid:77) d =0 S d V concentrated in degree 0 and with trivial differential, and the exterior algebra :(10.6) ∧ V = ∞ (cid:77) d =0 ∧ d V situated in degrees ..., − , − , and with trivial differential. By definition, the spaces (10.5)and (10.6) are quotients of the tensor algebra of V by the relations v ⊗ v (cid:48) ∓ v (cid:48) ⊗ v . Therefore,they are both particular cases of the symmetric algebra of a dg vector space:(10.7) SV • := (cid:32) ∞ (cid:77) n =0 V • ⊗ ... ⊗ V • (cid:33) (cid:46) (cid:16) v ⊗ v (cid:48) − ( − (deg v )(deg v (cid:48) ) v (cid:48) ⊗ v (cid:17) which inherits the differential from V • : d ( v ⊗ ... ⊗ v k ) = k (cid:88) i =1 ( − deg v + ... +deg v i − · v ⊗ ... ⊗ v i − ⊗ d ( v i ) ⊗ v i +1 ⊗ ... ⊗ v k LAG HILBERT SCHEMES, COLORED PROJECTORS AND KHOVANOV-ROZANSKY HOMOLOGY 73 By the very definition, (10.7) is a commutative dg algebra, which is concentrated in non-positive degrees as long as the original dg vector space V • is. In particular, when the dg vectorspace is concentrated in degree 0 (respectively -1), we obtain (10.5) (respectively (10.6). Example . A particularly important case of the construction (10.7) is when: V • = (cid:104) M s −→ N (cid:105) is concentrated in degrees − and 0. Then we have: SV • = (cid:104) ... d s −→ ∧ M ⊗ SN d s −→ M ⊗ SN d s −→ SN (cid:105) in degrees ..., − , − , , with differential given by:(10.8) d s ( m ∧ ... ∧ m k ⊗ n ) = ( − k − k (cid:88) i =0 m ∧ ... ∧ m i − ∧ m i +1 ∧ ... ∧ m k ⊗ s ( m i ) n for all m , ..., m k ∈ M and n ∈ SN .More generally, suppose that A is a dg algebra and M is a dg module for A . Define: S A M • = SM • (cid:46) ( am ⊗ m (cid:48) − m ⊗ am (cid:48) ) which will also be a dg module for A . The formalism above, as well as Example 10.3, apply.10.3. Affine dg schemes. Dg schemes can be defined as spectra of dg algebras with respect tothe ´etale topology, as detailed in [8]. We will not need the full theory, and instead follow theoriginal definition of Kontsevich. Definition 10.4. If X is an scheme with structure sheaf O X , an affine dg scheme supported on X is a sheaf A of dg algebras, concentrated in non-positive degrees, such that O X = H ( A ) .We will write Spec A for the affine dg scheme associated to A , to match this situation withthat of usual schemes. Philosophically, the approach of Definition 10.4 can be summarized bysaying that we ignore topological subtleties of dg schemes, and simply endow them with thetopology coming from O X . The natural definition of quasi-coherent sheaves is: QCoh(Spec A ) = A –Mod = (cid:110) P ∈ QCoh( X ) endowed with a dg module structure for A (cid:111) quasi–isomorphismAll of the dg schemes in this paper will be of finite type, meaning that A is finitely generatedover O X = H ( A ) . Since this is the case, it is natural to define coherent-sheaves as the fullsubcategory: A –mod = Coh(Spec A ) ⊂ QCoh(Spec A ) consisting of dg modules whose cohomology groups are coherent sheaves over O X = H ( A ) . Example . Suppose that A = S X [ N s → O X ] is the Koszul complex associated to a coherentsheaf N and a co-section s . Explicitly, we have: A = (cid:104) ... d s −→ ∧ N d s −→ N d s −→ O X (cid:105) The structure sheaf O X situated in degree 0, as in Example 10.3, upgraded to the situation ofmodules. If the co-section s is regular, then it is well-known that the Koszul complex is acyclic,and the dg algebra A becomes isomorphic to the usual commutative algebra O X /s . In this case,the dg scheme is simply the subscheme of X cut out by the section s . , , AND JACOB RASMUSSEN However, in general it may be that the section s is not regular (for example, s could be 0). Inthis case, the dg algebra A = ∧ • N has 0 differential but non-trivial grading. Explicitly: A –mod = (cid:110) graded coherent O X (cid:121) P • together with N ⊗ P • λ → P •− such that λ ◦ λ = 0 (cid:111) quasi–isomorphismIn particular, if N ∼ = O ⊕ nX is a free module, the choice of the datum λ corresponds to n com-muting degree − endomorphisms of P . Example . In general, the affine dg schemes we will encounter will combine the previousexample with the case of polynomial rings over ordinary algebras. Specifically, we will have: A = S X [ M s −→ N ] = (cid:104) ... d s −→ ∧ M ⊗ S X N d s −→ M ⊗ S X N d s −→ S X N (cid:105) where M s → N is a map of coherent sheaves of X . The differential d s is given by (10.8), andthe grading has ∧ i M ⊗ S N sitting in degree − i . But note that there is an extra grading on thealgebra A , given by placing ∧ i M ⊗ S j N in degree i + j . We will write this as: A • , ∗ = (cid:77) i,j ≥ A − i,i + j = (cid:77) i,j ≥ ∧ i M ⊗ S j N Since the ∗ = i + j grading is preserved by the differential d s , it descends to a grading onthe cohomology groups. For example, when the morphism s is regular (i.e. when the Koszulcomplex A is acyclic in negative degrees), the • grading collapses, and the ∗ grading matchesthe usual polynomial grading on the symmetric power S ∗ X ( N / M ) .10.4. Projective dg bundles. We do not wish to define projective dg schemes in completegenerality, but instead focus on projectivizations of dg vector bundles V • on a space X . Definition 10.7. A projective dg bundle is defined through its category of coherent sheaves: Coh(Proj S X V • ) = { graded S ∗ X V • dg modules } ( S ∗ V • /S ∗ > V • ) ∼ = 0 Let us make two remarks: first of all, an object in Coh(Proj S X V • ) has two gradings. Thefirst comes from the power ∗ of the symmetric power, and the second comes from the dg gradingon V • . Secondly, the difference between a projectivization and the affine cone Spec S X V • isthe same as in the classical case: there is, in the derived category of the former, an additionalquasi-isomorphism between the structure sheaf of the zero section and the zero module. Example . As in Example 10.8, let us study the case when V • = [ M s −→ N ] is a two stepcomplex of vector bundles, concentrated in degrees − and . In this case, we have a map:(10.9)where the map π is an actual projective bundle since N is a vector bundle on X . The symbol (cid:44) → emulates closed embeddings of schemes, because we tautologically have:(10.10) Coh (cid:16) Proj S X [ M s −→ N ] (cid:17) ∼ = LAG HILBERT SCHEMES, COLORED PROJECTORS AND KHOVANOV-ROZANSKY HOMOLOGY 75 ∼ = (cid:110) coherent sheaves on Proj S X N endowed with a dg action of ∧ • [ π ∗ M ( − s → O Proj S X N ] (cid:111) With this in mind, we think of Proj S X [ M s −→ N ] as the dg subscheme of Proj S X N cut outby the cosection s of the vector bundle π ∗ M ( − .Our main Example 10.8 should be interpreted as a dg version of the familiar notion of pro-jective bundles Proj S X V π −→ X , where V is a rank n locally free sheaf of X . In this case,recall the following formulas: π ∗ ( O ( k )) = S k V concentrated in degree 0 π ∗ ( O ( − k )) = S k − n V ∨ ⊗ ∧ top V ∨ concentrated in degree n − for all k ∈ N , where π ∗ denotes the derived pull-back. The second equality follows from thefirst one, together with relative Serre duality :(10.11) R • π ∗ ( A ) = R •− n +1 π ∗ ( A ∨ ⊗ ∧ top V ( − n )) ∨ for all A ∈ D b (Coh(Proj S X V )) . We now prove a similar formula in the dg setting Proposition 10.9. In the notation of Example 10.8, suppose rank M = m and rank N = n .Then: (10.12) R • π dg ∗ ( A ) = R •− n + m +1 π dg ∗ (cid:18) A ∨ ⊗ ∧ top N ( − n ) ∧ top M ( − m ) (cid:19) ∨ for all A ∈ D b (Coh(Proj S X [ M s −→ N ])) Proof. Implicitly in equation (10.10), one has the equation: R • π dg ∗ ( A ) = R • π ∗ (cid:16) A ⊗ ∧ • [ π ∗ M ( − s → O ] (cid:17) Applying (10.11) to the right hand side, we obtain R • π dg ∗ ( A ) = R •− n +1 π ∗ (cid:18) A ∨ ⊗ ∧ • (cid:104) π ∗ M ( − s → O (cid:105) ∨ ⊗ ∧ top N ∨ ( − n ) (cid:19) ∨ It is easy to see that ∧ • (cid:104) π ∗ M ( − s → O (cid:105) ∨ = ∧ • + m (cid:104) π ∗ M ( − s → O (cid:105) ⊗ ∧ top M ∨ ( m ) , hence: R • π dg ∗ ( A ) = R •− n + m +1 π ∗ (cid:18) A ∨ ⊗ ∧ • (cid:104) π ∗ M ( − s → O (cid:105) ⊗ ∧ top N ( − n ) ∧ top M ( − m ) (cid:19) ∨ which equals R •− n + m +1 π dg ∗ (cid:16) A ∨ ⊗ ∧ top N ( − n ) ∧ top M ( − m ) (cid:17) ∨ by another application of (10.10). (cid:3) R EFERENCES [1] M. Abel, M. Hogancamp. Stable homology of torus links via categorified Young symmetrizers II: one-column partitions. arXiv:1510.05330[2] M. Aganagic, S. Shakirov. Refined Chern-Simons theory and knot homology. String-Math 2011, 3–31, Proc.Sympos. Pure Math., 85, Amer. Math. 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