Calabi-Yau properties of 3 -dimensional DG Sklyanin algebras
aa r X i v : . [ m a t h . R A ] S e p CALABI-YAU PROPERTIES OF -DIMENSIONAL DGSKLYANIN ALGEBRAS X.-F. MAO, X.-T. WANG, Y.-N. YANG, AND M.-Y.ZHANG
Abstract.
In this paper, we compute all possible differential structures of a3-dimensional DG Sklyanin algebra A , which is a connected cochain DG alge-bra whose underlying graded algebra A is a 3-dimensional Sklyanin algebra S a,b,c . We show that there are three major cases depending on the parameters a, b, c of the underlying Sklyanin algebra S a,b,c : (1) either a = b or c = 0,then ∂ A = 0; (2) a = − b and c = 0, then the 3-dimensional DG Sklyaninalgebra is actually a DG polynomial algebra; and (3) a = b and c = 0, thenthe DG Sklyanin algebra is uniquely determined by a 3 × M . It isworthy to point out that case (2) has been systematically studied in [MGYC]and case (3) is just the DG algebra A O − ( k ) ( M ) in [MWZ]. Finally, we provethat all 3-dimensional DG Sklyanin algebras are Calabi-Yau DG algebras. introduction The theory of differential graded algebras (DG algebras, for short) and theirmodules has numerous applications in rational homotopy theory as well as algebraicgeometry. In particular, general results in DG homological algebra depend on theconstructions of some interesting families of DG algebras. In the literature, therehas been many papers on graded commutative DG algebras. Especially, the Sullivanalgebra and De Rham complex are fundamental DG algebra models in rationalhomotopy theory and differential geometry, respectively. Comparatively speaking,less attention has been paid to non-commutative DG algebras. To change thissituation, many attempts have been made to construct some interesting family ofnon-commutative cochain DG algebras with some nice homological properties suchas homologically smoothness, Gorensteinness and Calabi-Yau property. In [MHLX],[MGYC] and [MXYA], DG down-up algebras, DG polynomial algebras and DG freealgebras are introduced and systematically studied, respectively. It is exciting todiscover that non-trivial DG down-up algebras, non-trivial DG polynomial algebrasand DG free algebras with 2 degree 1 variables are Calabi-Yau DG algebras. SinceGinzburg introduced Calabi-Yau (DG) algebras in [Gin], they have been extensivelystudied due to their links to mathematical physics, representation theory and non-commutative algebraic geometry. In general, the homological properties of a DGalgebra are determined by the joint effects of its underlying graded algebra structureand differential structure. Although there have been some discriminating methods(cf.[HM, MYY]), it is still difficult in general to detect the Calabi-Yau propertyof a cochain DG algebra. The Calabi-Yau properties of DG down-up algebras,DG polynomial algebras and DG free algebras inspire us to construct cochain DGalgebra on some well-known Artin-Schelter regular algebras.The 3-dimensional Sklyanin algebras form the most important class of Artin-Schelter regular algebras of global dimension 3. Let k be an algebraically closedfield of characteristic 0 and D the subset of the projective plane P k consisting of Mathematics Subject Classification.
Primary 16E45, 16E65, 16W20,16W50.
Key words and phrases.
Calabi-Yau property, cochain DG algebra, Sklyanin algebra. the 12 points: D := { (1 , , , (0 , , , (0 , , } ⊔ { ( a, b, c ) | a = b = c } . Recall that the points ( a, b, c ) ∈ P k − D parametrize the 3-dimensional Sklyaninalgebras, S a,b,c = k h x, y, z i ( f , f , f ) , where f = ayz + bzy + cx f = azx + bxz + cy f = axy + byx + cz . We say that a cochain DG algebra A is a 3-dimensional Sklyanin DG algebra if itsunderlying graded algebra A is a 3-dimensional Sklyanin algebra S a,b,c , for some( a, b, c ) ∈ P k − D . We describe all possible differential structures on a 3-dimensionalSklyanin DG algebra by the following theorem (cf.Theorem 3.1): Theorem A.
Let A be a 3-dimensional DG Sklyanin algebra with A = S a,b,c ,( a, b, c ) ∈ P k − D . Then we have the following statements:(1) ∂ A = 0 if either | a | 6 = | b | or c = 0.(2) ∂ A is defined by ∂ A ( x ) = αx + βxy + γxz∂ A ( y ) = αyx + βy + γyz∂ A ( z ) = αxz + βyz + γz , for some ( α, β, γ ) ∈ A k , if a = − b, c = 0 . (3) ∂ A is defined by ∂ A ( x ) ∂ A ( y ) ∂ A ( z ) = M x y z , for some M ∈ M ( k ) , if a = b, c = 0 . One sees that the 3-dimensional DG Sklyanin algebra A is just a polynomial DGalgebra in three variables when a = − b and c = 0. Indeed, Theorem A (2) coincideswith [MGYC, Theorem 3.1]. Applying [MGYC, Corollary 6.3, Corollary 6.5], weshow that A is a Calabi-Yau DG algebra when a = − b and c = 0..When a = b and c = 0, the 3-dimensional DG Sklyanin algebra A in Theorem Ais just the DG algebra A O − ( k ) ( M ) in [MWZ]. Note that Theorem A (3) coincideswith [MWZ, Proposition 2.1]. The isomorphism problem, the automorphism group,the cohomology graded algebra and the Calabi-Yau properties of A O − ( k ) ( M ) havebeen systematically studied in [MWZ].For those 3-dimensional DG Sklyanin algebras with zero differential, we can showthat they are Calabi-Yau DG algebras by [MYY, Proposition 2.3 ]. Then we reachthe following conclusion (see Theorem 4.6). Theorem B.
All 3-dimensional DG Sklyanin algebras are Calabi-Yau DG algebras.———————————————————————-2.
Notations and conventions
Throughout this paper, k is an algebraically closed field of characteristic 0. Forany k -vector space V , we write V ∗ = Hom k ( V, k ). Let { e i | i ∈ I } be a basis of afinite dimensional k -vector space V . We denote the dual basis of V by { e ∗ i | i ∈ I } ,i.e., { e ∗ i | i ∈ I } is a basis of V ∗ such that e ∗ i ( e j ) = δ i,j . For any graded vector space W and j ∈ Z , the j -th suspension Σ j W of W is a graded vector space defined by(Σ j W ) i = W i + j . ALABI-YAU PROPERTIES OF 3-DIMENSIONAL DG SKLYANIN ALGEBRAS 3
A cochain DG algebra is a graded k -algebra A together with a differential ∂ A : A → A of degree 1 such that ∂ A ( ab ) = ( ∂ A a ) b + ( − | a | a ( ∂ A b )for all graded elements a, b ∈ A . For any DG algebra A , we denote A op as itsopposite DG algebra, whose multiplication is defined as a · b = ( − | a |·| b | ba for allgraded elements a and b in A .Let A be a cochain DG algebra. We denote by A i its i -th homogeneous compo-nent. The differential ∂ A is a sequence of linear maps ∂ i A : A i → A i +1 such that ∂ i +1 A ◦ ∂ i A = 0, for all i ∈ Z . If ∂ A = 0, A is called non-trivial. The cohomologygraded algebra of A is the graded algebra H ( A ) = M i ∈ Z ker( ∂ i A )im( ∂ i − A ) . For any cocycle element z ∈ ker( ∂ i A ), we write ⌈ z ⌉ as the cohomology class in H ( A ) represented by z . One sees that H ( A ) is a connected graded algebra if A is a connected cochain DG algebra. The derived category of left DG modulesover A (DG A -modules for short) is denoted by D( A ). A DG A -module M iscompact if the functor Hom D( A ) ( M, − ) preserves all coproducts in D( A ). By [MW1,Proposition 3.3], a DG A -module is compact if and only if it admits a minimal semi-free resolution with a finite semi-basis. The full subcategory of D( A ) consisting ofcompact DG A -modules is denoted by D c ( A ).A cochain algebra A is called connected if its underlying graded algebra A isa connected graded algebra. For any connected DG algebra A , we write m as themaximal DG ideal A > of A . Via the canonical surjection ε : A → k , k is both a DG A -module and a DG A op -module. It is easy to check that the enveloping DG algebra A e = A ⊗ A op of A is also a connected cochain DG algebra with H ( A e ) ∼ = H ( A ) e ,and m A e = m A ⊗ A op + A ⊗ m A op . We have the following list of homological properties for DG algebras.
Definition 2.1.
Let A be a connected cochain DG algebra.(1) If dim k H ( R Hom A ( k , A )) = 1, then A is called Gorenstein (cf. [FHT1]);(2) If A can be connected with the trivial DG algebra ( H ( A ) ,
0) by a zig-zag ←→← · · · → of quasi-isomorphisms, then A is called formal (cf.[Kal,Lunt]);(3) If A k , or equivalently A e A , has a minimal semi-free resolution with a semi-basis concentrated in degree 0, then A is called Koszul (cf. [HW]);(4) If A k , or equivalently the DG A e -module A is compact, then A is calledhomologically smooth (cf. [MW3, Corollary 2.7]);(5) If A is homologically smooth and R Hom A e ( A , A e ) ∼ = Σ − n A in the derived category D(( A e ) op ) of right DG A e -modules, then A is calledan n -Calabi-Yau DG algebra (cf. [Gin, VdB]).A connected cochain DG algebra A is called a 3-dimensional DG Sklyanin algebraif A = S a,b,c , for some ( a, b, c ) ∈ P k − D . The motivation of this paper is to studywhether 3-dimensional DG Sklyanian algebras have these homological propertiesmentioned in Definition 2.1. X.-F. MAO, X.-T. WANG, Y.-N. YANG, AND M.-Y.ZHANG differential structures In this section, we determine all possible differential structures of a 3-dimensionalDG Sklyanin algebra A . Let A = S a,b,c with ( a, b, c ) ∈ P k − D . Note that S a,b,c = k h x, y, z i ( f , f , f ) , where f = ayz + bzy + cx f = azx + bxz + cy f = axy + byx + cz . Since the differential ∂ A of A is a k -linear map of degree 1, we may let ∂ A ( x ) = ( x, y, z ) M x xyz ,∂ A ( y ) = ( x, y, z ) M y xyz ,∂ A ( z ) = ( x, y, z ) M z xyz , where M x = ( c x , c x , c x ) = ( m xij ) × = r x r x r x ,M y = ( c y , c y , c y ) = ( m yij ) × = r y r y r y and M z = ( c z , c z , c z ) = ( m zij ) × = r z r z r z are three 3 × A , we have the following system of equations ∂ A ( f ) = 0 Eq(1) ∂ A ( f ) = 0 Eq(2) ∂ A ( f ) = 0 Eq(3) ∂ A ( x ) = 0 Eq(4) ∂ A ( y ) = 0 Eq(5) ∂ A ( z ) = 0 Eq(6) . ALABI-YAU PROPERTIES OF 3-DIMENSIONAL DG SKLYANIN ALGEBRAS 5
The equation Eq(1) is0 = ∂ A [( x, y, z ) c a b xyz ]=( ∂ A ( x ) , ∂ A ( y ) , ∂ A ( z )) c a b xyz − ( x, y, z ) c a b ∂ A ( x ) ∂ A ( y ) ∂ A ( z ) =( x, y, z )[ M x xyz , M y xyz , M z xyz ] c a b xyz − ( x, y, z ) c a b ( x, y, z ) M x ( x, y, z ) M y ( x, y, z ) M z xyz =( x, y, z )( xc x + yc x + zc x , xc y + yc y + zc y , xc z + yc z + zc z ) c a b xzy − ( x, y, z ) c a b xr x + yr x + zr x xr y + yr y + zr y xr z + yr z + zr z xyz . Similarly, Eq(2) and Eq(3) are0 =( x, y, z )( xc x + yc x + zc x , xc y + yc y + zc y , xc z + yc z + zc z ) b c a xzy − ( x, y, z ) b c a xr x + yr x + zr x xr y + yr y + zr y xr z + yr z + zr z xyz and0 =( x, y, z )( xc x + yc x + zc x , xc y + yc y + zc y , xc z + yc z + zc z ) a b c xzy − ( x, y, z ) a b c xr x + yr x + zr x xr y + yr y + zr y xr z + yr z + zr z xyz , X.-F. MAO, X.-T. WANG, Y.-N. YANG, AND M.-Y.ZHANG respectively. For Eq(4),Eq(5) and Eq(6), we can also expand them similarly. Forexample, in A , Eq(4) is0 = ∂ A ◦ ∂ A ( x ) = ∂ A [( x, y, z ) M x xyz ]= ( ∂ A ( x ) , ∂ A ( y ) , ∂ A ( z )) M x xyz − ( x, y, z ) M x ∂ A ( x ) ∂ A ( y ) ∂ A ( z ) = ( x, y, z )[ M x xyz , M y xyz , M z xyz ] M x xyz − ( x, y, z ) M x ( x, y, z ) M x ( x, y, z ) M y ( x, y, z ) M z xyz =( x, y, z )[( xc x + yc x + zc x ) r x + ( xc y + yc y + zc y ) r x + ( xc z + yc z + zc z ) r x ] xyz − ( x, y, z )[ c x ( xr x + yr x + zr x ) + c x ( xr y + yr y + zr y ) + c x ( xr z + yr z + zr z )] xyz =( x, y, z )[( yc x + zc x ) r x + ( xc y + yc y + zc y ) r x + ( xc z + yc z + zc z ) r x ] xyz − ( x, y, z )[ c x ( yr x + zr x ) + c x ( xr y + yr y + zr y ) + c x ( xr z + yr z + zr z )] xyz . By similar computations, Eq(5) and Eq(6) are0 = ( x, y, z )[( xc x + yc x + zc x ) r y + ( xc y + zc y ) r y + ( xc z + yc z + zc z ) r y ] xyz − ( x, y, z )[ c y ( xr x + yr x + zr x ) + c y ( xr y + zr y ) + c y ( xr z + yr z + zr z )] xyz and0 = ( x, y, z )[( xc x + yc x + zc x ) r z + ( xc y + yc y + zc y ) r z + ( xc z + yc z ) r z ] xyz − ( x, y, z )[ c z ( xr x + yr x + zr x ) + c z ( xr y + yr y + zr y ) + c z ( xr z + yr z )] xyz , respectively. In order to study the solutions of Eq(1) ∼ Eq(6), we divide all 3-dimensional DG Sklyanin algebras into the following 4 case:Case 1 . a = 0 , b = 0 , c = 0;Case 2 . b = 0 , a = 0 , c = 0;Case 3 . a = 0 , b = 0 , c = 0;Case 4 . c = 0 , a = 0 , b = 0 . ALABI-YAU PROPERTIES OF 3-DIMENSIONAL DG SKLYANIN ALGEBRAS 7
Case . In this case, S a,b,c has a k -linear basis { x , x y, x z, xyx, xzx, xyz, yx , yxy, yzx, zxy } . Via some routine and tedious computations of Eq(1) , Eq(2) and Eq(3), we can seethat they are equivalent to m x = 0 m y = 0 m z = 0 m x = m z m x = m y m y = m z m x = cb ( m x − m x ) m x = cb ( m x − m x ) m y = cb ( m y − m y ) m y = cb ( m y − m y ) m z = cb ( m z − m z ) m z = cb ( m z − m z ) m x = 2 m z + cb m x m y = 2 m x + cb m y m z = 2 m y + cb m z . (1)Substituting (1) into the 30 equations obtained by Eq(4), Eq(5) and Eq(6), wesee that those equations are equivalent to m x = m x = m z = 0 . Therefore, theequations Eq(1) ∼ Eq(6) are equivalent to m x = m x = m x = m y = m y = m y = m z = m z = m z = 0 m x = cb m x m x = cb m x m x = cb m x m y = cb m y m y = cb m y m y = cb m y m z = cb m z m z = cb m z m z = cb m z . Then ∂ A is defined by ∂ A ( x ) = ( x, y, z ) α bc α bc α α bc α α xyz ∂ A ( y ) = ( x, y, z ) β bc β bc β β bc β β xyz ∂ A ( z ) = ( x, y, z ) γ bc γ bc γ γ bc γ γ xyz , X.-F. MAO, X.-T. WANG, Y.-N. YANG, AND M.-Y.ZHANG for some ( α , α , α , β , β , β , γ , γ , γ ) ∈ A k . Since bzy + cx = 0 bxz + cy = 0 byx + cz = 0in A , we have ∂ A ( x ) = ( x, y, z ) α bc α bc α α bc α α xyz = α x + bc α xz + bc α yx + α y + bc α zy + α z = α c ( cx + bzy ) + α c ( cy + bxz ) + α c ( cz + byx )= 0 . Similarly, we can show that ∂ A ( y ) = ∂ A ( z ) = 0. Hence ∂ A = 0.3.2. Case . In this case, b = 0 , a, c ∈ k × , one sees that S a,b,c has a k -linear basis { x , x y, x z, xy , xyx, xzy, yx , yxz, y x, zyx } . By computations of Eq(1) , Eq(2) and Eq(3), we can see that they are equivalent to m x = 0 m y = 0 m z = 0 m x = m y m x = m z m y = m z m x = ca ( m x − m x ) m x = ca ( m x − m x ) m y = ca ( m y − m y ) m y = ca ( m y − m y ) m z = ca ( m z − m z ) m z = ca ( m z − m z ) m x = 2 m y + ca m x m y = 2 m x + ca m y m z = 2 m x + ca m z . (2)Substituting (2) into the 30 equations obtained by Eq(4), Eq(5) and Eq(6), wesee that those equations are equivalent to m x = m x = m z = 0 . Therefore, the
ALABI-YAU PROPERTIES OF 3-DIMENSIONAL DG SKLYANIN ALGEBRAS 9 equations Eq(1) ∼ Eq(6) are equivalent to m x = m x = m x = m y = m y = m y = m z = m z = m z = 0 m x = ca m x m x = ca m x m x = ca m x m y = ca m y m y = ca m y m y = ca m y m z = ca m z m z = ca m z m z = ca m z . Then ∂ A is defined by ∂ A ( x ) = ( x, y, z ) α ac α α ac α ac α α xyz ∂ A ( y ) = ( x, y, z ) β ac β β ac β ac β β xyz ∂ A ( z ) = ( x, y, z ) γ ac γ γ ac γ ac γ γ xyz , for some ( α , α , α , β , β , β , γ , γ , γ ) ∈ A k . Since ayz + cx = 0 azx + cy = 0 axy + cz = 0in A , we have ∂ A ( x ) = ( x, y, z ) α ac α α ac α ac α α xyz = α x + ac α xy + ac α yz + α y + ac α zx + α z = α c ( cx + ayz ) + α c ( cy + azx ) + α c ( cz + axy )= 0 . Similarly, we can show that ∂ A ( y ) = ∂ A ( z ) = 0. Hence ∂ A = 0.3.3. Case . In this case, a, b, c ∈ k × , one sees that A = S a,b,c has a k -linear basis { x y, x z, xy , xz , yx , y x, y z, yz , xyz, xzy, yzx, yxz } . By computations, one sees that Eq(1) , Eq(2) and Eq(3) are equivalent to ( c − b ca ) m z − ( a + b a ) m y + c b m z + ( b + b a ) m y − c b m z = 0 − bc a m z + am x − bca m y − am y + ( c + b ca ) m y + am z − a c m x = 0 c a m z + b a m x − bca m y − am y + ( c + b ca ) m y + am z − a c m x = 0 bca m x − cm x + ( c + acb ) m z − ( acb + bca ) m z = 0 b − abc m z + bm x + cm y − am x − bca m y + ( a − b ) m z = 0 ab − a c m z + bm x − a b m x − bm z + a b m z + a − abc m x = 0 − am x + a b m x + am z − a b m z = 0 − bm x − cm y + bm x + bca m y + ab − b c m x = 0 bm z − ac + bcb m x + acb m x + cm y − a c m y − cm z + a b m x = 0 − c b m x + a + b a m y + c b m x − ( a + b ) m y + ac − bcb m x = 0 − am z + cm x − acb m y − cm z + b c m y + acb m z − bm x = 0 bca m y + ab − a b m y − am z − cm y + a b m z = 0 − b ca m y + bm z − c a m x − cm x + bc + ac a m x + abc m y − abc m z = 0 a − b a m y − ac + bca m x + cm y + b c + abca m x − acb m y = 0 bca m y + bm z − ( a + b a ) m z + cm x − c a m x − b ac m y + b ac m z = 0 cm z − bca m z + 2 am y − ( b a + a b ) m y + ( b a − a ) m z = 0 cm y − ( b + a b ) m z + am z + acb m x − a bc m x − c b m y + a bc m z = 0 − acb m z + 2 bm x + cm z − bm z − ( b a + a b ) m x + ( a b − b ) m z = 0 − cm y + am z − a cb m x − c b m y + abc m x + ( c b + ac b ) m y − abc m z = 0 cm x + ( b − a b ) m x − ( c + acb ) m y − bca m x + ( acb + a cb m y = 0 ab − b c m y + bca m z + am x − cm z − bm x + ( b − a ) m y = 0 a − abc m y + cm z − acb m x − am x + am y = 0 − cm z − acb m z − am y + am y = 0 − bca m z − bm x + cm z + b − abc m x + bm x = 0 b a m y + bm z − bm y − b a m z = 0 am y − am y − cm x + acb m x + ab − a c m y = 0 − bm y + am y + a − abc ) m z + cm x + ( b − a ) m z − acb m x = 0 − b a m y − am z + am y + ab − b c m z + b a m z + b − abc m y = 0 − bca m x + cm y − cm z − bm z + bca m z + a c m x − am y = 0( b − b a ) m x − bm z + acb m x + b a m z − cm x = 0 cm x + bca m y − ( c + bca ) m y + am z − cm z − b c m x + b a m y = 0 c a m y − c a m y + ( b + a b ) m x − ( a + b ) m x + ( bca − c ) m y = 0 am x − am y + a b m y + c b ) m z − a b m z − acb m x + a c m y = 0 − cm y + ( c + bca ) m z + acb m y − ( bca + acb ) m z = 0 − bm x + bm y − ac b m z − acb m x + bm z + ( c + a cb ) m x − b c m y = 0 c a m z + ( c − a cb ) m z − ( b + a b ) m x − c a m z + ( a + a b ) m x = 0 . ALABI-YAU PROPERTIES OF 3-DIMENSIONAL DG SKLYANIN ALGEBRAS 11
Note that the equations above can be divided into the following three systems ofequations: ( c − b ca ) m z − ( a + b a ) m y + c b m z + ( b + b a ) m y − c b m z = 0 − bc a m z + am x − bca m y − am y + ( c + b ca ) m y + am z − a c m x = 0 c a m z + b a m x − bca m y − am y + ( c + b ca ) m y + am z − a c m x = 0 bca m x − cm x + ( c + acb ) m z − ( acb + bca ) m z = 0 b − abc m z + bm x + cm y − am x − bca m y + ( a − b ) m z = 0 ab − a c m z + bm x − a b m x − bm z + a b m z + a − abc m x = 0 − am x + a b m x + am z − a b m z = 0 − bm x − cm y + bm x + bca m y + ab − b c m x = 0 bm z − ac + bcb m x + acb m x + cm y − a c m y − cm z + a b m x = 0 − c b m x + a + b a m y + c b m x − ( a + b ) m y + ac − bcb m x = 0 − am z + cm x − acb m y − cm z + b c m y + acb m z − bm x = 0 bca m y + ab − a b m y − am z − cm y + a b m z = 0 , (3) − b ca m y + bm z − c a m x − cm x + bc + ac a m x + abc m y − abc m z = 0 a − b a m y − ac + bca m x + cm y + b c + abca m x − acb m y = 0 bca m y + bm z − ( a + b a ) m z + cm x − c a m x − b ac m y + b ac m z = 0 cm z − bca m z + 2 am y − ( b a + a b ) m y + ( b a − a ) m z = 0 cm y − ( b + a b ) m z + am z + acb m x − a bc m x − c b m y + a bc m z = 0 − acb m z + 2 bm x + cm z − bm z − ( b a + a b ) m x + ( a b − b ) m z = 0 − cm y + am z − a cb m x − c b m y + abc m x + ( c b + ac b ) m y − abc m z = 0 cm x + ( b − a b ) m x − ( c + acb ) m y − bca m x + ( acb + a cb m y = 0 ab − b c m y + bca m z + am x − cm z − bm x + ( b − a ) m y = 0 a − abc m y + cm z − acb m x − am x + am y = 0 − cm z − acb m z − am y + am y = 0 − bca m z − bm x + cm z + b − abc m x + bm x = 0 , (4) b a m y + bm z − bm y − b a m z = 0 am y − am y − cm x + acb m x + ab − a c m y = 0 − bm y + am y + a − abc ) m z + cm x + ( b − a ) m z − acb m x = 0 − b a m y − am z + am y + ab − b c m z + b a m z + b − abc m y = 0 − bca m x + cm y − cm z − bm z + bca m z + a c m x − am y = 0( b − b a ) m x − bm z + acb m x + b a m z − cm x = 0 cm x + bca m y − ( c + bca ) m y + am z − cm z − b c m x + b a m y = 0 c a m y − c a m y + ( b + a b ) m x − ( a + b ) m x + ( bca − c ) m y = 0 am x − am y + a b m y + c b ) m z − a b m z − acb m x + a c m y = 0 − cm y + ( c + bca ) m z + acb m y − ( bca + acb ) m z = 0 − bm x + bm y − ac b m z − acb m x + bm z + ( c + a cb ) m x − b c m y = 0 c a m z + ( c − a cb ) m z − ( b + a b ) m x − c a m z + ( a + a b ) m x = 0 . (5) One sees that (3) is a system of linear equations with variables m z , m x , m y , m x , m y , m z , m y , m z and m x . Its solution is either m x = m x = m z = m z = 12 m y , m z = m y = m y = m x = 0 , or m x = ac m x , m x = bc m x , m z = m y = m y = m z = m y = m z = 0 . Similarly, (4) is a system of linear equations with variables m y , m z , m x , m z , m x , m y , m x , m y and m z . And (4) is equivalent to m z = ac m z , m z = bc m z , m y = m x = m x = m y = m x = m y = 0 . The last system of linear equations (5) has variables m x , m y , m z , m y , m z , m x , m z , m x and m y . By computations, its solution is either m y = m z = m y = m z = 12 m x , m z = m x = m x = m y = 0 , or m y = ac m y , m y = bc m y , m x = m z = m z = m x = m z = m x = 0 . Therefore, Eq(1), Eq(2) and Eq(3) implies one of the following systems of equa-tions: m x = m x = m z = m z = m y , m z = m y = m y = m x = 0 m z = ac m z , m z = bc m z , m y = m x = m x = m y = m x = m y = 0 m y = m z = m y = m z = m x , m z = m x = m x = m y = 0(6) m x = ac m x , m x = bc m x , m z = m y = m y = m z = m y = m z = 0 m z = ac m z , m z = bc m z , m y = m x = m x = m y = m x = m y = 0 m y = m z = m y = m z = m x , m z = m x = m x = m y = 0(7) m x = m x = m z = m z = m y , m z = m y = m y = m x = 0 m z = ac m z , m z = bc m z , m y = m x = m x = m y = m x = m y = 0 m y = ac m y , m y = bc m y , m x = m z = m z = m x = m z = m x = 0(8) m x = ac m x , m x = bc m x , m z = m y = m y = m z = m y = m z = 0 m z = ac m z , m z = bc m z , m y = m x = m x = m y = m x = m y = 0 m y = ac m y , m y = bc m y , m x = m z = m z = m x = m z = m x = 0 . (9) Conversely, if any one of (6),(7),(8) and (9) holds, then we can get Eq(1), Eq(2)and Eq(3).If Eq(1), Eq(2) and Eq(3) implies (6), then we substitute (6) into the 36 equationsobtained by Eq(4), Eq(5) and Eq(6). We see that those equations are equivalent to m kij = 0 , ∀ k ∈ { x, y, z } , ∀ i, j ∈ { , , } . It indicates ∂ A = 0.If Eq(1), Eq(2) and Eq(3) implies (7), then we substitute (7) into the 36 equationsobtained by Eq(4), Eq(5) and Eq(6). We see that those equations are equivalent to m x = ac m x ,m x = bc m x ,m z = ac m z ,m z = bc m z . ALABI-YAU PROPERTIES OF 3-DIMENSIONAL DG SKLYANIN ALGEBRAS 13
Then ∂ A is defined by ∂ A ( x ) = ( x, y, z ) ac α bc α α xyz ,∂ A ( y ) = ( x, y, z ) ac β bc β β xyz ,∂ A ( z ) = ( x, y, z ) ac γ bc γ γ xyz , for some ( α, β, γ ) ∈ A k . Since ayz + bzy + cx = 0 azx + bxz + cy = 0 axy + byx + cz = 0in A , we have ∂ A ( x ) = ( x, y, z ) ac α bc α α xyz = αc ( byx + axy + cz )= 0 . Similarly, we can show that ∂ A ( y ) = ∂ A ( z ) = 0. Hence ∂ A = 0.If Eq(1), Eq(2) and Eq(3) implies (8), then we substitute (8) into the 36 equationsobtained by Eq(4), Eq(5) and Eq(6). We see that those equations are equivalent to m z = ac m z , m z = bc m z , m y = ac m y , m y = bc m y . Then ∂ A is defined by ∂ A ( x ) = ( x, y, z ) ac α bc α α xyz ,∂ A ( y ) = ( x, y, z ) ac β bc β β xyz ,∂ A ( z ) = ( x, y, z ) ac γ bc γ γ xyz , for some ( α, β, γ ) ∈ A k . As above, we can show that ∂ A = 0.If Eq(1), Eq(2) and Eq(3) implies (9), then we substitute (9) into the 36 equationsobtained by Eq(4), Eq(5) and Eq(6). We see that those equations are equivalent to m x = ac m x , m x = bc m x , m z = ac m z , m z = bc m z , m y = ac m y , m y = bc m y . Then ∂ A is defined by ∂ A ( x ) = ( x, y, z ) ac α bc α α xyz ,∂ A ( y ) = ( x, y, z ) ac β bc β β xyz ,∂ A ( z ) = ( x, y, z ) ac γ bc γ γ xyz , for some ( α, β, γ ) ∈ A k . As above, we can get ∂ A = 0.By the discussion above, we can reach the conclusion that ∂ A = 0 in Case 3.3.4. Case . In this case, c = 0 , a, b ∈ k × , one sees that S a,b,c has a k -linear basis { x , x y, x z, xy , xyz, xz , y , y z, yz , z } . By computations, Eq(1) , Eq(2) and Eq(3) are equivalent to ( b − a b ) m z = 0( a − b a ) m y = 0( b + a b ) m z − ( a + a b ) m z = 0( a − b ) m y − ( a − a b ) m z − ( a − b ) m z + ( a − a b ) m y = 0( a + b a ) m y − ( b + b a ) m y = 0( b − a ) m z = 0( a − a b ) m y − am z + a b m z = 02 am y − a b m y + ( a b − a ) m z = 0 m y ( a − b ) = 0( a − b ) m z = 0 − ( b + a b ) m z + ( a + a b ) m z = 0( b − b a ) m x − bm z + b a m z = 0( − b + a b ) m z = 0( b − a ) m x − ( b − a ) m z + ( a − a b ) m z + ( a b − a ) m x = 02 bm x + ( b a − b ) m z − b a m x = 0( b − a b ) m x = 0( b + a b ) m x − ( a + a b ) m x = 0( b − a ) m x = 0( b − a ) m y = 0( a − a b ) m x − am y + a b m y = 0 − ( a + b a ) m y + ( b + b a ) m y = 02 am x + ( a b − a ) m y − a b m x = 0( b − a ) m y + ( a − a b ) m x + ( a b − a ) m y + ( a − b ) m x = 0( − a + b a ) m y = 0( a − b ) m x = 0 − ( b + a b ) m x + ( a + a b ) m x = 0( a b − b ) m x = 0 . (10) ALABI-YAU PROPERTIES OF 3-DIMENSIONAL DG SKLYANIN ALGEBRAS 15 If a = b , then (10) is equivalent to m x = m x = m y = m y = m z = m z = 0 m x = bb − a (2 m z − ab m z ) = bb − a (2 m y − ab m y ) m y = bb − a (2 m z − ab m z ) = bb − a (2 m x − ab m x ) m z = bb − a (2 m y − ab m y ) = bb − a (2 m x − ab m x ) m x = ab m x m y = ab m y m z = ab m z . (11)Substituting (11) into the 30 equations obtained by Eq(4), Eq(5) and Eq(6), we seethat those equations are equivalent to ( a + b ) m x m y = 0( a + b ) m x m x = 0( a + b )( m y ) = 0( a + b )( m z ) = 0( a + b )( m x ) = 0( a + b ) m x m y = 0( a + b ) m y m z = 0( a + b )( m z ) = 0( a + b )( m x ) = 0( a + b ) m x m z = 0( a + b )( m y ) = 0( a + b ) m y m z = 0 ⇔ a = − b or ( a = − bm x = m y = m z = 0 . Hence the equations Eq(1) ∼ Eq(6) are equivalent to m x = m x = m y = m y = m z = m z = 0 m x = m z + m z = m y + m y m y = m z + m z = m x + m x m z = m y + m y = m x + m x m x = − m x m y = − m y m z = − m z when a = − b = 0 , andthey are equivalent to m xii = m yii = m zii = 0 , ∀ i ∈ { , , } m x = ab m x m x = ab m x m y = ab m y m y = ab m y m z = ab m z m z = ab m z m x = ab m x m y = ab m y m z = ab m z when a, b ∈ k × , a = b . Now, let consider the case a = b . In this case, (10) is equivalent to m x = m x m x = m x m x = m x m y = m y m y = m y m y = m y m z = m z m z = m z m z = m z . (12)Substituting (12) into the 30 equations obtained by Eq(4), Eq(5) and Eq(6), onesees that all those equations hold. Therefore, the equations Eq(1) ∼ Eq(6) areequivalent to (12).By the discussion above, we can reach the following conclusions:(i) If a, b ∈ k × , a = b and c = 0, then ∂ A is defined by ∂ A ( x ) = ( x, y, z ) ab α α α ab α ab α α xyz ,∂ A ( y ) = ( x, y, z ) ab β β β ab β ab β β xyz ,∂ A ( z ) = ( x, y, z ) ab γ γ γ ab γ ab γ γ xyz , for some ( α , α , α , β , β , β , γ , γ , γ ) ∈ A k . Since ayz + bzy = azx + bxz = axy + byx = 0in A , we have ∂ A ( x ) = ( x, y, z ) ab α α α ab α ab α α xyz = α yx + ab α xy + α xz + ab α zx + α zy + ab α yz = 0 . Similarly, we can show that ∂ A ( y ) = ∂ A ( z ) = 0. Hence ∂ A = 0.(ii)If a = − b ∈ k × , c = 0, then ∂ A is defined by ∂ A ( x ) = ( x, y, z ) α α α β − α α γ − α − α xyz ,∂ A ( y ) = ( x, y, z ) β β α − β β β − β γ − β xyz ,∂ A ( z ) = ( x, y, z ) γ γ − γ γ α − γ β − γ γ xyz , ALABI-YAU PROPERTIES OF 3-DIMENSIONAL DG SKLYANIN ALGEBRAS 17 for some ( α , α , α , α , β , β , β , β , γ , γ , γ , γ ) ∈ A k . Since yz − zy = 0 zx − xz = 0 xy − yx = 0in A , we have ∂ A ( x ) = ( x, y, z ) α α α β − α α γ − α − α xyz = α x + α xy + ( β − α ) yx + α xz + ( γ − α ) zx + α yz − α zy = α x + β yx + γ zx = α x + β xy + γ xz Similarly, we can show that ∂ A ( y ) = α yx + β y + γ yz∂ A ( z ) = α xz + β yz + γ z . Let α = α , β = β and γ = γ . Then ∂ A is defined by ∂ A ( x ) = αx + βxy + γxz∂ A ( y ) = αyx + βy + γyz∂ A ( z ) = αxz + βyz + γz , ( α, β, γ ) ∈ A k . (iii)If a = b ∈ k × , c = 0, then ∂ A is defined by ∂ A ( x ) = ( x, y, z ) M x xyz ∂ A ( y ) = ( x, y, z ) M y xyz ∂ A ( z ) = ( x, y, z ) M z xyz , where M x = ( m xij ) × , M y = ( m yij ) × and M z = ( m zij ) × are 3 × yz + zy = zx + xz = xy + yx = 0in A , we have ∂ A ( x ) = ( x, y, z ) M x xyz = m x x + m x xy + m x yx + m x y + m x xz + m x zx + m x yz + m x zy + m x z = m x x + m x y + m x z . Similarly, we can show that ∂ A ( y ) = m y x + m y y + m y z and ∂ A ( z ) = m z x + m z y + m z z . Let m i = m xii , m i = m yii and m i = m zii , i = 1 , ,
3. Then ∂ A ( x ) ∂ A ( y ) ∂ A ( z ) = M x y z . By the computations and analysis above, the following theorem is immediate.
Theorem 3.1.
Let A be a -dimensional DG Sklyanin algebra with A = S a,b,c , ( a, b, c ) ∈ P k − D . Then we have the following statements:(1) ∂ A = 0 if either | a | 6 = | b | or c = 0 .(2) ∂ A is defined by ∂ A ( x ) = αx + βxy + γxz∂ A ( y ) = αyx + βy + γyz∂ A ( z ) = αxz + βyz + γz , for some ( α, β, γ ) ∈ A k , if a = − b, c = 0 . (3) ∂ A is defined by ∂ A ( x ) ∂ A ( y ) ∂ A ( z ) = M x y z , for some M ∈ M ( k ) , if a = b, c = 0 . Remark 3.2.
In the case of a = − b = 0 , one sees that S a,b, is the polynomialalgebra in degree one variables. It is easy to check that the result in Theorem 3.1(2)coincides with [MGYC, Theorem 3.1] . When a = b and c = 0 , the -dimensionalDG Sklyanin algebra A in Theorem 3.1 is just the DG algebra A O − ( k ) ( M ) in [MWZ] . Note that Theorem 3.1 (3) coincides with [MWZ, Proposition 2.1] . Calabi-Yau properties
In this section, we study the Calabi-Yau properties of 3-dimensional DG Sklyaninalgebras. When it comes to those trivial cases, we rely heavily on the followinglemma.
Lemma 4.1. [MYY, Proposition 3.2]
Let A be a connected cochain DG algebrasuch that H ( A ) = k h⌈ x ⌉ , ⌈ y ⌉ , ⌈ z ⌉i / a ⌈ y ⌉⌈ z ⌉ + b ⌈ z ⌉⌈ y ⌉ + c ⌈ x ⌉ a ⌈ z ⌉⌈ x ⌉ + b ⌈ x ⌉⌈ z ⌉ + c ⌈ y ⌉ a ⌈ x ⌉⌈ y ⌉ + b ⌈ y ⌉⌈ x ⌉ + c ⌈ z ⌉ , where ( a, b, c ) ∈ P k − D and x, y, z ∈ ker( ∂ A ) . Then A is a Calabi-Yau DG algebra. Proposition 4.2.
Let A be a connected cochain DG algebra such that A belongto one of the following cases: ( i ) A = S a, ,c , ( ii ) A = S ,b,c , ( iii ) A = S a,b,c with a, b, c ∈ k × . Then A is a Calabi-Yau DG algebra.Proof. By Theorem 3.1, we have ∂ A = 0. So H ( A ) = A belongs to one of thefollowing 3 cases( i ) H ( A ) = S a, ,c , ( ii ) H ( A ) = S ,b,c , ( iii ) H ( A ) = S a,b,c with a, b, c ∈ k × . By Lemma 4.1, A is a Calabi-Yau DG algebra. (cid:3) Proposition 4.3.
Let A be a connected cochain DG algebra such that A = S a, − a, with a ∈ k × . Then A is a Calabi-Yau DG algebra. ALABI-YAU PROPERTIES OF 3-DIMENSIONAL DG SKLYANIN ALGEBRAS 19
Proof.
In this case, A is actually a DG polynomial algebra. By [MGYC, Remark4.4], we have either ∂ A = 0 or A ∼ = A ′ with A ′ = k [ x , x , x ] , | x i | = 1 and ∂ A ′ ( x i ) = x x i , i = 1 , ,
3. If ∂ A = 0, then A is a Calabi-Yau DG algebra by[MGYC, Theorem 6.4]. If A ∼ = A ′ , then by [MGYC, Proposition 5.1], we have H ( A ) ∼ = H ( A ′ ) ∼ = k [ ⌈ x ⌉ , ⌈ x x ⌉ , ⌈ x ⌉ ] . By [MGYC, Proposition 6.2], A is a Calabi-Yau DG algebra. (cid:3) It remains to consider the case that A = S a,a, with a ∈ k × . By Theorem 3.1, ∂ A is determined by a matrix M = ( m ij ) × such that ∂ A ( x ) ∂ A ( y ) ∂ A ( z ) = M x y z . In this cases, the DG Sklyanin algebra is just the DG algebra A O − ( k ) ( M ) in[MWZ]. Let M and M ′ be two matrixes in M ( k ). By [MH, Theorem B], A O − ( k ) ( M ) ∼ = A O − ( k ) ( M ′ )if and only if there exists C = ( c ij ) × ∈ QPL ( k ) such that M ′ = C − M ( c ij ) × , where QPL ( k ) is the subgroup of GL ( k ) consisting of quasi-permutation matrixes.In spite of this, we are unable to list all the isomorphism class of such DG Sklyaninalgebras one by one. This makes the following proposition very difficult to prove. Proposition 4.4.
Let A be a connected cochain DG algebra such that A = S a,a, .Then A is a Calabi-Yau DG algebra. Remark 4.5.
The proof of Proposition 4.4 is quite long and complicated. It involvescomputations of cohomology, subtle technique in matrix analysis and classification.To make the paper more readable, we divide it into three independent section tocompute H ( A ) according to the rank of M . One will see the existence of someDG Sklyanin algebras, whose Calabi-Yau properties can’t be obtained from theircohomologies. For those cases, we just cite the proof in [MH] . We will give thefinal proof of 4.4 in Section 7. By Proposition 4.2, Proposition 4.3 and Proposition 4.4, we can get the followingtheorem.
Theorem 4.6.
All -dimensional DG Sklyanin algebras are Calabi-Yau DG alge-bras. In the rest of this paper, our goal is to prove Proposition 4.4. It is notoriouslyhard to accomplish. Comparatively speaking, it is much easier to compute thecohomology graded algebra H ( A ) of a Cochain DG algebra A . Moreover, it isproved in [MYY] that A is a Calabi-Yau DG algebra if the trivial DG algebra( H ( A ) ,
0) is Calabi-Yau. By [MH], we know that a connected cochain DG algebra A is a Kozul Calabi-Yau DG algebra if H ( A ) belongs to one of the following cases:( a ) H ( A ) ∼ = k ; ( b ) H ( A ) = k [ ⌈ z ⌉ ] , z ∈ ker( ∂ A );( c ) H ( A ) = k h⌈ z ⌉ , ⌈ z ⌉i ( ⌈ z ⌉⌈ z ⌉ + ⌈ z ⌉⌈ z ⌉ ) , z , z ∈ ker( ∂ A ) . Recently, it is proved in [MHLX, Proposition 6.5] that a connected cochain DGalgebra A is Calabi-Yau if H ( A ) = k [ ⌈ z ⌉ , ⌈ z ⌉ ] where z ∈ ker( ∂ A ) and z ∈ ker( ∂ A ). This motivates us to compute the cohomologes of corresponding DGSklyanin algebras case by case first. For this, we should make some preparations. some basic lemmas In this section, we give some simple lemmas, which will be used in the subsequentcomputations. If there is no special assumption is emphasized, we let A be a DGSklyanin algebra with A = S a,a, , and ∂ A is determined by a matrix M in M ( k ).For convenience, we let x = x, x = y, x = z in the proof of the following lemmas. Lemma 5.1.
For any t ∈ N , x t , y t , z t are cocycle central elements of A .Proof. One sees that x i is a central element of A since x i x j = x i x i x j = − x i x j x i = x j x i , when i = j . This implies that each x ti is a central element of A . By Theorem 3.1,we have ∂ A ( x i ) = ∂ A ( x i ) x i − x i ∂ A ( x i )= n X j =1 m ij x j x i − x i n X j =1 m ij x j = n X j =1 m ij ( x j x i − x i x j ) = 0 . Using this, we can inductively prove ∂ A ( x ti ) = 0. (cid:3) Lemma 5.2.
Let Ω be a coboundary element in A of degree d ≥ . (1) If d = 2 l + 1 is odd, then Ω = ∂ A [ xyf + xzg + yzh ] , where f, g and h are alllinear combinations of monomials with non-negative even exponents. (2) If d = 2 l is even, then Ω = ∂ A [ xf + yg + zh + xyzu ] , where f, g , h and u areall linear combinations of monomials with non-negative even exponents.Proof. By the assumption, we haveΩ = ∂ A [ X l + l + l = d − l ,l ,l ≥ C l ,l ,l x l x l x l ] . If d = 2 l + 1 is odd, then d = 2 l is even. Since X l + l + l = d − l ,l ,l ≥ C l ,l ,l x l x l x l = X l + l + l = d − l ,l ,l ≥ l ,l are odd , l is even C l ,l ,l x l x l x l + X l + l + l = d − l ,l ,l ≥ l ,l are odd , l is even C l ,l ,l x l x l x l + X l + l + l = d − l ,l ,l ≥ l ,l are odd , l is even C l ,l ,l x l x l x l + X l + l + l = d − l ,l ,l ≥ l ,l ,l are even C l ,l ,l x l x l x l , ALABI-YAU PROPERTIES OF 3-DIMENSIONAL DG SKLYANIN ALGEBRAS 21 we have Ω = ∂ A [ X l + l + l = d − l ,l ,l ≥ C l ,l ,l x l x l x l ]= ∂ A [ x x X l + l + l = d − l ,l ,l ≥ l ,l are odd , l is even C l ,l ,l x l − x l − x l ]+ ∂ A [ x x X l + l + l = d − l ,l ,l ≥ l ,l are odd , l is even C l ,l ,l x l − x l x l − ]+ ∂ A [ x x X l + l + l = d − l ,l ,l ≥ l ,l are odd , l is even C l ,l ,l x l x l − x l − ]by Proposition 5.1. Let f = X l + l + l = d − l ,l ,l ≥ l ,l are odd , l is even C l ,l ,l x l − x l − x l ,g = X l + l + l = d − l ,l ,l ≥ l ,l are odd , l is even C l ,l ,l x l − x l x l − ,h = X l + l + l = d − l ,l ,l ≥ l ,l are odd , l is even C l ,l ,l x l x l − x l − . Then we prove (1).If d = 2 l is even, then d − l − X l + l + l = d − l ,l ,l ≥ C l ,l ,l x l x l x l = X l + l + l = d − l ,l ,l ≥ l ,l are even , l is odd C l ,l ,l x l x l x l + X l + l + l = d − l ,l ,l ≥ l ,l are even , l is odd C l ,l ,l x l x l x l + X l + l + l = d − l ,l ,l ≥ l ,l are even , l is odd C l ,l ,l x l x l x l + X l + l + l = d − l ,l ,l ≥ l ,l ,l are odd C l ,l ,l x l x l x l , we have Ω = ∂ A [ X l + l + l = d − l ,l ,l ≥ C l ,l ,l x l x l x l ]= ∂ A [ x X l + l + l = d − l ,l ,l ≥ l ,l are even , l is odd C l ,l ,l x l x l x l − ]+ ∂ A [ x X l + l + l = d − l ,l ,l ≥ l ,l are even , l is odd C l ,l ,l x l x l − x l ]+ ∂ A [ x X l + l + l = d − l ,l ,l ≥ l ,l are even , l is odd C l ,l ,l x l − x l x l ]+ ∂ A [ x x x X l + l + l = d − l ,l ,l ≥ l ,l ,l are odd C l ,l ,l x l − x l − x l − ] . Let f = X l + l + l = d − l ,l ,l ≥ l ,l are even , l is odd C l ,l ,l x l − x l x l ,g = X l + l + l = d − l ,l ,l ≥ l ,l are even , l is odd C l ,l ,l x l x l − x l ,h = X l + l + l = d − l ,l ,l ≥ l ,l are even , l is odd C l ,l ,l x l x l x l − ,u = X l + l + l = d − l ,l ,l ≥ l ,l ,l are odd C l ,l ,l x l − x l − x l − . Then we prove (2). (cid:3)
Lemma 5.3.
Let M = ( m ij ) × be a matrix in GL ( k ) . Then x , y , z arecoboundary elements in A .Proof. For ∀ a , a , a ∈ k , we have ∂ A ( c x + c x + c x )= a ( m x + m x + m x ) + a ( m x + m x + m x )+ a ( m x + m x + m x )= ( a m + a m + a m ) x + ( a m + a m + a m ) x + ( a m + a m + a m ) x . So ∂ A ( a x + a x + a x ) = x if and only if a m + a m + a m = 1 a m + a m + a m = 0 a m + a m + a m = 0 ⇔ M T a a a = . ALABI-YAU PROPERTIES OF 3-DIMENSIONAL DG SKLYANIN ALGEBRAS 23
Since r ( M ) = 3, there exists a = m m − m m | M | a = m m − m m | M | a = m m − m m | M | such that ∂ A ( a x + a x + a x ) = x . Similarly, we can show there exist b = m m − m m | M | b = m m − m m | M | b = m m − m m | M | and c = m m − m m | M | c = m m − m m | M | c = m m − m m | M | such that ∂ A ( b x + b x + b x ) = x and ∂ A ( c x + c x + c x ) = x , respectively. (cid:3) Lemma 5.4.
Let M = ( m ij ) × be a matrix in GL ( k ) and m m − m m = 0 .If g (¯ y, ¯ z ) ∈ Z l +1 [ A / ( x )] and h (¯ y, ¯ z ) ∈ Z l [ A / ( x )] are sum of monomials invariables ¯ y and ¯ z with l ≥ . Then h (¯ y, ¯ z ) = l X i =0 r i ¯ y l − i ¯ z i with r i ∈ k, ≤ i ≤ l. Furthermore, there exist u ( y, z ) and v ( y, z ) , which are sums of monomials in vari-ables y and z , such that ( g (¯ y, ¯ z ) = ∂ A [ u ( y, z )] ,h (¯ y, ¯ z ) = ∂ A [ v ( y, z )] . Proof.
Let g ( ¯ x . ¯ x ) = l +1 P j =0 t j ¯ x l +1 − j ¯ x j and h ( ¯ x , ¯ x ) = l P j =0 r j ¯ x l − j ¯ x j , whereeach t j , r j ∈ k . Then0 = ∂ A ( l +1 X j =0 t j x l +1 − j x j )= ∂ A ( l X i =0 t i x l − − i x i + l +1 X i =1 t i − x l − i x i − )= l X i =0 [ t i ( m ¯ x + m ¯ x ) ¯ x l − i − ¯ x i + t i +1 ¯ x l − i − ¯ x i ( m ¯ x + m ¯ x )]= l X i =0 [( t i m + t i +1 m ) ¯ x l − i ¯ x i + ( t i m + t i +1 m ) ¯ x l − i − ¯ x i +2 ]and0 = ∂ A ( l X j =0 r j x l − j x j )= l X i =1 r i − [( m ¯ x + m ¯ x ) ¯ x l − i ¯ x i − − ¯ x l − i +1 ¯ x i − ( m ¯ x + m ¯ x )] . They imply t m + t m = 0 t m + t m + t m + t m = 0 t m + t m + t m + t m = 0 ........t l − m + t l − m + t l − m + t l − m = 0 t l m + t l +1 m + t l − m + t l − m = 0 t l m + t l +1 m = 0(13)and r m = 0 r m = 0 r m + r m = 0 r m + r m = 0 ........r l − m + r l − m = 0 r l − m + r l − m = 0 r l − m = 0 r l − m = 0 . (14)Since m m − m m = 0, the rank of the system matrix m m · · · m m m m · · · m m m m · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · m m m m · · · m m m m · · · m m of (13) is l + 2. Hence the space of the solutions of (13) is of dimension l . On theother hand, for any 1 ≤ i ≤ l , ∂ A ( x l − i +12 x i − ) is − m ¯ x l − i +3 ¯ x i − + m ¯ x l − i +2 ¯ x i − − m ¯ x l − i +1 ¯ x i + m ¯ x l − i ¯ x i +1 . Therefore, { − m m − m m , − m m − m m , · · · , − m m − m m , − m m − m m } is a k -basisof the space of the solutions of system (13). So there exists { s i − ∈ k | ≤ i ≤ l } such that ∂ A ( l P i =1 s i − x l − i +12 x i − ) = g ( ¯ x , ¯ x ). Take u ( x, y ) = l P i =1 s i − x l − i +12 x i − .. ALABI-YAU PROPERTIES OF 3-DIMENSIONAL DG SKLYANIN ALGEBRAS 25
Since (cid:12)(cid:12)(cid:12)(cid:12) m m m m (cid:12)(cid:12)(cid:12)(cid:12) = 0, we can conclude r = r = · · · = r l − = 0 from thesystem of equations (14). So h ( ¯ x , ¯ x ) = l P i =0 r i ¯ x l − i ¯ x i . Since ( ∂ A [ m m m − m m x − m m m − m m x ] = ¯ x ∂ A [ − m m m − m m x + m m m − m m x ] = ¯ x , we have h ( ¯ x , ¯ x ) = l X i =0 r i ¯ x l − i ¯ x i = ∂ A [ l − X i =0 r i ( m x m m − m m − m x m m − m m ) x l − i − x i ]+ ∂ A [ r l ( − m x m m − m m + m x m m − m m ) x l − ] . Take v ( x , x ) = l − X i =0 r i ( m x m m − m m − m x m m − m m ) x l − i − x i + r l ( − m x m m − m m + m x m m − m m ) x l − . Then we are done. (cid:3)
Remark 5.5.
Since y and z are cocycle elements in A , one sees that u ( y, z ) in Lemma 5.4 can be chosen as u ( y, z ) = l P i =1 s i − y l − i +1 z i − with s i − ∈ k , ≤ i ≤ l . Lemma 5.6.
Let M = ( m ij ) × be a matrix in GL ( k ) with m m − m m = 0 and m = 0 . Assume that I = ( x ) , I = ( x , y ) and I = ( x , y , z ) are thethree DG ideals generated by the subsets { x } , { x , y } and { x , y , z } of the DGalgebra A , respectively. Then H i ( I /I ) = k ⌈ ¯ y ⌉ , if i = 2 k ⌈ ¯ x ¯ y + ¯ y ( m m − m m m m − m m ¯ y + m m − m m m m − m m ¯ z ) ⌉ , if i = 30 , if i ≥ and H i ( I /I ) = k ⌈ ¯ z ⌉ , if i = 2 k ⌈− m ¯ x ¯ z + m ¯ z ⌉ ⊕ k ⌈− m ¯ y ¯ z + m ¯ z ⌉ , if i = 3 k ⌈ m ¯ x ¯ z − m ¯ y ¯ z − m ¯ x ¯ y ¯ z ⌉ , if i = 40 , if i ≥ . Proof.
By Lemma 5.1, each x i is a central cocycle element of A . So I , I and I areindeed DG ideals of A . Then H ( I /I ) = k ⌈ x ⌉ and H ( I /I ) = k ⌈ x ⌉ since I /I and I /I are concentrated in degrees ≥
2, ( I /I ) = kx and ( I /I ) = kx .Any graded cocycle element Ω of degree d in I /I can be written asΩ = ¯ x ¯ x f ( ¯ x , ¯ x ) + ¯ x g ( ¯ x , ¯ x ) , where f ( ¯ x , ¯ x ) and g ( ¯ x , ¯ x ) are sums of monomials in variables ¯ x and ¯ x . Wehave0 = ∂ I /I ( z )=( m ¯ x + m ¯ x ) ¯ x f ( ¯ x , ¯ x ) − ¯ x ¯ x ∂ A [ f ( x , x )] + ¯ x ∂ A [ g ( x , x )]= ¯ x { ( m ¯ x + m ¯ x ) f ( ¯ x , ¯ x ) + ∂ A [ g ( x , x )] } − ¯ x ¯ x ∂ A [ f ( x , x )] . Thus ( ∂ A [ f ( x , x )] = 0 ∂ A [ g ( x , x )] = − ( m ¯ x + m ¯ x ) f ( ¯ x , ¯ x ) . (15)When d = 3, we have | f ( x , x ) | = 0 and | g ( x , x ) | = 1. Let f ( x , x ) = c ∈ k and g ( x , x ) = c x + c x . Then − ( m ¯ x + m ¯ x ) c = ∂ A [ g ( x , x )]= ∂ A ( c x + c x )= c ( m x + m x + m x ) + c ( m x + m x + m x )= ( c m + c m ) ¯ x + ( c m + c m ) ¯ x . This implies that ( c m + c m = − cm c m + c m = − cm . And hence c = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − m m − m m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m m m m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) c = c ( m m − m m ) m m − m m c == (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m − m m − m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m m m m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) c = c ( m m − m m ) m m − m m Then Ω = ¯ x ¯ x c + ¯ x [ c ( m m − m m ) m m − m m ¯ x + c ( m m − m m ) m m − m m ¯ x ]and H ( I /I ) = k ⌈ ¯ x ¯ x + ¯ x ( m m − m m m m − m m ¯ x + m m − m m m m − m m ¯ x ) ⌉ since B ( I /I ) = 0.When d = 4, we have | f ( ¯ x , ¯ x ) | = 1 and | g ( ¯ x , ¯ x ) | = 2. Let f ( ¯ x , ¯ x ) = l ¯ x + l ¯ x and g ( ¯ x , ¯ x ) = t ¯ x + t ¯ x ¯ x + t ¯ x . Then by (15), we have0 = ∂ A [ f ( x , x )]= ∂ A ( l x + l x )= l ( m x + m x + m x ) + l ( m x + m x + m x )=( l m + l m ) ¯ x + ( l m + l m ) ¯ x , ALABI-YAU PROPERTIES OF 3-DIMENSIONAL DG SKLYANIN ALGEBRAS 27 which implies that ( l m + l m = 0 l m + l m = 0 . Since m m − m m = 0, we get l = l = 0 and hence f ( ¯ x , ¯ x ) = 0. Then by(15), we have0 = ∂ A [ g ( x , x )]= ∂ A [ t x + t x x + t x ]= t ( m x + m x + m x ) x − t x ( m x + m x + m x )= t m ¯ x ¯ x + t m ¯ x − t m ¯ x − t m ¯ x ¯ x . Thus t m = t m = t m = t m = 0. Since (cid:12)(cid:12)(cid:12)(cid:12) m m m m (cid:12)(cid:12)(cid:12)(cid:12) = 0, we get t = 0. So Ω = ¯ x ¯ x f ( ¯ x , ¯ x ) + ¯ x g ( ¯ x , ¯ x ) = ¯ x ( t ¯ x + t ¯ x ). By the proofof Lemma 5.3, there exist b = m m − m m | M | b = m m − m m | M | b = m m − m m | M | and c = m m − m m | M | c = m m − m m | M | c = m m − m m | M | such that ∂ A ( b x + b x + b x ) = x and ∂ A ( c x + c x + c x ) = x , respectively.Then z = ¯ x ( t ¯ x + t ¯ x )= ¯ x [ t ∂ A ( b x + b x + b x ) + t ∂ A ( c x + c x + c x )]= ∂ I /I { ¯ x [ t ( b ¯ x + b ¯ x + b ¯ x ) + t ( c ¯ x + c ¯ x + c ¯ x )] } . Hence H ( I /I ) = 0.When d = 2 l + 3 , l ≥
1, we have | f ( ¯ x , ¯ x ) | = 2 l and | g ( ¯ x , ¯ x ) | = 2 l + 1. Since ∂ A [ f ( x , x )] = 0 by (15), we get f ( ¯ x , ¯ x ) = l P i =0 r i ¯ x l − i ¯ x i by Lemma 5.4,where r i ∈ k, ≤ i ≤ l . Then by (15), we have ∂ A [ g ( x , x )] = − ( m ¯ x + m ¯ x ) f ( ¯ x , ¯ x )= − ( m ¯ x + m ¯ x )( l X i =0 r i ¯ x l − i ¯ x i )= ∂ A [( − m m m m − m m x + m m m m − m m x )( l X i =0 r i ¯ x l − i ¯ x i )]+ ∂ A [( m m m m − m m x − m m m m − m m x )( l X i =0 r i ¯ x l − i ¯ x i )]= ∂ A { l X i =0 r i [ m m − m m m m − m m x l − i +12 x i + m m − m m m m − m m x l − i x i +13 ] } Then, by Lemma 5.4, we may let g ( ¯ x , ¯ x )= l X i =0 r i [ m m − m m m m − m m ¯ x l − i +1 ¯ x i + m m − m m m m − m m ¯ x l − i ¯ x i +1 ]+ ∂ A [ u ( x , x )] , where u ( x , x ) is a sum of monomials in variables x and x . ThenΩ = ¯ x ¯ x f ( ¯ x , ¯ x ) + ¯ x g ( ¯ x , ¯ x )= l X i =0 r i ¯ x ¯ x l − i +2 ¯ x i + ¯ x ∂ A [ u ( x , x )]+ l X i =0 r i [ m m − m m m m − m m ¯ x l − i +3 ¯ x i + m m − m m m m − m m ¯ x l − i +2 ¯ x i +1 ]= l X i =0 r i [ ¯ x + ( m m − m m ) ¯ x + ( m m − m m ) ¯ x m m − m m ] ¯ x l − i +2 ¯ x i + ¯ x ∂ A [ u ( x , x )] . One sees that ω = x + ( m m − m m ) x +( m m − m m ) x m m − m m is a cocycle elementin A . Hence z = ∂ A [ − l − X i =0 r i ω ( b x + b x + b x ) x l − i x i − r l ωx x l − ( c x + c x + c x )]+ ¯ x ∂ A [ u ( x , x )]= ∂ I /I { [ − l − X i =0 r i ω ( b ¯ x + b ¯ x + b ¯ x ) ¯ x l − i − ¯ x i ] ¯ x } + ∂ I /I { [ − r l ω ( c ¯ x + c ¯ x + c ¯ x ) ¯ x l − + u ( ¯ x , ¯ x )] ¯ x } . Thus H l +3 ( I /I ) = 0.When d = 2 l + 4, we have | f ( ¯ x , ¯ x ) | = 2 l + 1 and | g ( ¯ x , ¯ x ) | = 2 l + 2. Since ∂ A [ f ( x , x )] = 0 by (15), we have f ( ¯ x , ¯ x ) = ∂ A [ l X i =1 s i − x l − i +12 x i − ]by Lemma 5.4 and Remark 5.5, where s i − ∈ k , 1 ≤ i ≤ l . Then by (15), we have ∂ A [ g ( x , x )] = − ( m ¯ x + m ¯ x ) f ( ¯ x , ¯ x )= − ( m ¯ x + m ¯ x ) ∂ A [ l X i =1 s i − x l − i +12 x i − ] . Then, by Lemma 5.4, we may let g ( ¯ x , ¯ x ) = − ( m ¯ x + m ¯ x )[ l X i =1 s i − ¯ x l − i +1 ¯ x i − ] + ∂ A [ v ( x , x )] . ALABI-YAU PROPERTIES OF 3-DIMENSIONAL DG SKLYANIN ALGEBRAS 29 where v ( x , x ) is a sum of monomials in variables x and x . ThenΩ = ¯ x ¯ x f ( ¯ x , ¯ x ) + ¯ x g ( ¯ x , ¯ x )= ¯ x ¯ x ∂ A [ l X i =1 s i − x l − i +12 x i − ] − ( m ¯ x + m ¯ x )[ l X i =1 s i − ¯ x l − i +3 ¯ x i − ]+ ¯ x ∂ A [ v ( x , x )]= − ∂ A [ x l X i =1 s i − x l − i +12 x i − − v ( x , x )] ¯ x = ∂ I /I [( v ( ¯ x , ¯ x ) − ¯ x l X i =1 s i − ¯ x l − i +1 ¯ x i − ) ¯ x ]and hence H l +4 ( I /I ) = 0.Since ( I /I ) = k ¯ x ¯ x ⊕ k ¯ x ¯ x ⊕ k ¯ x , any cocycle element in ( I /I ) canbe denoted by c ¯ x ¯ x + c ¯ x ¯ x + c ¯ x where c , c , c ∈ k . Then0 = ∂ I /I [ c ¯ x ¯ x + c ¯ x ¯ x + c ¯ x ]= c m ¯ x + c m ¯ x + c m ¯ x =( c m + c m + c m ) ¯ x and hence c m + c m + c m = 0, which has a basic solution system − m m , − m m So Z ( I /I ) = k ( − m ¯ x ¯ x + m ¯ x ) ⊕ k ( − m ¯ x ¯ x + m ¯ x ). Then H ( I /I ) = k ⌈− m ¯ x ¯ x + m ¯ x ⌉ ⊕ k ⌈− m ¯ x ¯ x + m ¯ x ⌉ since one sees easily that B ( I /I ) = 0. Any graded cocycle element z of degree d, d ≥ I /I can be written as χ = ¯ x ¯ x φ ( ¯ x ) + ¯ x ¯ x ϕ ( ¯ x ) + ¯ x ¯ x ¯ x ψ ( ¯ x ) + ¯ x λ ( ¯ x ) . We have0 = ∂ I /I ( χ ) = ∂ I /I [ ¯ x ¯ x φ ( ¯ x ) + ¯ x ¯ x ϕ ( ¯ x ) + ¯ x ¯ x ¯ x ψ ( ¯ x ) + ¯ x λ ( ¯ x )]= m ¯ x φ ( ¯ x ) − ¯ x ¯ x ∂ A [ φ ( x )] + m ¯ x ϕ ( ¯ x ) − ¯ x ¯ x ∂ A [ ϕ ( x )]+ m ¯ x ¯ x ψ ( ¯ x ) − m ¯ x ¯ x ψ ( ¯ x ) + ¯ x ¯ x ¯ x ∂ A [ ψ ( ¯ x )] + ¯ x ∂ A [ λ ( x )]= ¯ x [ m ¯ x φ ( ¯ x ) + m ¯ x ϕ ( ¯ x ) + ∂ A [ λ ( x )]] + ¯ x ¯ x ¯ x ∂ A [ ψ ( ¯ x )] − ¯ x [ ¯ x ∂ A [ φ ( x )] + m ¯ x ψ ( ¯ x )] + ¯ x [ m ¯ x ψ ( ¯ x ) − ¯ x ∂ A [ ϕ ( x )]] . Hence m ¯ x φ ( ¯ x ) + m ¯ x ϕ ( ¯ x ) + ∂ A [ λ ( x )] = 0¯ x ∂ A [ φ ( x )] + m ¯ x ψ ( ¯ x ) = 0 m ¯ x ψ ( ¯ x ) − ¯ x ∂ A [ ϕ ( x )] = 0 ∂ A [ ψ ( ¯ x )] = 0 . (16)When d = 4, we have | ψ ( ¯ x ) | = 0, | ϕ ( ¯ x ) | = | φ ( ¯ x ) | = 1 and | λ ( ¯ x ) | = 2. Let ψ ( ¯ x ) = c ∈ k . Then by (16), we get ϕ ( x ) = cm m x , φ ( x ) = − cm m x and λ ( ¯ x ) = c ′ ¯ x , for some c ′ ∈ k . So Z ( I /I ) = k ( − m ¯ x ¯ x + m ¯ x ¯ x + m ¯ x ¯ x ¯ x ) ⊕ k ¯ x . Then H ( I /I ) = k ⌈ m ¯ x ¯ x − m ¯ x ¯ x − m ¯ x ¯ x ¯ x ⌉ since B ( I /I ) = k ¯ x .When d = 2 l − ≥
5, we have | φ ( ¯ x ) | = 2 l − | ϕ ( ¯ x ) | = 2 l − | ψ ( ¯ x ) | = 2 l − | λ ( ¯ x ) | = 2 l −
3. Let ψ ( ¯ x ) = q ¯ x l − for some q ∈ k . Then 0 = ∂ A [ ψ ( ¯ x )] = qm ¯ x l − by (16). So q = 0 and ψ ( ¯ x ) = 0. Then we get ∂ A [ φ ( x )] = ∂ A [ ϕ ( x )] =0 by (16). Let φ ( x ) = px l − and ϕ ( x ) = rx l − , p, r ∈ k . Then ∂ A [ λ ( x )] = − m p ¯ x l − − m r ¯ x l − . So λ ( ¯ x ) = − ( m p + m r ) ¯ x l − m . Then χ = ¯ x ¯ x φ ( ¯ x ) + ¯ x ¯ x ϕ ( ¯ x ) + ¯ x ¯ x ¯ x ψ ( ¯ x ) + ¯ x λ ( ¯ x )= p ¯ x ¯ x l − + r ¯ x ¯ x l − − ( m p + m r ) ¯ x l − m = [ m ( p ¯ x + r ¯ x ) − ( pm + rm ) ¯ x ] m ¯ x l − = ∂ I /I { [ − m ( p ¯ x + r ¯ x ) + ( pm + rm ) ¯ x ] m ¯ x l − } . Thus H l − ( I /I ) = 0, for any l ≥
3. When d = 2 l ≥
6, we have | φ ( ¯ x ) | = 2 l − | ϕ ( ¯ x ) | = 2 l − | ψ ( ¯ x ) | = 2 l − | λ ( ¯ x ) | = 2 l −
2. So ∂ A [ λ ( x )] = 0 and ∂ A [ ψ ( x )] = 0. Then (16) is equivalent to m ¯ x φ ( ¯ x ) + m ¯ x ϕ ( ¯ x ) = 0¯ x ∂ A [ φ ( x )] + m ¯ x ψ ( ¯ x ) = 0 m ¯ x ψ ( ¯ x ) − ¯ x ∂ A [ ϕ ( x )] = 0 . Let λ ( ¯ x ) = s ¯ x l − and ψ ( x ) = t ¯ x l − . Then by the system of equations above,we get φ ( ¯ x ) = − m tm ¯ x l − and ϕ ( ¯ x ) = m tm ¯ x l − . Then χ = ¯ x ¯ x φ ( ¯ x ) + ¯ x ¯ x ϕ ( ¯ x ) + ¯ x ¯ x ¯ x ψ ( ¯ x ) + ¯ x λ ( ¯ x )= − m tm ¯ x ¯ x l − + m tm ¯ x ¯ x l − + t ¯ x ¯ x ¯ x l − + s ¯ x l = [ − m ¯ x ¯ x + m ¯ x ¯ x + m ¯ x ¯ x m ] t ¯ x l − + s ¯ x l = ∂ I /I { [ − m ¯ x ¯ x + m ¯ x ¯ x + m ¯ x ¯ x m ] t ¯ x l − + sm ¯ x l − } Hence H l ( I /I ) = 0 for any l ≥ (cid:3) Lemma 5.7.
Let M = ( m ij ) × and r ( M ) = 2 . Then r ( X ) = 5 , where X = m m m m m m m m m m m m m m m m m m m m m m m m m m m . Proof.
Since r ( M ) = 2, there exists ( s , s , s ) T = 0 such that M ( s , s , s ) T = 0,which is equivalent to s m m m + s m m m + s m m m = 0 . ALABI-YAU PROPERTIES OF 3-DIMENSIONAL DG SKLYANIN ALGEBRAS 31
Without the loss of generality, let s = 0. Then m m m , m m m are linearlyindependent and( m , m , m ) + s s ( m , m , m ) + s s ( m , m , m ) = 0 . For X , we can perform the following elementary row transformations X r + s s × r −−−−−−−→ r + s s × r s s m s s m s s m s s m s s m s s m m m m m m m m m m m m m m m m m m m m m m m m m r + s s × r −−−−−−−−→ r + s s s × r − s s m − s s m − s s m m m m m m m m m m m m m m m m m m m m m m m m m r + s s × r −−−−−−−→ m m m m m m m m m m m m m m m m m m m m m m m m . This indicates r ( X ) ≤ r ( X ) = r m m m m m m m m m m m m m m m m m m m m m m m m . Let l m m m m m m + l m m m m m m + l m m m m m m + l m m m + l m m m = 0 . Then l m + l m = 0 l m + l m = 0 l m + l m = 0 l m + l m + l m = 0 l m + l m + l m = 0 l m + l m + l m = 0 l m + l m + l m = 0 l m + l m + l m = 0 l m + l m + l m = 0 , which implies l = l = l = l = l = 0 since m m m , m m m are linearlyindependent. Thus r ( X ) = r m m m m m m m m m m m m m m m m m m m m m m m m = 5 . Similarly, we can show r ( X ) = 5 when s = 0 or s = 0. (cid:3) Lemma 5.8.
Let M = ( m ij ) × be a matrix in M ( k ) with r ( M ) = 2 . If r = m x + m y + m z ,r = m x + m y + m z ,r = m x + m y + m z , then the graded ideal ( r , r , r ) is a prime graded ideal of the polynomial gradedalgebra k [ x , y , z ] .Proof. Since r ( M ) = 2, there exist a non-zero solution vector ( t , t , t ) T of thehomogeneous linear equations M T X = 0. We have t r + t r + t r = ( t , t , t ) r r r = ( t , t , t ) M x x x = 0 . Since ( t , t , t ) T = 0, we may as well let t = 0. Then r = − t t r − t t r and hence( r , r , r ) = ( r , r ). Since m m m = − t t m m m − t t m m m , we have r (cid:18) m m m m m m (cid:19) = 2 , this indicates that there at least one non-zero minor among (cid:12)(cid:12)(cid:12)(cid:12) m m m m (cid:12)(cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12)(cid:12) m m m m (cid:12)(cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12)(cid:12) m m m m (cid:12)(cid:12)(cid:12)(cid:12) . ALABI-YAU PROPERTIES OF 3-DIMENSIONAL DG SKLYANIN ALGEBRAS 33
We may as well let (cid:12)(cid:12)(cid:12)(cid:12) m m m m (cid:12)(cid:12)(cid:12)(cid:12) = 0. Then one sees that k [ x , x , x ] / ( r , r ) ∼ = k [ x ]is a domain. So ( r , r , r ) = ( r , r ) is a graded prime ideal of k [ x , x , x ]. (cid:3) Lemma 5.9.
Assume that M = ( m ij ) × ∈ M ( k ) with r ( M ) = 2 , k ( s , s , s ) T and k ( t , t , t ) T are the solution spaces of homogeneous linear equations M X = 0 and M T X = 0 , respectively. We have the following statements.(1) If s t + s t + s t = 0 , then k [ ⌈ t x + t y + t z ⌉ ] is a subalgebra of H ( A ) ;(2) If s t + s t + s t = 0 , then k [ ⌈ t x + t y + t z ⌉ , ⌈ s x + s y + s z ⌉ ] / ( ⌈ t x + t y + t z ⌉ ) is a subalgebra of H ( A ) .Proof. Clearly, we have H ( A ) = k . Since r ( M T ) = 2 <
3, there is a non-zerosolution vector ( t , t , t ) T of the homogeneous linear equations M T X = 0. Forany c x + c x + c x ∈ Z ( A ), we have0 = ∂ A ( c x + c x + c x )= ( c , c , c ) ∂ A ( x ) ∂ A ( x ) ∂ A ( x ) = ( c , c , c ) M x x x , which implies that ( c , c , c ) M = 0 or equivalently M T c c c = 0. Thus H ( A ) = k ⌈ t x + t x + t x ⌉ .For any l x + l x x + l x x + l x + l x x + l x ∈ ker( ∂ A ), we have0 = ∂ A [ l x + l x x + l x x + l x + l x x + l x ]= l ( m x + m x + m x ) x − l x ( m x + m x + m x )+ l ( m x + m x + m x ) x − l x ( m x + m x + m x )+ l ( m x + m x + m x ) x − l x ( m x + m x + m x )= − ( l m + l m ) x + ( l m − l m ) x x + ( l m + l m ) x x − ( l m + l m ) x x + ( l m + l m ) x x + ( l m − l m ) x − ( l m + l m ) x x + ( l m − l m ) x x + ( l m + l m ) x . Hence l m + l m = 0 l m − l m = 0 l m + l m = 0 l m + l m = 0 l m + l m = 0 l m − l m = 0 l m + l m = 0 l m − l m = 0 l m + l m = 0 ⇔ l m + l m = 0 l m + l m = 0 l m + l m = 0 l m − l m = 0 l m − l m = 0 l m − l m = 0 l m + l m = 0 l m + l m = 0 l m + l m = 0 , which is equivalent to m m m m m m m m m l l l l l − l = 0 × . We claim that l = l = l = 0. Indeed, if any one of l , l , l is nonzero, thenthere are at least two non-zero linear independent vectors among l l , l − l , l l , which are all solutions of M X = 0. This contradicts with r ( M ) = 2. Henceker( ∂ A ) = kx ⊕ kx ⊕ kx . In A , we have( t x + t x + t x ) = t x + t x + t x . (1)If s t + s t + s t = 0, we claim that t x + t x + t x B ( A ). Indeed, if thereexist q x + q x + q x ∈ A such that ∂ A ( q x + q x + q x ) = t x + t x + t x ,then ( q , q , q ) M x x x = ∂ A ( q x + q x + q x )= t x + t x + t x = ( t , t , t ) x x x , which implies that ( q , q , q ) M = ( t , t , t ) and hence0 = ( q , q , q ) M s s s = ( t , t , t ) s s s = s t + s t + s t . This contradicts with the assumption that s t + s t + s t = 0. Then we get that t x + t x + t x B ( A ) if s t + s t + s t = 0. On the other hand, we havedim k B ( A ) = 2 since r ( M ) = 2. Therefore dim k H ( A ) = 1 and H ( A ) = k ⌈ t x + t x + t x ⌉ = k ⌈ t x + t x + t x ⌉ . In order to show k [ ⌈ t x + t x + t x ⌉ ] is a subalgebra of H ( A ), we need to show( t x + t x + t x ) n B n ( A ) for any n ≥
3. If this not the case, we have( t x + t x + t x ) n = ( ∂ A [ x x f + x x g + x x h ] , if n = 2 j + 1 is odd ∂ A [ x f + x g + x h + x x x u ] , if n = 2 j is evenwhere f, g, h and u are all linear combinations of monomials with non-negative evenexponents. When n = 2 j is even, we have( t x + t x + t x ) j = ( t x + t x + t x ) n = ∂ A [ x f + x g + x h + x x x u ]= ( m x + m x + m x ) f + ( m x + m x + m x ) g + ( m x + m x + m x ) h + ( m x + m x + m x ) x x u − x ( m x + m x + m x ) x g + x x ( m x + m x + m x ) u. ALABI-YAU PROPERTIES OF 3-DIMENSIONAL DG SKLYANIN ALGEBRAS 35
Considering the parity of exponents of the monomials that appear on both sidesthe equation above implies that( t x + t x + t x ) j = ( m x + m x + m x ) f + ( m x + m x + m x ) g + ( m x + m x + m x ) h = ∂ A ( x ) f + ∂ A ( x ) g + ∂ A ( x ) h and ∂ A ( x x x u ) = ( m x + m x + m x ) x x u − x ( m x + m x + m x ) x g + x x ( m x + m x + m x ) u = 0 . Therefore, ( t x + t x + t x ) j is in the graded ideal ( ∂ A ( x ) , ∂ A ( x ) , ∂ A ( x )) of k [ x , x , x ]. By Lemma 5.8, ( ∂ A ( x ) , ∂ A ( x ) , ∂ A ( x )) is a graded prime ideal of k [ x , x , x ]. So t x + t x + t x ∈ ( ∂ A ( x ) , ∂ A ( x ) , ∂ A ( x )). Hence there exist a , a and a in k such that t x + t x + t x = a ∂ A ( x ) + a ∂ A ( x ) + a ∂ A ( x )= ∂ A ( a x + a x + a x ) . But this contradicts with the fact that t x + t x + t x B ( A ), which we haveproved above. Thus ( t x + t x + t x ) n B n ( A ) when n is even.When n = 2 j + 1 is odd, we have( t x + t x + t x )( t x + t x + t x ) j = ( t x + t x + t x ) n = ∂ A [ x x f + x x g + x x h ]= ( m x + m x + m x ) x f − x ( m x + m x + m x ) f + ( m x + m x + m x ) x g − x ( m x + m x + m x ) g + ( m x + m x + m x ) x h − x ( m x + m x + m x ) h = − x [( m x + m x + m x ) f + ( m x + m x + m x ) g ]+ x [( m x + m x + m x ) f − ( m x + m x + m x ) h ]+ x [( m x + m x + m x ) h + ( m x + m x + m x ) g ]= x [ − ∂ A ( x ) f − ∂ A ( x ) g ] + x [ ∂ A ( x ) f − ∂ A ( x ) h ] + x [ ∂ A ( x ) h + ∂ A ( x ) g ] . This implies that t ( t x + t x + t x ) j = − ∂ A ( x ) f − ∂ A ( x ) g = ∂ A [ − x f − x g ] t ( t x + t x + t x ) j = ∂ A ( x ) f − ∂ A ( x ) h = ∂ A [ x f − x h ] t ( t x + t x + t x ) j = ∂ A ( x ) h + ∂ A ( x ) g = ∂ A [ x h + x g ] . Since ( t , t , t ) T = 0, there is at least one none-zero t i , i ∈ { , , } . Then weget ( t x + t x + t x ) j = ( t x + t x + t x ) j ∈ B j ( A ), which contradictswith proved fact that ( t x + t x + t x ) n B n ( A ) when n is even. Therefore,( t x + t x + t x ) n B n ( A ) when n is odd.Then we reach a conclusion that k [ ⌈ t x + t x + t x ⌉ ] is a subalgebra of H ( A )when s t + s t + s t = 0.(2)When s t + s t + s t = 0, we should show t x + t x + t x ∈ B ( A ) and s x + s x + s x B ( A ) first. In order to prove t x + t x + t x ∈ B ( A ), we need to show the existence of an element q x + q x + q x ∈ A such that ∂ A ( q x + q x + q x ) = ( q , q , q ) M x x x = ( t , t , t ) x x x , which is equivalent to M T q q q = t t t . Hence it suffices to show that the nonhomogeneous linear equations M T X = t t t has solutions. Let M = ( β , β , β ) and b = t t t . Since M s s s = 0, wehave P i =1 s i β i = 0 and hence P i =1 s i β Ti = 0. Hence r ( M T , b ) = r β T t β T t β T t = r β T t β T t s β T + s β T + s β T s t + s t + s t = r β T t β T t ≤ . On the other hand, we have r ( M T , b ) ≥ r ( M T ) = 2. So r ( M T , b ) = 2 = r ( M T )and then the nonhomogeneous linear equations M T X = t t t has solutions.Now, let us prove s x + s y + s z im( ∂ A ), which is equivalent to thenonhomogeneous linear equations M T X = s s s has no solutions. Let s = s s s . Then r ( M T , s ) = r β T s β T s β T s = r β T s β T s s β T + s β T + s β T s + s + s = r β T s β T s s + s + s = 3 = r ( M T ) = 2 . ALABI-YAU PROPERTIES OF 3-DIMENSIONAL DG SKLYANIN ALGEBRAS 37
Hence M T X = s has no solutions and H ( A ) = k ⌈ s x + s x + s x ⌉ . It remainsto show that ( s x + s x + s x ) j +1 B j +2 ( A )and ( t x + t x + t x )( s x + s x + s x ) j B j +1 ( A )for any j ≥
1. We will use a proof by contradiction.If ( s x + s x + s x ) j +1 ∈ B j +2 ( A ), then by Lemma 5.2, we have( s x + s x + s x ) j +1 = ∂ A [ x f + x g + x h + x x x u ] , where f, g , h and u are all linear combinations of monomials with non-negativeeven exponents. Considering the parity of exponents of the monomials that appearon both sides of the following equation( s x + s x + s x ) j +1 = ∂ A [ x f + x g + x h + x x x u ]= ( m x + m x + m x ) f + ( m x + m x + m x ) g + ( m x + m x + m x ) h + ( m x + m x + m x ) x x u − x ( m x + m x + m x ) x g + x x ( m x + m x + m x ) u implies that( s x + s x + s x ) j +1 = ( m x + m x + m x ) f + ( m x + m x + m x ) g + ( m x + m x + m x ) h = ∂ A ( x ) f + ∂ A ( x ) g + ∂ A ( x ) h and ∂ A ( x x x u ) = ( m x + m x + m x ) x x u − x ( m x + m x + m x ) x g + x x ( m x + m x + m x ) u = 0 . Therefore, ( s x + s x + s x ) j +1 is in the graded ideal ( ∂ A ( x ) , ∂ A ( x ) , ∂ A ( x ))of k [ x , x , x ]. By Lemma 5.8, ( ∂ A ( x ) , ∂ A ( x ) , ∂ A ( x )) is a graded prime ideal of k [ x , x , x ]. So s x + s x + s x ∈ ( ∂ A ( x ) , ∂ A ( x ) , ∂ A ( x )). Hence there exist b , b and b in k such that s x + s x + s x = b ∂ A ( x ) + b ∂ A ( x ) + b ∂ A ( x )= ∂ A ( b x + b x + b x ) . But this contradicts with the fact that s x + s x + s x B ( A ), which we haveproved above. Thus ( s x + s x + s x ) j +1 B j +2 ( A ), for any j ≥ t x + t x + t x )( s x + s x + s x ) j B j +1 ( A ), then by Lemma 5.2, wehave ( t x + t x + t x )( s x + s x + s x ) j = ∂ A [ x x f + x x g + x x h ] , where f, g and h are all linear combinations of monomials with non-negative evenexponents. Then( t x + t x + t x )( s x + s x + s x ) j = ∂ A [ x x f + x x g + x x h ]= ( m x + m x + m x ) x f − x ( m x + m x + m x ) f + ( m x + m x + m x ) x g − x ( m x + m x + m x ) g + ( m x + m x + m x ) x h − x ( m x + m x + m x ) h = − x [( m x + m x + m x ) f + ( m x + m x + m x ) g ]+ x [( m x + m x + m x ) f − ( m x + m x + m x ) h ]+ x [( m x + m x + m x ) h + ( m x + m x + m x ) g ]= x [ − ∂ A ( x ) f − ∂ A ( x ) g ] + x [ ∂ A ( x ) f − ∂ A ( x ) h ] + x [ ∂ A ( x ) h + ∂ A ( x ) g ] . This implies t ( s x + s x + s x ) j = − ∂ A ( x ) f − ∂ A ( x ) g = ∂ A ( − x f − x g ) t ( s x + s x + s x ) j = ∂ A ( x ) f − ∂ A ( x ) h = ∂ A ( x f − x h ) t ( s x + s x + s x ) j = ∂ A ( x ) h + ∂ A ( x ) g = ∂ A ( x h + x g ) . Since ( t , t , t ) T = 0, there is at least one non-zero t i , i ∈ { , , } . Then we getthat ( s x + s x + s x ) j ∈ B j ( A ). This contradicts with the proved fact that( s x + s x + s x ) j B j ( A ) for any j ≥ k [ ⌈ t x + t x + t x ⌉ , ⌈ s x + s x + s x ⌉ ] / ( ⌈ t x + t x + t x ⌉ )is a subalgebra of H ( A ). (cid:3) Computations of H ( A )In this section, we assume A is a DG Sklyanin algebra such tht A = S a,a, and ∂ A is determined by a matrix M ∈ M ( k ). According to the rank of M , we divideit into three subsections to compute separately.6.1. When r ( M ) = 3 . Clearly, H ( A ) = k . For any l x + l y + l z ∈ Z ( A ), wehave 0 = ∂ A ( l x + l y + l z ) = ( l , l , l ) M x y z , which implies that ( l , l , l ) M = 0 and hence M T l l l = 0. Then each l i = 0since r ( M T ) = 3. So Z ( A ) = 0 and H ( A ) = 0. Since ∂ A is a monomorphism, wehave dim k B ( A ) = 3 and B ( A ) = kx ⊕ ky ⊕ kz . We claim Z ( A ) = B ( A ). Itsuffices to show ( kx x ⊕ kx x ⊕ kx x ) T Z ( A ) = 0 since A = kx ⊕ ky ⊕ kz ⊕ kxy ⊕ kxz ⊕ kyz. ALABI-YAU PROPERTIES OF 3-DIMENSIONAL DG SKLYANIN ALGEBRAS 39
For any c xy + c xz + c yz ∈ Z ( A ), we have0 = ∂ A [ c xy + c xz + c yz ]= c ( m x + m y + m z ) y − c x ( m x + m y + m z )+ c ( m x + m y + m z ) z − c x ( m x + m y + m z )+ c ( m x + m y + m z ) z − c y ( m x + m y + m z )= ( − c m − c m ) x + ( c m − c m ) y + ( c m + c m ) z + ( c m − c m ) x y − ( c m + c m ) xy − ( c m + c m ) xz + ( c m + c m ) x z + ( c m + c m ) y z + ( c m − c m ) yz . Then c m + c m = 0 c m − c m = 0 c m + c m = 0 c m − c m = 0 c m + c m = 0 c m + c m = 0 c m + c m = 0 c m + c m = 0 c m − c m = 0 ⇔ c m + c m = 0 c m + c m = 0 c m + c m = 0 c m − c m = 0 c m − c m = 0 c m − c m = 0 c m + c m = 0 c m + c m = 0 c m + c m = 0 ⇔ c = 0 c = 0 c = 0since r ( M ) = 3. So ( kxy ⊕ kxz ⊕ kyz ) T Z ( A ) = 0. Thus H ( A ) = 0.Since x , y and z are central and cocycle elements in A , they generate a DGideal I = ( x , y , z ) of A . One sees that A /I = V ( x, y, z ) with ∂ A /I = 0. Thelong exact sequence of cohomologies induced from the shot exact sequence0 → I ι → A ε → A /I → ♣ ):0 → H ( A /I ) = k ( ⌈ x ∧ y ⌉ ) ⊕ k ( ⌈ x ∧ z ⌉ ) ⊕ k ( ⌈ y ∧ z ⌉ ) δ → H ( I ) H ( ι ) → H ( A ) H ( ε ) → H ( A /I ) = k ( ⌈ x ∧ y ∧ z ⌉ ) δ → H ( I ) H ( ι ) → H ( A ) → H ( A /I ) = 0 → H ( I ) H ( ι ) → H ( A ) → → · · · → H i ( I ) H i ( ι ) → H i ( A ) → → · · · . We claim that H ( I ) = k ⌈ ω ⌉ ⊕ k ⌈ ω ⌉ ⊕ k ⌈ ω ⌉ , where ω = − m x + m x y − m xy + m y − m xz + m yz ω = − m x + m x z − m xy + m y z − m xz + m z ω = − m x y + m x z − m y + m y z − m yz + m z . Any cocycle element Ω ∈ Z ( I ) can be written asΩ = ( q x + q y + q z ) x + ( q x + q y + q z ) y + ( q x + q y + q z ) z , where each q i ∈ k , 1 ≤ i ≤
9. Then0 = ∂ I ( z )= ( q , q , q ) M x y z x + ( q , q , q ) M x y z y + ( q , q , q ) M x y z z = ( q , q , q ) M x x y x z + ( q , q , q ) M x y y y z + ( q , q , q ) M x z y z z and hence ( q , q , q ) M = 0( q , q , q ) M = 0( q , q , q ) M = 0( q , q , q ) M + ( q , q , q ) M = 0( q , q , q ) M + ( q , q , q ) M = 0( q , q , q ) M + ( q , q , q ) M = 0 , which is equivalent to m m m m m m m m m m m m m m m m m m m m m m m m m m m q q q q q q q q q = 0 . Since r ( M ) = 3, one sees that r m m m m m m m m m m m m m m m m m m m m m m m m m m m = 6 . ALABI-YAU PROPERTIES OF 3-DIMENSIONAL DG SKLYANIN ALGEBRAS 41
Hence dim k Z ( I ) = 3. On the other hand, ∂ A ( xy ) = ω ∂ A ( xz ) = ω ∂ A ( yz ) = ω implies that ∂ I ( ω i ) = 0 , i = 1 , ,
3. Then Z ( I ) = kω ⊕ kω ⊕ kω and hence H ( I ) = k ⌈ ω ⌉ ⊕ k ⌈ ω ⌉ ⊕ k ⌈ ω ⌉ since B ( I ) = 0. The definition of connectinghomomorphism implies that δ ( ⌈ x ∧ y ⌉ ) = ⌈ ω ⌉ δ ( ⌈ x ∧ z ⌉ ) = ⌈ ω ⌉ δ ( ⌈ y ∧ z ⌉ ) = ⌈ ω ⌉ . Hence δ is a bijection. By the long exact sequence ( ♣ ), one sees that H ( A ) = 0.Since B ( A ) = kx ⊕ ky ⊕ kz , one sees that B ( I ) = kx ⊕ kx y ⊕ kx z ⊕ ky ⊕ ky z ⊕ kz . For any Ω ∈ Z ( I ) T ( I /B ( I )), we can write it asΩ = ( r xy + r xz + r yz ) x + ( r xy + r xz + r yz ) y + ( r xy + r xz + r yz ) z , where r i ∈ k, ≤ i ≤
9. Then0 = ∂ I (Ω) = [ r ( m x + m y + m z ) y − r x ( m x + m y + m z )] x + [ r ( m x + m y + m z ) z − r x ( m x + m y + m z )] x + [ r ( m x + m y + m z ) z − r y ( m x + m y + m z )] x + [ r ( m x + m y + m z ) y − r x ( m x + m y + m z )] y + [ r ( m x + m y + m z ) z − r x ( m x + m y + m z )] y + [ r ( m x + m y + m z ) z − r x ( m x + m y + m z )] y + [ r ( m x + m y + m z ) y − r x ( m x + m y + m z )] z + [ r ( m x + m y + m z ) z − r x ( m x + m y + m z )] z + [ r ( m x + m y + m z ) z − r y ( m x + m y + m z )] z = − ( r m + r m ) x + ( r m − r m ) y + ( r m + r m ) z + ( r m − r m ) x y + ( r m − r m + r m − r m ) x y + ( r m − r m + r m − r m ) x yz + ( r m + r m ) x z − ( r m + r m + r m + r m ) x y + ( r m − r m ) yz − ( r m + r m + r m + r m ) x z − ( r m + r m ) xy + ( r m + r m + r m + r m ) x y z + ( r m + r m ) y z + ( r m + r m + r m + r m ) x z − ( r m + r m ) xz − ( r m + r m + r m + r m ) xy z + ( r m − r m + r m − r m ) y z + ( r m + r m + r m + r m ) y z
32 X.-F. MAO, X.-T. WANG, Y.-N. YANG, AND M.-Y.ZHANG and hence r m + r m = 0 r m − r m = 0 r m + r m = 0 r m + r m = 0 r m − r m = 0 r m + r m = 0 r m + r m = 0 r m − r m = 0 r m + r m = 0 r m + r m + r m + r m = 0 r m − r m + r m − r m = 0 r m + r m + r m + r m = 0 r m − r m + r m − r m = 0 r m + r m + r m + r m = 0 r m + r m + r m + r m = 0 r m + r m + r m + r m = 0 r m − r m + r m − r m = 0 r m + r m + r m + r m = 0 . Since r ( M ) = 3, one sees that the rank of the coefficient matrix m m m − m m m m m m − m m m m m
00 0 0 0 0 0 m − m m m m m m m m − m m − m m m m m m − m m − m m m m m m m m m m m m m
00 0 0 m − m m − m m m m m is 8. Therefore, dim k [ Z ( I ) T ( I /B ( I ))] = 1. On the other hand, ∂ A ( xyz ) = ( m x + m y + m z ) yz − ( m x + m y + m z ) xz + xy ( m x + m y + m z )= x ( m yz − m xz + m xy ) + y ( m yz − m xz + m xy )+ z ( m yz − m xz + m xy ) ALABI-YAU PROPERTIES OF 3-DIMENSIONAL DG SKLYANIN ALGEBRAS 43
We have β = x ( m yz − m xz + m xy ) + y ( m yz − m xz + m xy )+ z ( m yz − m xz + m xy ) ∈ Z ( I ) \ ( I /B ( I ))and hence H ( I ) = k ⌈ β ⌉ . By the definition of connecting homomorphism, we have δ ( ⌈ x ∧ y ∧ z ⌉ ) = ⌈ β ⌉ 6 = 0 and hence δ is an isomorphism. By the cohomology longexact sequence ( ♣ ), we get H ( A ) = 0. Since H i ( A/I ) = 0 for any i ≥
4, we have H i +1 ( I ) ∼ = H i +1 ( A ) by the cohomology long exact sequence ( ♣ ).Since0 = | M | = m (cid:12)(cid:12)(cid:12)(cid:12) m m m m (cid:12)(cid:12)(cid:12)(cid:12) − m (cid:12)(cid:12)(cid:12)(cid:12) m m m m (cid:12)(cid:12)(cid:12)(cid:12) + m (cid:12)(cid:12)(cid:12)(cid:12) m m m m (cid:12)(cid:12)(cid:12)(cid:12) , there is at least one non-zero in (cid:26) (cid:12)(cid:12)(cid:12)(cid:12) m m m m (cid:12)(cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12)(cid:12) m m m m (cid:12)(cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12)(cid:12) m m m m (cid:12)(cid:12)(cid:12)(cid:12) (cid:27) . Without the loss of generality, we assume that (cid:12)(cid:12)(cid:12)(cid:12) m m m m (cid:12)(cid:12)(cid:12)(cid:12) = 0 and m = 0.Let Q = ( x , y ) / ( x ) and Q = I/ ( x , y ). By Lemma 5.6, we have H i ( Q ) = k ⌈ ¯ y ⌉ , if i = 2 k ⌈ ¯ x ¯ y + ¯ y ( m m − m m m m − m m ¯ y + m m − m m m m − m m ¯ z ) ⌉ , if i = 30 , if i ≥ H i ( Q ) = k ⌈ ¯ x ⌉ , if i = 2 k ⌈− m ¯ x ¯ x + m ¯ z ⌉ ⊕ k ⌈− m ¯ y ¯ z + m ¯ z ⌉ , if i = 3 k ⌈ m ¯ x ¯ z − m ¯ y ¯ z − m ¯ x ¯ y ¯ z ⌉ , if i = 40 , if i ≥ . The cohomology long exact sequence induced from the short exact sequence0 → ( x , y ) τ → I π → Q → · · · H ( π ) → H ( Q ) δ → H [( x , y )] H ( τ ) → H ( I ) H ( π ) → H ( Q ) = 0 δ → H [( x , y )] H ( τ ) → H ( I ) H ( π ) → H ( Q ) = 0 → · · · → H i [( x , y )] H i ( τ ) → H i ( I ) → → · · · . We have ∂ I ( m xz − m yz − m xyz )=( m m − m m ) x z + ( m m − m m ) x z +( m m − m m ) x yz + ( m m − m m ) y z +( m m − m m ) xy z + ( m m − m m ) y z = " (cid:12)(cid:12)(cid:12)(cid:12) m m m m (cid:12)(cid:12)(cid:12)(cid:12) x + (cid:12)(cid:12)(cid:12)(cid:12) m m m m (cid:12)(cid:12)(cid:12)(cid:12) x − (cid:12)(cid:12)(cid:12)(cid:12) m m m m (cid:12)(cid:12)(cid:12)(cid:12) y x z + " (cid:12)(cid:12)(cid:12)(cid:12) m m m m (cid:12)(cid:12)(cid:12)(cid:12) z + (cid:12)(cid:12)(cid:12)(cid:12) m m m m (cid:12)(cid:12)(cid:12)(cid:12) x − (cid:12)(cid:12)(cid:12)(cid:12) m m m m (cid:12)(cid:12)(cid:12)(cid:12) y y z
24 X.-F. MAO, X.-T. WANG, Y.-N. YANG, AND M.-Y.ZHANG and ∂ A " (cid:12)(cid:12)(cid:12)(cid:12) m m m m (cid:12)(cid:12)(cid:12)(cid:12) z + (cid:12)(cid:12)(cid:12)(cid:12) m m m m (cid:12)(cid:12)(cid:12)(cid:12) x − (cid:12)(cid:12)(cid:12)(cid:12) m m m m (cid:12)(cid:12)(cid:12)(cid:12) y x + ∂ A " (cid:12)(cid:12)(cid:12)(cid:12) m m m m (cid:12)(cid:12)(cid:12)(cid:12) z + (cid:12)(cid:12)(cid:12)(cid:12) m m m m (cid:12)(cid:12)(cid:12)(cid:12) x − (cid:12)(cid:12)(cid:12)(cid:12) m m m m (cid:12)(cid:12)(cid:12)(cid:12) y y = − | M | y x + | M | x y = 0 . So χ = " (cid:12)(cid:12)(cid:12)(cid:12) m m m m (cid:12)(cid:12)(cid:12)(cid:12) z + (cid:12)(cid:12)(cid:12)(cid:12) m m m m (cid:12)(cid:12)(cid:12)(cid:12) x − (cid:12)(cid:12)(cid:12)(cid:12) m m m m (cid:12)(cid:12)(cid:12)(cid:12) y x + " (cid:12)(cid:12)(cid:12)(cid:12) m m m m (cid:12)(cid:12)(cid:12)(cid:12) z + (cid:12)(cid:12)(cid:12)(cid:12) m m m m (cid:12)(cid:12)(cid:12)(cid:12) x − (cid:12)(cid:12)(cid:12)(cid:12) m m m m (cid:12)(cid:12)(cid:12)(cid:12) y y ∈ Z ( A ) . Since we have proved H ( A ) = 0, there exists ω ∈ A such that ∂ A ( ω ) = χ . Then ∂ I ( m xz − m yz − m xyz ) = χz = ∂ A ( ω ) z and hence δ ( ⌈ m xz − m yz − m xyz ⌉ ) = ⌈ ∂ A ( ω ) z ⌉ = 0 by the definitionof connecting homomorphism. So δ = 0. By the cohomology long exact sequenceabove, we have H i ( I ) ∼ = H i [( x , y )] , i ≥ . The cohomology long exact sequenceinduced from the short exact sequence0 → ( x ) τ → ( x , y ) φ → Q → · · · δ → H (( x )) H ( τ ) → H (( x , y )) H ( φ ) → H ( Q ) = 0 δ →· · · δ i − → H i (( x )) H i ( τ ) → H i (( x , y ))) H i ( φ ) → H i ( Q ) = 0 δ i → · · · . Hence H i (( x )) ∼ = H i (( x , y )) for any i ≥
5. Then we get H i (( x )) ∼ = H i (( x , y )) ∼ = H i ( I ) ∼ = H i ( A )for any i ≥
5. Since x is a central and cocycle element in A , one sees that H (( x )) = H ( A ) ⌈ x ⌉ . We have shown that H i ( A ) = 0, when i = 1 , , ,
4. Thenwe can inductively prove H i ( A ) = 0 for any i ≥ Proposition 6.1. If M = ( m ij ) × ∈ GL ( k ) , then H ( A ) = k . when r ( M ) = 2 . In this case, we claim dim k H ( A ) = 1. Indeed, any cocycleelement ξ = l x + l x y + l x z + l xy + l y + l y z + l xz + l yz + l z + l xyz in Z ( A ), we have0 = ∂ A ( ξ ) = l x ( m x + m y + m z ) + l x ( m x + m y + m z )+ l x ( m x + m y + m z ) + l ( m x + m y + m z ) y + l ( m x + m y + m z ) y + l y ( m x + m y + m z )+ l ( m x + m y + m z ) z + l ( m x + m y + m z ) z + l ( m x + m y + m z ) z + l ( m x + m y + m z ) yz − l x ( m x + m y + m z ) z + l xy ( m x + m y + m z ) . ALABI-YAU PROPERTIES OF 3-DIMENSIONAL DG SKLYANIN ALGEBRAS 45
This implies that l m + l m + l m = 0 l m + l m + l m + l m + l m + l m = 0 l m + l m + l m + l m + l m + l m = 0 l m + l m + l m + l m + l m + l m = 0 l m + l m + l m = 0 l m + l m + l m = 0 l = 0 . Hence m m m m m m m m m m m m m m m m m m m m m m m m m m m l l l l l l l l l = 0 . By Lemma 5.7, r m m m m m m m m m m m m m m m m m m m m m m m m m m m = 5 . So dim k Z ( A ) = 9 − ∂ A ( xy ) = ( m x + m y + m z ) y − x ( m x + m y + m z )= m x y + m y + m yz − m x − m xy − m xz ,∂ A ( xz ) = ( m x + m y + m z ) z − x ( m x + m y + m z )= m x z + m y z + m z − m x − m xy − m xz ,∂ A ( yz ) = ( m x + m y + m z ) z − y ( m x + m y + m z )= m x z + m y z + m z − m x y − m y − m yz are linearly independent, since0 = λ ∂ A ( xy ) + λ ∂ A ( xz ) + λ ∂ A ( yz )= λ ( m x y + m y + m yz − m x − m xy − m xz )+ λ ( m x z + m y z + m z − m x − m xy − m xz )+ λ ( m x z + m y z + m z − m x y − m y − m yz )= ( λ m − λ m ) x y + ( λ m − λ m ) y + ( λ m − λ m ) yz − ( λ m + λ m ) x − ( λ m + λ m ) xy − ( λ m + λ m ) xz + ( λ m + λ m ) x z + ( λ m + λ m ) y z + ( λ m + λ m ) z
36 X.-F. MAO, X.-T. WANG, Y.-N. YANG, AND M.-Y.ZHANG implies λ m − λ m = 0 λ m − λ m = 0 λ m − λ m = 0 λ m + λ m = 0 λ m + λ m = 0 λ m + λ m = 0 λ m + λ m = 0 λ m + λ m = 0 λ m + λ m = 0 ⇔ λ = λ = λ = 0since r ( M ) = 2. Then dim k B ( A ) = 3 and we show the claim dim k H ( A ) = 1.Let I = ( r , r , r ) be the DG ideal of A generated by the central coboundaryelements r = ∂ A ( x ) , r = ∂ A ( y ) and r = ∂ A ( z ). Then the DG quotient ring Q = A /I has trivial differential. Since each r i = m i x + m i y + m i z and r ( M ) = 2, we may assume without the loss of generality that r , r are linearlyindependent, which is equivalent to t = 0. Then r = t t r + t t r and I = ( r , r ).We have H i ( I ) = ( k ⌈ r ⌉ ⊕ k ⌈ r ⌉ , i = 2 ⌈ r ⌉ H i − ( A ) ⊕ ⌈ r ⌉ H i − ( A ) ⊕ ⌈ r y − xr ⌉ H i − ( A ) , i ≥ k H i ( Q ) = dim k Q i = , i < , i = 03 , i = 14 , i ≥ . The short exact sequence 0 → I ι → A π → Q → ♠ ):0 → H ( A ) H ( π ) → H ( Q ) δ → H ( I ) H ( ι ) → H ( A ) H ( π ) → H ( Q ) δ → H ( I ) H ( ι ) → H ( A ) H ( π ) → H ( Q ) δ → · · · δ i − → H i ( I ) H i ( ι ) → H i ( A ) H i ( π ) → H i ( Q ) δ i → · · · . Since r , r and r y − xr are coboundary elements in A , we have H i ( ι ) = 0 forany i ≥
3. The cohomology long exact sequence ( ♠ ) implies thatdim k H i ( A ) + dim k H i +1 ( I ) = dim k H i ( Q ) , i ≥ . By Lemma 5.9 and dim k H ( A ) = 1, we inductively get dim k H i ( A ) = 1 , i ≥ k H i ( A ) = 1 for any i ≥ k [ ⌈ t x + t y + t z ⌉ ] is a subalgebra of H ( A ) when s t + s t + s t = 0, and k [ ⌈ t x + t y + t z ⌈ , ⌈ s x + s y + s z ⌉ ] / ( ⌈ t x + t y + t z ⌉ )is a subalgebra of H ( A ) when s t + s t + s t = 0. Considering the dimension ofeach H i ( A ) gives that H ( A ) = k [ ⌈ t x + t y + t z ⌉ ] = H ( A ) when s t + s t + s t =0, and k [ ⌈ t x + t y + t z ⌉ , ⌈ s x + s y + s z ⌉ ] / ( ⌈ t x + t y + t z ⌉ ) = H ( A ) , when s t + s t + s t = 0.By computations and analysis above, we reach the following conclusion. ALABI-YAU PROPERTIES OF 3-DIMENSIONAL DG SKLYANIN ALGEBRAS 47
Proposition 6.2.
For M ∈ M ( k ) with r ( M ) = 2 , let k ( s , s , s ) T and k ( t , t , t ) T be the solution spaces of homogeneous linear equations M X = 0 and M T X = 0 ,respectively. Then H ( A ) = k [ ⌈ t x + t y + t z ⌉ ] if s t + s t + s t = 0 ; and H ( A ) equals to k [ ⌈ t x + t y + t z ⌉ , ⌈ s x + s y + s z ⌉ ] / ( ⌈ t x + t y + t z ⌉ ) when s t + s t + s t = 0 . when r ( M ) = 1 . We may let M = m m m l m l m l m l m l m l m with l , l ∈ k and ( m , m , m ) = 0. Indeed, one can see the reason by applying [ ? , TheoremB] and the following equations m m m l m l m l m l m l m l m = l m l m l m m m m l m l m l m m m m l m l m l m l m l m l m = l m l m l m m m m l m l m l m . Now, ∂ A ( x ) = m x + m y + m z ∂ A ( y ) = l [ m x + m y + m z ] ∂ A ( z ) = l [ m x + m y + m z ] . For any c x + c y + c z ∈ Z ( A ), we have0 = ∂ A ( c x + c y + c z ) = ( c + l c + l c )[ m x + m y + m z ] ⇒ c + l c + l c = 0 , which admits a basic solution system l − , l − . So Z ( A ) = k ( l x − y ) ⊕ k ( l x − z )and H ( A ) = k ⌈ l x − y ⌉ ⊕ k ⌈ l x − z ⌉ .For any c x + c xy + c xz + c y + c yz + c z ∈ Z ( A ), we have0 = ∂ A [ c x + c xy + c xz + c y + c yz + c z ]= c ( m x + m y + m z ) y − c xl ( m x + m y + m z )+ c ( m x + m y + m z ) z − c xl ( m x + m y + m z )+ c l ( m x + m y + m z ) z − c yl ( m x + m y + m z )= − ( c l + l c ) m x + ( c − c l ) m x y + ( c + c l ) m x z − ( c l + c l ) m xy − ( c l + c l ) m xz + ( c − c l ) m y +( c + c l ) m y z + ( c − c l ) m yz + ( c + c l ) m z . Since ( m , m , m ) = 0, we get c l + l c = 0 c − c l = 0 c + c l = 0 ⇔ l l
01 0 − l l c c c = 0 . We get c = tl , c = − tl , c = t , for some t ∈ k . Thus Z ( A ) = kx ⊕ ky ⊕ kz ⊕ k ( l xy − l xz + yz ) . Since B ( A ) = k ( m x + m y + m z ), we have H ( A ) = kx ⊕ ky ⊕ kz ⊕ k ( l xy − l xz + yz ) k ( m x + m y + m z ) . Moreover, we claim that dim k H i ( A ) = i + 1, for any i ≥
0. We prove this claimas follows. Let I = ( m x + m y + m z ) be the DG ideal of A generated bythe central coboundary elements ∂ A ( x ). Then the DG quotient ring Q = A /I hastrivial differential anddim k H i ( Q ) = dim k Q i = ( , i < i + 1 , i ≥ . The short exact sequence 0 → I ι → A π → Q → ♥ ):0 → H ( A ) H ( π ) → H ( Q ) δ → H ( I ) H ( ι ) → H ( A ) H ( π ) → H ( Q ) δ → H ( I ) H ( ι ) → H ( A ) H ( π ) → H ( Q ) δ → · · · δ i − → H i ( I ) H i ( ι ) → H i ( A ) H i ( π ) → H i ( Q ) δ i → · · · . Since m x + m y + m z = ∂ A ( x ) is a central coboundary elements in A , wehave H i ( I ) = ⌈ m x + m y + m z ⌉ H i − ( A ) and H i ( ι ) = 0 for any i ≥
2. Thecohomology long exact sequence ( ♥ ) implies thatdim k H i ( A ) + dim k H i +1 ( I ) = dim k H i ( Q ) = 2 i + 1 , i ≥ . Then dim k H i ( A ) + dim k H i − ( A ) = 2 i + 1 sincedim k H i +1 ( I ) = dim k {⌈ m x + m y + m z ⌉ H i − ( A ) } = dim k H i − ( A ) , i ≥ . Since dim k H ( A ) = 2, we can inductively get dim k H i ( A ) = i + 1, for any i ≥ H ( A ), we make a classification chartas follows: m l + m l = m , ( l l = 0; l l = 0; m l + m l = m , l l = 0; l = 0 , l = 0; l = 0 , l = 0; l = l = 0 . We will compute H ( A ) case by case according to this classification chart. Moreprecisely, we have the following proposition. Proposition 6.3.
Assume that A be a -dimensional DG Sklyanin algebra suchthat A = S a,a, , and ∂ A is defined by M = m m m l m l m l m l m l m l m with l , l ∈ k, ( m , m , m ) = 0 . Then we have the following statements. (1) If m l + m l = m and l l = 0 , then H ( A ) is k h⌈ l x − y ⌉ , ⌈ l x − z ⌉i ( m ⌈ l x − y ⌉ + m ⌈ l x − z ⌉ − ⌈ l x − y ⌉⌈ l x − z ⌉ + ⌈ l x − z ⌉⌈ l x − y ⌉ l l m l m l − m ) ; ALABI-YAU PROPERTIES OF 3-DIMENSIONAL DG SKLYANIN ALGEBRAS 49 (2) If m l + m l = m and l l = 0 , then H ( A ) = k h⌈ l x − y ⌉ , ⌈ l x − z ⌉i ( ⌈ l x − y ⌉⌈ l x − z ⌉ + ⌈ l x − z ⌉⌈ l x − y ⌉ ) ;(3) If m l + m l = m and l l = 0 , then H ( A ) = k h⌈ l x − y ⌉ , ⌈ l x − z ⌉i ( m ⌈ l x − y ⌉ + m ⌈ l x − z ⌉ ) ;(4) If m l + m l = m , l = 0 and l = 0 , then H ( A ) = k h⌈ l x − y ⌉ , ⌈ z ⌉ , ⌈ x ⌉i m ⌈ l x − y ⌉ + m ⌈ z ⌉ ⌈ x ⌉⌈ l x − y ⌉ − ⌈ l x − y ⌉⌈ x ⌉⌈ x ⌉⌈ z ⌉ − ⌈ z ⌉⌈ x ⌉⌈ l x − y ⌉⌈ z ⌉ + ⌈ z ⌉⌈ l x − y ⌉ ;(5) If m l + m l = m , l = 0 and l = 0 , then H ( A ) = k h⌈ l x − z ⌉ , ⌈ y ⌉ , ⌈ x ⌉i m ⌈ l x − z ⌉ + m ⌈ y ⌉ ⌈ x ⌉⌈ l x − z ⌉ − ⌈ l x − z ⌉⌈ x ⌉⌈ x ⌉⌈ y ⌉ − ⌈ y ⌉⌈ x ⌉⌈ l x − z ⌉⌈ y ⌉ + ⌈ y ⌉⌈ l x − z ⌉ ;(6) If m l + m l = m , l = 0 and l = 0 , then H ( A ) = k h⌈ z ⌉ , ⌈ y ⌉ , ⌈ x ⌉i m ⌈ y ⌉ + m ⌈ z ⌉ ⌈ x ⌉⌈ z ⌉ − ⌈ z ⌉⌈ x ⌉⌈ x ⌉⌈ y ⌉ − ⌈ y ⌉⌈ x ⌉⌈ z ⌉⌈ y ⌉ + ⌈ y ⌉⌈ z ⌉ . Proof. (1) In A , we have ( l x − y ) = l x + y , ( l x − z ) = l x + z , ( l x − y )( l x − z ) = l l x − l xz + l xy + yz. By the assumption that m l + m l = m , we get[ k ( l x − y ) ⊕ k ( l x − z ) ⊕ k ( l x − y )( l x − z )] \ B ( A ) = 0(17)and hence H ( A ) = k ⌈ l x − y ⌉ ⊕ k ⌈ l x − z ⌉ ⊕ k ⌈ ( l x − y )( l x − z ) ⌉ . We claim that k h⌈ l x − y ⌉ , ⌈ l x − z ⌉i ( m ⌈ l x − y ⌉ + m ⌈ l x − z ⌉ − ⌈ l x − y ⌉⌈ l x − z ⌉ + ⌈ l x − z ⌉⌈ l x − y ⌉ l l m l m l is a subalgebra of H ( A ). It suffices to show that ( l x − y ) n B n ( A )( l x − z ) n B n ( A )( l x − y ) i ( l x − z ) j B i + j ( A )for any n ≥ i, j ≥
1. Indeed, if ( l x − y ) n ∈ B n ( A ) then we have( l x − y ) n = ( ∂ A [ xyf + xzg + yzh ] , if n = 2 j + 1 is odd ∂ A [ xf + yg + zh + xyzu ] , if n = 2 j is even , where f, g, h and u are all linear combinations of monomials with non-negative evenexponents. When n = 2 j is even, we have( l x + y ) j = ( l x − y ) n = ∂ A [ xf + yg + zh + xyzu ]= ( m x + m y + m z ) f + l ( m x + m y + m z ) g + l ( m x + m y + m z ) h + ( m x + m y + m z ) yzu − xl ( m x + m y + m z ) zu + xyl ( m x + m y + m z ) u. Considering the parity of exponents of the monomials that appear on both sides ofthe equation above implies that( l x + y ) j = ( m x + m y + m z ) f + l ( m x + m y + m z ) g + l ( m x + m y + m z ) h = ∂ A ( x )[ f + l g + l h ]and ∂ A ( xyzu ) = ( m x + m y + m z ) yzu − l x ( m x + m y + m z ) zu + xyl ( m x + m y + m z ) u = 0 . Hence ( l x + y ) j is in the graded ideal ( ∂ A ( x )) of k [ x , y , z ]. By Lemma 5.8,( ∂ A ( x ) , ∂ A ( y ) , ∂ A ( z )) = ( ∂ A ( x )) is a graded prime ideal of k [ x , y , z ]. So l x + y ∈ ( ∂ A ( x )). Hence there exist a ∈ k such that l x + y = a ∂ A ( x ) = ∂ A ( a x ) . But this contradicts with the fact that l x + y B ( A ), which we have provedabove. Thus ( l x − y ) n B n ( A ) when n is even.When n = 2 j + 1 is odd, we have( l x − y )( l x + y ) j = ( l x − y ) n = ∂ A [ xyf + xzg + yzh ]= ( m x + m y + m z ) yf − l x ( m x + m y + m z ) f + ( m x + m y + m z ) zg − l x ( m x + m y + m z ) g + l ( m x + m y + m z ) zh − l y ( m x + m y + m z ) h = x ( m x + m y + m z )( f − l h ) − x ( m x + m y + m z )( l f + l g )+ z ( m x + m y + m z )( g + l h )= ( m x + m y + m z )[ y ( f − l h ) − x ( l f + l g ) + z ( g + l h )]= x [ − ∂ A ( y ) f − ∂ A ( z ) g ] + y [ ∂ A ( x ) f − ∂ A ( z ) h ] + z [ ∂ A ( y ) h + ∂ A ( x ) g ] . This implies that l ( l x + y ) j = − ( l f + l g )( m x + m y + m z )( l x + y ) j = ( m x + m y + m z )( l h − f )0 = g + l h. Then ( l x + y ) j = ( l x − y ) j ∈ B j ( A ), which contradicts with the proved factthat ( l x − y ) n B n ( A ) when n is even. Therefore, ( l x − y ) n B n ( A ) when n isodd. Then ( l x − y ) n B n ( A ) for any n ≥
3. Similarly, we can show that ( l x − z ) n B n ( A ) , for any n ≥ l x − y ) i +1 ( l x − z ) j B i +2 j +1 ( A ) , for any i, j ≥ l x − y ) i ( l x − z ) j +1 B i +2 j +1 ( A ) , for any i, j ≥ l x − y ) i ( l x − z ) j B i +2 j ( A ) , for any i, j ≥ . ALABI-YAU PROPERTIES OF 3-DIMENSIONAL DG SKLYANIN ALGEBRAS 51
It remains to prove ( l x − y ) i +1 ( l x − z ) j +1 B i +2 j +2 ( A ) for any i, j ≥ . If( l x − y ) i +1 ( l x − z ) j +1 ∈ B i +2 j +2 ( A ), then( l l x − l xz + l xy + yz )( l x + y ) i ( l x + z ) j = ( l x − y ) i +1 ( l x − z ) j +1 = ∂ A [ xf + yg + zh + xyzu ]= ( m x + m y + m z ) f + l ( m x + m y + m z ) g + l ( m x + m y + m z ) h + ( m x + m y + m z ) yzu − xl ( m x + m y + m z ) zu + xyl ( m x + m y + m z ) u. where f, g, h and u are all linear combinations of monomials with non-negative evenexponents. Hence l l x ( l x + y ) i ( l x + z ) j = ( m x + m y + m z ) f + l ( m x + m y + m z ) g + l ( m x + m y + m z ) h and ( l x + y ) i ( l x + z ) j = ( m x + m y + m z ) u ∈ ( ∂ A ( x )) . Since ( ∂ A ( x )) is a prime ideal in k [ x , y , z ], we conclude that ( l x + y ) ∈ ( ∂ A ( x ))or l x + z ∈ ( ∂ A ( x )). This contradicts with (17). By the discussion above, k h⌈ l x − y ⌉ , ⌈ l x − z ⌉i ( m ⌈ l x − y ⌉ + m ⌈ l x − z ⌉ − ⌈ l x − y ⌉⌈ l x − z ⌉ + ⌈ l x − z ⌉⌈ l x − y ⌉ l l m l m l )is a subalgebra of H ( A ). On the other hand, we have dim k H i ( A ) = i + 1. Thenwe can conclude that H ( A ) is k h⌈ l x − y ⌉ , ⌈ l x − z ⌉i ( m ⌈ l x − y ⌉ + m ⌈ l x − z ⌉ − ⌈ l x − y ⌉⌈ l x − z ⌉ + ⌈ l x − z ⌉⌈ l x − y ⌉ l l m l m l ) . (2) In this case, m l + m l = m and l l = 0. One sees that ( l x − y ) = l x + y , ( l x − z ) = l x + z , ( l x − y )( l x − z ) = − l xz + l xy + yz ( l x − z )( l x − y ) = − l xy + l xz − yz. Since m l + m l = m , we have[ k ( l x − y ) ⊕ k ( l x − z ) ⊕ k ( l x − y )( l x − z )] \ B ( A ) = 0and hence H ( A ) = k ⌈ l x − y ⌉ ⊕ k ⌈ l x − z ⌉ ⊕ k ⌈ ( l x − y )( l x − z ) ⌉ . Just as the proof of (1), we can show that k h⌈ l x − y ⌉ , ⌈ l x − z ⌉i ( ⌈ l x − y ⌉⌈ l x − z ⌉ + ⌈ l x − z ⌉⌈ l x − y ⌉ )is a subalgebra of H ( A ). On the other hand, we have dim k H i ( A ) = i + 1. Thenwe can conclude that H ( A ) = k h⌈ l x − y ⌉ , ⌈ l x − z ⌉i ( ⌈ l x − y ⌉⌈ l x − z ⌉ + ⌈ l x − z ⌉⌈ l x − y ⌉ ) . (3) In this case, m l + m l = m and l l = 0. So we have m ( l x − y ) + m ( l x − z ) = ( m l + m l ) x + m y + m z = m x + m y + m z = ∂ A ( x ) and ( ( l x − y )( l x − z ) + ( l x − z )( l x − y ) = 2 l l x ( l x − y )( l x − z ) − ( l x − z )( l x − y ) = 2[ yz − l xz + l xy ] . Hence H ( A ) is k ( l x − y )( l x − z ) ⊕ k ( l x − z )( l x − y ) ⊕ k ( l x − y ) ⊕ k ( l x − z ) k [ m ( l x + y ) + m ( l x + z )] . Just as the proof of (1), we can show that k h⌈ l x − y ⌉ , ⌈ l x − z ⌉i ( m ⌈ l x − y ⌉ + m ⌈ l x − z ⌉ )is a subalgebra of H ( A ). Since dim k H i ( A ) = i + 1, we can conclude that H ( A ) = k h⌈ l x − y ⌉ , ⌈ l x − z ⌉i ( m ⌈ l x − y ⌉ + m ⌈ l x − z ⌉ ) . (4) Since m l + m l = m , l = 0 and l = 0, we have m l = m , m ( l x − y ) + m z = m l x + m y + m z = m x + m y + m z = ∂ A ( x )and ( l x − y ) z + z ( l x − y ) = l ( xz + zx ) − ( yz + zy ) = 0. Thus H ( A ) = kx ⊕ k ( l x + y ) ⊕ k ( l x − y ) z ⊕ kx k [ m ( l x − y ) + m z ] . Just as the proof of Proposition 6.3, we can show that k h⌈ l x − y ⌉ , ⌈ z ⌉ , ⌈ x ⌉i m ⌈ l x − y ⌉ + m ⌈ z ⌉ ⌈ x ⌉⌈ l x − y ⌉ − ⌈ l x − y ⌉⌈ x ⌉⌈ x ⌉⌈ z ⌉ − ⌈ z ⌉⌈ x ⌉⌈ l x − y ⌉⌈ z ⌉ + ⌈ z ⌉⌈ l x − y ⌉ is a subalgebra of H ( A ). Since dim k H i ( A ) = i + 1, we get H ( A ) = k h⌈ l x − y ⌉ , ⌈ z ⌉ , ⌈ x ⌉i m ⌈ l x − y ⌉ + m ⌈ z ⌉ ⌈ x ⌉⌈ l x − y ⌉ − ⌈ l x − y ⌉⌈ x ⌉⌈ x ⌉⌈ z ⌉ − ⌈ z ⌉⌈ x ⌉⌈ l x − y ⌉⌈ z ⌉ + ⌈ z ⌉⌈ l x − y ⌉ . (5) In this case, we have m l + m l = m , l = 0 and l = 0. So m l = m , m ( l x − z ) + m y = m l x + m y + m z = m x + m y + m z = ∂ A ( x )and ( l x − z ) y + y ( l x − z ) = l ( xy + yx ) − ( yz + zy ) = 0. Thus H ( A ) = ky ⊕ k ( l x + z ) ⊕ k ( l x − z ) y ⊕ kx k [ m ( l x − z ) + m y ] . Just as the proof of (1), we can show that k h⌈ l x − z ⌉ , ⌈ y ⌉ , ⌈ x ⌉i m ⌈ l x − z ⌉ + m ⌈ y ⌉ ⌈ x ⌉⌈ l x − z ⌉ − ⌈ l x − z ⌉⌈ x ⌉⌈ x ⌉⌈ y ⌉ − ⌈ y ⌉⌈ x ⌉⌈ l x − z ⌉⌈ y ⌉ + ⌈ y ⌉⌈ l x − z ⌉ ALABI-YAU PROPERTIES OF 3-DIMENSIONAL DG SKLYANIN ALGEBRAS 53 is a subalgebra of H ( A ). Since dim k H i ( A ) = i + 1, we have H ( A ) = k h⌈ l x − z ⌉ , ⌈ y ⌉ , ⌈ x ⌉i m ⌈ l x − z ⌉ + m ⌈ y ⌉ ⌈ x ⌉⌈ l x − z ⌉ − ⌈ l x − z ⌉⌈ x ⌉⌈ x ⌉⌈ y ⌉ − ⌈ y ⌉⌈ x ⌉⌈ l x − z ⌉⌈ y ⌉ + ⌈ y ⌉⌈ l x − z ⌉ . (6) In this case m = 0, and hence ∂ A ( x ) = m y + m z ∂ A ( y ) = 0 ∂ A ( z ) = 0 . So H ( A ) = kx ⊕ ky ⊕ kz ⊕ kyzk ( m y + m z ) . Just as the proof of (1), we can show that k h⌈ z ⌉ , ⌈ y ⌉ , ⌈ x ⌉i m ⌈ y ⌉ + m ⌈ z ⌉ ⌈ x ⌉⌈ z ⌉ − ⌈ z ⌉⌈ x ⌉⌈ x ⌉⌈ y ⌉ − ⌈ y ⌉⌈ x ⌉⌈ z ⌉⌈ y ⌉ + ⌈ y ⌉⌈ z ⌉ is a subalgebra of H ( A ). Since dim k H i ( A ) = i + 1, we conclude H ( A ) = k h⌈ z ⌉ , ⌈ y ⌉ , ⌈ x ⌉i m ⌈ y ⌉ + m ⌈ z ⌉ ⌈ x ⌉⌈ z ⌉ − ⌈ z ⌉⌈ x ⌉⌈ x ⌉⌈ y ⌉ − ⌈ y ⌉⌈ x ⌉⌈ z ⌉⌈ y ⌉ + ⌈ y ⌉⌈ z ⌉ . (cid:3) A proof of proposition 4.4
In this section, we give a proof for Proposition 4.4. We rely heavily on thefollowing lemmas.
Lemma 7.1. [MH, Theorem A]
A connected cochain DG algebra A is a KozulCalabi-Yau DG algebra if H ( A ) belongs to one of the following cases: ( a ) H ( A ) ∼ = k ; ( b ) H ( A ) = k [ ⌈ z ⌉ ] , z ∈ ker( ∂ A );( c ) H ( A ) = k h⌈ z ⌉ , ⌈ z ⌉i ( ⌈ z ⌉⌈ z ⌉ + ⌈ z ⌉⌈ z ⌉ ) , z , z ∈ ker( ∂ A ) . Lemma 7.2. [MWZ, Proposition ]
Let A be a connected cochain DG algebra suchthat H ( A ) = k h⌈ y ⌉ , ⌈ y ⌉i / ( t ⌈ y ⌉ + t ⌈ y ⌉ + t ( ⌈ y ⌉⌈ y ⌉ + ⌈ y ⌉⌈ y ⌉ )) , where ( t , t , t ) ∈ P k − { (1 , , , (0 , , } . Then A is a Koszul Calabi-Yau DGalgebra. Lemma 7.3. [HM, Theorem 4.2]
Let A be a connected DG algebra. Then A is -Calabi-Yau if and only if (i) A is a Koszul DG algebra, and (ii) C = Tor A ( k A , A k ) is a symmetric coalgebra. Remark 7.4.
Lemma 7.3 indicates that a Koszul connected cochain DG algebra A is Calabi-Yau if and only if its Ext-algebra is a symmetric Frobenius algebra. Note that the DG algebra A in Proposition 4.4 is a DG Sklyanin algebra with A = S a,a, , and ∂ A is determined by a matrix M in M ( k ). By the computationsof H ( A ) in Section 6, one observes that(1) H ( A ) = k , when r ( M ) = 3;(2) H ( A ) = k [ ⌈ t x + t y + t z ⌉ ] if s t + s t + s t = 0, where k ( s , s , s ) T and k ( t , t , t ) T be the solution spaces of homogeneous linear equations M X = 0 and M T X = 0, respectively;(3) H ( A ) is k h⌈ l x − y ⌉ , ⌈ l x − z ⌉i ( m ⌈ l x − y ⌉ + m ⌈ l x − z ⌉ − ⌈ l x − y ⌉⌈ l x − z ⌉ + ⌈ l x − z ⌉⌈ l x − y ⌉ l l m l m l − m )when M = m m m l m l m l m l m l m l m with ( m , m , m ) = 0, l l = 0and m l + m l = m ;(4) H ( A ) = k h⌈ l x − y ⌉ , ⌈ l x − z ⌉i ( ⌈ l x − y ⌉⌈ l x − z ⌉ + ⌈ l x − z ⌉⌈ l x − y ⌉ ) , when M = m m m l m l m l m l m l m l m with ( m , m , m ) = 0, m l + m l = m and l l = 0;(5) H ( A ) = k h⌈ l x − y ⌉ , ⌈ l x − z ⌉i ( m ⌈ l x − y ⌉ + m ⌈ l x − z ⌉ ) , when M = m m m l m l m l m l m l m l m with ( m , m , m ) = 0, m l + m l = m and l l = 0.Applying Lemma 7.1 and Lemma 7.2, one sees that A is a Koszul Calabi-Yau DGalgebra in the 5 cases listed above. It remains to consider the Calabi-Yau propertiesof A in the following 4 cases:(1) Case 1: r ( M ) = 2 and s t + s t + s t = 0 where k ( s , s , s ) T and k ( t , t , t ) T be the solution spaces of homogeneous linear equations M X =0 and M T X = 0, respectively;(2) Case 2: M = m m m l m l m l m l m l m l m , with ( m , m , m ) = 0, m l + m l = m , l = 0 and l = 0;(3) Case 3: M = m m m l m l m l m l m l m l m , with ( m , m , m ) = 0, m l + m l = m , l = 0 and l = 0;(4) Case 4: M = m m m l m l m l m l m l m l m , with ( m , m , m ) = 0, m l + m l = m , l = 0 and l = 0.For their Calabi-Yau properties, one can’t judge from their cohomologies. The mainideas of the proof is to construct the minimal semi-free resolution of A k in each caseand compute the corresponding Ext-algebras. It involves further classifications andcomplicated matrix analysis. In order to avoid making it a very long paper. Werefer the reader to [MWZ] for a detailed proof. Note that the DG Sklyanin algebra A in Proposition 4.4 is just the DG algebra A O − ( k ) ( M ) in [MWZ]. ALABI-YAU PROPERTIES OF 3-DIMENSIONAL DG SKLYANIN ALGEBRAS 55
Acknowledgments
X.-F. Mao was supported by NSFC (Grant No.11871326), the Key Disciplines ofShanghai Municipality (Grant No.S30104) and the Innovation Program of ShanghaiMunicipal Education Commission (Grant No.12YZ031). X.-T. Wang was supportedby Simons Foundation Program: Mathematics and Physical Sciences-CollaborationGrants for Mathematician (Award No.688403). The authors thank Professor JamesZhang for his useful suggestions and comments on this paper.
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E-mail address : [email protected] Department of Mathematics, Shanghai University, Shanghai 200444, China
E-mail address : [email protected] Department of Mathematics, Shanghai University, Shanghai 200444, China
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