Holonomic D -Modules, the Dixmie Conjecture and the Jacobian - PDF document

Holonomic D -Modules, the Dixmie Conjecture and the Jacobian Conjecture Vladimir Bavula ∗ ∗ Talks/DOKBONN 1

Plan 1. Holonomic D -Modules and the Inequality of Bernstein. 2. The Jacobian Conjecture, the Dixmier Prob- lem and a Question of Rentschler. 3. Main Result: for Somewhat Commutative Algebras Holonomicity is Preserved under Restriction of Scalars. 2

1. Holonomic D -Modules and the Inequality of Bernstein Let K be a field of characteristic 0, X be a smooth irreducible algebraic affine variety of dim X = n , O ( X ) be the alg. of regular functions on X . The ring of differential operators D ( X ) on X is a simple Noetherian affine algebra of Gelfand- Kirillov dimension GK D ( X ) = 2 n . Ex . For X = K n , the ring of diff. operators with polynomial coefficients, ∂ ∂ A n := D ( K n ) = K [ x 1 , . . . , x n , , . . . , ] ∂x 1 ∂x n is called the Weyl algebra ; GK A n = 2 n and A n = A 1 ⊗ · · · ⊗ A 1 , n times. 3

Th 1.1. (The Inequality of Bernstein, 1971) Let M ̸ = 0 be a finitely generated A n -module. Then GK( M ) ≥ GK A n = n . 2 Th 1.2. Let M ̸ = 0 be a finitely generated D ( X )-module. Then GK( M ) ≥ GK D ( X ) = n . 2 Def . A finitely generated D ( X )-module M is called holonomic iff GK( M ) = n . Ex . O ( X ) is a holonomic D ( X )-module since GK O ( X ) = K . dim O ( X ) = dim X = n . In par- ticular, O ( K n ) = K [ x 1 , . . . , x n ] is a holonomic A n -module. L 1.3. Each holonomic module has finite length. A nonzero sub/factor module of a holonomic module is a holonomic module. If M ̸ = 0 is a simple A n -module then GK( M ) ∈ { GK A n = n, n +1 , . . . , 2 n − 1 = GK A n − 1 } . 2 4

Th 1.4. (Bernstein-Lunts, 1988-89) ”Gener- ically” GK( M ) = 2 n − 1 ( K is the field of com- plex numbers). 2. The Jacobian Conjecture, the Dixmier Problem and a Question of Rentschler The Dixmier Problem, 1968 : End K ( A n ) = Aut K ( A n )? Let P n = K [ x 1 , . . . , x n ] be a polynomial ring. For an alg. endomorphism f = ( f i ) ∈ End K ( P n ), f i := f ( x i ), the n × n matrix J ( f ) := ( ∂f i ∂x j ) is called the Jacobian (matrix) of f ; ∆( f ) := det J ( f ). For f, g ∈ End K ( P n ), J ( g ◦ f ) = g ( J ( f )) J ( g ) and ∆( g ◦ f ) = ∆( g ) g (∆( f )) . If f ∈ Aut K ( P n ) and g = f − 1 , then ∆( f ) ∈ K ∗ := K \{ 0 } . The Jacobian Conjecture : If, for f ∈ End K ( P n ), ∆( f ) ∈ K ∗ , then f ∈ Aut K ( P n ). 5

Th 2.1. (Bass, Connel and Wright, 1982) The Dixmier Problem ⇒ the Jacobian Conjec- ture. A Question of Rentschler (2 March, 2000) : for every simple A 1 -module M and for every σ ∈ End K ( A 1 ), the twisted A 1 -module σ M has finite lenght? If the Dixmier Problem has a positive solution then σ is an automorphism, hence A 1 -module σ M is simple. Th 2.2. Yes to the Question of Rentschler. Th 2.3. Let M be a holonomic A n -module and σ ∈ End K ( A n ). Then σ M is a holonomic A n -module, hence has finite length. Applications to the Dixmier Problem. The opposite algebra A n ≃ A op ∂ n , x i → x i , ∂x i → − ∂ ∂x i . 6

A n ⊗ A op ≃ A n ⊗ A n = A 2 n . n A n A nA n = A n ⊗ A op n A n = A 2 n A n , a simple holo- nomic A 2 n -module since 2 = GK( A 2 n ) GK A n = 2 n = 4 n . 2 For τ ∈ End K ( A n ), the twisted A n -bimodule ( ≡ the twisted A 2 n -module) τ A nτ contains the simple A n -subbimodule τ ( A n ) since A n · τ ( A n ) · A n = τ ( A n ) τ ( A n ) τ ( A n ) = τ ( A n A n A n ) = τ ( A n ), and τ ( A n ) ≃ A n . • the Dixmier Problem has a Positive So- lution iff A n = τ ( A n ) iff the τ A nτ is Sim- ple A 2 n -module (for all τ ) . 7

Th 2.4. For every τ ∈ End K ( A n ) the A 2 n - module τ A nτ is holonomic, hence has finite length. Proof . Since A n is a holonomic A 2 n -module, by Th 2.3 , τ A nτ is a holonomic A 2 n -module, hence has finite length. Comments. 1. Th 2.4 is a good argument in favour of the positive solution of the Dixmier Problem and the Jacobian Conjecture. 2. In case that the Dixmier Problem and the Jacobian Conjecture have negative solution Th 2.4 explains why it is difficult to give a counter- example ( τ ( A n ) is big ). 3. Main Result: for Somewhat Commuattive Algebras Holonomicity is Preserved under Restriction of Scalars 8

Def. An algebra A is called somewhat com- mutative if there exists a finite dimensional fil- tration, A = ∪ i ≥ 0 F i s.t. the associated graded algebra gr A := ⊕ F i /F i − 1 is commutative and affine . Ex. D ( X ), A n , the universal enveloping alge- bra U ( G ) of finite dimensional Lie algebra G . If M is a finitely generated A -module then GK M is a natural number. Def. For a somewhat commutative algebra A we define the holonomic number , h ( A ) := min { GK( M ) | M ̸ = 0 is a fin . gen . A − mod } . A finitely generated A -module M is called a holonomic A -module iff GK( M ) = h ( A ). Ex. h ( D ( X )) = dim X = n , h ( A n ) = n . 9

L 3.1. Each holonomic module has finite length. A nonzero sub/factor module of a holonomic module is a holonomic module. Th 3.2. (Holonomicity is Preserved under Restriction of Scalars) Let A and B be some- what commutative algebras such that h ( A ) = h ( B ). Then, for every algebra endomorphism τ : A → B and for every holonomic B -module M , the A -module τ M , obtained by resrtiction of scalars, is a holonomic A -module (hence has finite length). 10

C 3.3. Let X and Y be smooth irreducible al- gebraic affine varieties of dimension n . Then, for every algebra endomorphism τ : D ( X ) → D ( Y ) and for every holonomic D ( Y )-module M , the D ( X )-module τ M , obtained by resrtic- tion of scalars, is a holonomic D ( X )-module (hence has finite length). Proof . Algebras D ( X ), D ( Y ) are somewhat commutative and have the same holonomic number n . Now the result follows from Th 3.2 . C 3.4. Let X and Y be smooth irreducible algebraic affine varieties of dimension n and let σ, τ : D ( X ) → D ( Y ) be algebra homomor- Then the D ( X )-bimodule σ D ( Y ) τ is phisms. holonomic, and has finite length. 11

C 3.5. (Forward-Back) Let X and Y be smooth irreducible algebraic affine varieties of dimen- sion n and let σ : D ( X ) → D ( Y ) be an al- gebra homomorphism. For every holonomic D ( Y )-module M and for every nonzero element s ∈ O ( Y ), the localization M s of the module M at the powers of the element s is a holonomic D ( X )-module provided M s ̸ = 0. In particular, O ( Y ) s is a holonomic D ( X )-module for every nonzero s ∈ O ( Y ). Let f = ( f i ) ∈ End K ( P n ) be a momomorphism, f i := f ( x i ).The elements ∆ ij of the inverse matrix J ( f ) − 1 = (∆ ij ) belong to the localiza- tion P n, ∆ := ( P n ) ∆ of the polynomial ring P n at the powers of ∆ := det J ( f ) ̸ = 0. 12

The derivations of the algebra P n, ∆ , n ∑ δ i := ∆ ji ∂ j , i = 1 , . . . , n, j =1 satisfy, δ i ( f j ) = δ ij , i, j = 1 , . . . , n. Hence, the subalgebra W ( f ) of the localization A n, ∆ := ( A n ) ∆ of the Weyla lgebra A n , gen- erated by f 1 , . . . , f n , δ 1 , . . . , δ n , is isomorphic to the Weyl algebra A n . C 3.6. (Bass, 1989; van den Essen, 1991) The algebra P n, ∆ is a holonomic W ( f )-module. Proof . P n is a holonomic A n -module, now ap- ply C 3.5. 13