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
Vladimir Bavula
∗
∗Talks/DOKBONN
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lem and a Question of Rentschler.
Algebras Holonomicity is Preserved under Restriction of Scalars.
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Inequality of Bernstein Let K be a field of characteristic 0, X be a smooth irreducible algebraic affine variety
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) = 2n.
with polynomial coefficients, An := D(Kn) = K[x1, . . . , xn, ∂ ∂x1 , . . . , ∂ ∂xn ] is called the Weyl algebra; GK An = 2n and An = A1 ⊗ · · · ⊗ A1, n times.
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Th 1.1. (The Inequality of Bernstein, 1971) Let M ̸= 0 be a finitely generated An-module. Then GK(M) ≥ GK An
2
= n. Th 1.2. Let M ̸= 0 be a finitely generated D(X)-module. Then GK(M) ≥ GK D(X)
2
= n.
called holonomic iff GK(M) = n.
GK O(X) = K.dim O(X) = dim X = n. In par- ticular, O(Kn) = K[x1, . . . , xn] is a holonomic An-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 An-module then GK(M) ∈ {GK An 2 = n, n+1, . . . , 2n−1 = GK An−1}.
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Th 1.4. (Bernstein-Lunts, 1988-89) ”Gener- ically” GK(M) = 2n−1 (K is the field of com- plex numbers).
Problem and a Question of Rentschler The Dixmier Problem, 1968: EndK(An) = AutK(An)? Let Pn = K[x1, . . . , xn] be a polynomial ring. For an alg. endomorphism f = (fi) ∈ EndK(Pn), fi := f(xi), the n × n matrix J (f) := ( ∂fi
∂xj) is
called the Jacobian (matrix) of f; ∆(f) := det J (f). For f, g ∈ EndK(Pn), J (g ◦ f) = g(J (f)) J (g) and ∆(g ◦ f) = ∆(g) g(∆(f)). If f ∈ AutK(Pn) and g = f−1, then ∆(f) ∈ K∗ := K\{0}. The Jacobian Conjecture: If, for f ∈ EndK(Pn), ∆(f) ∈ K∗, then f ∈ AutK(Pn).
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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 A1-module M and for every σ ∈ EndK(A1), the twisted A1-module σM has finite lenght? If the Dixmier Problem has a positive solution then σ is an automorphism, hence A1-module
σM is simple.
Th 2.2. Yes to the Question of Rentschler. Th 2.3. Let M be a holonomic An-module and σ ∈ EndK(An). Then σM is a holonomic An-module, hence has finite length. Applications to the Dixmier Problem. The opposite algebra An ≃ Aop
n , xi → xi, ∂ ∂xi →
− ∂
∂xi.
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An ⊗ Aop
n
≃ An ⊗ An = A2n.
AnAnAn = An⊗Aop
n An = A2nAn, a simple holo-
nomic A2n-module since GK An = 2n = 4n
2 = GK(A2n) 2
. For τ ∈ EndK(An), the twisted An-bimodule (≡ the twisted A2n-module) τAnτ contains the simple An-subbimodule τ(An) since An·τ(An)·An = τ(An)τ(An)τ(An) = τ(AnAnAn) = τ(An), and τ(An) ≃ An.
lution iff An = τ(An) iff the τAnτ is Sim- ple A2n-module (for all τ).
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Th 2.4. For every τ ∈ EndK(An) the A2n- module τAnτ is holonomic, hence has finite length.
by Th 2.3, τAnτ is a holonomic A2n-module, hence has finite length.
favour of the positive solution of the Dixmier Problem and the Jacobian Conjecture.
Jacobian Conjecture have negative solution Th 2.4 explains why it is difficult to give a counter- example (τ(An) is big).
Commuattive Algebras Holonomicity is Preserved under Restriction of Scalars
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mutative if there exists a finite dimensional fil- tration, A = ∪i≥0 Fi s.t. the associated graded algebra gr A := ⊕Fi/Fi−1 is commutative and affine.
bra U(G) of finite dimensional Lie algebra G. If M is a finitely generated A-module then GK M is a natural number.
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).
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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 τM, obtained by resrtiction
finite length).
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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 τM, 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- phisms. Then the D(X)-bimodule σD(Y )τ is holonomic, and has finite length.
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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 Ms of the module M at the powers of the element s is a holonomic D(X)-module provided Ms ̸= 0. In particular, O(Y )s is a holonomic D(X)-module for every nonzero s ∈ O(Y ). Let f = (fi) ∈ EndK(Pn) be a momomorphism, fi := f(xi).The elements ∆ij of the inverse matrix J (f)−1 = (∆ij) belong to the localiza- tion Pn,∆ := (Pn)∆ of the polynomial ring Pn at the powers of ∆ := det J (f) ̸= 0.
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The derivations of the algebra Pn,∆, δi :=
n
∑
j=1
∆ji∂j, i = 1, . . . , n, satisfy, δi(fj) = δij, i, j = 1, . . . , n. Hence, the subalgebra W(f) of the localization An,∆ := (An)∆ of the Weyla lgebra An, gen- erated by f1, . . . , fn, δ1, . . . , δn, is isomorphic to the Weyl algebra An. C 3.6. (Bass, 1989; van den Essen, 1991) The algebra Pn,∆ is a holonomic W(f)-module.
ply C 3.5.
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