Poisson (co)homology of polynomial Poisson algebras in dimension four : Sklyanins case
In this paper, we compute the Poisson (co)homology of a polynomial Poisson structure given by two Casimir polynomial functions which define a complete intersection with an isolated singularity.
Authors: Serge Romeo Tagne Pelap
P oisson (o)homology of p olynomial P oisson algebras in dimension four : Skly anin's ase Serge Roméo T agne P elap Otob er 26, 2018 Abstrat In this pap er, w e ompute the P oisson (o)homology of a p olynomial P oisson struture giv en b y t w o Casimir p olynomial funtions whi h dene a omplete in tersetion with an iso- lated singularit y . Keyw ords : P oisson strutures, P oisson (o)homology , unimo dular P oisson struture, Casimir funtions, omplete in tersetion with an isolated singularit y . In tro dution The anonial or P oisson homology w as in tro dued indep enden tly b y Brylinsky ( [ 4℄) (as an imp or- tan t to ol in omputations of Ho hs hild and yli homology), and b y K oszul and Gelfand-Dorfman (inspired b y their algebrai approa h to the study of bi-hamiltonian strutures). This homology is dened as the homology of a dieren tial omplex degree -1 { Ω • ( M ) , ∂ π } on a P oisson manifold ( M , π ) , where π is a P oisson struture giv en either b y a 2 -tensor eld π ∈ H 0 ( M , ∧ 2 T M ) or, b y the orresp onding P oisson bra k et on algebra of funtions (smo oth, alge- brai,...) of M : { f , g } = < d f ∧ dg , π > . The ( − 1) − dieren tial ∂ π is giv en on the deomp osable dieren tial forms as ∂ k ( f 0 d f 1 ∧ ... ∧ d f k ) = X 1 ≤ i ≤ k ( − 1) i +1 { f 0 , f i } d f 1 ∧ ... ∧ c d f i ∧ ... ∧ d f k + X 1 ≤ i (6) has a nite dimension as a K -ve tor sp a e. The dimension of A sing ( P ) is alled the Milnor n um b er of the singular p oin t. W e shall no w giv e a denition of dimension for rings. F or this purp ose, note that the length of the hain P r ⊃ P r − 1 ⊃ · · · ⊃ P 0 in v olving r + 1 distint ideals of a giv en ring is tak en to b e r . Denition 2.4. The Krul l dimension of a ring R is the supr emum of the lengths of hains of distint prime ide als in R . Denition 2.5. L et R b e an asso iative and ommutative gr ade d K -algebr a. A system of homo- gene ous elements a 1 , ..., a d in R , wher e d is the Krul l dimension of R , is al le d a homo gene ous system of p ar ameters of R (h.s.o.p.) if R / < a 1 , ..., a d > is a nite dimensional K -ve tor sp a e. F or example, if w e onsider the K -algebra A = K [ x 1 , ..., x 4 ] , graded b y the w eigh t degree, w e ha v e a natural h.s.o.p. giv en b y the system x 1 , x 2 , x 3 , x 4 . Denition 2.6. A se quen e a 1 , ..., a n in a ommutative asso iative algebr a R is said to b e an R -r e gular se quen e if < a 1 , ..., a n > 6 = R and a i is not a zer o divisor of R / < a 1 , ..., a i − 1 > for i = 1 , 2 , ..., n . F or an y regular sequene a 1 , ..., a n , w e an dene a K oszul omplex whi h is exat (see W eib el [30 ℄) : 0 − → V 0 ( R n ) − → · · · − → V n − 2 ( R n ) ∧ ω − → V n − 1 ( R n ) ∧ ω − → V n ( R n ) 10 where ω = n X i =1 a i e i and ( e 1 , e 2 , · · · , e n ) is a basis of an R -mo dule free R n . In our partiular ase, R = K [ x 1 , x 2 , · · · , x n ] , using the iden tiations V p ( R n ) ⋍ Ω p ( R ) , the K oszul omplex asso iated to the sequene ∂ P ∂ x 1 , ∂ P ∂ x 2 , · · · , ∂ P ∂ x n ( P ∈ R ) ha v e the follo wing form : 0 − → A ∧ dP − → Ω 1 ( A ) − → · · · − → Ω n − 2 ( A ) ∧ dP − → Ω n − 1 ( A ) ∧ dP − → Ω n ( A ) Using the v etor notation for n = 4 , w e ha v e the follo wing omplex : 0 − → A − → ∇ P − → A 4 × − → ∇ P − → A 6 − → ∇ P ¯ × − → A 4 · − → ∇ P − → A Theorem 2.1. (Cohen-Maaula y) . L et R b e a no etherian gr ade d K -algebr a. If R has a h.s.o.p. whih is a r e gular se quen e, then any h.s.o.p. in R is a r e gular se quen e. Th us, for ea h P ∈ A = K [ x 1 , ..., x 4 ] whi h is a w eigh t homogeneous p olynomial with an isolated singularit y , the sequene ∂ P ∂ x 1 , ∂ P ∂ x 2 , ∂ P ∂ x 3 , ∂ P ∂ x 4 is regular, the asso iated K oszul omplex is exat. Denition 2.7. L et R b e a no etherian ommutation ring with unit. The depth, dpth ( I ) , of an ide al I of R is the maximal length q of an R -r e gular se quen e a 1 , · · · , a q ∈ I . Let M b e a free R -mo dule of nite rank n , where R is a a no etherian omm utativ e ring with unit. W e denote b y V p ( M ) the p - th exterior pro dut of M . By on v en tion V 0 ( M ) = R . Let η 1 , · · · , η k b e giv en elemen ts of M , and ( e 1 , · · · , e n ) b e a basis of M . η 1 ∧ · · · ∧ η k = X 1 ≤ i 1 < ··· i k ≤ n a i 1 , ··· ,i k e i 1 ∧ · · · ∧ e i k . W e denote b y A the ideal of R generated b y the o eien ts a i 1 , ··· , i k , 1 ≤ i 1 < · · · i k ≤ n. Then w e dene : Z p := { η ∈ V p ( M ) : η ∧ η 1 ∧ · · · ∧ η k = 0 } , p = 0 , 1 , 2 , · · · H p := Z p k X i =1 η i ∧ p − 1 ^ ( M ) , p = 0 , 1 , 2 , · · · W e ha v e the follo wing result from Ky o ji Saito : Theorem 2.2. ( [22 ℄) H p = 0 for 0 ≤ p < dpth ( A ) . Let us giv e an example. Supp ose A = K [ x 1 , x 2 , x 3 , x 4 ] and onsider P 1 , P 2 , t w o w eigh t homoge- neous p olynomials in A . W e sa y that ( P 1 , P 2 ) denes a omplete in tersetion if ( P 1 , P 2 ) is a regular sequene in A . And ( P 1 , P 2 ) has an isolated singularit y if A / h P 1 , P 2 , ∂ P 1 ∂ x i ∂ P 2 ∂ x j − ∂ P 1 ∂ x j ∂ P 2 ∂ x i , i < j = 1 , 2 , 3 , 4 i is a nite dimensional K -v etor spae. This dimension is also alled the Milnor n um b er of singularit y and denoted µ . Let ( P 1 , P 2 ) b e a omplete in tersetion with an isolated singularit y . W e denote b y η j = 4 X i =1 ∂ P j ∂ x i e i , j = 1 , 2 , where ( e 1 , e 2 , e 3 , e 4 ) is a basis of a free A -mo dule A 4 . Then η 1 ∧ η 2 = 4 X i
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