A ∞ -structures associated with pairs of 1-spherical objects and noncommutative orders over curves
aa r X i v : . [ m a t h . AG ] S e p A ∞ -STRUCTURES ASSOCIATED WITH PAIRS OF -SPHERICALOBJECTS AND NONCOMMUTATIVE ORDERS OVER CURVES ALEXANDER POLISHCHUK
Abstract.
We show that pairs (
X, Y ) of 1-spherical objects in A ∞ -categories, suchthat the morphism space Hom( X, Y ) is concentrated in degree 0, can be described bycertain noncommutative orders over (possibly stacky) curves. In fact, we establish amore precise correspondence at the level of isomorphism of moduli spaces which we showto be affine schemes of finite type over Z . Introduction
Main results.
The study of A ∞ -categories has become an important part of thestudy of derived categories in algebraic geometry, especially in connection with the homo-logical mirror symmetry. In [25, 26] we started to develop a systematic approach to themoduli spaces of minimal A ∞ -structures on a given graded vector space. In [25, 13, 26] werelated certain moduli spaces of A ∞ -stuctures to appropriate moduli spaces of curves, andin [14] this was applied to proving an arithmetic version of homological mirror symmetryfor n -punctured tori.Philosophically, when replacing a geometric object by the corresponding A ∞ -category,one enters the world of noncommutative geometry. Thus, it is natural that objects ofnoncommutative geometry should appear in descriptions of more general moduli spacesof A ∞ -structures.In the present paper we consider examples of such moduli spaces parametrizing A ∞ -categories generated by pairs of 1-spherical objects ( X, Y ) such that morphism spaceHom(
X, Y ) is concentrated in degree 0. Note that examples of such pairs come fromconsidering simple vector bundles on Calabi-Yau curves, as well as from Fukaya categoriesof punctured surfaces.Recall (see [32], [30, I.5]) that an object X of a k -linear A ∞ -category C , where k be afield, is called n -spherical if Hom i ( X, X ) = 0 for i = 0 , n , Hom ( X, X ) = Hom n ( X, X ) = k , and for any object Y of C the pairing between the morphism spaces in the cohomologycategory H ∗ C , Hom n − i ( Y, X ) ⊗ Hom i ( X, Y ) → Hom n ( X, X ) , induced by m , is perfect.Note that if we have a pair of 1-spherical objects ( X, Y ) in a minimal A ∞ -category, suchthat Hom( X, Y ) is concentrated in degree 0, then the only interesting double productsinvovling X and Y are the perfect pairingsHom ( Y, X ) ⊗ Hom ( X, Y ) → Hom ( X, X ) ≃ k, Hom ( X, Y ) ⊗ Hom ( Y, X ) → Hom ( Y, Y ) ≃ k. Supported in part by the NSF grant DMS-1700642 and by the Russian Academic Excellence Project‘5-100’. hus, up to an isomorphism, the graded associative algebra Hom( X ⊕ Y, X ⊕ Y ) is de-termined by a linear automorphism g of the finite-dimensional space Hom ( X, Y ), whichmeasures the difference between the above two pairings. More precisely, fixing trivializa-tions of Hom ( X, X ) and Hom ( Y, Y ) and a basis α , . . . , α n in Hom ( X, Y ), we get anidentification of graded associative algebrasHom( X ⊕ Y, X ⊕ Y ) ≃ S ( k n , g ) , where S ( g ) = S ( k n , g ) is a certain (2 n + 4)-dimensional algebra depending on g ∈ GL n ( k )(see Sec. 1.1). Furthermore, it is easy to see that replacing g by λ · g , where λ ∈ k ∗ , leadsto an isomorphic algebra.Since X and Y were objects of a minimal A ∞ -category, we get a minimal A ∞ -structureon S ( g ) extending the given m . Thus, the problem of describing pairs of 1-sphericalobjects ( X, Y ) as above with Hom ( X, Y ) of dimension n (in this case we refer to ( X, Y ) asan n -pair ) fits into the framework of [26, Sec. 2.2]. As in [26] we consider the moduli spaceof all minimal A ∞ -structures on the family of algebras S ( · ) over PGL n (extending the given m ). The corresponding moduli functor M ∞ (sph , n ) associates to a commutative ring R the set of gauge equivalence classes of minimal R -linear A ∞ -structures on an algebra ofthe form S ( R n , g ), where g ∈ PGL n ( R ) (see Sec. 1.1 for the precise definition).Note that by definition, we have a natural projection M ∞ (sph , n ) → PGL n , where we identify the affine scheme PGL n with the corresponding functor on commutativerings.In the case n = 2 we need to restrict possible elements g , so we consider the principalopen subscheme PGL [tr − ] ⊂ PGL where tr( g ) is invertible. We denote by M ∞ (sph , − ] ⊂ M ∞ (sph , A ∞ -structures in M ∞ (sph , n ) for n ≥ M ∞ (sph , − ] for n = 2) to certain noncommutative projective schemesin the sense of [2]. Recall that for a Noetherian graded algebra R one considers thequotient-category qgr R = grmod −R / tors of finitely generated graded R -modules bythe subcategory of torsion modules (it should be viewed as the category of coherentsheaves on the corresponding noncommutative scheme). Definition 0.1.1.
Let R be a commutative ring, V a finitely generated projective R -module, and L an invertible R -module. For g ∈ End( V ) ⊗ L we denote by End g ( V ) ⊂ End( V ) the R -submodule of transformations a such that tr( ga ) = 0. We define thesubalgebra in End( V )[ z ] by E ( V, g ) := { a + a z + . . . ∈ End( V )[ z ] | a ∈ R · id , a ∈ End g ( V ) } . (0.1.1)We view E ( V, g ) as a graded R -algebra, where deg( z ) = 1. Theorem A . For n ≥ , let us consider the functor M filt ( n ) associating with a commu-tative ring R the following data: a morphism g : Spec( R ) → PGL n and an isomorphism lass of filtered R -algebras ( A, F • ) equipped with an isomorphism gr F A ≃ E ( R n , g ) op , where we denote also by g the pull-back under g of the universal matrix in H (PGL n , Mat n ( O ) ⊗O (1)) . Then for n ≥ , there is an isomorphism of functors M ∞ (sph , n ) ≃ M filt ( n ) and each of these functors is representable by an affine scheme of finite type over Z .In the case n = 2 we have an isomorphism of modified functors M ∞ (sph , − ] ≃ M filt ( n )[tr − ] where we impose the condition that tr( g ) is invertible. These functors are still repre-sentable by an affine scheme of finite type over Z .In either case, if ( A, F • ) is the filtered S -algebra corresponding to an n -pair ( E, F ) of -spherical objects over a Noetherian commutative ring S , then there exists an A ∞ -functorfrom h E, F i to the A ∞ -enhancement of the derived category of qgr R ( A ) , where R ( A ) isthe Rees algebra of A , inducing a quasi-equivalence with its image. Note that in the case n = 1 the equivalence of Theorem A still holds if we restrictto working over Z [1 / M ∞ (sph ,
1) is identified with A . The corresponding pairs of spherical objects arerealized geometrically as ( O C , O p ), where C is an irreducible curve of arithmetic genus 1and p ∈ C is a smooth point.Another case that has a nice geometric realization is that of n = 2, g = id. Namely,working over C , Seidel showed in [31, Sec. 2] that pairs of spherical objects ( X, Y ) withdim Hom ( X, Y ) = 2 can be realized as ( O C , π ∗ O P (1)), where π : C → P is a (possiblysingular) double cover of P .It seems that in the case n = dim Hom ( X, Y ) > X, Y ). However, it is stillof a sufficiently simple kind. Namely, working over a field k , using the classical resultof Small-Warfield [33] on algebras of GK dimension one, we deduce that any filteredalgebra appearing in Theorem A is finite over its center. Using this we establish a naturalbijection between M ∞ (sph , n ) and certain orders over integral stacky curves. Here by an order over an integral stacky curve C we mean a coherent sheaf of O C -algebras, torsionfree as O C -module, whose stalk at the generic point of C is a simple k ( C )-algebra.We make the following definition concerning the types of stacky curves and orders weconsider. Definition 0.1.2. A neat pointed stacky curve over k is an integral 1-dimensional properstack C over k with a smooth stacky point of the form p = Spec( k ) /µ d , such that C \ { p } is an affine (non-stacky) curve. In addition we assume that the coarse moduli space C is a projective curve satisfying H ( C, O ) = k , and there exists an ´etale morphism of theform f : U/µ d → C , where U is a smooth affine curve with a µ d -action and k -point q ,such that µ d acts faithfully on the tangent space to q and f ( q ) = p .Note that a neat stacky curve either has a unique stacky point p (when d >
1) or is ausual curve (when d = 1). efinition 0.1.3. Let A be an order over an integral proper stacky curve C , such that h ( C, O ) = k . We say that A is spherical if A is a 1-spherical object in the perfectderived category of right A -modules, Perf( A op ), We say that A is weakly spherical if h ( C, A ) = h ( C, A ) = 1.It is easy to see that if A is spherical then it is weakly spherical. We will prove thatan order is spherical if and only if h ( C, A ) = 1 and there exists a morphism of coherentsheaves τ : A → ω C such that the pairing A ⊗ A → ω : x ⊗ y τ ( xy )is perfect (see Sec. 3.2). We say that a spherical order A is symmetric if the above pairing(which is uniquely defined up to a scalar) is symmetric. The importance of sphericalorders is due to the fact that they give rise to cyclic A ∞ -structures (see Corollary Cbelow). Theorem B . Fix a field k and a vector space V over k of dimension n ≥ . Let usconsider the following two groupoids:(1) filtered algebras ( A, F • ) with a fixed isomorphism gr F A ≃ E ( V, g ) op for some g ∈ P End( V ) (here morphisms exist only when the corresponding elements g ∈ P End( V ) areequal);(2) data ( C, p, v, A , τ, φ ) , where C is a neat pointed stacky curve, p ≃ Spec( k ) /µ d aunique (smooth) stacky point on C , such that A is a weakly spherical order over C withthe center O C , such that h ( C, A ( p )) = 0 ; v is a nonzero tangent vector at p ; and φ : A| p ≃ ρ ∗ End( V ) op is an isomorphism of algebras, where ρ : Spec( k ) → p is the projection.Then the map associating to ( C, p, A ) the algebra A = H ( C \ p, A ) with its naturalfiltration extends to an equivalence of groupoids (1) and (2).Furthermore, the element g in (1) is invertible if and only the corresponding order A isspherical. In this case, assuming in addition that either n ≥ or tr( g ) = 0 , we have anequivalence between the perfect derived category Perf( A op ) and the triangulated envelope ofthe A ∞ -category with an n -spherical pair ( X, Y ) associated with the data (1) by TheoremA. Under this equivalence the pair ( X, Y ) corresponds to the pair ( A , ρ ∗ V ) in Perf( A op ) .If either n ≥ or char( k ) = 2 , then a spherical order A is symmetric if and only if g isa scalar multiple of the identity matrix. Note that for all examples of spherical orders over stacky curves C that we were ableto construct, C is the usual nonstacky curve. However, we do not know whether this isalways the case.We are particularly interested in the case when A is symmetric because in this case weget a cyclic A ∞ -structure. Corollary C . Let k be a field. Assume that either n ≥ or char( k ) = 2 . Then everyminimal A ∞ -structure on the algebra S ( k n , id) can be realized by A ∞ -endomorphisms ofthe generator A ⊕ ρ ∗ V of Perf( A op ) , for a symmetric spherical order A over a neat pointedstacky curve ( C, p ) equipped with an isomorphism A| p ≃ ρ ∗ End( V ) op . If char( k ) = 0 thenevery such A ∞ -structure is gauge equivalent to a cyclic A ∞ -structure. As was shown in [15], cyclic A ∞ -structures on S ( k n , id) correspond to formal solutionsof set-theoretical Associative Yang-Baxter Equation (AYBE). Applying Corollary C, we et an algebro-geometric realizition of these solutions in the category of modules oversome symmetric spherical orders. Elsewhere we will present an explicit construction ofsuch orders giving rise to trigonometric nondegenerate solutions of the AYBE that wereclassified in [24].0.2. Organization of the paper.
In this paper we are dealing with the following threetypes of structures:(I) A ∞ -structures on S ( g ), or equivalently, n -pairs of 1-spherical objects;(II) filtered algebras as in Theorem A;(III) orders over neat stacky curves as in Theorem B.In Section 1 we study (I). Here the main result is the representability of the modulifunctor M ∞ (sph , n ) of A ∞ -structures by an affine scheme of finite type over PGL n . This isbased on the criterion from [26] which requires calculation of some Hochschild cohomology.The latter calculation is performed in Sec. 1.3 using Koszulness of algebras S ( g ) (withappropriate grading).In Section 2 we explain the passage from (I) to (II). First, in Sec. 2.1 and 2.2, we givesome background on spherical objects. In 2.3 we establish some properties of algebras E ( R n , g ). Then in Theorem 2.4.1 we give a construction of a filtered algebra correspondingto an n -pair of 1-spherical objects ( E, F ). The main idea is to consider the infinitesequence of objects E i = T iF ( E ) , where T F is the spherical twist functor associated with F , and then define a graded algebrastructure on R = L i Hom( E , E i ). It turns out that R has a canonical regular centralelement of degree 1, so it can be identified with the Rees algebra R ( A ) of a filtered algebra A . Note that this construction is inspired by a construction in the work of Van Roosmalen[28]. We also describe a natural R − R -bimodule structure on L i Hom ( E i , E ) in termsof a certain filtered automorphism φ of A (see Prop. 2.4.4).In Section 3, working over a field, we explain how to pass from (II) to (III) and discussspherical orders. Namely, in Sec. 3.1 we construct a noncommutative order over a stackycurve associated with a filtered algebra such that gr F A ≃ E ( R n , g ) op . In Sec. 3.2 we givea criterion for an order to be spherical (see Definition 0.1.3). In Sec. 3.4 we check theAS-Gorenstein property for the Rees algebras R ( A ) of our filtered algebras A . The proofuses a special order over the cuspidal cubic consructed in 3.3.In Section 4 we show how to go from (II) to (I). Namely, in Sec. 4.2 we consider thenoncommutative projective scheme in the sense of Artin-Zhang [2] associated with thegraded ring R ( A ) and construct a natural pair of 1-spherical objects in the derived cat-egory of qgr R ( A ), the category of coherent sheaves on this noncommutative projectivescheme (see Proposition 4.2.1). It is also straightforward to see that qgr R ( A ) is equiva-lent to the category of right modules over the corrresponding order A (see Prop. 3.1.3).However, for computations it is more convenient to use the presentation as qgr R ( A ) andto use results of [2].In Section 5 we prove a technical result characterizing minimal A ∞ -structures on thesubcategory of objects ( E i ) in terms of double products together with a single tripleproduct (see Cor. 5.1.4). This result is a slight generalization of the main theorem of [23],where a similar problem is studied for the subcategory of powers of a very ample line undle. The key computation of Hochschild cohomology is Theorem 5.1.2 generalizing[23, Sec. 3.1]. Then in 5.2 we perform a calculation of the relevant triple product in thecase of the A ∞ -structure coming from an n -pair of 1-spherical objects.In Section 6 we put everything together to prove our main results. In Sec. 6.1 weprove equivalence of functors corresponding to (I) and (II) over Noetherian rings. Themain difficulty is to prove that going from (I) to (II) and back one gets an A ∞ -structureequivalent to the original one. For this we use the characterization from Sec. 5.1. Anothertechnical detail is that double products on the subcategory ( E i ) are determined by thefiltered algebra A together with a filtered automorphism φ of A which we are able tocharacterize uniquely (see Prop. 2.4.5, Cor. 4.2.2).In Sec. 6.2 we prove representability of the functor corresponding to (II) by an affinescheme of finite type, and as a result, deduce the equivalence of (I) and (II) over arbitraryrings.In 6.4 we prove equivalence of (II) and (III). In addition to constructions of Section 3,we use a technical result on stacks discussed in 6.3.Finally, in Section 6.5 we show that symmetric spherical orders over curves give rise tocyclic A ∞ -structures and use this to prove Corollary C. Conventions . All A ∞ -algebras/categories are strictly unital. We denote by hom( · , · ) themorphism spaces in an A ∞ -category and by Hom( · , · ) their cohomology, For a commu-tative ring S , and a graded S -module M = L i M i , where each M i is a projective finitedimensional S -module, we denote by M ∗ the restricted dual of M , which is a graded S -module with components ( M ∗ ) i = ( M − i ) ∨ = Hom S ( M − i , S ). Acknowledgments . I am grateful to Yanki Lekili and Riley Casper for useful discussions,and to James Zhang for the help with proving finiteness of injective dimension in Sec. 3.4.I also thank the anonimous referee for valuable comments. Part of this work was donewhile the author was visiting the Korean Institute for Advanced Study, the ETH Zurichand the IHES. He would like to thank these institutions for hospitality and excellentworking conditions. 1.
A moduli space of A ∞ -structures The moduli functor.
For basics on A ∞ -structures we refer to [8]. We would liketo consider minimal (strictly unital) A ∞ -structures on a certain family of categories withtwo objects (which are generalizations of the category considered in [15]). Definition 1.1.1.
Let R be a commutative ring, V a finitely generated projective R -module, L an invertible R -module, and g : V → V ⊗ L an R -linear morphism. Thegraded category S ( V, g ) has two objects X and Y and the morphismsHom( X, Y ) = Hom ( X, Y ) = V, Hom(
Y, X ) = Hom ( Y, X ) = V ∨ = Hom R ( V, R ) , Hom ( X, X ) = R id X , Hom ( Y, Y ) = R id Y , Hom ( X, X ) = L , Hom ( Y, Y ) = R. The compositions are determined by v ∗ ◦ v = h v ∗ , g ( v ) i , v ◦ v ∗ = h v ∗ , v i , where v ∈ V , v ∗ ∈ V ∨ . n this paper we will consider minimal unital A ∞ -structures on the algebras S ( k n , g ),where g ∈ GL n ( k ), for fixed n ≥ m given above), up to a gauge equivalence.More precisely, this family of algebras can be viewed as a sheaf of associative algebras e S n on the scheme GL n (over Z ). Note that we have the standard action of the central G m on GL n and the universal element g can be viewed as a morphism of bundles over GL n , g : O n → O n ⊗ χ , compatible with the G m -action. Here χ is the identity character of G m , χ ( λ ) = λ . Thus, we can descend g to an isomorphism g : O n → O n ⊗ O (1)of vector bundles over PGL n = GL n / G m . Now it is easy to see that the sheaf of algebras e S n over GL n is isomorphic to the pull-back of the sheaf of algebras S n over PGL n associatedwith bundle O n and the morphism g as in Definition 1.1.1.As in [26], we consider the following functor over PGL n on affine schemes over PGL n . Definition 1.1.2.
The functor M ∞ ( S n ) associates to a pair ( R, g ), where R is a commuta-tive ring and g ∈ PGL n ( R ), the set of gauge equivalence classes of minimal A ∞ -structureson S ( R n , g ). Note that here g is viewed as a morphism R n → R n ⊗ g ∗ O (1).For each N ≥ M N of minimal A ′ N -structuresup to equivalence, where we consider ( m , m , . . . , m N ) and impose the A ∞ -identities upto [ m , m N ] + [ m , m N − ] + . . . In [26, Thm. 2.2.6] we gave a general criterion for thefunctors M N to be representable by an affine scheme and for the projection M ∞ → M to be a closed embedding. Note that there is a “bookkeeping” mistake in [26, Thm. 2.2.6]in that the results are stated for minimal A N -structures (which involve ( m , . . . , m N ) andthe identities up to [ m , m N − ] + . . . ), however, these results only become correct uponreplacing the A N -structures by A ′ N -structures (the same problem occurs in [25, Sec. 4.2]).In this paper we will apply general results on moduli of A ∞ -structures to our functor M ∞ ( S n ) over PGL n , for n ≥
3. In the case n = 2 we restrict to an open subscheme ofPGL and modify the functor accordingly. Namely, let us consider the closed embeddingSpec( Z / ֒ → PGL , Z / ֒ → PGL given by the unit over Z /
2, and let U ⊂ PGL be the complementary open subscheme.Let S ,U be the restriction of the sheaf of algebras S to U . Since U is non-affine, as in [26,Sec. 2.2], we first consider the functor f M ∞ ( S ,U ) of the set of gauge equivalence classes ofminimal A ∞ -structures, and then define M ∞ ( S ,U ) to be the sheafification of f M ∞ ( S ,U )with respect to the Zariski topology on the base.Note that when defining the functor M ∞ ( S n ) for n ≥
3, we do not need to pass to thesheafification with respect to the Zariski topology on the base PGL n since the base is affine(see [26, Thm. 2.2.6]). The same is true about the functor M ∞ ( S )[tr − ], correspondingto the affine subset of g in PGL with tr( g ) = 0. Theorem 1.1.3.
The functor M ∞ ( S n ) for n ≥ (resp., M ∞ ( S ,U ) ) is representable byan affine scheme of finite type over PGL n (resp., over U ). Furthermore, in both cases theprojection M ∞ → M to the moduli space of minimal A ′ -structures is a closed embedding. Note that the functor M ∞ (sph , n ) is the absolutization of M ∞ ( S n ): its point over anaffine scheme Spec( R ) consists of a morphism g : Spec( R ) → PGL n , together with an lement in M ∞ ( S n )( R, g ). Thus, we deduce the following corollary (in the case n = 2 werestrict to the open subset tr = 0 of U ⊂ PGL ). Corollary 1.1.4.
The functor M ∞ (sph , n ) for n ≥ ) (resp., M ∞ (sph , − ] ) is rep-resentable by an affine scheme of finite type over Z . We will give a proof of Theorem 1.1.3 in Sec. 1.3 after proving vanishing of somecomponents of the Hochschild cohomology of algebras S ( k n , g ). The computation is basedon the fact that these algebras are Koszul with respect to the grading deg V = deg V ∨ = 1:we use the well known Koszul resolution for the diagonal bimodule. The vanishing ofcertain components of the Hochschild cohomology translates into some statements aboutthe quadratic dual algebra S ! which we establish in Lemma 1.2.3.Note that in [15] we considered minimal A ∞ -structures on S ( R n , id), cyclic with respectto a natural pairing. We postpone the discussion of cyclic structures until Sec. 6.5.1.2. The dual quadratic algebra to S ( k n , g ) . Let us fix a field k and an element g ∈ GL n ( k ), where n ≥
2. Let us set for brevity S = S ( k n , g ). Note that we can view S as a K -algebra, where K = k ⊕ k , where the idempotents e X and e Y correspond tothe identity elements id X and id Y . We also denote by ξ X and ξ Y the basis elements inHom ( X, X ) and Hom ( Y, Y ).If ( α i ) are the elements of Hom( X, Y ) corresponding to the standard basis in k n and( β j ) are the dual elements of Hom( Y, X ), then the product in S is given by β i α j = a ij ξ X , α j β i = δ ij ξ Y , where g = ( a ij ). Note that the K -algebra S is generated by elements ( α i ) and ( β j ), so wecan view it as a quotient of the path algebra of the quiver with two vertices X and Y and n arrows in each direction corresionding to α i and β j . We will use two different gradings on S : deg given by deg( α i ) = 0, deg( β i ) = 1, and deg K given by deg K ( α i ) = deg K ( β i ) = 1.We denote by S j the graded components of S with respect to deg K .We are going to use the following convention about quadratic (and Koszul) duality over K = k · id X ⊕ k · id Y . For a quadratic K -algebra A with generators V XY and V Y X of degree1, and quadratic relations R XX ⊂ V Y X ⊗ V XY , R Y Y ⊂ V XY ⊗ V Y X , the dual quadraticalgebra has generators V ! XY = V ∨ Y X and V ! Y X = V ∨ XY and quadratic relations R ! XX = A ∨ ,XX ⊂ ( V Y X ⊗ V XY ) ∨ ≃ V ! Y X ⊗ V ! XY , and similarly for R ! Y Y .Thus, we think of the dual algebra S ! as the quotient of the path algebra of the quiverwith vertices X , Y , where the direction of α ∗ i (resp., β ∗ i ) is opposite to that of α i (resp., β i ). We denote by S ! j the graded components of S ! with respect to the grading deg K ( α ∗ i ) =deg K ( β ∗ i ) = 1. Lemma 1.2.1.
With respect to the grading deg K the algebra S is Koszul and the dualquadratic algebra S ! is generated by the dual generators ( α ∗ i ) and ( β ∗ i ) with the only rela-tions X ≤ i,j ≤ n a ij α ∗ j β ∗ i = 0 , n X i =1 β ∗ i α ∗ i = 0 . roof . The algebra S is obtained by folding from the following Z -algebra S Z = S Z ( g )(where we use the term Z -algebra in the sense of [4, Sec. 4]): S Z i,i +1 = ( V i even V ∨ i odd. , S Z i,i +2 = k, S Z i,j = 0 for j > i + 2 . Here V is the n -dimensional space with the basis ( α i ). The multiplication is given by thepairings V ⊗ V ∨ → k : v ⊗ v ∗
7→ h v ∗ , v i , V ∨ ⊗ V → k : v ∗ ⊗ v
7→ h v ∗ , g ( v ) i . There is a natural isomorphism S Z i +2 ,j +2 ≃ S Z i,j compatible with the product, and thealgebra S is the corresponding folding of S Z .The Koszul properties for S (with the grading deg K ) and for S Z are equivalent. Onthe other hand, it is easy to construct the isomorphism of Z -algebras between S Z and the Z -algebra corresponding to the Z -graded algebra B ⊕ B ⊕ B = k ⊕ V ⊕ k, where the multiplication V ⊗ V = B ⊗ B → B = k is given by a nondegeneratesymmetric bilinear form. It is well known that the algebra B is Koszul. Hence, S is alsoKoszul. (cid:3) Remark 1.2.2.
The algebra S ! for g = id is closely related to the noncommutativeprojective line P n (see [21], [17]). Namely, it is a folded version of the Z -algebra of anatural helix in the derived category of P n .Let e X , e Y ∈ K be the idempotents corresponding to the vertices X and Y , respectively. Lemma 1.2.3. (i) Let m ≥ . If x ∈ S ! m e X satisfies xα ∗ i = 0 for some i then x = 0 .(ii) Assume that either n ≥ , or g = λ · id (for any λ ∈ k ∗ ), or the characteristic of k is = 2 . If x ∈ S !2 e X , x ∈ S !2 e Y satisfy β ∗ i x = − x β ∗ i , α ∗ i x = x α ∗ i for all i , then x = 0 and x = 0 .(iii) If a collection of elements ( x i ) , where x i ∈ S !2 e Y , satisfies X i,j a ij x j β ∗ i = 0 , then there exists y ∈ S !1 e X such that x j = yα ∗ j for each j .Proof . (i) It is easy to see that the question does not depend on g , so we can assume g = id.In this case, we need to check that the element x in the algebra k h x , . . . , x n i / ( x + . . . + x n ) is not a right zero divisor. But in fact, the latter algebra is a domain by [38, Thm.0.2]. ii) It is easy to see that we can reformulate the question as follows. Given an n -dimensional vector space V (in our case the space spanned by ( β ∗ i )) and elements x ∈ V ∨ ⊗ V , x ∈ V ⊗ V ∨ , such that g ( v ) ⊗ x = − x ⊗ v mod(id ⊗ V + V ⊗ id) for any v ∈ V, (1.2.1) v ∗ ⊗ Ad( g − ) x = x ⊗ g ∗ ( v ∗ ) mod(id ⊗ V ∨ + V ∨ ⊗ id) for any v ∈ V ∨ , (1.2.2)we should deduce that x and x are proportional to id.Assume first that n ≥
3. Then we claim that (1.2.1) alone implies that x and x areproportional to id. Indeed, suppose we have g ( v ) ⊗ x + x ⊗ v = id ⊗ A ( v ) + B ( v ) ⊗ id ∈ V ⊗ V ∨ ⊗ V for all v , for some operators A, B ∈ End( V ). Taking the contraction in the third tensorcomponent with v ∗ ∈ V ∨ , we get the identity g ( v ) ⊗ h x , v ∗ i + h v, v ∗ i x = h A ( v ) , v ∗ i id + B ( v ) ⊗ v ∗ ∈ V ⊗ V ∨ . Thus, whenever h v, v ∗ i = 0, the operator h A ( v ) , v ∗ i id ∈ End( V ) is the sum of two opera-tors of rank 1. Since n ≥
3, this implies that h A ( v ) , v ∗ i = 0. Hence, A ( v ) is proportionalto v for any v ∈ V , i.e., A = λ · id for some λ ∈ k . A similar argument shows that B = µ · g for µ ∈ k . Thus, subtracting some multiples of id from x and x we reduce ourselves tothe situation when g ( v ) ⊗ x + x ⊗ v = 0 for any v ∈ V. Using contractions as above it is easy to deduce from this that x = 0 and x = 0.Next, assume that n = 2. Then we are going to rewrite condition (1.2.1) by fixing a sym-plectic isomorphism s : V → V ∨ and observing that ( s ⊗ id V )( V V ) and (id V ⊗ s )( V V )are precisely the lines spanned by the identity elements in V ∨ ⊗ V and V ⊗ V ∨ . Thus,defining y , y ∈ V ⊗ V by x = ( s ⊗ id)( y ) , x = (id ⊗ s )( y ) , we see that (1.2.1) is equivalent to the equation g ( v ) · q = − q · v in S V, where q i is the image of y i in S V . Assume first that g is not proportional to id. Then therelation above implies that q = q = 0 in S V . Indeed, let us pick v = 0 such that g ( v )is not a multiple of v , and assume q i are nonzero. Then we should have the followingequations in the algebra SV : q = v · v ′ , q = − g ( v ) · v ′ for some v ′ ∈ V , v ′ = 0. Then for any v we should have g ( v ) · v = g ( v ) · v in S V. But this is a contratiction as soon as v is not proportional to v .Finally, let us consider the case n = 2 and g = λ · id. Then the above argument gives q = − λq . imilarly, we can rewrite condition (1.2.2) for g = λ · id as v ⊗ y = λy ⊗ v, where v ∗ = s ( v ). Thus, we get q = λq . Since the characteristic is = 2, we deduce that q = q = 0.(iii) This follows from the exactness of the direct summand of the Koszul complex, . . . → e Y S !1 ⊗ S ∗ e X → e Y S !2 ⊗ S ∗ e X → e Y S !3 e X → S ! is Koszul). (cid:3) Calculation of Hochschild cohomology and proof of Theorem 1.1.3.
For a Z -graded algebra A , we use the following convention for the bigrading on its Hochschildcohomology HH ( A ). We denote by CH s + t ( A ) { t } the space of linear maps A ⊗ s → A ofdegree t . The corresponding bigraded piece in the Hochschild cohomology is denoted by HH s + t ( A ) { t } (then the upper grading is derived Morita invariant).Let us consider the algebra S = S ( k n , g ) as above. We denote by S{ m } (resp., S ! { m } )the graded components of S (resp., S ! ) with respect to the grading given by deg( α i ) =0, deg( β i ) = 1 (resp., deg( α ∗ i ) = 0, deg( β ∗ i ) = − S viewed as a graded algebra with this grading (note that this affectssome signs). Theorem 1.3.1.
For m ∈ Z , one has HH m ( S ) { < − m } = 0 . Assume in addition that either n ≥ , or g = λ · id (for any λ ∈ k ∗ ), or the characteristicof k is = 2 . Then HH ( S ) {− } = 0 . Proof . Recall that to compute the Hochschild cohomology of a Koszul K -algebra A (where K is commutative semisimple) we can use the Koszul resolution (see e.g.,[34, Sec.3]). More precisely, we have a natural embedding( A ! m ) ∗ ֒ → A ⊗ m (here and below all tensor products are over K ), so that the image consists of the inter-section of kernels of the partial multiplication maps a ⊗ . . . ⊗ a m a ⊗ . . . ⊗ a i a i +1 ⊗ . . . ⊗ a m . The corresponding subcomplex A ⊗ ( A ! • ) ∗ ⊗ A ⊂ A ⊗ T • ( A + ) ⊗ A in the standard bar-resolution of A by free A − A -bimodules is still exact. Thus, we geta resolution of the form[ . . . → A ⊗ ( A ! m ) ∗ ⊗ A d m ✲ A ⊗ ( A ! m − ) ∗ ⊗ A → . . . → A ⊗ ( A !1 ) ∗ ⊗ A → A ⊗ A ] → A. et ( v i ) be generators in A , ( v ∗ i ) the dual generators of A !1 . Then the differential is givenby d m ( r ⊗ φ ⊗ s ) = X i rv i ⊗ v ∗ i φ ⊗ s + ( − m X i r ⊗ φv ∗ i ⊗ v i s, where we use the A ! -bimodule structure on ( A ! ) ∗ given by the operators dual to the leftand right multiplication.Assume now that A has an additional grading deg, induced by some Z -grading on A .If we are interested in the Hochschild cohomology of A as a graded algebra with respectto this grading, at this point we need to be careful in using the appropriate signs. Namely,to compute the Hochschild cohomology HH ∗ ( A ) we apply the functor Hom A ⊗ A op (? , A ) tothe above resolution and use the identification A ! m ⊗ A ≃ Hom(( A ! m ) ∗ , A ) ≃ Hom A ⊗ A op ( A ⊗ ( A ! m ) ∗ ⊗ A, A )under which an element c ∈ Hom(( A ! m ) ∗ , A ) corresponds to the composition of id ⊗ c ⊗ idwith the multiplication µ in A . Now we have to use the Koszul sign convention: µ (id ⊗ c ⊗ id)( r ⊗ φ ⊗ s ) = ( − deg( c ) deg( r ) rc ( φ ) s. Thus, we get the complex computing the Hochschild cohomology of A , A → A !1 ⊗ A → . . . → A ! m ⊗ A δ m ✲ A ! m +1 ⊗ A → . . . with the differential δ m ( ψ ⊗ s ) = ( − (deg( ψ )+deg( s )) deg( v i ) X i ψv ∗ i ⊗ v i s + ( − m +1 X i v ∗ i ψ ⊗ sv i . Here we assume that the basis ( v i ) is homogeneous with respect to deg.We can apply this procedure in our case since S is Koszul, with the generators ( α i , β i )(see Lemma 1.2.1). We are interested in the components HH ∗ ( S ) { j } . As explained above,these spaces can be computed as cohomology of the complex ( S ! • ⊗ S ) { j } with respect tothe differential δ ( ψ ⊗ s ) = X i ( ψα ∗ i ⊗ α i s + ( − j ψβ ∗ i ⊗ β i s ) + ( − m +1 X i ( α ∗ i ψ ⊗ sα i + β ∗ i ψ ⊗ sβ i ) , where ψ ∈ S ! m .Since S{ j } = 0 for j = 0 ,
1, we have( S ! ⊗ S ) { j } = S ! { j } ⊗ S{ } ⊕ S ! { j − } ⊗ S{ } . Note also that because α ∗ i and β ∗ j have to alternate in any nonzero monomial in S ! m , wehave S ! m { j } = 0 unless m ∈ {− j − , − j, − j + 1 } . This immediately implies thevanishing HH m ( S ) { < − m − } = 0for any integer m .For m ≥ HH m ( S ) {− m − } is identified with the kernel of the map δ : S !2 m +1 {− m − } ⊗ S{ } → S !2 m {− m − } ⊗ S{ } . ut S !2 m +1 {− m − } ⊗ S{ } = S !2 m +1 e X ⊗ e X , and for x ∈ S !2 m +1 e X we have δ ( x ⊗ e X ) = X i xα ∗ i ⊗ α i . Thus, Lemma 1.2.3(i) implies that this kernel is zero.Next, HH ( S ) {− } is the cohomology in the middle term of S !1 {− } ⊗ S{ } → S !2 {− } ⊗ S{ } → S !3 {− } ⊗ S{ } ⊕ S !3 {− } ⊗ S{ } . (1.3.1)An element of S !2 {− } ⊗ S{ } has form x = x ⊗ e X + x ⊗ e Y + X j x j ⊗ α j , where x ∈ S !2 e X , x ∈ S !2 e Y and x i ∈ S !2 e Y . We have δ ( x ) = X i ( x α ∗ i ⊗ α i − x β ∗ i ⊗ β i ) − X i,j a ij x j β ∗ i ⊗ ξ − X i ( β ∗ i x ⊗ β i + α ∗ i x ⊗ α i ) . Thus, δ ( x ) = 0 if and only if β ∗ i x = − x β ∗ i , α ∗ i x = x α ∗ i for all i, X i,j a ij x j β ∗ i = 0 . The coboundaries come from elements S !1 {− } ⊗ S{ } = S !1 e X ⊗ e X and have form δ ( y ⊗ e X ) = X i yα ∗ i ⊗ α i . Thus, Lemma 1.2.3(ii)(iii) implies that (1.3.1) is exact (under our assumptions), andhence, HH ( S ) {− } = 0. (cid:3) Remark 1.3.2.
In [31, Sec. (2c)] the computation similar that of Theorem 1.3.1 is donein the case n = 2, g = id, k = C . In this case one also has HH ( S ) {− } = 0, and HH ( S ) {− } can be identified with the space of binary quartic polynomials. As explainedin [31, Sec. (2f)], this means that all minimal A ∞ -structures on the algebra S ( C , id) arerealized by double coverings of P . Proof of Theorem 1.1.3 . We apply [26, Thm. 2.2.6] to the family S n , n ≥ S ,U ).More precisely we use the following vanishing of components of Hochschild cohomologyfor any algebra S = S ( k n , g ) (implied by Theorem 1.3.1): HH ≤ ( S ) { < } = HH ( S ) { < − } = 0 . (cid:3) . Pairs of -spherical objects and noncommutative algebras Spherical objects and spherical twists.
Recall (see [32], [30, I.5]) that an object X of a k -linear A ∞ -category C , where k be a field, is called n -spherical if Hom i ( X, X ) = 0for i = 0 , n , Hom ( X, X ) = Hom n ( X, X ) = k , and for any object Y of C the pairingbetween the morphism spaces in the cohomology category H ∗ C ,Hom n − i ( Y, X ) ⊗ Hom i ( X, Y ) → Hom ( X, X ) , induced by m , is perfect.We need the following generalization of this notion to the case of an S -linear A ∞ -category, where S is a commutative ring (our definition is not the most general possible:we impose rather strong assumptions on the hom-complexes). Definition 2.1.1.
Let X be an object of an S -linear A ∞ -category C . Assume that for any Y the complexes hom( X, Y ) and hom(
Y, X ) are bounded complexes of finitely generatedprojective S -modules. Then X is called n -spherical if Hom i ( X, X ) = 0 for i = 0 , n ,Hom ( X, X ) = S · id X , L X := Hom n ( X, X ) is a locally free S -module of rank 1, andfor any Y in C the following composed chain map of complexes of S -modules is a quasi-isomorphism:hom( Y, X ) → hom( X, Y ) ∨ ⊗ S hom( X, X ) → hom( X, Y ) ∨ ⊗ τ ≥ n hom( X, X ) . (2.1.1)Here the first arrow induced by m , while the second comes from the natural maphom( X, X ) → τ ≥ n hom( X, X ), where τ ≥ n is the truncation functor. Also, P ∨ = Hom S ( P, S )is the dual of a finitely generated projective module P .Note that the complex τ ≥ n hom( X, X ) is bounded, has the only cohomology at the left-most term and all of its subsequent terms are finitely generated projective S -modules.This implies that the left-most term is also finitely generated projective and there existsa homotopy equivalence τ ≥ n hom( X, X ) → Hom n ( X, X )[ − n ] = L X [ − n ] . Fixing such an equivalence we can view the map (2.1.1) as a chain map s Y : hom( Y, X ) → hom( X, Y ) ∨ ⊗ L X [ − n ] (2.1.2)Let Tw( C ) denote the category of twisted complexes over C (see e.g., [8, Sec. 7.6]). Lemma 2.1.2. An n -spherical object X of C remains n -spherical in Tw( C ) .Proof . First, note that for any twisted complex Y we still have that hom( X, Y ) andhom(
Y, X ) are bounded complexes of finitely generated projective S -modules. Now as-sume we are given an exact triangle Y ′ → Y → Y ′′ → Y ′ [1] in Tw( C ), such that themaps (2.1.2) for Y ′ and Y ′′ are quasi-isomorphisms. Then we have a morphism of exact riangles of S -moduleshom( Y ′′ , X ) ✲ hom( Y, X ) ✲ hom( Y ′ , X ) → . . . hom( X, Y ′′ ) ∨ ⊗ L X [ − n ] s Y ′′ ❄ ✲ hom( X, Y ) ∨ ⊗ L X [ − n ] s Y ❄ ✲ hom( X, Y ′ ) ∨ ⊗ L X [ − n ] s Y ′ ❄ → . . . so the fact that s Y ′ and s Y ′′ are quasi-isomorphisms imply that s Y is also a quasi-isomorphism. Since every object of Tw( C ) is an iterated extension of shifts of objectsof C , the assertion follows. (cid:3) Given an n -spherical object E , we can define (see [32], [30, I.5]) the twist and theadjoint twist A ∞ -functors T E , T ′ E : Tw( C ) → Tw( C )by T E ( X ) = Cone(hom( E, X ) ⊗ E ev ✲ X ) , T ′ E ( X ) = Cone( X ev ′ ✲ hom( X, E ) ∨ ⊗ E )[ − . The proof of [32, Prop. 2.10] extends to our situation to prove that T ′ E T E and T E T ′ E areisomorphic to identity in the homotopy category of functors from Tw( C ) to itself.2.2. n -pairs of -spherical objects. A ∞ -structures we want to consider are related tothe following pairs of objects in A ∞ -categories. Definition 2.2.1.
Let C be a S -linear A ∞ -category, where S is a commutative ring.(i) We call a pair of 1-spherical objects ( E, F ) in C an n -pair if Hom ∗ ( E, F ) is concentratedin degree 0 and is isomorphic to S n . In addition we require that Hom ( F, F ) ≃ S . Notethat this implies that Hom ∗ ( F, E ) is a free S -module of rank n concentrated in degree1. We say that ( E, F ) a symmetric n -pair if in addition Hom ( E, E ) ≃ S and the twoperfect pairings Hom ( F, E ) ⊗ S Hom ( E, F ) → Hom ( E, E ) andHom ( E, F ) ⊗ S Hom ( F, E ) → Hom ( F, F ) (2.2.1)(coming from the conditions that E and F are 1-spherical) lead to two dualities Hom ( F, E ) ≃ Hom ( E, F ) ∨ that differ by a scalar in S ∗ .(ii) A weak n -pair in C is a pair of objects ( E, F ) in C such that F is 1-spherical withHom ( F, F ) ≃ S , E satisfies Hom ( E, E ) ≃ S , Hom =0 , ( E, E ) = 0, L E := Hom ( E, E ) isa locally free S -module of rank 1, and Hom ∗ ( E, F ) = Hom ( E, F ) ≃ S n . An enhancedweak n -pair is a weak n -pair ( E, F ) together with chosen isomorphisms Hom ( E, F ) ≃ S n and Hom ( F, F ) ≃ S .Note that for an enhanced weak n -pair, the second of the pairings (2.2.1) is perfect, soit defines an isomorphism Hom ( F, E ) ≃ V ∨ , where V := Hom ( E, F ). Then the first ofthe pairings (2.2.1) has formHom ( F, E ) ⊗ S Hom ( E, F ) ≃ V ∨ ⊗ V → L E : ( v ∗ , v )
7→ h v ∗ , gv i (2.2.2) or a unique g ∈ End S ( V ) ⊗ L E . Lemma 2.2.2.
Let ( E, F ) be a weak n -pair in C , and assume that C is split generated by ( E, F ) . Then ( E, F ) is an n -pair (i.e., E is -spherical) in C if and only if the element g ∈ End S ( V ) ⊗ L E defined by (2.2.2) is invertible.Proof . Let C ′ ⊂ C be the triangulated envelope of ( E, F ). It is clear that E is 1-sphericalin C if and only if it is 1-spherical in C ′ . But by Lemma 2.1.2, E is spherical in C ′ if andonly if the pairing (2.2.2) is perfect, which is equivalent to g being invertible. (cid:3) Example 2.2.3.
Let D b ( C ) be the (enhanced) derived category of coherent sheaves onan elliptic curve C over an algebraically closed field k . Then 1-spherical objects in D b ( C )are (up to shift) either simple vector bundles or the skyscraper sheaves O p . The group ofautoequivalences of D b ( C ) acts transitively on them, so any n -pair of 1-spherical objectscan be transformed by an autoequivalence into a pair ( E, O p ), where E is an simple vectorbundle of rank n . It is easy to see that any such n -pair is special.Given g ∈ PGL n ( S ) and a minimal A ∞ -structure on S = S ( S n , g ) we get an n -pair of 1-spherical objects in the corresponding full subcategory { e X ·S , e Y ·S} of right A ∞ -modulesover S .Conversely, starting with an enhanced n -pair ( E, F ) in an S -linear A ∞ -category, letus consider the full A ∞ -subcategory with the objects E and F . Due to our assumptionthat hom( E, E ), hom(
E, F ), hom(
F, F ) and hom(
F, E ) are bounded complexes of pro-jective modules, we can apply the homological perturbation to get an equivalent minimal A ∞ -structure on this subcategory. Furthermore, as was explained before, we can identifyHom ( F, E ) with V ∨ , where V = Hom ( E, F ) ≃ S n , so that the second of the composi-tions (2.2.1) becomes the standard pairing between V and V ∨ , while the first has the form(2.2.2) for some g ∈ GL n ( S ) ⊗ L E . Thus, we get a minimal A ∞ -structure on S ( S n , g ).2.3. Some properties of the algebra E ( S n , g ) . Let S be a commutative ring, L alocally free S -module of rank 1. For an element g ∈ Mat n ( S ) ⊗ L we consider the algebra E ( S n , g ) defined by (0.1.1). Lemma 2.3.1.
Assume that n ≥ and g ∈ Mat n ( S ) ⊗ L is such that there exists h ∈ Mat n ( S ) ⊗ L − with tr( gh ) = 1 .(i) Assume that either n ≥ or n = 2 and tr( g h ′ ) = 1 for some h ′ ∈ Mat ( S ) ⊗ L − .Then the algebra E ( S n , g ) is generated over S by degree elements.(ii) E ( S n , g ) is generated by degree and degree elements.(iii) Assume that g is invertible. Then the algebra E ( S n , g ) is Koszul.Proof . (i) Recall that E ( S n , g ) is the subspace End g ( S n ) of elements a ∈ Mat n ( S ) suchthat tr( ga ) = 0. The existence of h such that tr( gh ) = 1 implies that End g ( S n ) is a directsummand in Mat n ( S ).We have to prove the surjectivity of the mapEnd g ( S n ) ⊗ S End g ( S n ) → Mat n ( S ) (2.3.1)induced by the product in Mat n ( S ). irst, we claim that it is enough to prove the assertion in the case when S is a field.We can easily reduce to the case when S is local. Now let M denote the cokernel of theproduct map (2.3.1). Note that the construction of M is compatible with any change ofscalars S → S ′ . Thus, the case of the field implies that M/ m M = 0, where m ⊂ S is amaximal ideal. By Nakayama lemma, this gives that M = 0.Thus, it is enough to consider the case when S = k , where k is a field. In this case wewill prove a more general statement that for any pair of nonzero elements g , g one hasEnd g ( k n ) · End g ( k n ) = Mat n ( k ) , where in the case n = 2 we additionally require that g g = 0. Since the question isthat certain vectors generate Mat n ( k ) as a linear space, without loss of generality we canassume k to be algebraically closed.Note that for any a, b ∈ GL n ( k ) one has a · End g ( k n ) · End g ( k n ) · b = End g a − ( k n ) · End b − g ( k n ) . Thus, we can replace g by g a − and g by b − g . In the case when g and g are invertiblethis reduces the statement to the case g = g = 1, which is easy to check.Next, we observe that for any nonzero g ∈ Mat n ( k ) there exists a ∈ GL n ( k ) such thattr( ag ) = 0. Indeed, otherwise, the entire hyperplane End g ( k n ) would be contained in theirreducible hypersurface det( a ) = 0. Thus, we can assume that tr( g ) = tr( g ) = 0. Inthis case we have 1 ∈ End g ( k n ) and 1 ∈ End g ( k n ). Hence,End g ( k n ) + End g ( k n ) ⊂ End g ( k n ) · End g ( k n ) . Thus, the only case when this subspace is not the entire Mat n ( k ) is when End g ( k n ) =End g ( k n ), i.e., g and g are proportional. In this case we get that End g ( k n ) is asubalgebra. As was observed above, we can assume that g is degenerate. Let us choosea basis in which the last row of g vanishes. Let g ′ denote the ( n − × ( n − g obtained by deleting the last row and last column.Assume first that g ′ = 0. Then End g ( k n ) contains the maximal parabolic subalgebra ofendomorphisms preserving the hyperplane spanned by the first n − n ≥ g ( k n ) cannot be a subalgebra, since it would be strictlybigger than the maximal parabolic subalgebra. In the case n = 2 if g ′ = 0 then g = 0which contradicts the assumption that g g = 0 (recall that g and g are proportional)Next, consider the case g ′ = 0. Let e ij denote the standard matrices with 1 as the ( i, j )-entry. Then for every i we have e in ∈ End g ( k n ), and for every j ≤ n − A j with zero last row such that e nj + A j ∈ End g ( k n ). But then e ij = e in · ( e nj + A j ),so we deduce that End g ( k n ) · End g ( k n ) is Mat n ( k ).(ii) We have to show that the product mapEnd g ( S n ) ⊗ S Mat n ( S ) → Mat n ( S )is surjective. As before, it is enough to consider the case when S = k is a field. Further-more, as in part (i) we reduce to the case when tr( g ) = 0, so that 1 ∈ End g ( S n ), in whichcase the assertion is clear.(iii) It is easy to check that the algebra E ( S n , g ) is quadratic dual to the second Veronesesubalgebra of the algebra S ( S n , g ), corresponding to the vertex X (i.e., one considers aths of even length starting from X ). Since the latter algebra is Koszul by Lemma 1.2.1,the result follows (see [27, Prop. 2.2(i)]). (cid:3) In the next result we consider derivations of E ( S n , g ) as an ungraded algebra (i.e.,there is no Koszul sign in the Leibnitz rule). This result will be used later in studyingautomorphisms of a filtered algebra whose associated graded algebra is isomorphic to E ( S n , g ). Proposition 2.3.2.
Assume that one of the following two conditions holds: • n ≥ and g is invertible; • tr( g ) is a generator of L , and there exists h ∈ GL n ( S ) with tr( gh ) = 0 .Then any derivation E ( S n , g ) → E ( S n , g ) of degree m ≤ − is zero.Proof . The problem is local, so we can assume that L = S .Let us set for brevity E := E ( S n , g ), E ′ := End( V )[ z, z − ]. By the assumption, we canfix an element h ∈ E such that the operators E ′ i → E ′ i +1 of left and right multiplicationby h are invertible for i ∈ Z . Now assume we have a derivation D : E → E of degree m ≤ −
1. First, we are going to extend D to a derivation D ′ : E ′≥ → E ′ of degree m . Forthis we compose D with the embedding E ֒ → E ′ and then set for x ∈ E ′≥ , D ′ ( x ) = D ( xh ) h − − xD ( h ) h − , where we use the operation of multiplication by h − as a degree − E ′≥ → E ′ . Notethat the expression in the right-hand side is well-defined since h ∈ E and xh ∈ E ′≥ = E ≥ . Also we have D ′ = D on E . Before checking that D ′ is a derivation we observe thatfor x ∈ E ′ i , i ≥
1, one has D ′ ( x ) = D ′′ ( x ) = h − D ( h x ) − h − D ( h ) x. Indeed, this can be checked by applying the Leibnitz identity to write D ( h xh ) in twoways (note that h x ∈ E and xh ∈ E ). Now we can prove the Leibnitz identity for D ′ = D ′′ . Namely, it is enough to check that for x , x ∈ E ′≥ one has D ( x x ) = D ′′ ( x ) x + x D ′ ( x ) . It is easy to see that this is equivalent to the identity D ( h x ) x h + h x D ( x h ) = D ( h ) x x h + h D ( x x ) h + h x x D ( h ) , obtained by writing D ( h x x h ) in two ways.Next, we claim that there exists a ∈ End( V ) and s ∈ S such that D ′ ( x ) = ad( az m ) + s · z m +1 ddz . Indeed, as above, we can extend D ′ to a derivation on End( V )[ z, z − ] of degree m . UsingMorita equivalence of the latter ring with S [ z, z − ] we get the result.Let us first assume that m = −
1. Our goal is to show that s = 0 and a is proportionalto the identity. To this end we investigate the condition D ′ (End g ( V ) z ) ⊂ S · id , hich means that for any x ∈ End g ( V ) one has( a + s ) x − xa ∈ S · id , Equivalently, for any y ∈ End( V ) with tr( y ) = 0 and any x ∈ End g ( V ), one hastr(( a + s ) xy − xay ) = tr( x [ y ( a + s ) − ay ]) = 0 . Equivalently, we should have y ( a + s ) − ay ∈ S · g whenever tr( y ) = 0 . (2.3.2)Assume first n ≥ g is invertible. Then substituting y = e i j with i = j in(2.3.2) we get an equality of the form e i j ( a + s ) − ae i j = λ · g. We claim that this is possible only when λ = 0. Indeed, for every j = j we get λ · e j = µ j · g − e i for some µ j ∈ S . Let g − e i = P b i e i . Then we have a system of equations µ j b j = λ, µ j b i = 0 for i = j, j = j . Since g · g − = id, we have some ( a i ) in S with P b i a i = 1. Thus, we deduce µ j = λa j . Plugging back in the above equation, we get that λ · I = 0, where I ⊂ S is the idealgenerated by ( a j b j − j = j and ( a j b i ) i = j,j = j . Now choosing a pair i = j in [1 , n ] \ { j } (this is possible since n ≥ I ⊃ ( a i b i − , a j b j − , a j b i ) = (1) , and hence, λ = 0.Thus, we derive that for every i = j one has e ij ( a + s ) − ae ij = 0This immediately implies that a is diagonal, and the diagonal entries ( a ii ) satisfy a jj − a ii = s for i = j . Using again the assumption n ≥
3, we obtain that s = 0 and a is proportionalto id.Next, let us consider the case when tr( g ) is invertible (but g is not necessarily invertible).Considering traces of both sides of (2.3.2) we derive that y ( a + s ) − ay = 0 whenever tr( y ) = 0 . Now substituting y = e ij for i = j one can easily derive that a has to be a diagonalmatrix. As we have above, in the case n ≥ s = 0 and a is proportionalto id. If n = 2 then taking into account the equation for the diagonal y with entries 1and −
1, we get the same conclusion.In the case m ≤ − D ′ (End g ( V ) z ) = 0 , i.e., ( a + s ) x − xa = 0 for any x ∈ End g ( V ). As we have seen above, this implies that a = s = 0. (cid:3) emark 2.3.3. One can also check that there are no nonzero derivations E ( k , g ) →E ( k , g ) of degree − k is a field of characteristic = 2 and g is invertible. Example 2.3.4.
Assume that n = 2 and 2 = 0 in S , and let us take g = id. Then thereexist nontrivial derivations of E ( S , id) of degree −
1. More precisely, for any 2 × a the derivation ad( az − ) + tr( a ) ddz of Mat ( S )[ z, z − ] restricts to a derivation of E ( S , id).This essentially amounts to the identity[ y, a ] = tr( a ) y + tr( ay ) idfor 2 × a and y such that tr( y ) = 0 (it only holds because 2 = 0 in S ).2.4. The algebra associated with a pair of -spherical objects. Let (
E, F ) be aweak n -pair in a minimal S -linear A ∞ -category with Hom ( E, F ) = V ≃ S n , equippedwith the trivialization Hom ( F, F ) ≃ S. We denote by ξ F ∈ Hom ( F, F ) the corresponding generator.We are going to associate to these data an S -algebra with an increasing exhaustivefiltration ( F n A ), together with an isomorphism of graded S -algebrasgr F A = M n ≥ F n A/F n − A ≃ E ( V, g ) op , (2.4.1)where g ∈ End( V ) ⊗ L E is defined as before. Namely, we use the second of the pairings(2.2.1) to identify Hom ( F, E ) with V ∨ , and define g so that the first pairing takes theform (2.2.2).We will see also that (under some mild assumptions) the filtered algebra ( A, F • A ) isdetermined by the following higher products: m : V ⊗ V ∨ ⊗ V → V ≃ Hom ( E, F ) ⊗ Hom ( F, E ) ⊗ Hom ( E, F ) → Hom ( E, F ) =
V,m : V ∨ ⊗ V ⊗ V ∨ ≃ Hom ( F, E ) ⊗ Hom ( E, F ) ⊗ Hom ( F, E ) → Hom ( F, E ) ≃ V ∨ ,m : V ∨ ⊗ V ≃ Hom ( F, E ) ⊗ Hom ( F, F ) ⊗ Hom ( E, F ) → Hom ( E, E ) = L E ,m : V ∨ ⊗ V ⊗ V ∨ ⊗ V ≃ Hom ( F, E ) ⊗ Hom ( E, F ) ⊗ Hom ( F, E ) ⊗ Hom ( E, F ) → Hom ( E, E ) = S. (2.4.2)Let us define the maps r, r ′ : End( V ) → End( V ) by r ( v ⊗ v ∗ ) = X i m ( e i , v ∗ , v ) ⊗ e ∗ i , r ′ ( v ⊗ v ∗ ) = e i ⊗ X i m ( v ∗ , v, e ∗ i ) , (2.4.3)for v ∈ V , v ∗ ∈ V ∨ , where ( e i ) and ( e ∗ i ) are dual bases of V and V ∨ . Similarly, we define s : End( V ) → L E by s ( v ⊗ v ∗ ) = m ( v ∗ , ξ F , v ) . (2.4.4) Theorem 2.4.1.
Let ( E, F ) be a weak n -pair in a minimal S -linear A ∞ -category C , suchthat Hom ( E, F ) = V ≃ S n , where n ≥ . Let us fix a trivialization Hom ( F, F ) ≃ S ,and let g ∈ End( V ) ⊗ L E be the corresponding element defined using pairings (2.2.1) .Assume in addition that there exists h ∈ End( V ) ⊗ L − E such that tr( gh ) = 1 . Set E i = T i ( E ) ∈ Tw( C ) , here T = T F is the spherical twist with respect to F . Let us consider the graded associa-tive algebra R = R T,E := M n ≥ Hom( E , E n ) , (2.4.5) with the product ab = T i ( a ) ◦ b , where b ∈ Hom( E , E i ) , a ∈ Hom( E , E j ) . Then(i) R is canonically isomorphic to the Rees algebra of a filtered algebra ( A, F • A ) equippedwith an isomorphism gr F ( A ) ≃ E ( V, g ) op ≃ E ( V ∨ , g ∗ ) . In addition, Hom =0 ( E , E n ) = 0 for n > .(ii) There exist embeddings End g ( V ) ֒ → R and End( V ) ֒ → R , such that R = End g ( V ) ⊕ S · t, R = End( V ) ⊕ End g ( V ) · t ⊕ S · t , where t is the central element of degree corresponding to the isomorphism with the Reesalgebra. With respect to these decompositions, for a, b ∈ End g ( V ) ⊂ R one has a · b = ba + [ r ( b ) a + br ′ ( a ) + s ( ba ) h ] t + m ( a ⊗ b ) t . (2.4.6) Here we use the higher products (2.4.2) and the corresponding maps r, r ′ , s (see (2.4.3) , (2.4.4) ).Proof . It will be notationally convenient for a while not to use the trivialization ofHom ( F, F ), so let us set L := Ext ( F, F ).For an S -module M , we set M L i := M ⊗ L ⊗ i . Note that the second of the pairings(2.2.1) induces an identification Hom ( F, E ) ≃ V ∨ L . Step 1 . We start by finding explicit twisted complexes representing E i . Namely, let usdenote by E i , for i ≥
1, the following twisted complex:Hom ( F, E ) L i − ⊗ F δ i ✲ Hom ( F, E ) L i − ⊗ F δ i − ✲ . . . δ ✲ Hom ( F, E ) ⊗ F δ ✲ E. (2.4.7)Here the differentials δ i with i > L ⊗ F → F [1],while the differential δ : Hom ( F, E ) ⊗ F → E [1] is also the evaluation map.We are going to construct the homotopy equivalences T ( E i ) ≃ E i +1 . Note that for i = 0 we have T ( E ) = T ( E ) = E by the definition of the twist functor T = T F . Thecomplex hom( F, E i ) has formHom ( F, E ) L i − ⊗ id F Hom ( F, E ) L i − ⊗ id F · · · Hom ( F, E ) ⊗ id F Hom ( F, E ) L i Hom ( F, E ) L i − ✲ · · · ✲ Hom ( F, E ) L ✲ Hom ( F, E ) ✲ with the first row in degree 0 and the second row in degree 1 (note that higher productsdo not appear since we assume our A ∞ -structures to be strictly unital). Since all thecomponents of the differential are isomorphisms, the natural embedding and the projectiongive a homotopy equivalencehom( F, E i ) ≃ Hom ( F, E ) L i [ − . ence, we deduce a homotopy equivalence E i +1 = Cone(Hom ( F, E ) L i ⊗ F [ − δ i +1 ✲ E i ) ≃ Cone(hom(
F, E i ) ⊗ F ev ✲ E i ) = T ( E i )(2.4.8)as claimed.For what follows we need to know explicitly the maps between E i +1 and T ( E i ). Notethat the embedding of Hom ( F, E ) L i ⊗ F [ −
1] into hom(
F, E i ) ⊗ F commutes with themaps to E i used to form the above cones, however, the projection in the other directiononly commutes up to homotopy. Namely, we have a homotopy h between the map ev :hom( F, E i ) ⊗ F → E i and the compositionhom( F, E i ) ⊗ F → Hom ( F, E ) L i ⊗ F δ i +1 ✲ E i , with the nonzero components h j : Hom ( F, E ) L j ⊗ F id ✲ Hom ( F, E ) L j ⊗ F ֒ → E i , for 0 ≤ j ≤ i −
1. Hence, the map E i +1 → T ( E i ) is given by the obvious embedding ofcomplexes, while the map T ( E i ) → E i +1 is the identity on the summands E i and all thesummands Hom ( F, E ) L j ⊗ F of hom ( F, E i ) ⊗ F . Step 2 . The complex hom( E , E i ) = hom( E, E i ) has form i − M j =0 Hom ( F, E ) L j ⊗ Hom ( E, F ) ! ⊕ Hom ( E, E ) → Hom ( E, E ) , with the differential given by d (id E ) = 0, d ( e ⊗ ξ ⊗ j ⊗ x ) = m j +2 ( e, ξ, . . . , ξ, x ) . Recall that the map m : Hom ( F, E ) ⊗ Hom(
E, F ) → Hom ( E, E ) = L E can beidentified with the map( V ∨ ⊗ L ) ⊗ V = End( V ) ⊗ L → L E : a tr( ga )(note that canonically g is an element of End( V ) ⊗ L − ⊗ L E ). Thus, we immediately seethat for i ≥ =0 ( E , E i ) = 0, while Hom ( E , E i ) fits into an exact sequence0 → End g ( V ) L ⊕ Hom ( E, E ) → Hom ( E , E i ) → i M j =2 End( V ) L j ! → , where we use the identification Hom ( F, E ) ≃ V ∨ L . Step 3 . For 0 ≤ i < j let us consider the map of complexeshom( E i , E j ) T ✲ hom( T ( E i ) , T ( E j )) → hom( E i +1 , E j +1 ) , here the second arrow is induced by the maps E i +1 → T ( E i ) and T ( E j ) → E j +1 de-scribed in Step 1. Then in the case i > ( E i , E j ) ✲ Hom (Hom ( F, E ) L i − ⊗ F, Hom ( F, E ) L j − ⊗ F )hom ( E i +1 , E j +1 ) T ❄ ✲ Hom (cid:0) Hom ( F, E ) L i ⊗ F, Hom ( F, E ) L j ⊗ F (cid:1) ⊗ id L ❄ while in the case i = 0 the following square is commutativehom ( E , E j ) ✲ Hom ( E, Hom ( F, E ) L j − ⊗ F )hom ( E , E j +1 ) T ❄ ✲ Hom (cid:0) Hom ( F, E ) ⊗ F, Hom ( F, E ) L j ⊗ F (cid:1) ❄ where the horizontal arrows are the natural projections, while the right vertical arrowin the second diagram sends a ∈ Hom ( F, E ) L j − ⊗ V ≃ End( V ) L j to a ∗ ⊗ id F ∈ End( V ∨ ) L j ⊗ Hom ( F, F ). Step 4 . For each i ≥
1, let us consider the natural projection (see Step 2) π i : Hom ( E , E i ) → V ∨ L i ⊗ Hom(
E, F ) ≃ End( V ) L i . Note that for i ≥ i = 1 its image is End g ( V ) L . We claim thatthe map π = ( π i ) is a homomorphism of graded algebras R → E ( V, g ) op . To prove this let us consider elements a ∈ Hom ( E , E i ) and b ∈ Hom( E , E j ), where i > j >
0, and set a = π i ( a ), b = π j ( b ). Iterating Step 3 we see the component of T j ( a ) ∈ hom ( E j , E i + j ) in Hom (cid:0) Hom ( F, E ) L j − ⊗ F, Hom ( F, E ) L i + j − ⊗ F (cid:1) is a ∗ ⊗ id F .It is easy to see that the pi i + j ( T j ( a ) ◦ b ) is obtained by composing the above componentof T j ( a ) with b ∈ End( V ) L j ≃ Hom ( E, Hom ( F, E ) L j − ⊗ F ). Thus, if we view b as anelement of V ∨ ⊗ V L j then we get π i + j ( T j ( a ) ◦ b ) = ( a ∗ ⊗ id V )( b ) = ba ∈ End( V ) L i + j . Step 5 . Let us define t ∈ Hom ( E , E ) to be the element represented by the elementid E ∈ Hom ( E, E ) in the complex hom(
E, E ), so that we have a decompositionHom( E , E ) ≃ End g ( V ) L ⊕ S · t. (2.4.9) e claim that for i ≥ T i ( t ) ∈ Hom( E i , E i +1 ) is represented by the closedmap of twisted complexes V ∨ L i ⊗ F δ ✲ V ∨ L i − ⊗ F δ ✲ . . . δ ✲ V ∨ L ⊗ F δ ✲ EV ∨ L i +1 ⊗ F δ ✲ V ∨ L i ⊗ F id ❄ δ ✲ V ∨ L i − ⊗ F id ❄ δ ✲ . . . δ ✲ V ∨ L ⊗ F id ❄ δ ✲ E id ❄ Indeed, this can be easily checked by induction by computing the effect of the twist functor T on the above map and taking into account the homotopy equivalence (2.4.8). Step 6 . It follows easily from Step 5 that the map of the left multiplication by t in ouralgebra, Hom( E , E i ) t · ✲ Hom( E , E i +1 )is injective and its image is given by the classes that have representatives in hom( E , E i +1 )with zero component in V ∨ L i +1 ⊗ Hom(
E, F ) ≃ V ∨ ⊗ V L i +1 ≃ End( V ) L i +1 . Thus, for i ≥ → Hom( E , E i − ) t · ✲ Hom( E , E i ) π i ✲ E ( V, g ) i L i → . (2.4.10)From these exact sequences we deduce that the algebra R is generated by degree 1 anddegree 2 elements. Indeed, since π is a homomorphism, this follows from the similarproperty of E ( V, g ) op (see Lemma 2.3.1). Step 7 . Next, let us consider an element a ∈ End g ( V ) L , and let us view it as a cochainin the term V ∨ L ⊗ Hom(
E, F ) ≃ End( V ) L of the complex hom( E , E ). We would liketo calculate T ( a ). By definition, T ( a ) is obtained as the composition E = Cone(hom( F, E ) ⊗ F ev ✲ E ) → Cone(hom(
F, E ) ⊗ F ev ✲ E ) → E , where the arrow between the cones is induced by a : E → E , and the last arrow is theprojection T ( E ) → E that we computed before. Now the map hom( F, E ) a ✲ hom( F, E )has two nonzero components: the map Ext ( F, E ) → V ∨ L induced by by the compositionwith a and the map µ a : Ext ( F, E ) → Ext ( F, E ) : x m ( δ , a, x ) . (2.4.11)It follows that T ( a ) is represented by the map V ∨ L ⊗ F δ ✲ EV ∨ L ⊗ Fa ∗ ⊗ id F ❄ ✲ V ∨ L ⊗ Fa ❄ ✲ ✲ E (2.4.12) here the diagonal arrow is µ a ⊗ id F . Step 8 . Now let us check that t is central in R . By Step 6, it is enough to check t commutes with elements of degree 1 and 2, lifting arbitrary elements in E ( V, g ) and E ( V, g ) under the homomorphism π . First, let us check that at = ta in R , for a ∈ End g ( V ) L ⊂ R . Note that here at = T ( a ) ◦ t , ta = T ( t ) ◦ a . From our description of T ( t ), we immediately get that ta is represented by the map EV ∨ L ⊗ F ✲ V ∨ L ⊗ Fa ❄ ✲ E (2.4.13)On the other hand, from the description (2.4.12) of T ( a ) it follows immediately that at = T ( a ) ◦ t is represented by the same chain map (2.4.13) as ta .Similarly, let us consider an element A ∈ R represented by a closed map EV ∨ L ⊗ FA ❄ δ ✲ V ∨ L ⊗ F δ ✲ A ✲ E where m ( δ, δ, A ) + m ( δ, A ) = 0. Then one can easily check that T ( A ) ◦ t and T ( t ) ◦ A are both represented by the map EV ∨ L ⊗ F ✲ V ∨ L ⊗ FA ❄ ✲ V ∨ L ⊗ F ✲ A ✲ E Since t is a central nonzero divisor, it follows that the algebra R is the Rees algebra ofa filtered algebra ( A, F • A ). Furthermore, by Steps 4 and 6 we havegr F ( A ) ≃ R / ( t ) ≃ E ( V, g ) op , which proves (i). Step 9 . For a, b ∈ End g ( V ) L ⊂ R , the product ab = T ( a ) ◦ b in R can be easilycomputed using the representation (2.4.12) for T ( a ): we get T ( a ) ◦ b = ( A , A , A ) ∈ hom ( E , E ) = End( V ) L ⊕ End( V ) L ⊕ Hom ( E, E ) , with A = ba, A = m ( a, δ , b ) + ( µ a ⊗ id F ) ◦ b, A = m ( δ , a, δ , b ) , where µ a is given by (2.4.11). rom this point on we use the trivialization L = S · ξ F . Assume for simplicity ofnotation that a = v a ⊗ v ∗ a , b = v b ⊗ v ∗ b , for v a , v b ∈ V , v ∗ a , v ∗ b ∈ V ∨ (the general case isproved similarly).The evaluation map δ : Hom ( F, E ) ⊗ F → E [1] corresponds to the identity element P e i ⊗ e ∗ i in Hom ( F, E ) ∨ ⊗ Hom ( F, E ) ≃ V ⊗ V ∨ . Thus, the map µ a : V ∨ → V ∨ sends v ∗ to X i h v ∗ a , e i i · m ( e ∗ i , v a , v ∗ ) = m ( v ∗ a , v a , v ∗ ) . Similarly, m ( a, δ , b ) = m ( v a , v ∗ b , v b ) ⊗ v ∗ a ,m ( δ , a, δ , b ) = m ( v ∗ a , v a , v ∗ b , v b ) . Thus, A = m ( v a , v ∗ b , v b ) ⊗ v ∗ a + v b ⊗ m ( v ∗ a , v a , v ∗ b ) , A = m ( v ∗ a , v a , v ∗ b , v b ) . Using the operators r and r ′ we can rewrite the formula for A as A = r ( b ) a + br ′ ( a ) . Now recall that R = Hom ( E , E ) is the subspace of hom ( E , E ) consisting of( A , A , A ) such that s ( A ) + tr( gA ) = 0 , where s ( v ⊗ v ∗ ) = m ( v ∗ , ξ F , v ). Thus, we can define the splitting σ of the projection π : R = Hom ( E , E ) → End( V ) by setting σ ( A ) = ( A, − s ( A ) h, ∈ Hom ( E , E ) ⊂ hom ( E , E ) , for A ∈ End( V ). Rewriting the element ( A , A , A ) ∈ Hom ( E , E ) as σ ( A ) + ( A + s ( A ) h ) t + A t we get (2.4.6). (cid:3) Corollary 2.4.2.
Under the assumptions of Theorem 2.4.1, if in addition g is invertible,then the isomorphism class of the corresponding filtered algebra ( A, F • A ) is determined bythe higher products (2.4.2) .Proof . Indeed, in this case by Lemma 2.3.1(iii), E ( V, g ) is generated in degree 1 and hasquadratic defining relations. Hence, the same is true for R . But, by Theorem 2.4.1(ii),the product R ⊗ R → R is determined by the higher products (2.4.2). (cid:3) Until the end of this section we fix an n -pair ( E, F ) and keep the notation of Theorem2.4.1. In particular, R is the graded S -algebra defined by (2.4.5). Let us define thestructure of a graded R − R -bimodule on M i ∈ Z Hom ( E i , E ) = M i ≥ Hom ( E i , E )as follows. The grading of Hom ( E i , E ) is set to be − i . The left and right multiplicationof x ∈ Hom ( E i , E ) by a ∈ Hom ( E , E j ), with j ≤ i , are given by a · x = T − j ( a ◦ x ) , x · a = x ◦ T i − j ( a ) . e would like to calculate this R − R -bimodule, which together with the algebra R completely determines the full subcategory with the objects ( E i ). This is important forproving Theorem A since, as we will show in 5.1 and 5.2, this subcategory can be equippedwith a unique minimal A ∞ -structure up to rescaling (and gauge equivalence). Lemma 2.4.3.
Let B be a non-negatively graded S -algebra, such that B = S and allthe graded components B i are finitely generated projective S -modules. Let M be a graded B − B -bimodule such that M is isomorphic to B ∗ (the restricted dual of B ) as a graded left B -module and as a graded right B -module. Then there exists an automorphism φ : B → B ,preserving the grading, such that M is isomorphic to ( id B φ ) ∗ as a graded B − B -bimodule.Proof . Note that M − i ≃ B ∨ i , so all the graded components of M are finitely generatedprojective modules. Consider the restricted dual M ∗ . Then M ∗ is isomorphic to B as aleft and as a right graded B -module. This easily implies the claim. (cid:3) Proposition 2.4.4.
Under the assumptions of Theorem 2.4.1, assume in addition that g is invertible. Then there is an isomorphism of graded R − R -bimodules M i ≥ Hom ( E i , E ) ≃ ( id R φ ) ∗ ⊗ S L E , (2.4.14) where φ is a graded automorphism of R , such that φ ( t ) = t and the automorphism φ of R / ( t ) ≃ E ( V, g ) op , induced by φ , is equal the restriction of the automorphism Ad( g − ) : x g − xg of End( V )[ z ] op .Proof . Without loss of generality we can assume that C is generated by ( E, F ). Then byLemma 2.2.2, the object E = E is 1-spherical. It follows that all the objects E i = T i ( E )are 1-spherical in C . As before, we fix a trivialization Hom ( F, F ) = S · ξ F , and considerthe identification Hom ( F, E ) ≃ V ∨ , such that the second of the pairings (2.2.1) getsidentified with the the natural pairing V ⊗ V ∨ → S , while the first of these pairings isgiven by (2.2.2). We also use the isomorphisms Hom ( E i , E i ) ≃ Hom ( E, E ) = L E . Step 1 . There exists an isomorphism of bimodules (2.4.14) for some φ .We can use the perfect pairing (given by the composition)Hom ( X, E ) ⊗ Hom ( E , X ) → Hom ( E , E ) = L E to define an isomorphismHom ( X, E ) ⊗ L − E ∼ ✲ Hom ( E , X ) ∨ . These isomorphisms are functorial in X , which immediately implies that summing theseisomorphisms over X = E i , we get an isomorphism of right R -modules M i ≥ Hom ( E i , E ) ⊗ L − E ≃ M i ≥ Hom ( E , E i ) ∨ = R ∗ . (2.4.15)On the other hand, using the perfect pairingsHom ( X, E i ) ⊗ Hom ( E i , X ) → Hom ( E i , E i ) ≃ L E e similarly get an isomorphism of right R -modules R = M i ≥ Hom ( E , E i ) ≃ M i ≥ Hom ( E i , E ) ∨ ⊗ L E . Dualizing we get another isomorphism of the form (2.4.15) which is compatible with theleft R -module structures. By Lemma 2.4.3, there exists an automorphism φ : R → R such that (2.4.14) holds.It remains to check that φ ( t ) = t and to calculate the action of φ on R / ( t ). To this endwe will calculate some compositions of morphisms between E = E and E = [ V ∨ ⊗ F → E ]. Recall that we have a canonical decompositionHom( E , E ) = R = End g ( V ) ⊕ S · t. Step 2 . We construct canonical identificationsHom ( E , E ) = End( V ) / ( S · id) ⊕ L E , (2.4.16) τ : Hom ( E , E ) ∼ ✲ L E , such that the compositionHom ( E, E ) ⊕ Hom ( V ∨ ⊗ F, E ) ⊕ Hom ( V ∨ ⊗ F, V ∨ ⊗ F ) = hom ( E , E ) → Hom ( E , E ) τ ✲ S, where the first arrow is the natural projection to cohomology, is given by( ξ, x, y ⊗ y ∗ ) ξ + h y ∗ , gy i , (2.4.17)where ξ ∈ L E , y ∈ V , y ∗ ∈ V ∨ .By definition, the complex hom( E , E ) has the formHom ( E, E ) → Hom ( V ∨ ⊗ F, E ) ⊕ Hom ( E, E ) , where the differential maps id E to the evaluation morphism in Hom ( V ∨ ⊗ F, E ). Thisimmediately leads to the identification (2.4.16).We havehom ( E , E ) = Hom ( E, E ) ⊕ Hom ( E, V ∨ ⊗ F ) ⊕ Hom ( V ∨ ⊗ F, V ∨ ⊗ F )and the part of the differential hom ( E , E ) → hom ( E , E ) that maps to the summandsHom ( E, E ) ⊕ Hom ( V ∨ ⊗ F, V ∨ ⊗ F ) is the mapHom ( E, V ∨ ⊗ F ) ( − (? ◦ δ ) ,δ ◦ ?) ✲ Hom ( E, E ) ⊕ Hom ( V ∨ ⊗ F, V ∨ ⊗ F ) , with both components induced by the evaluation map V ∨ ⊗ F → E [1]. We can identifythis map with the map V ⊗ V ∨ → L E ⊕ V ⊗ V ∨ : y ⊗ y ∗ ( −h y ∗ , gy i , y ⊗ y ∗ ) . Thus, the map hom ( E , E ) → S given by (2.4.17) descends to a map on cohomology, τ : Hom ( E , E ) → S . It is clear from the definition that τ is surjective. Since E is1-spherical, we deduce that τ is an isomorphism. Step 3 . Let T : L E = Hom ( E, E ) → Hom ( E , E ) be the map induced by the sphericaltwist T . Then τ ◦ T is the identity map of L E . Hence, the mapHom ( E , E ) ⊗ Hom ( E , E ) → Hom ( E , E ) = S : a ⊗ x a · x = T − ( a ◦ x ) , ends a ⊗ x to τ ( a ◦ x ).Indeed, it is easy to see that T ( ξ ) is given by the element ξ ∈ Hom ( E, E ) ⊂ hom ( E , E ). Step 4 . For B ∈ End( V ) / ( S · id) ⊂ Hom ( E , E ) one has B ◦ t = 0 in Hom ( E , E )and t ◦ B = 0 in Hom ( E , E ). Also, viewing ξ ∈ L E as an element of Hom ( E , E ) (see(2.4.16)), we get ξ ◦ t = ξ and τ ( t ◦ ξ ) = ξ . Hence, for any x ∈ Hom ( E , E ), we have t · x = x · t in the bimodule L i ≥ Hom ( E i , E ).Indeed, recall that t corresponds to the element id E ∈ Hom ( E, E ) ⊂ hom ( E, E ).The vanishing of the composition B ◦ t = 0 is clear from the composition rule: EV ∨ ⊗ F δ ✲ E id E ❄ EB ❄ The composition t ◦ B is calculated by the diagram V ∨ ⊗ F δ ✲ EEB ❄ V ∨ ⊗ F δ ✲ E id E ❄ Thus, it belongs to the summand Hom ( V ∨ ⊗ F, E ) ⊂ hom ( E , E ), which is annihilatedby τ , hence zero in cohomology. he composition ξ ◦ t is calculated by the diagram EV ∨ ⊗ F δ ✲ E id E ❄ Eξ ❄ Finally, the composition t ◦ ξ is calculated by the diagram V ∨ ⊗ F δ ✲ EEξ ❄ V ∨ ⊗ F δ ✲ E id E ❄ so it is given by the element ξ ∈ Hom ( E, E ) ⊂ hom ( E , E ), which implies the assertion. Step 5 . For A ∈ End g ( V ) ⊂ Hom ( E , E ) and B ∈ End( V ) / ( S · id), we have B ◦ A = tr( BgA ) ,τ ( A ◦ B ) = tr( AgB ) . Indeed, the composition B ◦ A is calculated by the diagram EV ∨ ⊗ FA ❄ δ ✲ EEB ❄ e claim that the composition of the vertical arrows is tr( BgA ). Indeed, it is easy tosee using our conventions, that for A = v ∗ ⊗ v and B = w ∗ ⊗ w , this composition will be h v ∗ , w i · h w ∗ , gv i , which gives our claim.Finally, the composition A ◦ B is calculated by the diagram V ∨ ⊗ F δ ✲ EEB ❄ V ∨ ⊗ FA ❄ δ ✲ E Note that this composition will have a component in Hom ( V ∨ ⊗ F, V ∨ ⊗ F ) given by thecomposition m ( A, B ) of the vertical arrows, as well, as a component in Hom ( V ∨ ⊗ F, E )given by m ( δ, A, B ). However, the latter component does not give any contribution tothe cohomology class, due to formula (2.4.17). For A = v ∗ ⊗ v and B = w ∗ ⊗ w , we have m ( A, B ) = h w ∗ , v i · w ⊗ v ∗ , so applying τ we get τ ( m ( A, B )) = h w ∗ , v i · h v ∗ , gw i = tr( AgB ) . Step 6 . Since R is generated in degree 1, the automorphism φ is uniquely determinedby its restriction to R , which is in turn uniquely determined by the equation x · φ ( a ) = a · x in L E = Hom ( E , E ), where a ∈ R , x ∈ Hom ( E , E ). Hence, by Step 4, we deducethat φ ( t ) = t . Furthermore, still by Step 4, for a ∈ R , one has B · a = 0 for all B ∈ End( V ) / ( S · id) ⊂ Hom ( E , E ) if and only if a ∈ S · t . Therefore, for A ∈ End g ( V ),the element φ ( A ) mod S · t is determined by the products B · φ ( A ) in Hom ( E , E ). Nowthe calculation of Step 5 implies that φ ( A ) ≡ g − Ag mod S · t . (cid:3) Recall that R is identified with the Rees algebra R ( A ) (see Theorem 2.4.1). Thus, agraded automorphism φ of R such that φ ( t ) = t is the same thing as a filtered automor-phism of A . In view of Proposition 2.4.4 it is important to study such automorphisms.The next result shows that in some cases φ is uniquely determined by the induced auto-morphism of gr F A . Proposition 2.4.5.
Let V = S n , where n ≥ , and let g ∈ End S ( V ) ⊗ L be an invertibleelement (where L is a locally free S -module of rank ). Assume that either n ≥ or tr( g ) is a generator of L . Let ( A, F • A ) be a filtered S -algebra, such that gr F A ≃ E ( V, g ) op . (2.4.18) Suppose φ and φ are automorphisms of A , preserving the filtration (i.e., φ i F j A = F j A ),such that the induced automorphisms of gr F A are the same. Then φ = φ . roof . Let us consider the automorphism φ = φ − φ of A . Then φ still preserves thefiltration and induces the identity on gr F A . It is easy to check that setting for a ∈ F n AD ( a ) = φ ( a ) − a mod F n − A we get a well defined derivation D : gr F A → gr F A of degree −
1. Using (2.4.18) andProposition 2.3.2, we deduce that D = 0. In particular, φ ( a ) = a for a ∈ F A . But Lemma2.3.1(i) implies that A is generated by F A (since this is true for gr F A ≃ E ( V, g ) op ). Hence,it follows that φ ( a ) = a for all a ∈ A . (cid:3) Connection with noncommutative orders over stacky curves
Filtered algebras and orders.
In this section we work over a fixed ground field k . Let A be a filtered algebra over k equipped with an isomorphism (2.4.1) for some g ∈ P End( V ), and let Z ⊂ A be its center. We equip Z with the induced filtration. Lemma 3.1.1.
The algebra A is finitely generated, prime and of GK-dimension . Hence, A is Noetherian and finite over its center Z , which is a -dimensional domain, finitelygenerated as k -algebra. Also, A is an order in a central simple algebra over the quotientfield of Z .Proof . Since the algebra E ( V, g ) is generated by degree 1 and degree 2 elements (seeLemma 2.3.1(ii)), we deduce that A is generated by F A , hence, it is finitely generated.Given a nonzero ideal I ⊂ A , let I ⊂ End( V ) be the set of all elements x such that xz n appears as an initial form of an element of I for some n . Then I is a nonzero ideal,hence, I = End( V ). Hence, for a pair of nonzero ideals I, J ⊂ A we have I J = 0,so IJ = 0, which shows that A is prime. We have dim F i A/F i − A = n for i >
1, sothe GK-dimension of A is one. Now the results of [33] and [29] imply that A and Z areNoetherian, A is finite over Z , and Z has dimension 1.Note that the center of gr F ( A ) ≃ E ( V, g ) op is either k [ z , z ] ⊂ k [ z ], in the case whentr( g ) = 0, or k [ z ], when tr( g ) = 0. Thus, gr F ( Z ) is a graded k -subalgebra in k [ z ], i.e., agroup algebra of a subsemigroup in natural numbers. This easily implies that the algebra R ( Z ) is a domain, finitely generated as a k -algebra. Next, the fact that gr F ( A ) ≃ E ( V, g ) op is torsion free as a module over gr F ( Z ) ⊂ k [ z ] implies that A is torsion free as a Z -module.Let K be the quotient field of Z . Then A ⊗ Z K is a finite-dimensional prime algebra over K with the center K , so it is a central simple algebra over K . (cid:3) Next, we would like to extend A to a sheaf of algebras over a projective curve compact-ifying Spec( Z ). The first obvious choice is to consider the Rees algebras R ( A ) = M m ≥ F m A and R ( Z ) = M m ≥ F m Z and to consider the corresponding Proj-construction. However, the resulting structuresare not always easy to analyze. Namely, the problem arises when gr F ( Z ) is contained in k [ t d ] for some d ≥
2. It turns out that a better behaved construction is provided by thestacky version of Proj, which we denote by Proj st . amely, for any commutative non-negatively graded k -algebra B = L n ≥ B n , where B = k , one can define a stackProj st ( B ) := Spec( B ) \ { } / G m , (3.1.1)where 0 is the point corresponding to the augmentation ideal B + . Assuming in additionthat B is finitely generated, we have an equivalence of the category Coh(Proj st ( B )) withthe quotient of the category of finitely generated graded B -modules by the subcategoryof finite-dimensional modules. Note that we have a natural line bundle O (1) on Proj st ( B )such that elements of B n can be viewed as global sections of O ( n ). The coarse moduli forProj st ( B ) is the usual scheme Proj( B ).Now starting with an algebra A as above we define the stacky curve C by C := Proj st R ( Z ) . Let d ≥ F ( Z ) ⊂ k [ t d ]. We will see below that d measuresthe “stackiness” of C (see Lemma 3.1.2(i)). In particular, d = 1 if and only if C is theusual curve.Let us denote by t the element 1 ∈ R ( Z ) = F A ∩ Z . Note that t is a non-zero-divisor,and R ( A ) /t R ( A ) ≃ gr F ( A ), R ( Z ) /t R ( Z ) ≃ gr F ( Z ). Since gr F ( A ) ≃ E ( V, g ) op if finitelygenerated as a gr F ( Z )-module (see the proof of Lemma 3.1.1), we deduce that R ( A ) isfinitely generated as an R ( Z )-module.Thus, localizing R ( A ) we get a sheaf of coherent O -algebras A on C . More precisely, ^ R ( A ) is a G m -equivariant sheaf of coherent O -algebras on Spec( R ( Z )). The sheaf ofalgebras A is obtained by restricting ^ R ( A ) to Spec( R ( Z )) \ { } and descending the resultto C .Note that the complement to the divisor ( t = 0) in C is identified with the affine curveSpec( R ( Z ) / ( t − Z ), the restriction of A to it gets identified with the coherentsheaf of algebras corresponding to A . Lemma 3.1.2. (i) The pointed stacky curve C is neat, and the divisor ( t = 0) is theunique stacky point p = e p/µ d (where e p = Spec( k ) ). Thus, we have O C (1) ≃ O C ( p ) .There is a natural identification of the cotangent space T ∗ e p C with χ , the -dimensionalspace on which µ d acts with weight .(ii) The sheaf A is an order on C , i.e., a torsion-free coherent sheaf of O -algebras, whosestalk at the generic point is a central simple K ( X ) -algebra. Furthermore, the center of A is O C .(iii) One has a natural isomorphism of algebras on p , A| p ≃ ρ ∗ End( V ) op , (3.1.2) where ρ : e p → p is the natural morphism. Similarly, we have a natural isomorphism A ( mp ) | p ≃ ρ ∗ End( V ) for every m ∈ Z , compatible with the above isomorphism via theidentification O ( mp ) | p ≃ χ − m . The rank of A is equal to dn .(iv) The natural map F m A = R m ( A ) → H ( C, A ( mp )) is an isomorphism for m ∈ Z (where F − A = 0 ), so we have an isomorphism of graded algebras R ( A ) ≃ M n ≥ H ( C, A ( n )) . (3.1.3) ne has h ( A ) = 1 and h ( A ( p )) = 0 . In particular, A is a weakly spherical order in thesense of Definition 0.1.3.Proof . (i) We have seen that R ( Z ) is a domain, so C is integral. Also, the coarse moduliis Proj R ( Z ) which is a projective curve. The divisor p := ( t = 0) ⊂ C can be identified with Proj st (gr F ( Z )). Since gr F ( Z ) ⊂ k [ z d ], and these algebras agree inall sufficiently high degrees, we see that Spec(gr F ( Z )) is an affine line with the pinchedorigin. In particular, p = (Spec(gr F ( Z )) \ / G m ≃ Bµ d . Set S = Spec( R ( Z )) \ { } . We can view t as a map S → A and the fiber over 0, D ⊂ S is a closed G m -orbit with the stabilizer µ d . Since D = Spec(gr F ( Z )) \ { } is smooth, thesurface S is smooth near D . By the argument of Luna’s ´etale slice theorem (see [16]),there exists a smooth µ d -invariant locally closed curve Σ ⊂ S through the point z = 1 of D such that the induced map of stacks Σ /µ d → S/ G m is ´etale.The identification of the cotangent space at e p with χ comes from the fact that p is givenby the equation t = 0, where t is a section of O C (1).(ii) This first assertion follows from Lemma 3.1.1. Also, we know that R ( Z ) is the centerof R ( A ), so O C is the center of A .(iii) Note that A / A ( − p ) is the localization of the graded module gr F ( A ) ≃ E ( V, g ) op over gr F ( Z ). Up to finite-dimensional pieces, this is the same as considering End( V ) op [ z ]as a k [ z d ]-module, which easily implies the isomorphism (3.1.2). If we identify sheaveson p with µ d -representations then the functor ρ ∗ is given by tensoring with the regularrepresentation of µ d , so the image of ρ ∗ is stable under tensoring with χ .Finally, since p is a smooth (stacky) point of C , the sheaf A is locally free near p . Thus,by considering ranks in the isomorphism (3.1.2), we obtain that the rank of A is dn .(iv) Note that the natural isomorphism α : A ∼ ✲ H ( C \ { p } , A )sends F m A to H ( C, A ( m )). It is easy to check that the induced map of the associatedgraded spaces has as components the natural maps F m A/F m − A ≃ E ( V, g ) m ֒ → End( V ) = H ( p, ρ ∗ End( V )) ≃ H ( p, A ( mp ) | p ) , where we use (iii). It follows that α − H ( C, A ( mp )) = F m A for every m ∈ Z . Inparticular, H ( C, A ) = k .Note that for m ≥ F m A = mn . Hence, for sufficiently large m wehave χ ( A ( mp )) = h ( A ( mp )) = mn . Hence, χ ( A ) = 0 and χ ( A ( p )) = n . Thus, since h ( A ) = 1 and h ( A ( p )) = n , we get h ( A ) = 1 and h ( A ( p )) = 0. (cid:3) We will now start using the framework of noncommutative projective geometry asdeveloped in [2]. Thus, for a (non-negatively) graded Noetherian algebra B we considerthe category qgr B , the quotient of the category of finitely generated right B -modulesby the subcategory of torsion modules. This category is equipped with a grading shiftfunctor M M (1) and a special object O (the image of B ). We refer to these data asProj nc B , the noncommutative Proj of B . roposition 3.1.3. The category
Coh( A op ) of coherent right A -modules is equivalent tothe category qgr R ( A ) .Proof . This is proved similarly to [3, Prop. 2.3]. Let ^ R ( A ) be the coherent sheaf of O -algebras on Spec( R ( Z )) associated with R ( A ), viewed as an R ( Z )-algebra. The cate-gory Coh( A op ) can be identified with the category of G m -equivariant coherent sheaves of ^ R ( A ) op -modules on Spec( R ( Z )) \ { } . The latter category is the quotient of the categoryof G m -equivariant coherent sheaves of ^ R ( A ) op -modules on Spec( R ( Z )) by the subcategoryof sheaves with support in { } . This immediately leads to the required equivalence. (cid:3) Below we refer to the condition χ introduced in [2, Def. 3.7] which is useful in the contextof noncommutative projective geometry. We also use the notion of the cohomologicaldimension of Proj nc of a graded Noetherian algebra B defined in terms of the cohomologyfunctor H i ( · ) := Ext i qgr B ( O , · )(see [2, Sec. 7]). Note that for i ≥
1, one has an isomorphism H i ( M ) ≃ lim m →∞ Ext i +1 B op ( B/B ≥ m , M ) (3.1.4)(see [2, Prop. 7.2]). Thus, finiteness of the cohomological dimension of Proj nc B is equiv-alent to finiteness of the cohomological dimension of the functorΓ B + = lim m →∞ Hom B op ( B/B ≥ m , M ) . Corollary 3.1.4.
The algebra R ( A ) is right Noetherian, satisfies the condition χ , and Proj nc R ( A ) has cohomological dimension ≤ . The same is true for the algebra R ( A ) op .Proof . We can rewrite isomorphism (3.1.3) as R ( A ) ≃ M n ≥ H ( C, A ( n )) ≃ M n ≥ Hom A op ( A , A ( n )) . Hence, from Proposition 3.1.3, by [2, Thm. 4.5], we get that R ( A ) is right Noetherianand satisfies χ .Next, we claim that for every coherent right A -module M the spaces Ext j A op ( A , M )are finite-dimensional, Ext > A op ( A , M ) = 0, and Ext j A op ( A , M ( i )) = 0 for i ≫
0. Indeed,this immediately follows from the identification Ext j A op ( A , M ) ≃ H j ( C, M ) (note thatthe latter cohomology is isomorphic to the cohomology of the push-forward of M to thecoarse moduli space of C ).By Proposition 3.1.3, we deduce a similar statement for the cohomology functor H ∗ on the category qgr R ( A ). In particular, we see that the cohomological dimension ofProj nc R ( A ) is ≤
1. Now the fact that R ( A ) satisfies χ follows from [2, Thm. 7.4(2)].The last assertion follows from the fact that gr F ( A op ) ≃ E ( V, g ) ≃ E ( V ∨ , g ∗ ) op , so wecan repeat the argument with A replaced by A op . (cid:3) .2. Spherical orders and duality.
Let A be an order over a proper stacky curve C with a stacky point p ≃ Bµ d such that A| p ≃ ρ ∗ End( V ) op , where ρ : e p → p is the µ d -covering of p by e p ≃ Spec( k ). Then we can view ρ ∗ V as a right A -module supported at p . Note that if d = 1 then this module is V ⊗ O p . Lemma 3.2.1.
Let A be an order over a neat pointed stacky curve C with the uniquestacky point p ≃ Bµ d ∈ C , such that A| p ≃ ρ ∗ End( V ) op , where V is a finite-dimensionalvector space. Then the pair of A op -modules ( A , ρ ∗ V ) (resp., ( A , A ( − p )) ) split generates Perf( A op ) .Proof . First, we note that the A -module A| p is the direct sum of several copies of ρ ∗ V .In particular, ρ ∗ V is in Perf( A op ), and the pairs ( A , ρ ∗ V ) and ( A , A ( − p )) split generatethe same subcategory. Next, using the exact sequences0 → A ( − ( m + 1) p ) → A ( − mp ) → A| p → m ≥
0, we see that the subcategory hA , ρ ∗ V i generated by our objects contains all A ( − mp ) for m ≥ A -module M there exists a surjection of the form L Ni =1 A ( − n i ) → M for some n i ≥
0. Indeed, let p = Bµ d . This is a unique stacky point,and µ d acts on the fiber of O C ( − p ) at p by the identity character. Hence, using [19, Prop.5.2], we see that the bundle E = L di =1 O ( − ip ) over C has the property that the map π ∗ π ∗ Hom( E , F ) ⊗ E → F , where π : C → C is the coarse moduli map, is surjective for every quasicoherent sheaf F on C . Let p ∈ C be the image of p . Then π ∗ O C ( p ) ≃ O C ( dp ). Thus, viewing a coherentright A -module M as a coherent sheaf of O -modules, we get a surjection of the form E ( − dmp ) ⊕ N ≃ π ∗ E ( − mp ) ⊕ N → π ∗ Hom( E , M ) ⊗ E → M. Hence, the induced map of right A -modules E ( − dmp ) ⊕ N ⊗ A → M is also surjective, which proves our claim.Now we can repeat the well known argument for the category of perfect O -modules(see e.g., the proof of [20, Thm. 4]): starting with any perfect complex of A -modules E ,we can find bounded above complex P • , where each P i is a direct sum of modules of theform A ( − mp ), and a quasi-isomorphism P • → E . Now we consider brutal truncation σ ≥− n P • for sufficiently large n . The cone of the composition σ ≥− n P • → P • → E will be isomorphic in the derived category to F [ n + 1], where F is a coherent right A -module. Furthermore, for sufficiently large n , we will have Hom( E, F [ n + 1]) = 0, so wededuce that E is a direct summand in σ ≥− n P • . (cid:3) We say that a pairing
F ⊗ G → H , where F , G and H are coherent sheaves on a scheme, is perfect on the left (resp. on theright ) if the induced map F →
Hom( G , H ) (resp., G →
Hom( F , H )) is an isomorphism. e say that such a paring is perfect in the derived category (on the left or on the right)if the similar statements hold with Hom replaced by R Hom.
Proposition 3.2.2.
Let A be an order over an integral proper stacky curve C , which issmooth near all stacky points and satisfies H ( C, O ) = k .(i) A is spherical if and only if h ( C, A ) = 1 and there is an isomorphism of left A -modules A ≃
Hom( A , ω C ) , (3.2.1) where ω C is the dualizing sheaf on C (equivalently, one can ask for an existence of anisomorphism of right A -modules above). In particular, A is spherical if and only if A op is spherical.Furthermore, if A is spherical then h ( C, A ) = h ( C, A ) = 1 and for a nonzero mor-phism τ : A → ω C (which is unique up to rescaling) the pairing A ⊗ A → ω C : ( x, y ) τ ( xy ) is perfect in the derived category (on both sides).(ii) Assume now that ( C, p ) is a neat pointed stacky curve, and A is a spherical order overit, such that A| p ≃ ρ ∗ End( V ) . Let g ∈ End( V ) be the element such that the morphism End( V ) ≃ H ( A ( p ) | p ) τ | p ✲ H ( ω C ( p ) | p ) ≃ k is of the form x tr( gx ) . Then g is invertible, and the boundary homomorphism End( V ) ≃ H ( A ( p ) | p ) → H ( A ) ≃ k, associated with the exact sequence → A → A ( p ) → A ( p ) | p → , is also of the form x tr( gx ) , for an appropriate choice of an isomorphism H ( A ) ≃ k . In addition, onehas h ( C, A ( p )) = 0 .Proof . (i) For any vector bundles V , V ′ over C we have an isomorphismHom A ( A ⊗ V , A ⊗ V ′ ) ≃ Hom( V , A ⊗ V ′ ) , whereas Ext i vanish for i >
0. Hence, we have isomorphismsExt i A ( A , A ⊗ V ) ≃ H i ( C, A ⊗ V ) , Ext i A ( A ⊗ V , A ) ≃ Ext i ( V , A ) . In particular, Ext i ( A , A ) ≃ H i ( C, A ). Furthermore, the canonical pairingsExt − i A ( A ⊗ V , A ) ⊗ Ext i A ( A , A ⊗ V ) → Ext A ( A , A )get identified with the natural composed mapsExt − i ( V , A ) ⊗ H i ( A ⊗ V ) → H ( A ⊗ A ) → H ( A ) , (3.2.2)where the second arrow is induced by the multiplication on A . Since the modules of theform A⊗V split generate Perf( A ), we deduce that A is 1-spherical as an object of Perf( A )(i.e., the order A op is spherical) if and only if h ( A ) = h ( A ) = 1 and all the pairings(3.2.2) are perfect.Now assume that A is 1-spherical in Perf( A ). The Serre duality on C gives us perfectpairings Ext − i ( A ⊗ V , ω C ) ⊗ H i ( A ⊗ V ) → H ( ω C ) . n particular, we have a nonzero generator τ in the 1-dimensional space Hom( A , ω C ) suchthat the induced map H ( A ) H ( τ ) ✲ H ( ω C ) is an isomorphism. It is easy to check thatthe map (3.2.2) for i = 1 fits into a commutative diagramHom( V , A ) ⊗ H ( A ⊗ V ) ✲ H ( A )Hom( V , Hom( A , ω C )) ⊗ H ( A ⊗ V ) ❄ ✲ H ( ω C ) H ( τ ) ❄ (3.2.3)where the bottom arrow is the Serre duality pairing combined with the isomorphismHom( V , Hom( A , ω C )) ≃ Hom(
A ⊗ V , ω C ) , and the left vertical arrow comes from the morphism of left A -modules ν = ν τ : A →
Hom( A , ω C ) : a ( x τ ( xa )) . Since both horizontal arrows give perfect pairing and H ( τ ) is an isomorphism, we deducethat the map Hom( V , A ) → Hom( V , Hom( A , ω C )) , induced by ν , is an isomorphism for all vector bundles V . It follows that ν is an isomor-phism.Note for an order A with h ( A ) = 1, the biduality morphism A →
Hom(Hom( A , ω C ) , ω C )is an isomorphism. Indeed, since ω C is a dualizing sheaf on C , it suffices to check thatExt > ( A , ω C ) = 0. Equivalently, we have to check thatExt i ( A , ω C ⊗ L m ) = H ( C, Ext i ( A , ω C ) ⊗ L m ) = 0for i > m ≫
0, where L be an ample line bundle on C (see [6, Prop. 6.9]). BySerre duality, this reduces to the vanishing of H ( C, A ⊗ L − m ), which is clear since everyglobal section of A is a scalar multiple of the unit.The above biduality statement easily implies that the bilinear pairing A ⊗ A → ω C : a ⊗ a ′ τ ( aa ′ )for some τ : A → ω C induces an isomorphism ν of left A -modules as above if and only ifthe corresponding morphism of right A -modules ν ′ : A →
Hom( A , ω C ) : a ( x τ ( ax ))is an isomorphism.Now let us start with an order A such that h ( A ) = 1 and there exists an isomorphismof left A -modules A ≃
Hom( A , ω C ). Let τ ∈ Hom( A , ω C ) be the element corresponding tothe unit global section of A under this isomorphism. Then the isomorphism is equal to ν τ .Since τ generates the space Hom( A , ω C ), by Serre duality, the map H ( A ) H ( τ ) ✲ H ( ω C ) s an isomorphism. Hence, the diagram 3.2.3 implies that the pairing (3.2.2) for i = 0 isperfect. Now the pairing (3.2.2) for i = 1 fits into a similar diagramExt ( V , A ) ⊗ H ( A ⊗ V ) ✲ H ( A )Ext ( V , Hom( A , ω C )) ⊗ H ( A ⊗ V ) ❄ ✲ H ( ω C ) H ( τ ) ❄ (3.2.4)Now we have an isomorphismExt ( V , Hom( A , ω C )) ≃ H (Hom( V , Hom( A , ω C ))) ≃ H (Hom( A⊗V , ω C )) ≃ Ext ( A⊗V , ω C )since Ext ( A ⊗ V , ω C ) = 0. Thus, the pairing given by the bottom horizontal arrow indiagram (3.2.4) is perfect, hence, so is the pairing given by the top horizontal arrow.For the last assertion, we use the isomorphism H ( C, A ) ∗ ≃ H ( C, Hom( A , ω C )) ≃ H ( C, A ) , together with the vanishing of Ext > ( A , ω C ) observed before (which holds since h ( C, A ) =1).(ii) We can think of ρ ∗ End( V ) as the algebra End( V ) ⊗ R µ d in the category of µ d -representations, where R µ d = d − M i =0 χ i is the regular representation of µ d . The restriction τ | p : A| p → ω C | p can be viewed as amorphism of µ d -representations, End( V ) ⊗ R µ d → χ, whose unique non-trivial component, End( V ) ⊗ χ → χ , corresponds to the functional x tr( gx ) on End( V ). The fact that the induced pairing τ | p ( xy ) on End( V ) ⊗ R µ d isnondegenerate easily implies that g is invertible.Now let us consider the morphism of exact sequences induced by τ ,0 ✲ A ✲ A ( p ) ✲ A ( p ) | p ✲ ✲ ω C τ ❄ ✲ ω C ( p ) τ ❄ ✲ ω C ( p ) | p τ | p ❄ ✲ assing to the corresponding exact sequences of cohomology, we get a commutative square H ( A ( p ) | p ) ✲ H ( A ) H ( ω C ( p ) | p ) τ | p ❄ ✲ H ( ω C ) τ ❄ in which the horizontal arrows are boundary homomorphisms. The non-degeneracy of thepairing Hom( A , ω C ) ⊗ H ( A ) → H ( ω C )implies that the right vertical arrow is an isomorphism. Since the bottom horizontal arrowis also an isomorphism, we deduce that the top horizontal arrow can be identified with τ | p . (cid:3) Special spherical orders over the cuspidal cubic.
Let C cusp be a cuspidal curveof arithmetic genus 1 over a field k , q a singular point, p a smooth point. Note that thenormalization map is a homeomorphism, so we can identify C cusp with P as a topologicalspace. We assume that p corresponds to ∞ ∈ P , while q corresponds to 0 ∈ P . For an n -dimensional vector space V and g ∈ GL( V ), let us define an order A cusp g over C cusp asthe subsheaf of algebras A cusp g ⊂ End( V ) ⊗ O P , consisting of the elements that have anexpansion a ( z ) = c · I + a z + . . . near 0 ∈ P , with c ∈ k and tr( ga ) = 0.Note that functions on C cusp have expansion f ( z ) = a + a z + . . . near 0 ∈ P , so theorder A cusp g indeed contains O C cusp . It contains O P precisely when tr( g ) = 0.Let us denote by tr g : A cusp → O C cusp the homomorphism induced by the mapEnd( V ) ⊗ O P → O P : A tr( gA ) . Lemma 3.3.1.
The O -bilinear form tr g ( aa ′ ) on A cusp induces an isomorphism of right A cusp -modules A cusp ∼ ✲ H om O C cusp ( A cusp , O C cusp ) : a ( a ′ tr g ( aa ′ )) . (3.3.1) Proof . Away from q this is clear, so it is enough to consider the completions of thestalks at q . Since the localization [ A cusp q [ z − ] is just the matrix algebra over k (( z )), weknow that any functional [ A cusp q → k [[ z ]] has form a tr g ( ab ), for some b = b − n z − n + b − n +1 z − n +1 + . . . Now considering the condition that tr g ( ab ) has to be a formal series ofthe form c + c z + . . . , we easily deduce that b has to be in [ A cusp q . (cid:3) Remark 3.3.2.
We also have an isomorphism of left A cusp -modules like (3.3.1), given by a ′ ( a tr g ( aa ′ )), which is in general different from (3.3.1).The order A cusp g is closely related to the algebra E ( V, g ) defined by (0.1.1). emma 3.3.3. Under the construction of Sec. 3.1, the order ( A cusp g ) op over C cusp comesfrom the algebra E ( V, g ) op , viewed as a filtered algebra. In particular, we have an equiva-lence mod −A cusp g ≃ qgr E ( V, g )[ t ] and an isomorphism of graded algebras E ( V, g )[ t ] ≃ M n ∈ Z H ( C cusp , A cusp g ( n )) . Also, the algebra E ( V, g ) op [ t ] is right Noetherian and satisfies χ .Proof . First, we need to identify A cusp g with the sheafification of E ( V, g )[ t ], viewed as agraded module over k [ z , z ][ t ]. For this we observe that A cusp g is the subsheaf in End( V ) ⊗O P , which coincides with End( V ) ⊗ O P over the complement to q . The same is true forthe sheafification of E ( V, g )[ t ]. Hence, it is enough to compare the restrictions of the twosheaves to the affine open subset C cusp \ { p } (i.e., the open subset t = 0). It remains toobserve that H ( C cusp \ { p } , A cusp g ) = E ( V, g ) ⊂ End( V )[ z ] . This proves the first first assertion. The other assertions follow from Lemma 3.1.2 andCorollary 3.1.4. (cid:3)
AS-Gorenstein condition over a field.
For graded modules M and N over agraded algebra B we use the notationExt iB ( M, N ) := M j ∈ Z Ext i B− gr ( M, N ( j ))where B − gr is the category of (left) B -modules.Recall that a connected graded algebra B over a field k is called left Artin-SchelterGorenstein ( AS-Gorenstein ) with the parameter ( d, m ) if B has a finite left injectivedimension and Ext B ( k, B ) is 1-dimensional, concentrated in degree d and internal degree m . Similarly one defines the notion of right AS-Gorenstein. Proposition 3.4.1.
For any g ∈ GL( V ) the algebra E ( V, g )[ x ] (where deg( x ) = 1 ) is leftand right AS-Gorenstein with the parameter (2 , .Proof . Let us set B = E ( V, g )[ x ]. It is easy to see that E ( V, g ) op ≃ E ( V ∨ , g ∗ ), so it isenough to check that B is right Gorenstein. We have a natural identification of gradedalgebras B = M n ∈ Z H ( C cusp , A cusp g ( n )) , where O (1) = O ( p ) (see Lemma 3.3.3). Next, by Lemma 3.3.1, for any n ∈ Z the map A cusp ( n ) ∼ ✲ H om O C cusp ( A cusp ( − n ) , O C cusp ) ,a ( a ′ tr g ( aa ′ )), is an isomorphism of right A cusp -modules. Since the C cusp is Goren-stein with ω C cusp ≃ O C cusp , by Serre duality, we deduce an isomorphism M n ∈ Z H ( C cusp , A cusp ( n )) ∼ ✲ M n ∈ Z H ( C cusp , A cusp ( − n )) ∗ f right B -modules. In other words, we have an isomorphism of left B -modules M n ∈ Z H ( C cusp , A cusp ( n )) ≃ B ∗ , (3.4.1)where B ∗ is the restricted dual of B .Next, let us consider the bar-resolution of k by the complex of free right B -modules, . . . → B + ⊗ B + ⊗ B → B + ⊗ B → B (3.4.2)Localizing this sequence on C cusp and twisting, we get for each m ∈ Z an exact sequenceof left A cusp -modules, . . . → B + ⊗ B + ⊗ A cusp ( m ) → B + ⊗ A cusp ( m ) → A cusp ( m ) → E -term given by the cohomology of the terms ofthis complex. Thus, the E -term has two rows, corresponding to H and H . The rowof H ’s is the degree m component of the complex (3.4.2), which is exact for m = 0,and has 1-dimensional cohomology in degree 0 for m = 0. On the other hand, using theisomorphism (3.4.1) of left B -modules we can identify the row of H ’s with the degree m component of the complex . . . → B + ⊗ B + ⊗ B ∗ → B + ⊗ B ∗ → B ∗ (3.4.3)Since the spectral sequence abuts to zero, the row of H ’s should be exact for m = 0 andhas one-dimensional cohomology in the term of degree − m = 0.Now the resolution (3.4.2) for k as a right B -module shows that the complex (3.4.3)computes Tor B ( k, B ∗ ), while its restricted dual computes Ext ∗ B op ( k, B ).In other words, Ext ∗ B op ( k, B ) has the one-dimenisonal cohomology concentrated in co-homological degree 2 and internal degree 0.To conclude that B is right Gorenstein it remains to check that B has finite injectivedimension as a right module over itself. To this end we use [5, Thm. 4.5] together with[35, Thm. 6.3]. More precisely, by Lemma 3.3.3, both B and B op are right Noetherian,satisfy χ , and their Proj nc has finite cohomological dimension. Hence, [35, Thm. 6.3] givesexistence of a balanced dualizing complex over A . Now the same method as in [5, Thm.4.5] can be used to prove that B has finite injective dimension (see also [37, Thm. 0.3(3)]for a similar proof in the case of local rings). (cid:3) Proposition 3.4.2.
Let k be a field. For any filtered k -algebra ( A, F • A ) satisfying (2.4.1) for some g ∈ GL n ( k ) , the Rees algebra R ( A ) is left and right AS-Gorenstein with param-eters (2 , .Proof . We observe that due to the nature of the Rees algebra construction there is aflat family R u ( A ) of graded algebras over A such that specializing u to a nonzero valuegives an algebra isomorphic to R ( A ), while R ( A ) is gr F ( A )[ x ] ≃ E ( V, g )[ x ]. Namely, R u ( A ) = R ( A )[ u, x ] / ( t − xu ), where t ∈ R ( A ) is the canonical central element (heredeg x = 1). herefore, since the algebra E ( V, g )[ x ] is AS-Gorenstein with parameters (2 ,
0) (seeProposition 3.4.1), we get that Ext i ( k, R ( A )) = 0 for i = 2 and Ext ( k, R ( A )) is at mostone-dimensional, concentrated in the internal degree 0.Now let us check that Ext ∗ ( k, R ( A )) cannot be entirely zero. Indeed, if it were zerothen using (3.1.4), we would get that all higher cohomology of O ( i ) on Proj nc vanishes.Now we use the identification of qgr R ( A ) with coherent modules over the correspondingorder A (see Proposition 3.1.3) and get a contradiction with the fact that H ( C, A ) ≃ Ext A op ( A , A ) is 1-dimensional (see Lemma 3.1.2(iv)).Finally, by Corollary 3.1.4, R ( A ) and R op are both Noetherian, satisfy χ and theirProj nc has finite cohomological dimension. Thus, as in the proof of Proposition 3.4.1 wecan deduce that R ( A ) has finite injective dimension, so R ( A ) is AS-Gorenstein. (cid:3) From filtered algebras to spherical n -pairs Gorenstein condition over a base ring.
Now we are going to switch to workingover an arbitrary Noetherian commutative ring S . Throughout this section we fix afiltered S -algebra ( A, F • A ) equipped with an isomorphism (2.4.1) for some invertible g ∈ End S ( V ) ⊗ L (where V ≃ S n ), and let R = R ( A ) be the corresponding Rees algebra.Note that the graded components of R are locally free S -modules of finite rank. Lemma 4.1.1.
The algebra R is right and left Noetherian.Proof . First, we note that the ring R / ( t ) ≃ E ( V, g ) op is right and left Noetherian, since E ( V, g ) op is finitely generated over its central subring S [ z ]. Since t is a regular centralelement, the assertion follows. (cid:3) Proposition 4.1.2.
Let R ∗ be the restricted dual of R . Then for i = 2 , one has Tor R i ( S, R ∗ ) = Tor R i ( R ∗ , S ) = 0 , Ext i R ( S, R ) = Ext i R op ( S, R ) = 0 , and Tor R ( S, R ∗ ) , Tor R ( R ∗ , S ) , Ext R ( S, R ) and Ext R op ( S, R ) are locally free S -modulesof rank , concentrated in the internal degree .Proof . Step 1 . First, we observe that the assertion is true when S = k is a field.Indeed, this follows from Proposition 3.4.2 and from the duality between Tor B ( k, B ∗ ) andExt B op ( k, B ). Step 2 . In the general case, since R is Noetherian, we can find a free resolution . . . → P → P → P → S, where P i are free graded R -modules of finite rank. Let us set Q • = ( P • ) ⊗ R R ∗ , so that H i ( Q • ) ≃ Tor R i ( S, R ∗ ), and the graded components of Q i are free S -modules of finite rank.Let us assume that S is local with the maximal ideal M , and set k := S/M , R k := R⊗ S k .We are going to prove H i Q • = 0 for i = 2, while H Q • ≃ S is concentrated in internaldegree 0.We have Q • ⊗ S k ≃ ( P • ⊗ S k ) ⊗ R k R ∗ k . ote that P • ⊗ S k is a free graded resolution of k over R k , so that H i ( Q • ⊗ S k ) ≃ Tor R k i ( k, R ∗ k ) . Note that we know from Step 1 that the latter spaces are zero for i = 2 and are isomorphicto k in degree 0 for i = 2.Now let us consider the third quadrant spectral sequence E p,q = Tor S − q ( H − p Q, k ) = ⇒ E ∞ n = Tor S − n ( Q • , k ) , with the differentials d r : E rp,q → E rp − r +1 ,q + r . Note that since the terms of the complex Q • are free S -modules we have Tor Si ( Q • , k ) = H i ( Q • ⊗ S k ), so E ∞ n = 0 for n = − E ∞− is k in the internal degree 0. It follows that the term E , survives in thespectral sequence, so we get H Q ⊗ S k = 0. Since H Q has finitely generated gradedcomponents, by Nakayama lemma, this implies that H Q = 0. Therefore, E ,q = 0, sothe term E − , survives, and we get H Q ⊗ S k = 0. Hence, H Q = 0, and so E − ,q = 0.Thus, the terms E − , and E − , − survive, and we deduce H Q ⊗ S k ≃ k (in degree 0) andTor S ( H Q, k ) = 0. Hence, H Q ≃ S (sitting in degree 0). Now the similar argument willprove by induction in n ≥ H n Q = 0 (for the base we use the vanishing of E − ,q for q ≤ − Step 3 . For arbitrary R , since the construction of the complex Q • is compatible withlocalization, we deduce that H i Q • = 0 for i = 2, while H Q • is a projective S -moduleof rank 1 sitting in internal degree 0. This finishes the computation of Tor R i ( S, R ∗ ).Since the complex Hom S ( Q • , S ) computes Ext i R ( S, R ), and since H i Q • are projective, wededuce that Ext i R ( S, R ) ≃ Hom S (Tor R i ( S, R ∗ )) , and the assertion about Ext ∗R ( S, R ) follows. It remains to apply the same argument to R op . (cid:3) Noncommutative projective scheme associated with a filtered algebra.
As before, we consider the noncommutative projective scheme over S associated with R = R ( A ), i.e., the category qgr R , defined as the quotient of the category of gradedfinitely-generated right R -modules by the subcategory of torsion modules. We denote by O ( j ) the object of qgr R corresponding to the module R ( j ). Recall also that H i (?) :=Ext ∗ qgr R ( O , ?). Proposition 4.2.1. (i) In the category qgr R ( A ) one has H i ( O ( j )) = 0 for i = 0 , H ( O ( j )) = 0 for j > and there is a natural isomorphism of graded algebras M j H ( O ( j )) ≃ R ( A ) . (ii) Let F be the object of qgr R ( A ) corresponding to the graded right R ( A ) -module V [ z ] ,with the module structure induced by the homomorphism R ( A ) → R ( A ) /t R ( A ) ≃ E ( V, g ) op ֒ → End( V ) op [ z ] . hen the multiplication by z induces an isomorphism F ≃ F (1) . We have a natural exactsequence in qgr R ( A ) : → O ( − t ✲ O ✲ V ∨ ⊗ F → and canonical isomorphisms H ( F ) ≃ V , Ext ( F, O ) ≃ V ∨ . Also, Hom(
F, F ) = S · id F , H > ( F ) = 0 and Ext i ( F, O ) = 0 for i = 1 .(iii) There exist canonical isomorphisms H ( O ) ≃ L and Ext ( F, F ) ≃ S such that thecompositions Ext ( F, O ) ⊗ H ( F ) → H ( O ) and H ( F ) ⊗ Ext ( F, O ) → Ext ( F, F ) getidentified the pairings h v ∗ , gv i and h v, v ∗ i , where v ∈ V , v ∗ ∈ V ∨ . Hence, the pair ( O , F ) is an n -pair of -spherical objects with the corresponding element g ∈ End S ( V ) ⊗ L .(iv) For every n ∈ Z we have isomorphisms O ( n + 1) ≃ T ( O ( n )) , where T = T F is thespherical twist associated with F . Hence, the graded algebra R T, O equipped with its naturalcentral element of degree (see Theorem 2.4.1) is isomorphic to ( R ( A ) , t ) .Proof . (i) Let us set R = R ( A ). By [2, Prop. 7.2], we have H i ( O ( j )) = lim m →∞ Ext i +1 ( R / R ≥ m , R ( j )) for i ≥ , and there is an exact sequence0 → τ ( R ( j )) → R j → H ( O ( j )) → lim m →∞ Ext ( R / R ≥ m , R ( j )) → , where τ ( M ) denotes the torsion submodule of M . Note that τ ( R ( j )) = 0 since t is anonzero divisor. On the other hand, by Proposition 4.1.2, we haveExt =2 ( S ( m ) , R ( j )) = 0 , Ext ( S ( m ) , R ( j )) = 0 for m = j. Hence, the above exact sequence implies that the natural map R j → H ( O ( j ))is an isomorphism. The compatibility of these maps with the products is well known (see[2, Thm. 4.5(2)]. Similarly, using the above formula for H i ( O ( j )) with i > H > ( O ( j )) = 0 and that H ( O ( j )) = 0 for j > z gives an injection V [ z ] → V [ z ](1) with finite-dimensionalcokernel, hence, it induces an isomorphism F ≃ F (1). The exact sequence (4.2.1) isinduced by the sequence of graded R -modules0 → R ( − · t ✲ R → E ( V, g ) → E ( V, g ) ≥ ≃ End( V )[ z ] ≥ . Twisting (4.2.1) by (2) and using the identifica-tion F ≃ F (2), we get an exact sequence0 → O (1) → O (2) → V ∨ ⊗ F → H > ( F ) = 0. Note that the sequence (4.2.2) also immediately implies that Hom( V ∨ ⊗ F, O ) = 0, hence, Hom( F, O ) = 0. ext, we claim that the natural map V = V [ z ] → H ( F ) is an isomorphism. Notethat there is a morphism of exact sequences0 ✲ R ✲ R ✲ V ∨ ⊗ V ✲ ✲ H ( O (1)) ❄ ✲ H ( O (2)) ❄ ✲ V ∨ ⊗ H ( F ) ❄ ✲ H , and all vertical maps arethe natural maps of the form M → H ( f M ), where f M is the object of qgr R associatedwith a graded module M . Note that the exactness of the bottom row follows the vanishingof H ( O (1)) proved in (i). Since the two left vertical arrows are isomorphisms (again by(i)), we deduce that rightmost vertical arrow is also an isomorphism, which proves ourclaim.Another useful observation is that we have a morphism of functors M · t ✲ M (1), whichvanishes on F . Hence, the natural morphismsHom( G, F ) → Hom( G ( − , F ) , Ext ( F, G ) → Ext ( F, G (1))induced by t , are zero. Thus, applying the functor Hom(? , F ) to the exact sequence (4.2.1),we deduce the isomorphism Hom( V ∨ ⊗ F, F ) ∼ ✲ H ( F ), induced by the projection O → V ∨ ⊗ F . Using the isomorphism V → H ( F ), this implies that Hom( F, F ) = S · id.Now let us consider another twist of (4.2.1):0 → O → O (1) → V ∨ ⊗ F → ( V ∨ ⊗ F, O ) ≃ V ⊗ Ext ( F, O ),so it gives a canonical morphism V ∨ → Ext ( F, O ). In other words, this is precisely theconnecting homomorphism in the long exact sequence of Ext ∗ ( F, ?) applied to (4.2.3):0 = Hom( F, O (1)) → V ∨ ⊗ Hom(
F, F ) → Ext ( F, O ) t ✲ Ext ( F, O (1)) → . . . Since the map on Ext induced by t is zero, we see that the map V ∨ → Ext ( F, O ) is anisomorphism.Finally, the vanishing of Ext ≥ ( F, O ) follows from the long exact sequence of Ext ∗ (? , O )applied to (4.2.1).(iii) First, the long exact sequence of cohomology applied to (4.2.3) has form0 → R → R → V ∨ ⊗ V → H ( O ) → H ( O (1)) = 0Furthermore, the induced map R / R → V ∨ ⊗ V is exactly the embedding E ( V, g ) ⊂ End( V ), so its image is the S -submodule End( V ) g ⊂ End( V ). Thus, there is a uniqueisomorphism H ( O ) ≃ L , such that the above map V ∨ ⊗ V → H ( O ) gets identifiedwith v ∗ ⊗ v
7→ h v ∗ , gv i . Note that this map corresponds to the composition Ext ( F, O ) ⊗ H ( F ) → H ( O ) using the identifications Ext ( F, O ) ≃ V ∨ , H ( F ) ≃ V defined in (ii). ext, we observe that since the map Ext ( F, O ( − t ✲ Ext ( F, O ) is zero, the longexact sequence of Ext ∗ ( F, ?) associated with (4.2.1) gives an isomorphismExt ( F, O ) ∼ ✲ Ext ( F, V ∨ ⊗ F ) ≃ V ∨ ⊗ Ext ( F, F ) . (4.2.4)In particular, this implies that Ext ( F, F ) is a locally free S -module of rank 1. We havea split exact sequence0 → End ( V ) ⊗ F → V ⊗ V ∨ ⊗ F tr ⊗ id F ✲ F → V ⊗ O → F corresponding to the indentification V = H ( F ), is the composition V ⊗ O id V ⊗ p ✲ V ⊗ V ∨ ⊗ F tr ⊗ id F ✲ F, where p : O → V ∨ ⊗ F is the map from the sequence (4.2.1). Thus, the map Ext ( F, V ⊗O ) → Ext ( F, F ) can be identified with the compositionExt ( F, V ⊗ O ) ∼ ✲ Ext ( F, V ⊗ V ∨ ⊗ F ) = V ⊗ V ∨ ⊗ Ext ( F, F ) tr ⊗ id ✲ Ext ( F, F ) , where the first arrow is obtained from (4.2.4) by tensoring with V .Thus, we deduce the surjectivity of the composition map V ⊗ V ∨ ≃ Hom( O , F ) ⊗ Ext ( F, O ) → Ext ( F, F ) . (4.2.5)We claim that End ( V ) ⊂ End( V ) = V ⊗ V ∨ is contained in the kernel of this map. Indeed,it is enough to check that for any v ∈ V ≃ Hom( O , V ) and any v ∗ ∈ V ∨ ≃ Ext ( F, O ),such that h v ∗ , v i = 0, the composition of v ∗ and v in Ext ( F, F ) vanishes. Let us considerthe push-out of (4.2.3) by v : O → F :0 ✲ O t ✲ O (1) ✲ V ∨ ⊗ F ✲ ✲ Fv ❄ ✲ E v ❄ ✲ V ∨ ⊗ F id ❄ ✲ E ′ v := ker( t : E v → E v (1)) ⊂ E v . We can represent E v by the graded R -module V [ z ] ≥ ⊕ R (1) ≥ / { ( − v ∗ r, tr ) | r ∈ R ≥ } , so that the embedding F → E v corresponds to the embedding of the summand V [ z ] ≥ .Here for r ∈ R m we denote by v ∗ r ∈ V · z m the result of the right action of the imageof r in R m / R m − ⊂ End( V ) op · z m on v . Hence, E ′ v corresponds to the submodule ofpairs ( x, r ) such that v ∗ r = 0. This easily implies that the image of the projection E ′ v → O (1) / O · t ≃ V ∨ ⊗ F coincides with h v i ⊥ ⊗ F . Thus, the bottom sequence indiagram (4.2.6) contains as a subsequence the exact sequence0 → F → E ′ v → h v i ⊥ ⊗ F → f objects in the subcategory ker( t ) ⊂ qgr R . Note that the latter subcategory is naturallyidentified with qgr End( V )[ z ] and that F is a projective object in this subcategory. Hence,the sequence (4.2.7) splits which proves the required vanishing in Ext ( F, F ).It follows that the composition map (4.2.5) factors through a surjective map S ≃ End( V ) / End ( V ) → Ext ( F, F ) . Since Ext ( F, F ) is a locally free S -module of rank 1, this map is in fact an isomorphism.(iv) As we have seen above, the exact sequence (4.2.3) induces an isomorphism V ∨ → Ext ( F, O ). Hence, it gives a canonical isomorphism O (1) ≃ T F ( O ) . On the other hand, for any n ∈ Z we have an isomorphism F ( n ) ≃ F . Hence, applyingthe autoequivalence M M ( n ) to the above isomorphism we get an isomorphism O ( n + 1) ≃ T F ( n ) ( O ( n )) ≃ T F ( O ( n )) . (cid:3) Corollary 4.2.2.
There exists a graded automorphism φ of R , such that φ ( t ) = t , theinduced automorphism of R / ( t ) ≃ E ( V, g ) op is Ad( g − ) , and there is an isomorphism ofgraded R − R -bimodules H ( O ) := M i ∈ Z H ( O ( i )) ≃ ( id R φ ) ∗ ⊗ S L . Proof . This follows by combining Proposition 4.2.1 with Proposition 2.4.4. (cid:3)
Next, we are going to construct a certain short exact sequence in qgr R ( A ) (it willbe used in Sec. 5.2 to characterize A ∞ -structures). Namely, let us consider an element hz ∈ End g ( V )[ z ] ≃ F A/F A , where h is any invertible element of End( V ) ⊗ L − , suchthat tr( gh ) = 0. Note that we get take h = g − h where h is an element of GL( V ) withtr( h ) = 0 (it exists since we assume that V ≃ S n and n ≥ e h ∈ F A ⊗ L − be any lifting of hz to F A ⊗ L − = R ( A ) ⊗ L − . The right multiplication by hz induces an injective map E ( V, g ) → E ( V, g )(1) ⊗ L − (which we can view as a map ofright R ( A )-modules) with finite-dimensional cokernel, hence, an isomorphism V ∨ ⊗ F → V ∨ ⊗ F (1) ⊗ L − in qgr R ( A ). Since t ∈ R ( A ) is central, we have a commutative diagramin qgr R ( A ) with exact rows0 ✲ O t · ✲ O (1) ✲ V ∨ ⊗ F (1) ✲ ✲ O (1) ⊗ L − e h · ❄ t · ✲ O (2) ⊗ L − e h · ❄ ✲ V ∨ ⊗ F (2) ⊗ L − · hz ❄ ✲ → O α ✲ O (1) ⊕ O (1) ⊗ L − β ✲ O (2) ⊗ L − → here α = ( t · , e h · ) , β = ( e h · , ( − t ) · ) . (4.2.9) Lemma 4.2.3.
Under the isomorphism
Ext ( O (2) , O ) ⊗ L ∼ ✲ Hom S ( R , L ) ⊗ L : c ( r c · r ) , where we use the identification of H ( O ) ≃ L from Proposition 4.2.1(iii), the class γ ∈ Ext ( O (2) , O ) ⊗ L of extension (4.2.8) corresponds to the functional R → L : r tr( p ( r ) z h − g ) , where p : R → R / ( t ) ≃ E ( V, g ) op is the natural projection (note that p ( r ) is an elementof End( V ) z ).Proof . Let γ ′ ∈ Ext ( V ∨ ⊗ F, O ( − ≃ Ext ( V ∨ ⊗ F (1) , O ) be the class of the extension(4.2.1), and let p : O → V ∨ ⊗ F be the natural projection. We claim that γ is equal tothe composition O (2) ⊗ L − p ✲ V ∨ ⊗ F (2) ⊗ L − · ( hz ) − ✲ V ∨ ⊗ F (1) γ ′ ✲ O [1] . Indeed, this follows immediately from the commutative diagram in which rows and columnsextend to short exact sequences, O t ✲ O (1) p ✲ V ∨ ⊗ F (1) O = ✻ α ✲ O (1) ⊕ O (1) ⊗ L − ✻ β ✲ O (2) ⊗ L − ( · ( hz ) − ) ◦ p ✻ ✻ ✲ O (1) ⊗ L − ✻ = ✲ O (1) ⊗ L − − t ✻ Next, for any r ∈ R m ( A ) the composition O r ✲ O ( m ) p ✲ V ∨ ⊗ F ( m ) correspondsto an element p ( r ) ∈ E ( V ) m ≃ End( V ) z m ≃ H ( V ∨ ⊗ F ( m )). Finally, the extensionclass γ ′ corresponds to the identity element in V ⊗ V ∨ under an isomorphism Ext ( V ∨ ⊗ F (1) , O ) ≃ V ⊗ V ∨ , and the composition Ext ( F, O ) ⊗ Hom( O , F ) → H ( O ) is given by v ∗ ⊗ v
7→ h v ∗ , gv i . This easily implies the assertion. (cid:3) Characterization of A ∞ -structures Minimally non-formal algebras.
Let A be a graded algebra over a commutativering S . We can consider structures of ( S -linear) minimal A ∞ -algebras on A extending thegiven m , up to (strict) gauge equivalences. If every such structure is gauge equivalentto the one with m i = 0 for i >
2, then A is called intrinsically formal . We are going to escribe a class of algebras A for which gauge equivalence classes of minimal A ∞ -structuresare classified by elements of a certain S -module.Recall that the set of minimal A ∞ -structures on A is governed by the Hochschild co-homology HH ∗ ( A ) of the underlying graded associative algebra. More precisely, if wealready have products m i for i ≤ n −
1, forming an A n − -structure, then the set of m n extending these to an A n -structure is a torsor over HH ( A ) − n (we follow the gradingconvention in which m n is a Hochschild 2-cochain of internal degree 2 − n ). So the van-ishing of HH ( A ) < implies intrinsic formality. The next simplest case after that whichoccurs in some situations is when HH ( A ) − n is nonzero for the unique value n = d ≥ A ∞ -structure is gauge equivalent to the one with m i = 0 for 2 < i < d .Furthermore, m d , calculated for such a representative, gives a class in HH − d , whichuniquely determines the gauge equivalence class of the A ∞ -structure. Definition 5.1.1.
Let B = L n ≥ B n be a graded S -algebra with B = S , and let M = L n M n be a graded B − B -bimodule. We define the bigraded algebra A ( B, M, d )to be B ⊕ M , where M · M = 0, with the natural internal grading and the homologicalgrading given by deg( S ) = 0, deg( M ) = d .The next theorem is a slight generalization of the results in [23, Sec. 3.1]. As in [23,Sec. 3.1], for graded B − B -bimodules M , . . . , M n let us consider the bar-complexBar • ( M , . . . , M n ) := M ⊗ S T ( B + ) ⊗ S M ⊗ S . . . ⊗ S T ( B + ) ⊗ S M n , where T ( B + ) is the tensor algebra of B + = L n ≥ B n as an S -module. The grading isgiven byBar − m ( M , . . . , M n ) := M m + ... + m n − = m M ⊗ S T m ( B + ) ⊗ S M ⊗ S . . . ⊗ S T m n − ( B + ) ⊗ S M n . Note that the cohomology H − m of the complex Bar • ( S, M ) (resp., Bar • ( M, S )) is isomor-phic to Tor Bm ( S, M ) (resp., Tor Bm ( M, S )).
Theorem 5.1.2.
Assume that M bounded above, i.e., M n = 0 for n > n , and that P l := Tor B ∗ ( S, M ) and P r := Tor B ∗ ( M, S ) are both finitely generated projective S -modules,concentrated in degree d + 1 and the internal degree . Let us fix an embedding P r ϕ ✲ M ⊗ S T d +1 ( B + ) inducing the isomorphism of P r with the cohomology of Bar • ( M, S ) . Then for the algebra A = A ( B, M, d ) one has HH i − md ( A ) = 0 for m ≥ and i < m , and there is an embedding HH − d ( A ) ֒ → Hom S ( P r , S ) , induced by the evaluation of a Hochschild cochain on the image of ϕ . Here the lower indexdenotes the grading on Hochschild cohomology induced by the homological grading on A .Proof . Below we consider graded B − B -bimodules M i which are equipped with a pairof isomorphisms l : M i → M ⊗ S P i , r : M i → P ′ i ⊗ S M , for some finitely generatedprojective S -modules P i , P ′ i , where l (resp., r ) is compatible with the left (resp., right)graded B -module structures. tep 1 . H i (Bar • ( M , M )) = 0 for i = − d − B -modules H − d − Bar • ( M , M ) ≃ P ′ ⊗ S P r ⊗ S M and an isomorphism of left B -modules H − d − Bar • ( M , M ) ≃ M ⊗ S P l ⊗ S P . To prove the first assertion we consider the spectral sequence associated with the fil-tration on Bar • ( M , M ) induced by the Z -grading on M . The corresponding E -termis E ≃ H • Bar • ( M , S ) ⊗ S M ≃ P ′ ⊗ S P r ⊗ S M . Hence, the spectral sequence degenerates and we obtain the first assertion. Similarly,the second assertion is obtained by considering the spectral sequence associated with thefiltration induced by the Z -grading on M . Step 2 . H i Bar • ( M , . . . , M n ) = 0 for i > − ( n − d + 1), n ≥ n . For n = 2 the assertion follows from Step 1. Now for n > • ( M , . . . , M n ) as the total complex associated with a bicomplex, byconsidering the bigrading given by the sums of the tensor degrees in even and odd factors T ( B + ). This leads to a spectral sequence abutting to H ∗ Bar • ( M , . . . , M n ) with the E -term H ∗ Bar • ( M , M ) ⊗ S T ( B + ) ⊗ S H ∗ Bar • ( M , M ) ⊗ S T ( B + ) ⊗ S . . . where the last tensor factor is either M n or H ∗ Bar • ( M n − , M n ). Thus, the E -term isisomorphic to the complex of the formBar • ( M ′ , . . . , M ′ n ′ )[( n − n ′ )( d + 1)]where M ′ = H ∗ Bar • ( M , M ), M ′ = H ∗ Bar • ( M , M ), etc., satisfy the same assumptionsas ( M i ) by Step 1. Applying the induction assumption we deduce the result. Step 3 . H i Bar • ( S, M , M , . . . , M n , S ) = 0 for i > − n ( d + 1), and we have isomorphisms H − d − Bar • ( S, M , S ) ≃ P l ⊗ S P ≃ P ′ ⊗ S P r of graded S -modules.Consider first the complex Bar • ( S, M , S ) = T ( B + ) ⊗ S M ⊗ S T ( B + ). We can viewit naturally as the total complex associated with a bicomplex (where the bigrading isinduced by two tensor degrees). Considering the corresponding two spectral sequenceswe immediately deduce the vanishing of H i Bar • ( S, M , S ) for i > − d − H − d − Bar • ( S, M , S ).Now we use induction on n . As before, we equip Bar • ( S, M , . . . , M n , S ) with thebigrading using sums of tensor degrees in even and odd factors T ( B + ). Thus, we get aspectral sequence of a bicomplex abutting to H ∗ Bar • ( S, M , . . . , M n , S ) with the E -termeither of the form T ( B + ) ⊗ H ∗ Bar • ( M , M ) ⊗ T ( B + ) ⊗ . . . H ∗ Bar • ( M n − , M n ) ⊗ T ( B + ) =Bar • ( S, M ′ , . . . , M ′ n/ , S )[ n ( d + 1) / f n is even, or of the form H ∗ Bar • ( S, M ) ⊗ T ( B + ) ⊗ H ∗ Bar • ( M , M ) ⊗ T ( B + ) ⊗ . . . ⊗ H ∗ Bar • ( M n − , M n ) ⊗ T ( B + ) ≃ P l ⊗ S P ⊗ S Bar • ( S, M ′ , . . . , M ′ ( n − / , S )[( n + 1)( d + 1) / n is odd. In both cases the required cohomology vanishing follows from the inductionassumption. Step 4.
Let us fix m ≥ C i − md ⊂ Hom S ( A ⊗ i + , A )the submodule of degree − md with respect to the homological grading on A and of degree0 with respect to the internal grading on A . Note that the Hochschild differential maps C i − md to C i +1 − md , and HH i − md ( A ) is the ( i + md )-th cohomology of the complex C •− md . Wehave an exact sequence of complexes0 → C •− md ( M ) → C •− md → C • ( B ) → C i − md ( M ) ⊂ C i − md (resp., C i − md ( B ) ⊂ C i − md ) consists of maps A ⊗ i + → M (resp., A ⊗ i + → B ). Note that C i − md ( B ) (resp., C i − md ( M )) consists of S -linear maps[ T ( B + ) ⊗ S M ⊗ S T ( B + ) ⊗ S . . . ⊗ S M ⊗ T ( B + )] i → B (resp., [ T ( B + ) ⊗ S M ⊗ S T ( B + ) ⊗ S . . . ⊗ S M ⊗ T ( B + )] i → M )preserving the internal grading, where there are m (resp., m + 1) factors of M in thesource and the index i refers to the total number of tensor factors (of M and B + ).We claim that H i ( C •− md ( M )) = H i ( C •− md ( B )) = 0 for i < m ( d + 2)and in addition, H d +2 ( C •− d ( M )) = 0and there is an embedding H d +2 ( C •− d ( B )) ֒ → Hom S ( P r , S )induced by the evaluation on the image of ϕ .For the proof, let us consider the decomposition C •− md ( B ) = Q j ≥ C •− md ( B j ), where C •− md ( B j ) ⊂ C •− md ( B ) denote the maps with the image contained in B j . Let us con-sider the corresponding decreasing filtration on C •− md ( B ). Note that the correspondingassociated graded complex is M j ≥ Hom S (Bar • ( S, ( M ) m , S ) j , B j )[ − m ] , where the lower index denotes the internal grading (and we grade the dual complex usingthe convention Hom( K • , ?) i = Hom( K − i , ?)). By Step 3, we have the vanishing H i Hom S (Bar • ( S, ( M ) m , S ) j , B j ) = 0 for i < m ( d + 1) ,H d +1 Hom S (Bar • ( S, M, S ) j , B j ) = 0 for j > , nd in addition, H d +1 Hom S (Bar • ( S, M, S ) , B ) ≃ Hom S ( P r , S ) . Hence, using [23, Lem. 3.2], we obtain the required statements about cohomology of C •− md ( B ).To deal with the cohomology of C •− md ( M ) we consider the decreasing filtration on C •− md ( M ) associated with the grading induced by the sum of the tensor degrees in thefirst and last factors of the tensor product T ( B + ) ⊗ M ⊗ T ( B + ) ⊗ . . . ⊗ M ⊗ T ( B + ). Theassociated graded complex can be identified withHom gr − S − mod ( T ( B + ) ⊗ S Bar • (( M ) m +1 ) ⊗ S T ( B + ) , M )[ − m − . Thus, the required vanishing follows from Step 2. (cid:3)
Corollary 5.1.3.
Under the conditions of Theorem 5.1.2, the map ( m • ) m d +2 in-duces a bijection between the set of gauge-equivalence classes of minimal ( S -linear) A ∞ -structures on A = A ( B, M, d ) with given m and the S -module HH − d ( A ) .Proof . Since the grading on A is concentrated in degrees 0 and d , the only potentiallynonzero higher products are m n with n ≡ d ). Thus, the A ∞ -identities implythat m d +2 is a Hochschild cocycle. Now the vanishing of HH − md for m ≥ m • ) is determined by thecohomology class of m d +2 . Furthermore, since HH − md ( A ) = 0 for m ≥
2, by [25, Lem.3.1.2(ii)], every Hochschild cocycle m d +2 extends to an A ∞ -structure on A . (cid:3) Corollary 5.1.4.
Under the conditions of Theorem 5.1.2 with d = 1 , assume that wehave a finitely generated projective S -module Q and a locally free S -module L of rank ,maps of S -modules α : Q → B n , β : Q ∨ → B n ⊗ L − and an element γ ∈ M − n − n ⊗ L ,where n > , n > , such that m ( β ⊗ α )(id Q ) = 0 in B n + n ⊗ L − and m ( γ ⊗ β ) = 0 in Q ∨ ⊗ M − n − , where m is induced by the product in B and the B -bimodule structureon M . Assume in addition that the S -module P r = Tor ( M, S ) is locally free of rank .Then up to a gauge equivalence, there exists at most one minimal S -linear A ∞ -structureon A = A ( B, M, with the given m and with m (( γ ⊗ β ⊗ α )(id Q )) = 1 . (5.1.1) Furthermore, if such an A ∞ -structure exists then the gauge equivalence classes of allminimal A ∞ -structures with the given m are classified by elements of S via the map ( m • ) m (( γ ⊗ β i ⊗ α )(id Q )) . Proof . As in Corollary 5.1.3, we see that a minimal A ∞ -structure on A with the given m is determined by the class of m in HH − ( A ), and we have an embedding HH − ( A ) ֒ → Hom S ( P r , S ) . Next, we have a morphism S → P r = Tor ( M, S ) given by the cycle ( γ ⊗ β i ⊗ α )(id Q )in the complex Bar • ( M, S ). Assume that there exists an A ∞ -structure on A with thegiven m such that (5.1.1) holds. Then the composition HH − ( A ) → Hom S ( P r , S ) f f (( γ ⊗ β ⊗ α )(id Q )) ✲ S ends the class of m to 1. It follows that the morphism Hom S ( P r , S ) → S is surjective.Since P r is locally free of rank 1, we deduce that this morphism is an isomorphism. Hence,the map HH − ( A ) → S , given by the evaluation on ( γ ⊗ β ⊗ α )(id Q ), is an isomorphism,which implies our assertion. (cid:3) Triple product calculation.
Now we return to the situation of Section 4.2, so S is a Noetherian commutative ring, ( A, F • A ) a filtered S -algebra equipped with anisomorphism (2.4.1) for some invertible g ∈ End S ( V ) ⊗ L (where V ≃ S n ), and R = R ( A )be the corresponding Rees algebra.We would like to show that Corollary 5.1.4 is applicable to minimal S -linear A ∞ -structures on the algebra A ( R , ( id R φ ) ∗ ⊗ L ) ,
1) (see Def. 5.1.1), where φ is the automor-phism of R defined in Corollary 4.2.2. Recall that we have an isomorphism of bimodulesover R , H ( O ) ≃ ( id R φ ) ∗ ⊗ L .As in Lemma 4.2.3, let us fix an invertible element h ∈ End( V ) ⊗ L − such thattr( gh ) = 0 and its lifting e h ∈ F A ⊗ L − = R ⊗ L − , and let us consider the extensionclass γ ∈ Ext ( O (2) , O ) ⊗ L = H ( O ( − ⊗ L of the exact sequence (4.2.8) in qgr R .We can view the maps α and β from this exact sequence as maps of S -modules α : Q → R , β : Q ∨ → R ⊗ L − , where Q = S ⊕ L , satisfying m ( β ⊗ α )(id Q ) = 0 and m ( γ ⊗ β ) = 0. Lemma 5.2.1.
Up to a gauge equivalence, there is at most one minimal S -linear A ∞ -structure on A ( R , ( id R φ ) ∗ , with given m and satisfying m (( γ ⊗ β ⊗ α )(id Q )) = 1 . (5.2.1) Proof . Note that ( id R φ ) ∗ ⊗ L is isomorphic to R ⊗ L as a left and as a right R -module,so by Proposition 4.1.2, the conditions of Theorem 5.1.2 with d = 1 are satisfied, with P r and P l locally free over S of rank 1. Thus, the result follows from Corollary 5.1.4. (cid:3) Remark 5.2.2.
Using the compatibility of the higher products with exact triangles(see e.g., [30, Lem. 3.7]) one can check that (5.2.1) holds for the minimal A ∞ -structureon A ( R , H ( O ) ,
1) coming from the standard A ∞ -enhancement of the derived category D (qgr R ) (defined uniquely up to a gauge equivalence). This explains why this is a naturalcondition to consider.In the case when the filtered algebra ( A, F • A ) is associated with a pair of 1-spherical ob-jects as in Theorem 2.4.1, we get an A ∞ -structure on the algebra of the form A ( R , ( id R φ ′ ) ∗ ⊗L , A ∞ -structure on the subcategory of twistedcomplexes ( E i ), obtained by homological perturbation, and use Proposition 2.4.4 to iden-tify the resulting algebra with A ( R , ( id R φ ′ ) ∗ ⊗ L , S -modules Hom( E i , E j ) are finitely generatedprojective. Furthermore, if we assume in addition that either n ≥ g ) is a generatorof L , then we deduce that φ ′ = φ , since such an automorphism is uniquely determined byits action on R / ( t ) ≃ E ( V, g ) op by Proposition 2.4.5.We need to check that this A ∞ -structure satisfies our normalization condition on m . roposition 5.2.3. Assume that g is invertible, and let φ be the unique automorphism of R , such that φ ( t ) = t and the induced automorphism of gr F A ≃ E ( V, g ) op is Ad( g − ) (seeProposition 2.4.5 and Corollary 4.2.2). Now assume that R arises as the graded algebraassociated with an n -pair ( E, F ) as in Theorem 2.4.1. Then the corresponding minimal A ∞ -structure on A ( R , ( id R φ ) ∗ ⊗ L , satisfies m ( γ, β, α ) = id , where α : E → E ⊕ E ⊗ L − , β : E ⊕ E ⊗ L − → E ⊗ L − and γ : E ⊗ L − → E correspond to the elements (4.2.9) and to the class of the exact sequence (4.2.8) .Proof . (i) It is enough to compute the relevant triple Massey product in the A ∞ -categoryof twisted complexes over the A ∞ -category C generated by our n -pair ( E, F ), i.e., be-fore applying the homological perturbation (this follows from the functoriality of Masseyproducts, see [22, Prop. 1.1].We use our presentations for E and E as twisted complexes from the proof of Theorem2.4.1 (see (2.4.7)). Thus, the complex hom( E , E ) has formHom ( E, E ) → (cid:0) Hom ( V ∨ L ⊗ F, E ) ⊕ Hom ( V ∨ L ⊗ F, E ) ⊕ Hom ( E, E ) (cid:1) with the differential induced by the maps δ and δ (see notation in (2.4.7)). Step 1 . We claim that the map γ is equal to the class γ ′ ∈ Hom ( E , E ) ⊗L E of the closedelement h − ∈ End( V ) ⊗ L E ≃ Hom ( V ∨ L ⊗ F, E ) ⊗ L E ⊂ hom ( E , E ) ⊗ L E . Indeed, byLemma 4.2.3, it is enough to check that the map Hom ( E, E ) → Hom ( E, E ) ⊗L E = L E ,given by postcomposing with γ ′ , coincides with the compositionHom( E, E ) → Hom(
E, V ∨ L ⊗ F ) ≃ End( V ) A tr( Ah − g ) ✲ L . Indeed, this amounts to checking that the composition E → V ∨ L F h − ✲ E ⊗ L E [1]is given by A tr( Ah − g ) which follows from the identification of the compositionHom ( F, E ) ⊗ Hom ( E, F ) → Hom ( E, E ) with v ∗ ⊗ v
7→ h v ∗ , gv i . Step 2 . Using the computations from the proof of Theorem 2.4.1 we see that the com-ponents of α and β are represented by the following closed maps. The maps E t ✲ E and E t ✲ E are given by EV ∨ ⊗ F δ ✲ E id E ❄ V ∨ ⊗ F δ ✲ EV ∨ ⊗ F δ ✲ V ∨ ⊗ F id E ❄ δ ✲ E id E ❄ lso, for a certain choice of e h , the maps E e h ✲ E ⊗ L − E and E e h ✲ E ⊗ L − E aregiven by EV ∨ ⊗ F ⊗ L − E h ❄ δ ✲ E ⊗ L − E V ∨ ⊗ F ✲ EV ∨ ⊗ F ⊗ L − E h ∗ ⊗ id F ❄ δ ✲ V ∨ ⊗ F ⊗ L − E h ❄ δ ✲ ✲ E ⊗ L − E where the diagonal arrow is µ h ⊗ id F (recall that µ a is defined by (2.4.11)).Note that any other choice of e h is of the form e h + c · t , for c ∈ L − . Hence, for adifferent choice of e h , the sequence of maps α, β would change by an automorphism of E ⊕ E ⊗ L − , so the Massey product is unaffected by such a change. Step 3 . Now one easily checks that m ( β, α ) = 0 and that m ( γ, β, α ) = 0. However, theproduct m ( γ, β ) is not zero on the cochain level. In fact, its only nonzero component isgiven by the composition V ∨ ⊗ F δ ✲ EV ∨ ⊗ F ⊗ L − E h ∗ ⊗ id F ❄ δ ✲ V ∨ ⊗ F ⊗ L − E h ❄ δ ✲ ✲ E ⊗ L − E E h − ✲ It is easy to check that the composition h − ◦ ( h ∗ ⊗ id F ) in this diagram is equal to δ .Hence, we obtain m ( γ, β ) = d (id E ) , where we view id E as an element of hom ( E , E ). Thus, by the definition of the Masseyproduct, we get M P ( γ, β, α ) = m (id E , t ) = id E . (cid:3) . Proofs of the main results
Proof of Theorem A for Noetherian rings.
For a commutative ring S we canthink of an S -point of PGL n as isomorphism classes of pairs ( g, L ), where L is an invertible S -module and g ∈ End S ( V ) ⊗ L (where V = S n ) is an invertible element. Recall thatfor a graded S -algebra B and a graded B − B -bimodule M , we define A ( B, M,
1) as thetrivial square-zero extension B ⊕ M (see Def. 5.1.1).Let us consider the functors on the category of Noetherian commutative rings, thatassociate to S the set of(1) ( g, L ) ∈ PGL n ( S ) and minimal A ∞ -structures on S ( V, g ) up to a gauge equivalence;(2) ( g, L ) ∈ PGL n ( S ) and isomorphism classes of ( A, F • A, ι, φ ; m • ), where ( A, F • A ) afiltered algebra equipped with an isomorphism ι : gr F A ≃ E ( V, g ) op and an automorphism φ : A → A such that the induced automorphism φ of gr F A ≃ E ( V, g ) op is equal to Ad( g − );and m • is a minimal A ∞ -structure on A ( R ( A ) , ( id R φ ( A )) ∗ ⊗ L ,
1) with given m and suchthat m ( γ, β, α ) = 1 (viewed up to a gauge equivalence);(3) ( g, L ) ∈ PGL n ( S ) and isomorphism classes of ( A, F • A, ι ) as in (2).In the case n = 2 we always assume in addition that tr( g ) = 0. Step 1 . Construction of an injective map from (1) to (2).Starting from a minimal A ∞ -structure on S n ( V, g ), we consider the corresponding n -pair of spherical objects ( E, F ) (see Sec. 2.2). Now we consider twisted objects ( E i ) inTheorem 2.4.1 and use this Theorem and Proposition 2.4.4 to identify the correspondingalgebra M i Hom( E , E i ) ⊕ M i Ext ( E i , E )with A ( R ( A ) , ( id R φ ( A )) ∗ ⊗ L , A, F • A ) and an automor-phism φ satisfying the conditions in (2). Applying homological perturbation (see [8, Sec.3.3]) to the A ∞ -structure on the subcategory ( E i ) we get a minimal A ∞ -structure m • on A ( R ( A ) , ( id R φ ( A )) ∗ ⊗ L , m holds due to Proposition5.2.3.Next, let us show injectivity of this map. For a minimal A ∞ -structure m • on S n ( V, g )let us denote by S ( m • ) the corresponding A ∞ -algebra, which can be also viewed as an A ∞ -category with two objects ( E, F ). Let Π
T w S ( m • ) denote the A ∞ split-closure of thecategory of twisted complexes over S ( m • ). The exact triangle E → E → V ∨ ⊗ F → E [1] (6.1.1)coming from the definition of E = T F ( E ), shows that Π T w S ( m • ) is split-generated by E and E . Thus, by [30, Cor. 4.9], the inclusion of the full subcategory on objects ( E i ) { E i | i ≥ } ֒ → Π T w S ( m • ) , extends to a quasi-equivalenceΠ T w { E i | i ≥ } ∼ ✲ Π T w S ( m • ) . Thus, if two minimal A ∞ -structures on A n , ( m • ) and ( m ′• ) induce gauge-equivalent A ∞ -structures on h E i | i ≥ i , then there exists a quasi-equivalenceΦ : Π T w S ( m • ) ≃ Π T w S ( m ′• ) uch that H Φ is the identity on Hom ∗ ( E i , E j ). Since the functor H Φ is triangulated,the exact triangle (6.1.1) shows that V ∨ ⊗ Φ( F ) ≃ Φ( V ∨ ⊗ F ) ≃ V ∨ ⊗ F. Such an isomorphism is induced by a unique isomorphism Φ( F ) ≃ F ⊗ M , for somelocally free S -module of rank 1 equipped with an isomorphism V ∨ ⊗ M ≃ V ∨ .Localizing over an open affine covering Spec( S ), we can trivialize L E and M . Then Φgives an A ∞ -autoequivalence of the subcategory { E, F } , identical on objects, and suchthat the induced autoequivalence of { E i | i ≥ } is isomorphic to the identity. To provethat ( m • ) and ( m ′• ) are gauge equivalent, it is enough to have that H Φ | { E,F } is isomorphicto the identity. Just using the condition the the endomorphism of Hom ( E, E ) inducedby H Φ is the identity, it is easy to see that H Φ should have the following form: it isgiven by the maps h : Hom ( E, F ) = V → V = Hom ( E, F ) , ( h − ) ∗ : V ∨ = Hom ( F, E ) → Hom ( F, F ) = V ∨ , Hom ( F, F ) λ · ? ✲ Hom ( F, F ) , where h ∈ GL( V ) and λ ∈ S ∗ satisfy h − gh = λg. Now the condition that the induced automorphism of the graded algebra R = L i ≥ Hom( E , E i )is the identity, implies that the automorphism Ad( h − ) of R / ( t ) ≃ E ( V, g ) op is the iden-tity. But this is possible only when h is a scalar matrix, h = c · id. In this case we canchange Φ by an isomorphic A ∞ -equivalence (rescaling at F ), so that H Φ becomes theidentity.Thus, we deduce that ( m • ) and ( m ′• ) are gauge equivalent locally over Spec( S ). Ap-plying [26, Thm. 2.2.6(ii)], we deduce that they are globally gauge equivalent. Step 2 . The forgetful map from (2) to (3) is injective. Indeed, the uniqueness of theautomorphism φ follows from Prop. 2.4.5 (here we use the assumption that tr( g ) = 0 if n = 2), while the uniqueness of the A ∞ -structure follows from Lemma 5.2.1. Step 3 . We have a natural map from (3) to (1): starting from (
A, F • A, ι ) we constructan n -pair ( E, F ) by considering the derived category D b (qgr R ( A )) and considering theobjects E = O and F ∈ qgr R ( A ) defined in Proposition 4.2.1(ii). Step 4 . We claim that the composition (3) → (1) → (2) → (3) is the identity. Togetherwith the injectivity proved in Steps 1 and 2, this would imply that our arrows givebijections between data (1), (2) and (3).Thus, we start from ( A, F • A, ι ), consider the corresponding n -pair of spherical objects( E, F ) as in Step 3, then look at the twisted complexes E i = T iF ( E ) and consider the cor-responding Hom-algebra R T F ,E . By [30, Lem. 3.34], the inclusion of the full subcategory { E, F } extends to an A ∞ -functorΦ : T w { E, F } → D (qgr R ( A )) , which is a quasi-equivalence with its image. Thus, Proposition 4.2.1(iv) gives the requiredisomorphism of the algebra R T F ,E with R ( A ), preserving the natural central elements t .Note that the last assertion of the theorem follows from the fact that the composition(1) → (2) → (3) → (1) is the identity. (cid:3) emark 6.1.1. Note that going from (3) to (1) and then to (2) equips any filteredalgebra A as above with a canonical filtered automorphism φ as in (2). This is preciselythe automorphism constructed in Cor. 4.2.2.6.2. Moduli spaces and the proof of Theorem A.
We would like to show that thefunctor associating to S an isomorphism class of the data( L , g, A, F • A, ι : gr F ( A ) ≃ E ( V, g ) op ) (6.2.1)as before (where L is a locally free S -module of rank 1, g ∈ End( S n ) ⊗ L is an invertibleelement, such that tr( g ) is a generator of L if n = 2), is representable by an affine schemeSpec S filt of finite type over Z .For n ≥
3, let S be the algebra of functions on PGL n (which is the degree 0 part inthe localization Z [ x ij ][det − ]), and let g ∈ End( S n ) ⊗ O S (1) be the universal invertibleelement. In the case n = 2, we define S as the algebra of functions on the open subsetof PGL n given by nonvanishing of tr( g ) n / det( g ).Recall that the algebra E := E ( S n , g ) op is Koszul (see Lemma 2.3.1(iii)). This meansthat the filtered algebras A we would like to study are given by nonhomogeneous quadraticrelations whose homogeneous quadratic parts are the quadratic relations in E . Let I E ⊂E ⊗ S E denote the space of quadratic relations. Lemma 6.2.1.
The natural morphism ∇ : E ∨ → Hom S ( I E , E ) : ξ ( e ⊗ e ′ ξ ( e ) e ′ + ξ ( e ′ ) e ) (6.2.2) is a split embedding of S -modules.Proof . Since both E ∨ and Hom S ( I E , E ) are finitely generated projective modules over S ,it is enough to check that for any S -algebra S , the morphism ∇ S , obtained from ∇ byextension of scalars, is injective. Now we note that since E S = E ⊗ S S is quadratic, theelements of ker( ∇ S ) are precisely derivations of E S of degree −
1. By Proposition 2.3.2,any such derivation is zero. (cid:3)
Proposition 6.2.2.
The functor associating to S the set of isomorphism classes of triples (6.2.1) is representable by an affine scheme Spec S filt of finite type over S (and hence,over Z ).Proof . As was discussed above, our functor associates to S the set of nonhomogeneousquadratic algebras deforming E S = E ⊗ S S . For such an algebra we can always choose asplitting s : E S, → F A of the projection F A → F A/F A ≃ E S, . Set U := s ( E S, ) ⊂ F A . Then algebra A can be given by some submodule of nonhomogeneous quadraticrelations I A ⊂ U ⊗ S U ⊕ U ⊕ S, such that the projection to U ⊗ S U ≃ E S, ⊗ S E S, induces an isomorphism of I A with thesubmodule I E ,S of quadratic relations in E S . Thus, I A is the graph of an S -linear map( φ, θ ) : I E ,S → U ⊕ S. Conversely, starting from such data we can construct the algebra T ( U ) / ( I A ) which isequipped with a map E → gr F ( A ). Since the algebra E is Koszul, by [27, Sec. V.2], his correspondence gives a bijection between the set of quadruples ( A, F • A, ι, s ), where s : E S, → F A is a splitting, and pairs of maps ( φ, θ ) satisfying certain quadratic equations(analogs of Jacobi identity). Now we can change a splitting s to s + ξ , where ξ ∈ Hom S ( E S, ( S ) , S ). It is easy to seethat this corresponds to a certain action of Hom S ( E S, ( S ) , S ) (viewed as an additive group)on pairs ( φ, θ ). Furthermore, φ gets changed to φ + ∇ ( ξ ), with ∇ given by (6.2.2). Thus,if we fix a complementary S -submodule K ⊂ Hom S ( I E , E ) to the image of ∇ , which ispossible by Lemma 6.2.1, then every orbit of the above action has a unique representative( φ, θ ) with θ ∈ K S . The set of such ( φ, θ ), satisfying the quadratic equations mentionedabove, is the required affine scheme of finite type over S . (cid:3) End of proof of Theorem A . As we have seen in Proposition 6.2.2 and Corollary 1.1.4, wehave two finitely generated Z -algebras, S filt and S A ∞ , that represent the functors (3) and(1) from Sec. 6.1. Since both S A ∞ and S filt are Noetherian, by the Noetherian case ofTheorem A(i) and by Yoneda lemma, we obtain an isomorphism S filt ≃ S A ∞ . This gives the required isomorphism of functors. (cid:3)
Polarizing line bundles on cyclotomic stacks.
We need a technical result char-acterizing stacks obtained as Proj st of a graded algebra in terms of polarizing line bundles(see (3.1.1) for the definition of the stacky version of the Proj-construction). This resultis based on [1, Sec. 2.4.2] (in fact, all the needed arguments are already in [1], just notthe statement itself).Let X be a proper algebraic stack over a field k . Following [1], we say that X is cyclo-tomic if it has cyclotomic stabilizers (i.e., each geometric fiber of I X → X is isomorphic to µ d for some d ). A line bundle L over X is called uniformizing if for each geometric point p in X , the action of Aut( p ) on the fiber of L is effective. Let us consider the principal G m -bundle over X associated with L : P L := Spec X ( M i ∈ Z L i ) → X . By [1, Prop. 2.3.10], L is uniformizing if and only if the stack P L is representable.Now let X be the coarse moduli space of X , and let π : X → X be the projection.A polarizing line bundle L over X is a uniformizing line bundle such that some positivepower of L is isomorphic to the pull-back of an ample line bundle on X . Proposition 6.3.1. If L is a polarizing line bundle on a proper cyclotomic stack X thenthere is a natural isomorphism ( X , L ) ≃ (Proj st ( R L ) , O (1)) , where R L is the graded algebra given by R L := M j ≥ H ( X , L j ) . In [27] we work over a field, however, the argument still applies in the case when all the gradedcomponents E S,i are finitely generated projective modules over S , and E S, = S . roof . Let M be an ample line bundle on X such that L N ≃ π ∗ M . Interpreting elementsof R L as functions on P L we get a regular morphism P L → Spec( R L ) \ { } (6.3.1)compatible with the G m -action. On the other hand, the homomorphism of rings L M i → L j π ∗ L j gives rise to a finite morphism P L → P M . Let us consider the commutativediagram P L ✲ Spec( R L ) P M ❄ ✲ Spec( R M ) ❄ in which the lower horizontal arrow is an open embedding with a dense image (since M isample). By Zariski’s main theorem, the composed map P L → Spec( R M ) can be factoredas the composition P L j ✲ ˆ P L f ✲ Spec( R M )where j is an open embedding with a dense image and f is finite. As shown in the proofof [1, Cor. 2.4.4], in fact, ˆ P L = Spec( R L ). Thus, the morphism (6.3.1) is a dense openembedding, and so is the induced map of quotients by G m , X →
Proj st ( R L ) . Since X is proper, this map is an isomorphism. (cid:3) Proof of Theorem B.
We have discussed in Sec. 3.1 the construction of the orderon a neat stacky pointed curve associated with a filtered algebra (
A, F • ) (see especiallyLemma 3.1.2).To go from a neat stacky curve ( C, p ) with an order A to a filtered algebra, let usconsider the filtration F i A = H ( C, A ( ip )) on the algebra A = H ( C \ p, A ). From thetrivialization of O ( p ) | p (given by the tangent vector) we get a natural injective homomor-phism of graded algebras gr F A → End( V ) op ⊗ k [ z ] . We claim that its image is E ( V, g ) op for some g ∈ P End( V ) (see (0.1.1)). Indeed, theexact sequence0 → H ( C, A ) → H ( C, A ( p )) → H ( p, A ( p ) | p ) → H ( C, A ) → F A → H ( p, A ( p ) | p ) ≃ End( V ) is of codi-mension 1. Hence, it has form End g ( V ) for a unique g ∈ P End( V ). On the other hand,since H ( C, A ( ip )) = 0 for i ≥
1, the restriction maps F m A = H ( C, A ( mp )) → H ( p, A ( mp ) | p ) ≃ End( V )are surjective for m ≥
2, which proves our claim. Thus, we get an isomorphism of gr F A with E ( V, g ) op . sing Lemma 3.1.2(iv) it is easy to check that starting from a filtered algebra ( A, F • )and constructing an order A over a stacky curve C , we then recover the original filteredalgebra by the above construction.Conversely, if we start with an order A over a neat pointed stacky curve ( C, p ) andconsider the filtered algebra (
A, F • ) with F i A = H ( C, A ( ip )), then we recover ( C, p, A )by the Proj st construction, described in Sec. 3.1. More precisely, since µ d acts faithfullyon the fiber of O C ( p ) at p , this line bundle is uniformizing. Furthermore, O C ( dp ) is a pull-back of the ample line bundle O C ( p ) on the coarse moduli C (where p ∈ C is the imageof p ). Hence, O C ( p ) is polarizing, and by Proposition 6.3.1, we have an isomorphism( C, O C ( p )) ≃ (Proj st ( R ( Z )) , O (1)) , where Z = ∪ i ≥ H ( C, O C ( ip )) is the center of A (recall that R ( · ) denotes the Reesconstruction). Furthermore, any coherent sheaf F on Proj st ( R ( Z )) is identified with thelocalization of the corresponding graded R ( Z )-module, L i H ( F ( i )). Applying this to A , we see that it corresponds to the localization of R ( A ).Next, by Lemma 3.2.1, the pair ( A , ρ ∗ V ) generates Perf( A op ). Thus, by Proposition3.1.3, we have an equivalence of Perf( A op ) with a full subcategory in D qgr R ( A ), whichsends ( A , ρ ∗ V ) to the pair ( O , F ).If g is invertible then, by Proposition 4.2.1(iii), the pair ( O , F ) in D qgr R ( A ) is an n -pair of 1-spherical objects, so A is a spherical order. Conversely, if A is spherical then g is invertible by Proposition 3.2.2(ii).Finally, let us prove that A is symmetric if and only if g is scalar. Note that for anyspherical order we have a canonical Nakayama automorphism κ defined by the equation τ ( yx ) = τ ( xκ ( y )) , where τ : A → ω C is a nonzero morphism. Indeed, we have two isomorphisms of coherentsheaves, ν : A →
Hom( A , ω C ) : a ( x τ ( xa )) , ν ′ : A →
Hom( A , ω C ) : a ′ ( x τ ( a ′ x ))(see Proposition 3.2.2), and we set κ = ν − ◦ ν ′ . The fact that κ is an automorphism ofalgebras follows from the defining identity.By Proposition 3.2.2(ii), the restricted functional τ | p : A| p ≃ End( V ) ⊗ R µ d → χ is given by x tr( gx ) on End( V ) ⊗ χ . Hence, we have κ | p = Ad( g ) ⊗ id : End( V ) ⊗ R µ d → End( V ) ⊗ R µ d , where R µ d is the regular representation of µ d . This immediately shows that if A issymmetric, i.e., κ = id, then g is scalar. Conversely, assume that g is scalar. Then κ | p = id. Now κ induces a filtered automorphism of the algebra A = H ( C \ { p } , A ), andthe condition that κ | p = id implies that the induced automorphism of gr F A is equal tothe identity. Hence, by Proposition 2.4.5, this automorphism of A is equal to the identity,and so κ = id. (cid:3) .5. A criterion for cyclic A ∞ -structures. Recall that an A ∞ -algebra over a field k is called cyclic if it is equipped with a bilinear form h· , ·i such that h m n ( a , . . . , a n ) , a n +1 i = ( − n (deg( a )+1) h a , m n ( a , . . . , a n +1 ) i . Kontsevich and Soibelman give a general criterion [10, Thm. 10.2.2] in the case whenchar( k ) = 0 stating that such a cyclic structure exists on a minimal model of an A ∞ -algebra A with finite dimensional cohomology H ∗ ( A ), provided there is a functional θ : HC N ( A ) → k (where HC ∗ ( A ) is the cyclic homology of A ) such that the induced pairingon H ∗ ( A ), h x, y i = θ ( ι ( xy )) , where ι : H ∗ ( C ) → HC ∗ ( A ) is the natural map, is perfect.In the case of algebras of the form H ∗ ( C, A ), where A is a sheaf of coherent algebrasover a curve C , we can provide a more direct construction of a cyclic structure, whichonly uses the assumption that char( k ) = 2, and relies instead on a cyclic version of thehomological perturbation considered in [11]. Proposition 6.5.1.
Let B = B ⊕ B be a dg-algebra over a field k , concentrated indegrees [0 , , h· , ·i a pairing of degree on B satisfying h x, y i = ( − deg( x ) deg( y ) h y, x i , h dx, y i + ( − deg( x ) h x, dy i = 0 . Assume also that H ∗ ( B ) is finite-dimensional and the induced pairing on H ∗ ( B ) is perfect.Then the data for the homological perturbation can be chosen in such a way that theresulting minimal A ∞ -structure on H ∗ ( B ) is cyclic with respect to the pairing induced by h· , ·i .Proof . Let A ⊂ ker( d ) ⊂ B be any (graded) subspace of cohomology representatives.We claim that there exists a subspace C ⊂ B , complementary to ker( d ), such that h C, A i = 0. Indeed, let us start with an arbitrary such complement C ⊂ B . Thenthe pairing C ⊗ A → k can be interpreted as a map C → A ∗ ≃ A (where the latterisomorphism is given by the pairing between A and A ). Correcting C by this map,we get a new subspace in C ⊕ A , which is still complementary to ker( d ), and which isorthogonal to A .Note that we have orthogonalities h C, A i = 0, h C, C i = 0. Hence, the standard ho-motopy operator Q : B → B associated with the decomposition B = im( d ) ⊕ A ⊕ C satsifies h Qx, y i = ( − deg( x ) h x, Qy i . As was observed in [11, Sec. 3.3], this implies that the minimal A ∞ -structure on H ∗ ( B )given by the tree formula from [9] is cyclic. (cid:3) We apply this general result in the following geometric setup.
Proposition 6.5.2.
Let C be a tame proper DM-stacky curve over a field k of charac-teristic = 2 , with a Cohen-Macaulay coarse moduli space C such that H ( C, O ) = k . Let A be a coherent sheaf of O C -algebras, equipped with a morphism τ : A → ω C . Assume hat we have a morphism τ : A → ω C such that τ ( xy ) = τ ( yx ) . Assume that the inducedpairing A ⊗ A → ω C induced by τ ( xy ) , is perfect in the derived category (either on the left or on the right).Then the minimal A ∞ -structure on H ∗ ( C, A ) obtained by the homological perturbationcan be chosen to be cyclic with respect to the pairing θ ( xy ) , where θ : H ( C, A ) → H ( C, ω C ) → k is the functional induced by τ .Proof . First of all, since C is tame, we have an isomorphism of algebras H ∗ ( C, A ) ≃ H ∗ ( C, A ) , where A := π ∗ A and π : C → C to the coarse moduli map. Also, we have an isomorphism π ∗ ω C ≃ ω C (see [18, Prop. 2.3.1]). Thus, we can view the morphism π ∗ τ as a morphism τ : A → ω C . Furthermore, the induced pairing
A ⊗ A → ω C , given by τ ( xy ), factors through the natural projection A ⊗ A → π ∗ ( A ⊗ A ) and hence, isstill symmetric. Finally, by the relative duality we have an isomorphism π ∗ R Hom( A , ω C ) ≃ π ∗ R Hom( A , π ! ω C ) ≃ R Hom( A , ω C ) , which implies the τ still satisfies the required non-degeneracy condition. Thus, replacing( C, A , τ ) with ( C, A , τ ), we can assume that C is a usual (non-stacky) curve.We can compute H ∗ ( C, A ) using the Cech resolution C ( A ) : A ( U ) ⊕ A ( U ) δ ✲ A ( U ) , with respect to a covering C = U ∩ U , where U i are open affine subsets, U = U ∩ U .Here δ ( f , f ) = f − f . Since char( k ) = 2, we can equip C ( A ) with the following dg-algebra structure: the product on A ( U ) ⊕ A ( U ) is the one on direct sum of algebras,while for ( f , f ) ∈ A ( U ) ⊕ A ( U ), g ∈ A ( U ∩ U ), we set( f , f ) g = ( f + f ) | U g , g ( f , f ) = g ( f + f ) | U . Note that with respect to this product we have[( f , f ) , g ] = 12 ([ f | U , g ] + [ f | U , g ]) . Since the map τ vanishes on the commutators, the induced map of Cech complexes C ( A ) → C ( ω C )also does, with respect to the above product. Composing this map with a map C ( ω C ) → k [ − H ( C, ω C ) → k , we get a map θ : C ( A ) → , vanishing on the image of the differential and on the commutators. Furthermore, Serreduality implies that the map H i ( C, A ) ⊗ H − i ( C, A ) → H ( ω C ) → k, induced by τ ( xy ) is a perfect pairing. Thus, using Proposition 6.5.1 we get a cyclic A ∞ -structure with respect to θ ( xy ). (cid:3) Remark 6.5.3.
In characteristic zero there is a generalization of Proposition 6.5.2 tocoherent sheaves of dg-algebras over higher-dimensional schemes over a field of character-istic zero. In this case one has to use Thom-Sullivan construction (see [7, Sec. 5.2], [36,App. A,B]) to get a multiplicative structure on derived global sections, and then applythe criterion of Kontsevich-Soibelman [10, Thm. 10.2.2].One more observation is that the assumptions of Proposition 6.5.2 are preserved whenpassing from A to the endomorphism sheaf of a locally projective A -module. Lemma 6.5.4.
Let A be a coherent sheaf of O C -algebras, together with a morphism τ : A → ω C , satisfying the assumptions of of Proposition 6.5.2, and let P be a locallyprojective finitely generated A -module. Consider the sheaf of algebras e A := End A ( P ) .Then the assumptions of Proposition 6.5.2 still hold for e A and e τ : e A → ω C defined as thecomposition of τ and the trace morphism tr : End A ( P ) → A / [ A , A ] .Proof . Note that as O -module, e A is locally a summand in A ⊕ n for some n . Hence, theassumption that Ext > ( A , ω C ) = 0 implies that the same holds for e A , so we only need tocheck that e τ vanishes on [ e A , e A ] and that the pairing e τ ( xy ) is perfect. The former is thestandard fact about traces. For the latter we can assume P to be a direct summand of A ⊕ n . First, we observe that the pairing τ (tr( xy )) : Mat n ( A ) ⊗ O Mat n ( A ) → O is a direct sum of pairings A · e ij ⊗ A · e ji → O , which are perfect by assumption.Next, consider a direct sum decomposition A ⊕ n = P ⊕ Q , and let e P and e Q be thecorresponding idempotents in Mat n ( A ). Then to deduce that the restriction of τ (tr( xy ))to e P Mat n ( A ) e P it is enough to check that the decompositionMat n ( A ) = e P Mat n ( A ) e P ⊕ (cid:0) e Q Mat n ( A ) ⊕ e P Mat n ( A ) e Q (cid:1) is orthogonal with respect to our form. But this immediately follows from the identities e P e Q = 0 and tr( yx ) ≡ tr( xy ) mod[ A , A ]. (cid:3) Proof of Corollary C . The first part follows immediately from Theorem B: we can realizeevery A ∞ -structure on S ( k n , id) by the one coming from a symmetric spherical order A .For the last assertion, we use the fact that for such A , a nonzero morphism τ : A → ω C induces a symmetric pairing A ⊗ A → ω C which is perfect in derived category (seeProposition 3.2.2(ii)). Recall that we want to construct a cyclic minimal A ∞ -structure onExt ∗ ( G, G ), where G = A ⊕ ρ ∗ V . Let L be a sufficiently positive power of an ample linebundle on C . Then twisting G through the spherical object L − ⊗ A gives an A -module P fitting into an exact sequence0 → P → Hom A ( L − ⊗ A , G ) ⊗ L − ⊗ A → G → . ince the local projective dimension of G is 1, this immediately implies that P is locallyprojective. Furthermore, since the spherical twist can be defined on a dg-level, we canreplace G by P when studying the minimal A ∞ -structure on Ext ∗A ( G, G ) ≃ Ext ∗A ( P , P )obtained by the homological perturbation. Now, combining Proposition 6.5.2 with Lemma6.5.4, we get that the minimal A ∞ -structure on H ∗ ( C, End A ( P , P )) obtained by the ho-mological perturbation can be chosen to be cyclic. (cid:3) References [1] D. Abramovich, B. Hassett,
Stable varieties with a twist , in
Classification of algebraic varieties , 1–38,Eur. Math. Soc., Z¨urich, 2011.[2] M. Artin, J. J. Zhang,
Noncommutative projective schemes , Advances Math. 109 (1994), 228–287.[3] D. Auroux, L. Katzarkov, D. Orlov,
Mirror symmetry for weighted projective planes and their non-commutative deformations , Annals of Math. 167 (2008), 867–943.[4] A. Bondal, A. Polishchuk,
Homological properties of associative algebras: method of helices . RussianAcad. Sci., Izvestia Math. 42 (1994), 219–260.[5] P. Jorgensen,
Local cohomology for non-commutative graded algebras , Communications in Algebra 25(1997), 575–591.[6] R. Hartshorne,
Algebraic Geometry , Springer, 1997.[7] V. Hinich, V. Schechtman,
Deformation theory and Lie algebra homology, I,II , Algebra Colloq. 4(1997), 213–240, 291–316.[8] B. Keller,
Introduction to A -infinity algebras and modules , Homology Homotopy Appl. 3 (2001), 1–35.[9] M. Kontsevich, Y. Soibelman, Homological mirror symmetry and torus fibration , in
Symplectic Ge-ometry and Mirror Symmetry (Seoul, 2000) , 203–263, World Sci. Publishing, River Edge, NJ, 2001.[10] M. Kontsevich, Y. Soibelman,
Notes on A ∞ -algebras, A ∞ -categories and non-commutative geometry.I , in Homological mirror symmetry , 153–219, Springer, Berlin, 2009.[11] C. I. Lazaroiu,
Generating the superpotential on a D-brane category: I , preprint arXiv:hep-th/0610120.[12] Y. Lekili, T. Perutz,
Arithmetic mirror symmetry for the -torus , preprint arXiv:1211.4632.[13] Y. Lekili, A. Polishchuk, A modular compactification of M ,n from A ∞ -structures , arXiv:1408.0611,to appear in J. Reine Angew. Math.[14] Y. Lekili, A. Polishchuk, Arithmetic mirror symmetry for genus 1 curves with n marked points ,Selecta Math. 23 (2017), 1851–1907.[15] Y. Lekili, A. Polishchuk, Associative Yang-Baxter equation and Fukaya categories of square-tiledsurfaces , Advances in Math. 343 (2019), 273–315.[16] D. Luna,
Slices ´Etales , Bull. Soc. Math. de France 33 (1973), 81–105.[17] H. Minamoto,
A noncommutative version of Beilinson’s theorem , J. Algebra 320 (2008), 238–252.[18] F. Nironi,
Grothendieck duality for Deligne-Mumford stacks , preprint arXiv:0811.1955.[19] M. Olsson, J. Starr,
Quot functors for Deligne-Mumford stacks , Comm. Algebra 31 (2003), no. 8,4069–4096.[20] D. Orlov,
Remarks on generators and dimensions of triangulated categories , Mosc. Math. J. 9 (2009),153–159.[21] D. Piontkovski,
Coherent algebras and noncommutative projective lines , J. Algebra 319 (2008), 3280–3290.[22] A. Polishchuk,
Classical Yang-Baxter equation and the A ∞ -constraint , Adv. Math. 168 (2002), 56–95.[23] A. Polishchuk, Extensions of homogeneous coordinate rings to A ∞ -algebras , Homology, Homotopyand Applications 5 (2003), 407–421.[24] A. Polishchuk, Massey products on cycles of projective lines and trigonometric solutions of the Yang-Baxter equations , in
Algebra, Arithmetic and Geometry, Vol.II: in Honor of Yu. I. Manin , 573–618,Birkh¨auser, Boston, 2009.[25] A. Polishchuk,
Moduli of curves as moduli of A ∞ -structures , Duke Math J. 166 (2017), 2871–2924.
26] A. Polishchuk,
Moduli of curves with nonspecial divisors and relative moduli of A ∞ -structures ,arXiv:1511.03797, to appear in Journal of the Inst. Math. Jussieu.[27] A. Polishchuk, L. Positselskii, Quadratic algebras , Amer. Math. Soc., Providence, RI, 2005.[28] A.-C. Van Roosmalen,
Abelian -Calabi-Yau categories , IMRN 2008, no. 6, Art. ID rnn003.[29] W. Schelter, On the Krull-Akizuki theorem , J. London Math. Soc. (2) 13 (1976), 263–264.[30] P. Seidel,
Fukaya categories and Picard-Lefschetz theory , European Math. Soc., Z¨urich, 2008.[31] P. Seidel,
Abstract analogues of flux as symplectic invariants , M´em. Soc. Math. Fr. (N.S.) No. 137(2014).[32] P. Seidel, R. Thomas,
Braid group actions on derived categories of coherent sheaves , Duke Math. J.108 (2001), 37–108.[33] L. W. Small, R. B. Warfield, Jr.,
Prime affine algebras of Gelfand-Kirillov dimension one , J. Algebra91 (1984), 386–389.[34] M. Van den Bergh,
Non-commutative homology of some three-dimensional quantum spaces , in
Pro-ceedings of Conference on Algebraic Geometry and Ring Theory in honor of Michael Artin, Part III(Antwerp, 1992) , K-Theory 8 (1994), 213–230.[35] M. Van den Bergh,
Existence theorems for dualizing complexes over non-commutative graded andfiltered rings , J. Algebra 195 (1997), 662–679.[36] M. Van den Bergh,
On global deformation quantization in the algebraic case , J. Algebra 315 (2007),no. 1, 326–395.[37] Q. S. Wu and J. J. Zhang,
Dualizing complexes over noncommutative local rings , J. Algebra 239(2001), 513–548.[38] J. J. Zhang,
Non-Noetherian regular rings of dimension
2, Proc. AMS 126 (1998), 1645–1653.