The Weyl algebra Modules over the Weyl algebra
Francisco J. Castro Jimenez´ Department of Algebra - University of Seville
Dmod2011: School on D-modules and applications in Singularity Theory (First week: Seville, 20 - 24 June 2011) IMUS (U. of Seville) - ICMAT (CSIC, Madrid)
The Weyl algebra – p. Notice
These slides differ slightly to the ones used in my course on the Weyl algebra and its modules.
The Weyl algebra – p. Notice
These slides differ slightly to the ones used in my course on the Weyl algebra and its modules. I have added to the previous version the answers to some of the questions of the participants in the School Dmod2011.
The Weyl algebra – p. Notice
These slides differ slightly to the ones used in my course on the Weyl algebra and its modules. I have added to the previous version the answers to some of the questions of the participants in the School Dmod2011. I thank very much the participants for their comments and suggestions.
The Weyl algebra – p. Notice
These slides differ slightly to the ones used in my course on the Weyl algebra and its modules. I have added to the previous version the answers to some of the questions of the participants in the School Dmod2011. I thank very much the participants for their comments and suggestions. More comments can be sent to [email protected]
The Weyl algebra – p. Main references I
• Coutinho, S. C., A primer of algebraic D-modules, London Mathematical Society Student Texts, 33. Cambridge University Press, Cambridge, 1995.
The Weyl algebra – p. Main references I
• Coutinho, S. C., A primer of algebraic D-modules, London Mathematical Society Student Texts, 33. Cambridge University Press, Cambridge, 1995. • Bernstein J., Modules over the ring of differential operators. Study of the fundamental solutions of equations with constant coefficients. Funkcional. Anal. i Priložen. 5 (1971), no. 2, 1–16. and Analytic continuation of generalized functions with respect to a parameter. Funkcional. Anal. i Priložen. 6 (1972), no. 4, 26–40.
The Weyl algebra – p. Main references I
• Coutinho, S. C., A primer of algebraic D-modules, London Mathematical Society Student Texts, 33. Cambridge University Press, Cambridge, 1995. • Bernstein J., Modules over the ring of differential operators. Study of the fundamental solutions of equations with constant coefficients. Funkcional. Anal. i Priložen. 5 (1971), no. 2, 1–16. and Analytic continuation of generalized functions with respect to a parameter. Funkcional. Anal. i Priložen. 6 (1972), no. 4, 26–40. • Björk J-E., Rings of Differential Operators. North-Holland, Amsterdam 1979.
The Weyl algebra – p. Main references II
• Castro-Jiménez F.J., Modules Over the Weyl Algebra, in Le Dung˜ Trang (ed.), Algebraic approach to differential equations, papers from the School held in Bibliotheca Alexandrina, Alexandria, Egypt, November 12-24, 2007. World Scientific, New Jersey, 2010.
The Weyl algebra – p. Main references II
• Castro-Jiménez F.J., Modules Over the Weyl Algebra, in Le Dung˜ Trang (ed.), Algebraic approach to differential equations, papers from the School held in Bibliotheca Alexandrina, Alexandria, Egypt, November 12-24, 2007. World Scientific, New Jersey, 2010. • Ehlers F., The Weyl algebra in Borel A. et al. Algebraic D-modules. Perspectives in Mathematics, 2. Academic Press, Inc., Boston, MA, 1987. xii+355 pp.
The Weyl algebra – p. Main references II
• Castro-Jiménez F.J., Modules Over the Weyl Algebra, in Le Dung˜ Trang (ed.), Algebraic approach to differential equations, papers from the School held in Bibliotheca Alexandrina, Alexandria, Egypt, November 12-24, 2007. World Scientific, New Jersey, 2010. • Ehlers F., The Weyl algebra in Borel A. et al. Algebraic D-modules. Perspectives in Mathematics, 2. Academic Press, Inc., Boston, MA, 1987. xii+355 pp.
The Weyl algebra – p. Main references II
• Castro-Jiménez F.J., Modules Over the Weyl Algebra, in Le Dung˜ Trang (ed.), Algebraic approach to differential equations, papers from the School held in Bibliotheca Alexandrina, Alexandria, Egypt, November 12-24, 2007. World Scientific, New Jersey, 2010. • Ehlers F., The Weyl algebra in Borel A. et al. Algebraic D-modules. Perspectives in Mathematics, 2. Academic Press, Inc., Boston, MA, 1987. xii+355 pp.
The Weyl algebra – p. Exercises
María-Cruz Fernández-Fernández. Some exercises on the Weyl algebra. Monday 20th, June. 17:30-19:00; Room 2.1
The Weyl algebra – p. Exercises
María-Cruz Fernández-Fernández. Some exercises on the Weyl algebra. Monday 20th, June. 17:30-19:00; Room 2.1
José-María Ucha-Enríquez. Computer Algebra in D–module theory. Tuesday 21st, June. 17:30-19:00. Computer Room
The Weyl algebra – p. The Weyl algebra
C[x] = C[x1,...,xn] ring of polynomials.
(informal definition) The Weyl algebra An(C) of order n over the complex numbers C is the set
The Weyl algebra – p. The Weyl algebra
C[x] = C[x1,...,xn] ring of polynomials.
(informal definition) The Weyl algebra An(C) of order n over the complex numbers C is the set
∂ α1 ∂ αn p (x) · · · | p (x) ∈ C[x] α ∂x ∂x α Nn 1 n α=(α1,...,αXn)∈
The Weyl algebra – p. The Weyl algebra
C[x] = C[x1,...,xn] ring of polynomials.
(informal definition) The Weyl algebra An(C) of order n over the complex numbers C is the set
∂ α1 ∂ αn p (x) · · · | p (x) ∈ C[x] α ∂x ∂x α Nn 1 n α=(α1,...,αXn)∈
An(C) is a ring (and a C-algebra). The addition + is the natural one.
The Weyl algebra – p. The Weyl algebra
C[x] = C[x1,...,xn] ring of polynomials.
(informal definition) The Weyl algebra An(C) of order n over the complex numbers C is the set
∂ α1 ∂ αn p (x) · · · | p (x) ∈ C[x] α ∂x ∂x α Nn 1 n α=(α1,...,αXn)∈
An(C) is a ring (and a C-algebra). The addition + is the natural one. The product · is defined using Leibniz’s rule: ∂ · f(x) = f(x) ∂ + ∂f(x) for f(x) ∈ C[x] ∂xi ∂xi ∂xi
The Weyl algebra – p. The Weyl algebra
∂ ∂xi ∂ C[x] −→ C[x] is a C–linear map (i.e. ∈ EndC(C[x])) ∂xi
The Weyl algebra – p. The Weyl algebra
∂ ∂xi ∂ C[x] −→ C[x] is a C–linear map (i.e. ∈ EndC(C[x])) ∂xi The multiplication by f ∈ C[x]
φ C[x] −→f C[x]
(i.e. φf (g) = fg for all g ∈ C[x]) is also a C–linear map (i.e. φf ∈ EndC(C[x]))
The Weyl algebra – p. The Weyl algebra
∂ ∂xi ∂ C[x] −→ C[x] is a C–linear map (i.e. ∈ EndC(C[x])) ∂xi The multiplication by f ∈ C[x]
φ C[x] −→f C[x]
(i.e. φf (g) = fg for all g ∈ C[x]) is also a C–linear map (i.e. φf ∈ EndC(C[x])) Denote ∂ = ∂ for i = 1,...,n. i ∂xi
The Weyl algebra – p. The Weyl algebra
EndC(C[x]) the ring of endomorphisms of the C-vector space C[x]. The product in this ring is the composition of endomorphisms. EndC(C[x]) is a noncommutative ring (if the number of variables n ≥ 1). The unit of this ring is the identity map (i.e. φ1 : C[x] → C[x])
The Weyl algebra – p. The Weyl algebra
EndC(C[x]) the ring of endomorphisms of the C-vector space C[x]. The product in this ring is the composition of endomorphisms. EndC(C[x]) is a noncommutative ring (if the number of variables n ≥ 1). The unit of this ring is the identity map (i.e. φ1 : C[x] → C[x])
(Formal definition) An(C) is the subring (the subalgebra) of EndC(C[x]) generated by the endomorphisms
φx1 ,...,φxn ,∂1,...,∂n.
The Weyl algebra – p. The Weyl algebra
EndC(C[x]) the ring of endomorphisms of the C-vector space C[x]. The product in this ring is the composition of endomorphisms. EndC(C[x]) is a noncommutative ring (if the number of variables n ≥ 1). The unit of this ring is the identity map (i.e. φ1 : C[x] → C[x])
(Formal definition) An(C) is the subring (the subalgebra) of EndC(C[x]) generated by the endomorphisms
φx1 ,...,φxn ,∂1,...,∂n.
Convention: A0(C) = C. To simplify we will write An = An(C).
The Weyl algebra – p. The Weyl algebra
Exercise.- Prove the following equalities in An:
∂i ◦ φxi = φxi ◦ ∂i + 1
∂i ◦ φxj = φxj ◦ ∂i if i =6 j ∂i ◦ ∂j = ∂j ◦ ∂i for all pairs (i, j).
φxi ◦ φxj = φxj ◦ φxi for all pairs (i, j). An is a non-commutative ring (for n ≥ 1).
The Weyl algebra – p. The Weyl algebra
Exercise.- Prove the following equalities in An:
∂i ◦ φxi = φxi ◦ ∂i + 1
∂i ◦ φxj = φxj ◦ ∂i if i =6 j ∂i ◦ ∂j = ∂j ◦ ∂i for all pairs (i, j).
φxi ◦ φxj = φxj ◦ φxi for all pairs (i, j). An is a non-commutative ring (for n ≥ 1).
Proposition.- The map C[x] −→ An defined by f 7→ φf is an injective morphism of rings (and of C-algebras).
The Weyl algebra – p. The Weyl algebra
Exercise.- Prove the following equalities in An:
∂i ◦ φxi = φxi ◦ ∂i + 1
∂i ◦ φxj = φxj ◦ ∂i if i =6 j ∂i ◦ ∂j = ∂j ◦ ∂i for all pairs (i, j).
φxi ◦ φxj = φxj ◦ φxi for all pairs (i, j). An is a non-commutative ring (for n ≥ 1).
Proposition.- The map C[x] −→ An defined by f 7→ φf is an injective morphism of rings (and of C-algebras).
Notation.- We will write xi instead of φxi . We will write P Q instead of P ◦ Q, for P, Q ∈ An. α α1 αn x = x1 · · · xn β β1 βn ∂ = ∂1 · · · ∂n .
The Weyl algebra – p. The product in An a) f ∈ C[x] and β ∈ Nn. Then
β ∂βf = ∂σ(f)∂β−σ σ ! σX≪β where σ ≪ β stands for σi ≤ βi for i = 1,...,n,
β β! = σ!(β−σ)! and β! = β1! · · · βn! σ !
The Weyl algebra – p. 10 The product in An a) f ∈ C[x] and β ∈ Nn. Then
β ∂βf = ∂σ(f)∂β−σ σ ! σX≪β where σ ≪ β stands for σi ≤ βi for i = 1,...,n,
β β! = σ!(β−σ)! and β! = β1! · · · βn! σ ! For n = 1 the equality j j − ∂j f = ∂k(f)∂j k 1 X t 1 k=0 k can be proved by induction on j. The general case follows from the previous one and the distributivity of the product in An with respect to the sum.
The Weyl algebra – p. 10 The product in An a) f ∈ C[x] and β ∈ Nn. Then
β ∂βf = ∂σ(f)∂β−σ σ ! σX≪β where σ ≪ β stands for σi ≤ βi for i = 1,...,n,
β β! = σ!(β−σ)! and β! = β1! · · · βn! σ ! For n = 1 the equality j j − ∂j f = ∂k(f)∂j k 1 X t 1 k=0 k can be proved by induction on j. The general case follows from the previous one and the distributivity of the product in An with respect to the sum.
The Weyl algebra – p. 10 The product in An
γ γ b) β,γ ∈ Nn then ∂β(xγ) = β! xγ−β where = 0 β ! β ! if the relation β ≪ γ doesn’t hold.
The Weyl algebra – p. 11 The product in An
γ γ b) β,γ ∈ Nn then ∂β(xγ) = β! xγ−β where = 0 β ! β ! if the relation β ≪ γ doesn’t hold. c) α,α′,β,β′ ∈ Nn then
′ ′ ′ ′ xα∂βxα ∂β = xα+α ∂β+β +
β α′ ′ ′ σ! xα+α −σ∂β+β −σ. ′ σ ! σ ! σ≪β,σX≪α ,σ6=0
The Weyl algebra – p. 11 The Weyl algebra
Proposition.- : The set of monomials α β n B = {x ∂ | α, β ∈ N } is a basis of the C-vector space An.
The Weyl algebra – p. 12 The Weyl algebra
Proposition.- : The set of monomials α β n B = {x ∂ | α, β ∈ N } is a basis of the C-vector space An.
Each nonzero element P in An can be written in an unique way as a finite sum
α β P = pαβx ∂ Xα,β for some nonzero complex numbers pαβ.
The Weyl algebra – p. 12 The Weyl algebra
Proposition.- : The set of monomials α β n B = {x ∂ | α, β ∈ N } is a basis of the C-vector space An.
Each nonzero element P in An can be written in an unique way as a finite sum
α β P = pαβx ∂ Xα,β for some nonzero complex numbers pαβ. β α Moreover, P = β pβ(x)∂ with pβ(x) = α pαβx . P P
The Weyl algebra – p. 12 An(K)
K be a field of characteristic zero. Polynomial ring K[x] = K[x1,...,xn].
The Weyl algebra – p. 13 An(K)
K be a field of characteristic zero. Polynomial ring K[x] = K[x1,...,xn]. Definition.- Let n ≥ 1 be an integer number. The n-th Weyl algebra over K, denoted by An(K), is the subalgebra of EndK(K[x]) generated by the endomorphisms
φx1 ,...,φxn ,∂1,...,∂n.
The Weyl algebra – p. 13 An(K)
K be a field of characteristic zero. Polynomial ring K[x] = K[x1,...,xn]. Definition.- Let n ≥ 1 be an integer number. The n-th Weyl algebra over K, denoted by An(K), is the subalgebra of EndK(K[x]) generated by the endomorphisms
φx1 ,...,φxn ,∂1,...,∂n.
All the results given so far for An(C) are also valid for An(K). But the case char(K) = p > 0 is different (see e.g. the book by S.C. Coutinho, page 17).
The Weyl algebra – p. 13 Modules over An
C[x] is a left An–module.
The Weyl algebra – p. 14 Modules over An
C[x] is a left An–module. β P = β pβ(x)∂ ∈ An, f ∈ C[x], P ∂β1+···+βn (f) P • f = P (f) = p (x) β β1 βn ∂x1 · · · ∂xn Xβ
The Weyl algebra – p. 14 Modules over An
C[x] is a left An–module. β P = β pβ(x)∂ ∈ An, f ∈ C[x], P ∂β1+···+βn (f) P • f = P (f) = p (x) β β1 βn ∂x1 · · · ∂xn Xβ
An • f = {P (f) | P ∈ An}⊂ C[x]. C[x] = An • 1.
The Weyl algebra – p. 14 Modules over An
C[x] is a left An–module. β P = β pβ(x)∂ ∈ An, f ∈ C[x], P ∂β1+···+βn (f) P • f = P (f) = p (x) β β1 βn ∂x1 · · · ∂xn Xβ
An • f = {P (f) | P ∈ An}⊂ C[x]. C[x] = An • 1. C[x] ≃ An AnnAn (1)
AnnAn (1) = {P ∈ An | P (1) = 0} = An(∂1,...,∂n)
The Weyl algebra – p. 14 Systems of LPDE
P1(u) = 0 . . S ≡ . . (0.0.0) Pℓ(u) = 0 where Pi is a linear differential operator Pi ∈ An = An(C).
The Weyl algebra – p. 15 Systems of LPDE
P1(u) = 0 . . S ≡ . . (0.0.0) Pℓ(u) = 0 where Pi is a linear differential operator Pi ∈ An = An(C). Assume u(x) ∈ C[x] is a solution of S.
The Weyl algebra – p. 15 Systems of LPDE
P1(u) = 0 . . S ≡ . . (0.0.0) Pℓ(u) = 0 where Pi is a linear differential operator Pi ∈ An = An(C). Assume u(x) ∈ C[x] is a solution of S.
Then ( i QiPi)(u(x))=0 for all Qi ∈ An. P
The Weyl algebra – p. 15 Systems of LPDE
P1(u) = 0 . . S ≡ . . (0.0.0) Pℓ(u) = 0 where Pi is a linear differential operator Pi ∈ An = An(C). Assume u(x) ∈ C[x] is a solution of S.
Then ( i QiPi)(u(x))=0 for all Qi ∈ An. One saysP then that u(x) is a solution of the left ideal An(P1,...,Pℓ) generated by the Pi.
The Weyl algebra – p. 15 Systems LPDE → An–Modules
To the system S we associate the left An–module A M = M(S) = n An(P1,...,Pℓ)
The Weyl algebra – p. 16 Systems LPDE → An–Modules
To the system S we associate the left An–module A M = M(S) = n An(P1,...,Pℓ)
Example.- 1 variable x = x1. 2 d To the equation (x∂x + ∂x)(u) = 0 (here ∂x = dx) we associate the (left) A1–module A M = 1 . A1P
The Weyl algebra – p. 16 Systems LPDE → An–Modules
To the system S we associate the left An–module A M = M(S) = n An(P1,...,Pℓ)
Example.- 1 variable x = x1. 2 d To the equation (x∂x + ∂x)(u) = 0 (here ∂x = dx) we associate the (left) A1–module A M = 1 . A1P
The Weyl algebra – p. 16 Solutions
P1(u) = 0 . . S ≡ . . (0.0.0) Pℓ(u) = 0
The Weyl algebra – p. 17 Solutions
P1(u) = 0 . . S ≡ . . (0.0.0) Pℓ(u) = 0 Sol(S; C[x]) (vector space) polynomial solutions of the system S.
The Weyl algebra – p. 17 Solutions
P1(u) = 0 . . S ≡ . . (0.0.0) Pℓ(u) = 0 Sol(S; C[x]) (vector space) polynomial solutions of the system S. Any solution u ∈ C[x] of S induces a morphism
An ψu : M = −→ C[x] An(P1,...,Pℓ) given by ψu(P ) = P (u).
The Weyl algebra – p. 17 Solutions
Any morphism
A ψ : M = n −→ C[x] An(P1,...,Pℓ) induces a solution of S (just by considering uψ = ψ(1)).
The Weyl algebra – p. 18 Solutions
Any morphism
A ψ : M = n −→ C[x] An(P1,...,Pℓ) induces a solution of S (just by considering uψ = ψ(1)). Exercise.- The mappings
u ∈ Sol(S; C[x]) 7→ ψu ∈ HomAn (M, C[x]) and
ψ ∈ HomAn (M, C[x]) 7→ uψ ∈ Sol(S; C[x]) are both C–linear maps and moreover they are inverse to each other.
The Weyl algebra – p. 18 Solutions
Any morphism
A ψ : M = n −→ C[x] An(P1,...,Pℓ) induces a solution of S (just by considering uψ = ψ(1)). Exercise.- The mappings
u ∈ Sol(S; C[x]) 7→ ψu ∈ HomAn (M, C[x]) and
ψ ∈ HomAn (M, C[x]) 7→ uψ ∈ Sol(S; C[x]) are both C–linear maps and moreover they are inverse to each other.
The Weyl algebra – p. 18 Solutions
The example before justifies the following
The Weyl algebra – p. 19 Solutions
The example before justifies the following
Definition.- Let F be a left An–module and M be a finitely generated An–module. The solution space of M in F is the C–vector space
HomAn (M, F)
The Weyl algebra – p. 19 Solutions
The example before justifies the following
Definition.- Let F be a left An–module and M be a finitely generated An–module. The solution space of M in F is the C–vector space
HomAn (M, F)
There are several algorithms computing a basis of the vector space HomAn (An/I, C[x]) [for An/I holonomic]. These algorithms use Groebner basis in the ring An. T. Oaku-N. Takayama (2000) and H. Tsai-U. Walther (2000)
The Weyl algebra – p. 19 Order and total order of a LDO
n Denote |β| = i βi = β1 + · · · + βn for β ∈ N . P
The Weyl algebra – p. 20 Order and total order of a LDO
n Denote |β| = i βi = β1 + · · · + βn for β ∈ N . Definition.- P[order and total order]
α β β P = pαβx ∂ = pβ(x)∂ ∈ An Xαβ Xβ
ord(P ) = max{|β| s.t. pβ(x) =6 0}
The Weyl algebra – p. 20 Order and total order of a LDO
n Denote |β| = i βi = β1 + · · · + βn for β ∈ N . Definition.- P[order and total order]
α β β P = pαβx ∂ = pβ(x)∂ ∈ An Xαβ Xβ
ord(P ) = max{|β| s.t. pβ(x) =6 0}
T ord (P ) = max{|α| + |β| s.t. pαβ =6 0} Convention: ord(0) = ordT (0) = −∞.
The Weyl algebra – p. 20 Order and total order of a LDO
Example.- (n=2)
2 3 3 2 2 2 P = x1x2∂1 − 5x2∂1∂2 + ∂1∂2 − 1
The Weyl algebra – p. 21 Order and total order of a LDO
Example.- (n=2)
2 3 3 2 2 2 P = x1x2∂1 − 5x2∂1∂2 + ∂1∂2 − 1
ord(P ) = 4
The Weyl algebra – p. 21 Order and total order of a LDO
Example.- (n=2)
2 3 3 2 2 2 P = x1x2∂1 − 5x2∂1∂2 + ∂1∂2 − 1
ord(P ) = 4
ordT (P ) = 6
The Weyl algebra – p. 21 Symbols
Definition.- [principal symbol]
α β β P = pαβx ∂ = pβ(x)∂ Xαβ Xβ β σ(P ) = pβ(x)ξ ∈ C[x][ξ] = C[x][ξ1,...,ξn]. |β|=ord(X P )
The Weyl algebra – p. 22 Symbols
Definition.- [principal symbol]
α β β P = pαβx ∂ = pβ(x)∂ Xαβ Xβ β σ(P ) = pβ(x)ξ ∈ C[x][ξ] = C[x][ξ1,...,ξn]. |β|=ord(X P )
Definition.- [principal total symbol]
T α β σ (P ) = pαβx ξ ∈ C[x][ξ]. T |α+β|X=ord (P )
The Weyl algebra – p. 22 Symbols
Definition.- [principal symbol]
α β β P = pαβx ∂ = pβ(x)∂ Xαβ Xβ β σ(P ) = pβ(x)ξ ∈ C[x][ξ] = C[x][ξ1,...,ξn]. |β|=ord(X P )
Definition.- [principal total symbol]
T α β σ (P ) = pαβx ξ ∈ C[x][ξ]. T |α+β|X=ord (P )
The Weyl algebra – p. 22 Symbols
Example.- (n=2)
2 3 3 2 2 2 P = x1x2∂1 + x2∂1∂2 + ∂1∂2
The Weyl algebra – p. 23 Symbols
Example.- (n=2)
2 3 3 2 2 2 P = x1x2∂1 + x2∂1∂2 + ∂1∂2
2 2 σ(P ) = ξ1ξ2
The Weyl algebra – p. 23 Symbols
Example.- (n=2)
2 3 3 2 2 2 P = x1x2∂1 + x2∂1∂2 + ∂1∂2
2 2 σ(P ) = ξ1ξ2
T 2 3 3 2 σ (P ) = x1x2ξ1 + x2ξ1ξ2
The Weyl algebra – p. 23 Symbols
σ(P ) homogeneous polynomial of degree ord(P ) in the ξ-variables
The Weyl algebra – p. 24 Symbols
σ(P ) homogeneous polynomial of degree ord(P ) in the ξ-variables σT (P ) homogeneous polynomial of degree ordT (P ) in the variables (x, ξ).
The Weyl algebra – p. 24 Symbols
σ(P ) homogeneous polynomial of degree ord(P ) in the ξ-variables σT (P ) homogeneous polynomial of degree ordT (P ) in the variables (x, ξ). In general σ(P ) =6 σT (P ). In general we do not have neither σ(P + Q) = σ(P ) + σ(Q) nor σT (P + Q) = σT (P ) + σT (Q).
The Weyl algebra – p. 24 Symbols
σ(P ) homogeneous polynomial of degree ord(P ) in the ξ-variables σT (P ) homogeneous polynomial of degree ordT (P ) in the variables (x, ξ). In general σ(P ) =6 σT (P ). In general we do not have neither σ(P + Q) = σ(P ) + σ(Q) nor σT (P + Q) = σT (P ) + σT (Q). e.g. P = ∂1 + x1 + 1, Q = −∂1 − x1. T •)σ(P ) = ξ1 =6 ξ1 + x1 = σ (P ). Moreover •)1 = σ(P + Q) =6 0 = σ(P ) + σ(Q) •)1 = σT (P + Q) =6 0 = σT (P ) + σT (Q).
The Weyl algebra – p. 24 Properties of σ and σT
Exercise.- For P, Q ∈ An one has • ord(P Q) = ord(P ) + ord(Q) and σ(P Q) = σ(P )σ(Q).
The Weyl algebra – p. 25 Properties of σ and σT
Exercise.- For P, Q ∈ An one has • ord(P Q) = ord(P ) + ord(Q) and σ(P Q) = σ(P )σ(Q). • ordT (P Q) = ordT (P ) + ordT (Q) and σT (P Q) = σT (P )σT (Q).
The Weyl algebra – p. 25 Properties of σ and σT
Exercise.- For P, Q ∈ An one has • ord(P Q) = ord(P ) + ord(Q) and σ(P Q) = σ(P )σ(Q). • ordT (P Q) = ordT (P ) + ordT (Q) and σT (P Q) = σT (P )σT (Q). • ord(P Q − QP ) ≤ ord(P ) + ord(Q) − 1 and ordT (P Q − QP ) ≤ ordT (P ) + ordT (Q) − 2.
The Weyl algebra – p. 25 Properties of σ and σT
Exercise.- For P, Q ∈ An one has • ord(P Q) = ord(P ) + ord(Q) and σ(P Q) = σ(P )σ(Q). • ordT (P Q) = ordT (P ) + ordT (Q) and σT (P Q) = σT (P )σT (Q). • ord(P Q − QP ) ≤ ord(P ) + ord(Q) − 1 and ordT (P Q − QP ) ≤ ordT (P ) + ordT (Q) − 2. • ord(P + Q) ≤ max{ord(P ), ord(Q)} (and similarly for ordT ). If ord(P ) = ord(Q) and σ(P ) + σ(Q) =6 0 then σ(P + Q) = σ(P ) + σ(Q) (and similarly for ordT and σT ).
The Weyl algebra – p. 25 An is an integral domain.
Proposition.- An is an integral domain.
The Weyl algebra – p. 26 An is an integral domain.
Proposition.- An is an integral domain.
Proof.- Assume P, Q ∈ An are both non-zero. Then ord(P Q) = ord(P ) + ord(Q) ≥ 0. Then P Q is non-zero.
The Weyl algebra – p. 26 An is an integral domain.
Proposition.- An is an integral domain.
Proof.- Assume P, Q ∈ An are both non-zero. Then ord(P Q) = ord(P ) + ord(Q) ≥ 0. Then P Q is non-zero.
The Weyl algebra – p. 26 Graded ideals
Definition.- For each left (or right) ideal I ⊂ An we call the graded ideal associated with I
gr(I) = C[x][ξ]{σ(P ) | P ∈ I}.
The Weyl algebra – p. 27 Graded ideals
Definition.- For each left (or right) ideal I ⊂ An we call the graded ideal associated with I
gr(I) = C[x][ξ]{σ(P ) | P ∈ I}.
Definition.- We call the total graded ideal associated with I the ideal grT (I) of C[x][ξ] generated by the family of principal total symbols of elements in I
grT (I) = C[x][ξ]{σT (P ) | P ∈ I}.
The Weyl algebra – p. 27 Graded ideals
Definition.- For each left (or right) ideal I ⊂ An we call the graded ideal associated with I
gr(I) = C[x][ξ]{σ(P ) | P ∈ I}.
Definition.- We call the total graded ideal associated with I the ideal grT (I) of C[x][ξ] generated by the family of principal total symbols of elements in I
grT (I) = C[x][ξ]{σT (P ) | P ∈ I}.
Both gr(I) and grT (I) are polynomial ideals in 2n variables
The Weyl algebra – p. 27 Graded ideals
I = AnP a principal left ideal.
The Weyl algebra – p. 28 Graded ideals
I = AnP a principal left ideal. Exercise.- gr(I) = C[x, ξ]σ(P )
The Weyl algebra – p. 28 Graded ideals
I = AnP a principal left ideal. Exercise.- gr(I) = C[x, ξ]σ(P ) Remember gr(I) = C[x, ξ]{σ(Q) | Q ∈ I} Since gr(I) is an ideal and P ∈ I we have C[x, ξ]σ(P ) ⊂ gr(I).
The Weyl algebra – p. 28 Graded ideals
I = AnP a principal left ideal. Exercise.- gr(I) = C[x, ξ]σ(P ) Remember gr(I) = C[x, ξ]{σ(Q) | Q ∈ I} Since gr(I) is an ideal and P ∈ I we have C[x, ξ]σ(P ) ⊂ gr(I).
If Q ∈ I then Q = RP for some R ∈ An. Then σ(Q) = σ(R)σ(P ) ∈ C[x, ξ]σ(P ).
The Weyl algebra – p. 28 Graded ideals
I = AnP a principal left ideal. Exercise.- gr(I) = C[x, ξ]σ(P ) Remember gr(I) = C[x, ξ]{σ(Q) | Q ∈ I} Since gr(I) is an ideal and P ∈ I we have C[x, ξ]σ(P ) ⊂ gr(I).
If Q ∈ I then Q = RP for some R ∈ An. Then σ(Q) = σ(R)σ(P ) ∈ C[x, ξ]σ(P ). Similarly grT (I) = C[x, ξ]σT (P ).
The Weyl algebra – p. 28 Graded ideals
I = AnP a principal left ideal. Exercise.- gr(I) = C[x, ξ]σ(P ) Remember gr(I) = C[x, ξ]{σ(Q) | Q ∈ I} Since gr(I) is an ideal and P ∈ I we have C[x, ξ]σ(P ) ⊂ gr(I).
If Q ∈ I then Q = RP for some R ∈ An. Then σ(Q) = σ(R)σ(P ) ∈ C[x, ξ]σ(P ). Similarly grT (I) = C[x, ξ]σT (P ).
I = An(P1,...,Pℓ). In general C[x, ξ](σ(P1),...,σ(Pℓ)) gr(I)
The Weyl algebra – p. 28 Graded ideals
Problem.- Input: P1,...,Pℓ ∈ An
The Weyl algebra – p. 29 Graded ideals
Problem.- Input: P1,...,Pℓ ∈ An Output.- A finite system of generators of gr(I) (similarly for grT (I))
The Weyl algebra – p. 29 Graded ideals
Problem.- Input: P1,...,Pℓ ∈ An Output.- A finite system of generators of gr(I) (similarly for grT (I)) If ℓ = 1 then Problem is solved (previous exercise). The general case can be solved using Groebner basis theory in An. (see, Castro F. Théorème de division pour les opérateurs différentiels et calcul des multiplicités, Thèse de 3eme cycle, Univ. of Paris VII, (1984); see also Saito M., Sturmfels B. and Takayama N., Gröbner deformations of hypergeometric differential equations. Algorithms and Computation in Mathematics, 6. Springer-Verlag, Berlin, (2000).)
The Weyl algebra – p. 29 An is Noetherian
The Weyl algebra – p. 30 An is Noetherian
Remember: A ring R is left (resp. right) Noetherian if any left (resp. right) ideal of R is finitely generated. A ring R is Noetherian if it is left and right Noetherian.
The Weyl algebra – p. 30 An is Noetherian
Remember: A ring R is left (resp. right) Noetherian if any left (resp. right) ideal of R is finitely generated. A ring R is Noetherian if it is left and right Noetherian.
Hilbert's basis Theorem: Let K be a field. K[y1,...,ym] is Noetherian (for all m ≥ 0).
The Weyl algebra – p. 30 An is Noetherian
Remember: A ring R is left (resp. right) Noetherian if any left (resp. right) ideal of R is finitely generated. A ring R is Noetherian if it is left and right Noetherian.
Hilbert's basis Theorem: Let K be a field. K[y1,...,ym] is Noetherian (for all m ≥ 0).
Proposition.- An is left and right Noetherian.
The Weyl algebra – p. 30 An is a simple ring
An is a simple ring (i.e. the only two-sided ideals in An are (0) and An)
The Weyl algebra – p. 31 An is a simple ring
An is a simple ring (i.e. the only two-sided ideals in An are (0) and An)
The Weyl algebra – p. 31 An is a simple ring
An is a simple ring (i.e. the only two-sided ideals in An are (0) and An)
The Weyl algebra – p. 31 Filtrations on An
[Order filtration]
Fk(An) = {P ∈ An | ord(P ) ≤ k} for k ∈ Z. We will simply write Fk = Fk(An)
The Weyl algebra – p. 32 Filtrations on An
[Order filtration]
Fk(An) = {P ∈ An | ord(P ) ≤ k} for k ∈ Z. We will simply write Fk = Fk(An)
1. Fk = {0} for k ≤−1.
2. Fk ⊂ Fk+1 for k ∈ Z.
3. FkFℓ ⊂ Fk+ℓ for k,ℓ ∈ Z.
4. An = k Fk. 5. 1 ∈ F0S= C[x]. n + k 6. Each Fk is a free C[x]-module with rank . k !
The Weyl algebra – p. 32 Filtrations on An
[Order filtration]
Fk(An) = {P ∈ An | ord(P ) ≤ k} for k ∈ Z. We will simply write Fk = Fk(An)
1. Fk = {0} for k ≤−1.
2. Fk ⊂ Fk+1 for k ∈ Z.
3. FkFℓ ⊂ Fk+ℓ for k,ℓ ∈ Z.
4. An = k Fk. 5. 1 ∈ F0S= C[x]. n + k 6. Each Fk is a free C[x]-module with rank . k !
The Weyl algebra – p. 32 Filtrations on An
[Total order filtration]
T Bk(An) = {P ∈ An | ord (P ) ≤ k} for k ∈ Z. We will simply write Bk = Bk(An).
The Weyl algebra – p. 33 Filtrations on An
[Total order filtration]
T Bk(An) = {P ∈ An | ord (P ) ≤ k} for k ∈ Z. We will simply write Bk = Bk(An).
1. Bk = {0} for k ≤−1.
2. Bk ⊂ Bk+1 for k ∈ Z.
3. BkBℓ ⊂ Bk+ℓ for k,ℓ ∈ Z.
4. An = k Bk. 5. 1 ∈ B0S= C. 2n + k 6. dimC(Bk) = . k !
The Weyl algebra – p. 33 Filtrations on An
[Total order filtration]
T Bk(An) = {P ∈ An | ord (P ) ≤ k} for k ∈ Z. We will simply write Bk = Bk(An).
1. Bk = {0} for k ≤−1.
2. Bk ⊂ Bk+1 for k ∈ Z.
3. BkBℓ ⊂ Bk+ℓ for k,ℓ ∈ Z.
4. An = k Bk. 5. 1 ∈ B0S= C. 2n + k 6. dimC(Bk) = . k !
The Weyl algebra – p. 33 Classical characteristic vectors
β P (u)=( β pβ(x)∂ )(u) = 0 with pβ(x) ∈ R[x] P
The Weyl algebra – p. 34 Classical characteristic vectors
β P (u)=( β pβ(x)∂ )(u) = 0 with pβ(x) ∈ R[x] n n ξ0 ∈ R isPcharacteristic for P at x0 ∈ R if σ(P )(x0, ξ0) = 0.
The Weyl algebra – p. 34 Classical characteristic vectors
β P (u)=( β pβ(x)∂ )(u) = 0 with pβ(x) ∈ R[x] n n ξ0 ∈ R isPcharacteristic for P at x0 ∈ R if σ(P )(x0, ξ0) = 0. The classical characteristic variety of the operator P is by definition the set
n n Char(P ) = {(x0, ξ0) ∈ R × R | σ(P )(x0, ξ0) = 0}.
The Weyl algebra – p. 34 Classical characteristic vectors
β P (u)=( β pβ(x)∂ )(u) = 0 with pβ(x) ∈ R[x] n n ξ0 ∈ R isPcharacteristic for P at x0 ∈ R if σ(P )(x0, ξ0) = 0. The classical characteristic variety of the operator P is by definition the set
n n Char(P ) = {(x0, ξ0) ∈ R × R | σ(P )(x0, ξ0) = 0}.
Assume ord(P ) ≥ 1. P is elliptic if Char(P ) ⊂ Rn × {0}.
The Weyl algebra – p. 34 Classical characteristic vectors
β P (u)=( β pβ(x)∂ )(u) = 0 with pβ(x) ∈ R[x] n n ξ0 ∈ R isPcharacteristic for P at x0 ∈ R if σ(P )(x0, ξ0) = 0. The classical characteristic variety of the operator P is by definition the set
n n Char(P ) = {(x0, ξ0) ∈ R × R | σ(P )(x0, ξ0) = 0}.
Assume ord(P ) ≥ 1. P is elliptic if Char(P ) ⊂ Rn × {0}. 2 2 Laplace operator ∂1 + ... + ∂n is elliptic.
The Weyl algebra – p. 34 Classical characteristic vectors
β P (u)=( β pβ(x)∂ )(u) = 0 with pβ(x) ∈ R[x] n n ξ0 ∈ R isPcharacteristic for P at x0 ∈ R if σ(P )(x0, ξ0) = 0. The classical characteristic variety of the operator P is by definition the set
n n Char(P ) = {(x0, ξ0) ∈ R × R | σ(P )(x0, ξ0) = 0}.
Assume ord(P ) ≥ 1. P is elliptic if Char(P ) ⊂ Rn × {0}. 2 2 Laplace operator ∂1 + ... + ∂n is elliptic. 2 2 2 Wave operator ∂1 − ∂2 − ... − ∂n is not elliptic.
The Weyl algebra – p. 34 Classical characteristic vectors
The classical characteristic variety of the operator P gives us some information about the structure of the solution space of the equation P (u) = 0 nearby a given point in the space.
The Weyl algebra – p. 35 Classical characteristic vectors
The classical characteristic variety of the operator P gives us some information about the structure of the solution space of the equation P (u) = 0 nearby a given point in the space. e.g.: P = x∂ + 1 ∈ A1(R) (x = x1 is just one variable)
The Weyl algebra – p. 35 Classical characteristic vectors
The classical characteristic variety of the operator P gives us some information about the structure of the solution space of the equation P (u) = 0 nearby a given point in the space. e.g.: P = x∂ + 1 ∈ A1(R) (x = x1 is just one variable) V := Char(P ) = {(a, b) ∈ R × R | σ(P )(a, b) = ab = 0}. This (affine) variety V is the union of the two coordinates lines in R2.
The Weyl algebra – p. 35 Classical characteristic vectors
The classical characteristic variety of the operator P gives us some information about the structure of the solution space of the equation P (u) = 0 nearby a given point in the space. e.g.: P = x∂ + 1 ∈ A1(R) (x = x1 is just one variable) V := Char(P ) = {(a, b) ∈ R × R | σ(P )(a, b) = ab = 0}. This (affine) variety V is the union of the two coordinates lines in R2. Around any non-zero x = a ∈ R, the equation x∂(u) + u = 0 has a non-zero analytic solution u(x) = 1/x. The same equation has no analytic solution around x = 0 (and (0, 0) is the intersection of the two components of V ).
The Weyl algebra – p. 35 Characteristic variety
To define the analog to the classical characteristic variety for a general system of LPDE
The Weyl algebra – p. 36 Characteristic variety
To define the analog to the classical characteristic variety for a general system of LPDE
P11(u1) + · · · + P1m(um) = 0 . . . . S . . . . Pℓ1(u1) + · · · + Pℓm(um) = 0
The Weyl algebra – p. 36 Characteristic variety
To define the analog to the classical characteristic variety for a general system of LPDE
P11(u1) + · · · + P1m(um) = 0 . . . . S . . . . Pℓ1(u1) + · · · + Pℓm(um) = 0 is more involved and in general the naive approach of simply considering the principal symbols of the equations turns out to be unsatisfactory. We will use graded ideals and graded modules
The Weyl algebra – p. 36 Characteristic variety
To the system S we associate the An–module Am M(S) = n An(P1,...,Pℓ)
The Weyl algebra – p. 37 Characteristic variety
To the system S we associate the An–module Am M(S) = n An(P1,...,Pℓ)
where Pi =(Pi1,...,Pim) and An(P1,...,Pℓ) denote the m submodule of An generated by P1,...,Pℓ. If m = 1 (only one unknown) M(S) = An/An(P1,...,Pℓ).
The Weyl algebra – p. 37 Characteristic variety
To the system S we associate the An–module Am M(S) = n An(P1,...,Pℓ)
where Pi =(Pi1,...,Pim) and An(P1,...,Pℓ) denote the m submodule of An generated by P1,...,Pℓ. If m = 1 (only one unknown) M(S) = An/An(P1,...,Pℓ).
Definition.- I ⊂ An left ideal. The characteristic variety of the left An–module An/I is defined as
2n Char(An/I) = VC(gr(I)) ⊂ C .
The Weyl algebra – p. 37 Characteristic variety
2n VC(gr(I)) ⊂ C .
The Weyl algebra – p. 38 Characteristic variety
2n VC(gr(I)) ⊂ C .
2n VC(gr(I)) = {(x0, ξ0) ∈ C | σ(P )(x0, ξ0) = 0, ∀P ∈ I}.
The Weyl algebra – p. 38 Characteristic variety
2n VC(gr(I)) ⊂ C .
2n VC(gr(I)) = {(x0, ξ0) ∈ C | σ(P )(x0, ξ0) = 0, ∀P ∈ I}.
The Weyl algebra – p. 38 Characteristic variety
Example.- I = AnP principal ideal.
The Weyl algebra – p. 39 Characteristic variety
Example.- I = AnP principal ideal.
gr(I) = C[x, ξ]σ(P )
Char(An/I) = VC(gr(I)) = VC(σ(P )) = Char(P )
The Weyl algebra – p. 39 Characteristic variety
Example.- I = AnP principal ideal.
gr(I) = C[x, ξ]σ(P )
Char(An/I) = VC(gr(I)) = VC(σ(P )) = Char(P )
When An(P1,...,Pℓ) is not principal (or we have more than 1 unknowns the computation is more involved. We need filtrations on An-modules.
The Weyl algebra – p. 39 Graded rings
B The graded ring gr (An) Exercise.- Bk/Bk−1 is a C–vector space with dimension 2n + k − 1 . 2n − 1 !
The Weyl algebra – p. 40 Graded rings
B The graded ring gr (An) Exercise.- Bk/Bk−1 is a C–vector space with dimension 2n + k − 1 . 2n − 1 ! Quick answer.- The residue classes
α β α β x ∂ = x ∂ + Bk−1 with |α + β| = k generate the quotient vector space Bk/Bk−1 and, moreover, they are linearly independent since a linear combination
α β λαβx ∂ |α+Xβ|=k belongs to Bk−1 if and only if all λαβ are 0. The Weyl algebra – p. 40 Graded rings
B The graded ring gr (An) Exercise.- Bk/Bk−1 is a C–vector space with dimension 2n + k − 1 . 2n − 1 ! Quick answer.- The residue classes
α β α β x ∂ = x ∂ + Bk−1 with |α + β| = k generate the quotient vector space Bk/Bk−1 and, moreover, they are linearly independent since a linear combination
α β λαβx ∂ |α+Xβ|=k belongs to Bk−1 if and only if all λαβ are 0. The Weyl algebra – p. 40 Graded rings
Proposition.- The Abelian group
B Bk gr (An) := Z Bk−1 Mk∈ has a natural structure of commutative ring with unit.
The Weyl algebra – p. 41 Graded rings
Proposition.- The Abelian group
B Bk gr (An) := Z Bk−1 Mk∈ has a natural structure of commutative ring with unit. Proof.- k,ℓ ∈ Z,
Bk Bℓ Bk+ℓ µkℓ : × → Bk−1 Bℓ−1 Bℓ+k−1 defined by
µkℓ(P, Q) = P Q for P ∈ Bk, Q ∈ Bℓ.
The Weyl algebra – p. 41 Graded rings
Proposition.- The Abelian group
B Bk gr (An) := Z Bk−1 Mk∈ has a natural structure of commutative ring with unit. Proof.- k,ℓ ∈ Z,
Bk Bℓ Bk+ℓ µkℓ : × → Bk−1 Bℓ−1 Bℓ+k−1 defined by
µkℓ(P, Q) = P Q for P ∈ Bk, Q ∈ Bℓ.
The Weyl algebra – p. 41 Graded rings
Proof continued.- The map µkℓ is well defined. Since P Q − QP ∈ Bk+ℓ−1 we have P Q = QP .
The Weyl algebra – p. 42 Graded rings
Proof continued.- The map µkℓ is well defined. Since P Q − QP ∈ Bk+ℓ−1 we have P Q = QP . We define
′ B B B µ : gr (An) × gr (An) → gr (An) by bilinearity:
′ µ ( Pk, Qℓ) = PkQℓ Xk Xℓ Xk,ℓ where Pk ∈ Bk and Qℓ ∈ Bℓ for all k,ℓ.
The Weyl algebra – p. 42 Graded rings
Proof continued.- The map µkℓ is well defined. Since P Q − QP ∈ Bk+ℓ−1 we have P Q = QP . We define
′ B B B µ : gr (An) × gr (An) → gr (An) by bilinearity:
′ µ ( Pk, Qℓ) = PkQℓ Xk Xℓ Xk,ℓ where Pk ∈ Bk and Qℓ ∈ Bℓ for all k,ℓ. ′ B µ defines a product on gr (An)
The Weyl algebra – p. 42 Graded rings
B Proposition.- The graded ring gr (An) is isomorphic to the polynomial ring C[x, ξ] := C[x1,...,xn, ξ1,...,ξn] endowed with the natural grading.
The Weyl algebra – p. 43 Graded rings
B Proposition.- The graded ring gr (An) is isomorphic to the polynomial ring C[x, ξ] := C[x1,...,xn, ξ1,...,ξn] endowed with the natural grading. Proof.- k ∈ N,
ηk : Bk/Bk−1 → C[x, ξ]k defined
η p xα∂β + B = p xαξβ. k αβ k−1 αβ |α+Xβ|≤k |α+Xβ|=k
The Weyl algebra – p. 43 Graded rings
B Proposition.- The graded ring gr (An) is isomorphic to the polynomial ring C[x, ξ] := C[x1,...,xn, ξ1,...,ξn] endowed with the natural grading. Proof.- k ∈ N,
ηk : Bk/Bk−1 → C[x, ξ]k defined
η p xα∂β + B = p xαξβ. k αβ k−1 αβ |α+Xβ|≤k |α+Xβ|=k
The Weyl algebra – p. 43 Graded rings
Proof continued.- C[x, ξ]k homogeneous polynomials of degree k.
The Weyl algebra – p. 44 Graded rings
Proof continued.- C[x, ξ]k homogeneous polynomials of degree k.
The family ηk yields by bilinearity a natural isomorphism
B η : gr (An) → C[x, ξ] of graded rings.
The Weyl algebra – p. 44 Graded rings
Proof continued.- C[x, ξ]k homogeneous polynomials of degree k.
The family ηk yields by bilinearity a natural isomorphism
B η : gr (An) → C[x, ξ] of graded rings.
The Weyl algebra – p. 44 Graded rings
We can prove similar results for
F Fk gr (An) := Z Fk−1 Mk∈
The Weyl algebra – p. 45 Graded rings
We can prove similar results for
F Fk gr (An) := Z Fk−1 Mk∈
F gr (An) has a natural structure of commutative ring with unit.
The Weyl algebra – p. 45 Graded rings
We can prove similar results for
F Fk gr (An) := Z Fk−1 Mk∈
F gr (An) has a natural structure of commutative ring with unit. F gr (An) is isomorphic to the polynomial ring C[x, ξ] endowed with the ξ–grading.
The Weyl algebra – p. 45 Filtrations on An–modules
An F –filtration on a An–module M is a family Γ=(Mk)k∈N of finitely generated C[x]–submodules of M such that:
The Weyl algebra – p. 46 Filtrations on An–modules
An F –filtration on a An–module M is a family Γ=(Mk)k∈N of finitely generated C[x]–submodules of M such that:
(i) Mk ⊂ Mk+1 for all k ∈ N.
The Weyl algebra – p. 46 Filtrations on An–modules
An F –filtration on a An–module M is a family Γ=(Mk)k∈N of finitely generated C[x]–submodules of M such that:
(i) Mk ⊂ Mk+1 for all k ∈ N.
(ii) k Mk = M. S
The Weyl algebra – p. 46 Filtrations on An–modules
An F –filtration on a An–module M is a family Γ=(Mk)k∈N of finitely generated C[x]–submodules of M such that:
(i) Mk ⊂ Mk+1 for all k ∈ N.
(ii) k Mk = M. (iii)SFkMℓ ⊂ Mk+ℓ for all (k,ℓ).
The Weyl algebra – p. 46 Filtrations on An–modules
1.- (Fk(An))k is an F –filtration on An.
The Weyl algebra – p. 47 Filtrations on An–modules
1.- (Fk(An))k is an F –filtration on An. 2.- k ∈ N, Fk(C[x]) = C[x] for k ≥ 0. The family (Fk(C[x]))k is an F –filtration on C[x].
The Weyl algebra – p. 47 Filtrations on An–modules
1.- (Fk(An))k is an F –filtration on An. 2.- k ∈ N, Fk(C[x]) = C[x] for k ≥ 0. The family (Fk(C[x]))k is an F –filtration on C[x].
3.- Let I ⊂ An be an ideal and denote Fk(I) = Fk(An) ∩ I for k ∈ N. The family (Fk(I))k is an F –filtration on I.
The Weyl algebra – p. 47 Filtrations on An–modules
1.- (Fk(An))k is an F –filtration on An. 2.- k ∈ N, Fk(C[x]) = C[x] for k ≥ 0. The family (Fk(C[x]))k is an F –filtration on C[x].
3.- Let I ⊂ An be an ideal and denote Fk(I) = Fk(An) ∩ I for k ∈ N. The family (Fk(I))k is an F –filtration on I. 4.- Let I ⊂ An be an ideal and denote A F (A ) + I F n = k n k I I for k ∈ N. The family (Fk(An/I))k is an F –filtration on An/I. It will be called the induced F –filtration on An/I
The Weyl algebra – p. 47 Graded modules
M, an An–module. Γ=(Mk)k an F –filtration on M.
The Weyl algebra – p. 48 Graded modules
M, an An–module. Γ=(Mk)k an F –filtration on M.
M grΓ(M) := k Mk−1 Mk is also an Abelian group (and a vector space).
The Weyl algebra – p. 48 Graded modules
M, an An–module. Γ=(Mk)k an F –filtration on M.
M grΓ(M) := k Mk−1 Mk is also an Abelian group (and a vector space). Γ An element in gr (M) is a finite sum k mk where each mk belongs to Mk. P
The Weyl algebra – p. 48 Graded modules
Proposition.- The Abelian group grΓ(M) has a natural F structure of gr (An)–module. [The graded module associated with Γ on M]
The Weyl algebra – p. 49 Graded modules
Proposition.- The Abelian group grΓ(M) has a natural F structure of gr (An)–module. [The graded module associated with Γ on M] F Γ Γ Proof.- ν : gr (An) × gr (M) → gr (M) defined by bilinearity from the maps
B M M k × ℓ → k+ℓ Bk−1 Mℓ−1 Mk+ℓ−1 defined by
Pk mℓ = Pkmℓ.
The Weyl algebra – p. 49 Graded modules
Proposition.- The Abelian group grΓ(M) has a natural F structure of gr (An)–module. [The graded module associated with Γ on M] F Γ Γ Proof.- ν : gr (An) × gr (M) → gr (M) defined by bilinearity from the maps
B M M k × ℓ → k+ℓ Bk−1 Mℓ−1 Mk+ℓ−1 defined by
Pk mℓ = Pkmℓ.
F Γ ν defines a natural action of gr (An) on gr (M).
The Weyl algebra – p. 49 grΓ(M) f.g. ⇒ M f.g.
Proposition.- M an An–module. Γ=(Mk)k a filtration on M. If grΓ(M) if finitely generated over C[x, ξ] then M is finitely generated.
The Weyl algebra – p. 50 grΓ(M) f.g. ⇒ M f.g.
Proposition.- M an An–module. Γ=(Mk)k a filtration on M. If grΓ(M) if finitely generated over C[x, ξ] then M is finitely generated.
Idea of the Proof.- Assume {m1,... mℓ} is a homogeneous generating system of grΓ(M). Then (using induction on the Γ–order of the elements in M one proves that) M is generated by {m1,...,mℓ}
The Weyl algebra – p. 50 grΓ(M) f.g. ⇒ M f.g.
Proposition.- M an An–module. Γ=(Mk)k a filtration on M. If grΓ(M) if finitely generated over C[x, ξ] then M is finitely generated.
Idea of the Proof.- Assume {m1,... mℓ} is a homogeneous generating system of grΓ(M). Then (using induction on the Γ–order of the elements in M one proves that) M is generated by {m1,...,mℓ}
Definition.- A filtration Γ=(Mk)k on M is said to be good if grΓ(M) is finitely generated over C[x, ξ].
The Weyl algebra – p. 50 Good filtrations (I)
Exercise.- I ⊂ An left ideal. Prove that: (i) The induced F –filtration on I is a good filtration. (ii) The induced F –filtration on An/I is a good filtration.
The Weyl algebra – p. 51 Good filtrations (I)
Exercise.- I ⊂ An left ideal. Prove that: (i) The induced F –filtration on I is a good filtration. (ii) The induced F –filtration on An/I is a good filtration.
(iii) Any finitely generated An–module M admits a good F –filtration. (Hint for (iii). If m1,...,mr is a generating system for M, define Mk := j Fkmj for k ∈ N. The family (Mk)k is a good filtration on M) P
The Weyl algebra – p. 51 Good filtrations (I)
Exercise.- I ⊂ An left ideal. Prove that: (i) The induced F –filtration on I is a good filtration. (ii) The induced F –filtration on An/I is a good filtration.
(iii) Any finitely generated An–module M admits a good F –filtration. (Hint for (iii). If m1,...,mr is a generating system for M, define Mk := j Fkmj for k ∈ N. The family (Mk)k is a good filtration on M) P
The Weyl algebra – p. 51 Good filtrations (II)
Proposition.- M a (left) An–module. Γ=(Mk)k a F –filtration on M. The following conditions are equivalents
The Weyl algebra – p. 52 Good filtrations (II)
Proposition.- M a (left) An–module. Γ=(Mk)k a F –filtration on M. The following conditions are equivalents i) Γ is a good F –filtration on M.
The Weyl algebra – p. 52 Good filtrations (II)
Proposition.- M a (left) An–module. Γ=(Mk)k a F –filtration on M. The following conditions are equivalents i) Γ is a good F –filtration on M. ii) There exists k0 ∈ N s.t. Mk+ℓ = FℓMk for all ℓ and all k ≥ k0.
The Weyl algebra – p. 52 Good filtrations (III)
Proposition.- M a (left) An–module. ′ ′ Γ=(Mk)k, and Γ =(Mk)k two F –filtrations on M. Then we have:
The Weyl algebra – p. 53 Good filtrations (III)
Proposition.- M a (left) An–module. ′ ′ Γ=(Mk)k, and Γ =(Mk)k two F –filtrations on M. Then we have: i) If Γ is a good F –filtration then there exists k1 ∈ N s.t.
′ Mk ⊂ Mk+k1 for all k ∈ N.
The Weyl algebra – p. 53 Good filtrations (III)
Proposition.- M a (left) An–module. ′ ′ Γ=(Mk)k, and Γ =(Mk)k two F –filtrations on M. Then we have: i) If Γ is a good F –filtration then there exists k1 ∈ N s.t.
′ Mk ⊂ Mk+k1 for all k ∈ N. ii) If Γ and Γ′ are both good filtrations then there exists k2 ∈ N s. t. ′ ′ Mk−k2 ⊂ Mk ⊂ Mk+k2 for all k ∈ N.
The Weyl algebra – p. 53 Good filtrations (IV)
Proposition.- M a finitely generated An–module. ′ ′ Γ=(Mk)k, and Γ =(Mk)k two good F –filtrations on M. Then we have:
The Weyl algebra – p. 54 Good filtrations (IV)
Proposition.- M a finitely generated An–module. ′ ′ Γ=(Mk)k, and Γ =(Mk)k two good F –filtrations on M. Then we have:
′ F Γ F Γ Anngr (An)(gr (M)) = Anngr (An)(gr (M)) q q
The Weyl algebra – p. 54 Good filtrations (IV)
Proposition.- M a finitely generated An–module. ′ ′ Γ=(Mk)k, and Γ =(Mk)k two good F –filtrations on M. Then we have:
′ F Γ F Γ Anngr (An)(gr (M)) = Anngr (An)(gr (M)) q q
The Weyl algebra – p. 54 Characteristic variety
Definition.- M finitely generated An–module. The characteristic variety of M is defined as
Γ F Char(M) = V(Anngr (An)(gr (M))) for a good F –filtration Γ on M.
The Weyl algebra – p. 55 Characteristic variety
Definition.- M finitely generated An–module. The characteristic variety of M is defined as
Γ F Char(M) = V(Anngr (An)(gr (M))) for a good F –filtration Γ on M. The definition is independent of the good F –filtration.
The Weyl algebra – p. 55 Characteristic variety
Definition.- M finitely generated An–module. The characteristic variety of M is defined as
Γ F Char(M) = V(Anngr (An)(gr (M))) for a good F –filtration Γ on M. The definition is independent of the good F –filtration. This definition coincides with the previous one when M = An/I.
The Weyl algebra – p. 55 Characteristic variety
Definition.- M finitely generated An–module. The characteristic variety of M is defined as
Γ F Char(M) = V(Anngr (An)(gr (M))) for a good F –filtration Γ on M. The definition is independent of the good F –filtration. This definition coincides with the previous one when M = An/I. Characteristic varieties are not easy to compute. There are algorithms computing characteristic varieties that use Groebner basis in the Weyl algebra An. These algorithms have been implemented: Macaulay2, Risa/Asir, Singular ...
The Weyl algebra – p. 55 Dimension
We associate to any f.g. An–module M its dimension which is a positive integer number (unless for M = (0); we write dim((0)) = −1. Definition.- M finitely generated An–module. The dimension of M (denoted dim(M)) is the Krull dimension of Char(M) the characteristic variety of M.
The Weyl algebra – p. 56 Dimension
We associate to any f.g. An–module M its dimension which is a positive integer number (unless for M = (0); we write dim((0)) = −1. Definition.- M finitely generated An–module. The dimension of M (denoted dim(M)) is the Krull dimension of Char(M) the characteristic variety of M.
M = An/I for I a left ideal in An. dim(M) is the Krull dimension of the quotient ring C[x, ξ]/gr(I).
The Weyl algebra – p. 56 Dimension
We associate to any f.g. An–module M its dimension which is a positive integer number (unless for M = (0); we write dim((0)) = −1. Definition.- M finitely generated An–module. The dimension of M (denoted dim(M)) is the Krull dimension of Char(M) the characteristic variety of M.
M = An/I for I a left ideal in An. dim(M) is the Krull dimension of the quotient ring C[x, ξ]/gr(I). If W ⊂ C2n is an affine linear variety its dimension coincides with its Krull dimension.
The Weyl algebra – p. 56 Dimension
Example.- I = AnP , for P ∈ An.
The Weyl algebra – p. 57 Dimension
Example.- I = AnP , for P ∈ An.
• if P is a nonzero constant then An/I = (0) and its dimension is -1.
The Weyl algebra – p. 57 Dimension
Example.- I = AnP , for P ∈ An.
• if P is a nonzero constant then An/I = (0) and its dimension is -1. n n • If P = 0 then Char(An/I) = Char(An) = C × C and then its dimension is 2n. • If P ∈ An \ C then σ(P ) ∈ C[x][ξ] is a non-constant polynomial and the Krull dimension of VC(σ(P )) is 2n − 1. So in this case dim(An/I) = 2n − 1.
The Weyl algebra – p. 57 Dimension. Bernstein inequality
[Bernstein’s inequality] Let M be a nonzero finitely generated An-module. Then 2n ≥ dim(M) ≥ n.
The Weyl algebra – p. 58 Dimension. Bernstein inequality
[Bernstein’s inequality] Let M be a nonzero finitely generated An-module. Then 2n ≥ dim(M) ≥ n.
A finitely generated An–module M is said to be holonomic if either M = (0) or dim(M) = n.
The Weyl algebra – p. 58 Dimension. Bernstein inequality
[Bernstein’s inequality] Let M be a nonzero finitely generated An-module. Then 2n ≥ dim(M) ≥ n.
A finitely generated An–module M is said to be holonomic if either M = (0) or dim(M) = n.
For P ∈ An \ C the quotient An/AnP is holonomic if and only if n = 1.
The Weyl algebra – p. 58 Dimension. Hilbert polynomials
M finitely generated An–module. Γ=(Mk)k a B–filtration on M.
The Weyl algebra – p. 59 Dimension. Hilbert polynomials
M finitely generated An–module. Γ=(Mk)k a B–filtration on M.
Definition.- Let M be an An-module. A B–filtration on M is a family Γ=(Mk)k∈N of finitely dimensional C–vector subspaces of M such that: (i) Mk ⊂ Mk+1 for all k ∈ N. (ii) k Mk = M. (iii) BkMℓ ⊂ Mk+ℓ for all (k,ℓ). S
The Weyl algebra – p. 59 Dimension. Hilbert polynomials
M finitely generated An–module. Γ=(Mk)k a B–filtration on M.
Definition.- Let M be an An-module. A B–filtration on M is a family Γ=(Mk)k∈N of finitely dimensional C–vector subspaces of M such that: (i) Mk ⊂ Mk+1 for all k ∈ N. (ii) k Mk = M. (iii) BkMℓ ⊂ Mk+ℓ for all (k,ℓ). S Denote by HFM,Γ : N → N the map
The Weyl algebra – p. 59 Dimension. Hilbert polynomials
M finitely generated An–module. Γ=(Mk)k a B–filtration on M.
Definition.- Let M be an An-module. A B–filtration on M is a family Γ=(Mk)k∈N of finitely dimensional C–vector subspaces of M such that: (i) Mk ⊂ Mk+1 for all k ∈ N. (ii) k Mk = M. (iii) BkMℓ ⊂ Mk+ℓ for all (k,ℓ). S Denote by HFM,Γ : N → N the map
HFM,Γ(ν) = dimC(Mν) for ν ∈ N. HFM,Γ is called the Hilbert function associated with the B–filtration Γ on M.
The Weyl algebra – p. 59 Dimension. Hilbert polynomials
M finitely generated An–module. Γ=(Mk)k a good B–filtration on M.
The Weyl algebra – p. 60 Dimension. Hilbert polynomials
M finitely generated An–module. Γ=(Mk)k a good B–filtration on M. Proposition.- There exists a unique polynomial HPM,Γ(t) ∈ Q[t] such that
HFM,Γ(ν) = HPM,Γ(ν) for ν big enough.
The Weyl algebra – p. 60 Dimension. Hilbert polynomials
M finitely generated An–module. Γ=(Mk)k a good B–filtration on M. Proposition.- There exists a unique polynomial HPM,Γ(t) ∈ Q[t] such that
HFM,Γ(ν) = HPM,Γ(ν) for ν big enough.
HPM,Γ(t) is called the Hilbert polynomial associated with the B–filtration Γ on M.
The Weyl algebra – p. 60 Dimension. Hilbert polynomials
M finitely generated An–module. Γ=(Mk)k a good B–filtration on M. Proposition.- There exists a unique polynomial HPM,Γ(t) ∈ Q[t] such that
HFM,Γ(ν) = HPM,Γ(ν) for ν big enough.
HPM,Γ(t) is called the Hilbert polynomial associated with the B–filtration Γ on M.
Proposition.- The leading term of HPM,Γ(f) has the form ad n d! t (with ad ∈ N) and it is independent of the good B–filtration Γ on M.
The Weyl algebra – p. 60 Dimension. Hilbert polynomials
(J. Bernstein, 1971) M finitely generated An–module. Γ=(Mk)k a good B–filtration on M.
The Weyl algebra – p. 61 Dimension. Hilbert polynomials
(J. Bernstein, 1971) M finitely generated An–module. Γ=(Mk)k a good B–filtration on M.
dim(M) = deg(HPM,Γ(t))
The Weyl algebra – p. 61 Dimension. Hilbert polynomials
(J. Bernstein, 1971) M finitely generated An–module. Γ=(Mk)k a good B–filtration on M.
dim(M) = deg(HPM,Γ(t))
The Weyl algebra – p. 61 Multiplicities. Hilbert polynomials
M finitely generated An–module.
The Weyl algebra – p. 62 Multiplicities. Hilbert polynomials
M finitely generated An–module.
Definition.- The multiplicity of M is e(M) := add!.
The Weyl algebra – p. 62 Multiplicities. Hilbert polynomials
M finitely generated An–module.
Definition.- The multiplicity of M is e(M) := add!. d where adt is the leading term of the Hilbert polynomial HPM,Γ(t) for a (or any) good B–filtration Γ on M.
The Weyl algebra – p. 62 Multiplicities. Hilbert polynomials
M finitely generated An–module.
Definition.- The multiplicity of M is e(M) := add!. d where adt is the leading term of the Hilbert polynomial HPM,Γ(t) for a (or any) good B–filtration Γ on M. e(M) ∈ N. If M =6 (0) then e(M) > 0.
The Weyl algebra – p. 62 Induced filtrations
Γ=(Mk)k∈N a filtration on an An–module M (i.e. either an F –filtration or a B–filtration on M).
The Weyl algebra – p. 63 Induced filtrations
Γ=(Mk)k∈N a filtration on an An–module M (i.e. either an F –filtration or a B–filtration on M). N ⊂ M a sub-module. k ∈ N denote Nk := Mk ∩ N and M M + N = k N N k .
The Weyl algebra – p. 63 Induced filtrations
Γ=(Mk)k∈N a filtration on an An–module M (i.e. either an F –filtration or a B–filtration on M). N ⊂ M a sub-module. k ∈ N denote Nk := Mk ∩ N and M M + N = k N N k . ′ Proposition.- The family Γ =(Nk)k∈N is a filtration on N. The family Γ′′ = M is a filtration on M/N. [They are N k k∈N called the induced filtrations on and respectively] N M/N
The Weyl algebra – p. 63 Graded exact sequence
′ N ⊂ M (An–modules). Γ=(Mk)k a filtration on M. Γ and Γ′′ the induced filtrations on N and M/N.
The Weyl algebra – p. 64 Graded exact sequence
′ N ⊂ M (An–modules). Γ=(Mk)k a filtration on M. Γ and Γ′′ the induced filtrations on N and M/N. There is a canonical exact sequence of graded modules
′ ′′ 0 → grΓ (N) → grΓ(M) → grΓ (M/N) → 0.
The Weyl algebra – p. 64 Graded exact sequence
′ N ⊂ M (An–modules). Γ=(Mk)k a filtration on M. Γ and Γ′′ the induced filtrations on N and M/N. There is a canonical exact sequence of graded modules
′ ′′ 0 → grΓ (N) → grΓ(M) → grΓ (M/N) → 0.
Γ If Γ=(Mk)k is a good filtration (i.e. gr (M) is a f.g. C[x, ξ]–module) then Γ′ and Γ′′ are also good filtrations (on N and M/N respectively), since C[x, ξ] is a noetherian ring.
The Weyl algebra – p. 64 Graded exact sequence
′ N ⊂ M (An–modules). Γ=(Mk)k a filtration on M. Γ and Γ′′ the induced filtrations on N and M/N. There is a canonical exact sequence of graded modules
′ ′′ 0 → grΓ (N) → grΓ(M) → grΓ (M/N) → 0.
Γ If Γ=(Mk)k is a good filtration (i.e. gr (M) is a f.g. C[x, ξ]–module) then Γ′ and Γ′′ are also good filtrations (on N and M/N respectively), since C[x, ξ] is a noetherian ring.
• HFM,Γ(ν) = HFN,Γ′ (ν) + HFM/N,Γ′′ (ν) for all ν ∈ N. • dim(M) = max{ dim(N), dim(M/N)}. • If dim(N) = dim(M/N) then e(M) = e(N) + e(M/N).
The Weyl algebra – p. 64 Holonomic An–modules
N ⊂ M (M finitely generated An–module).
The Weyl algebra – p. 65 Holonomic An–modules
N ⊂ M (M finitely generated An–module). Proposition.- M is holonomic if and only if N and M/N are holonomic.
The Weyl algebra – p. 65 Holonomic An–modules
N ⊂ M (M finitely generated An–module). Proposition.- M is holonomic if and only if N and M/N are holonomic. If M is holonomic then e(M) = e(N) + e(M/N).
The Weyl algebra – p. 65 Holonomic ⇒ Artinian
Proposition.- M holonomic An–module. Then M is artinian and length(M) ≤ e(M)
The Weyl algebra – p. 66 Holonomic ⇒ Artinian
Proposition.- M holonomic An–module. Then M is artinian and length(M) ≤ e(M) Sketch of the proof.- Assume M =6 (0). M = M0 ⊃ M1 ⊃ M2 ⊃··· a decreasing chain of An–submodules of M. Each Mi is holonomic.
The Weyl algebra – p. 66 Holonomic ⇒ Artinian
Proposition.- M holonomic An–module. Then M is artinian and length(M) ≤ e(M) Sketch of the proof.- Assume M =6 (0). M = M0 ⊃ M1 ⊃ M2 ⊃··· a decreasing chain of An–submodules of M. Each Mi is holonomic.
If there exists i such that Mi = (0) then the chain is stationary. Assume the chain non-stationary and that each Mi is nonzero.
The Weyl algebra – p. 66 Holonomic ⇒ Artinian
Proof continued.-
0 −→ Mi+1 −→ Mi −→ Mi/Mi+1 −→ 0
The Weyl algebra – p. 67 Holonomic ⇒ Artinian
Proof continued.-
0 −→ Mi+1 −→ Mi −→ Mi/Mi+1 −→ 0
Then e(Mi) = e(Mi+1) + e(Mi/Mi+1).
The Weyl algebra – p. 67 Holonomic ⇒ Artinian
Proof continued.-
0 −→ Mi+1 −→ Mi −→ Mi/Mi+1 −→ 0
Then e(Mi) = e(Mi+1) + e(Mi/Mi+1). r Then e(M) = e(Mr+1) + i=0 e(Mi/Mi+1) ≥ r + 1, for each r ≥ 0. This is a contradiction. Moreover, the length of M should be less or equal thanP e(M).
The Weyl algebra – p. 67 Rational functions f nonzero polynomial in C[x]
The Weyl algebra – p. 68 Rational functions f nonzero polynomial in C[x]
C[x]f rational functions with poles along f = 0, or the localization of C[x] with respect to the multiplicative set k {f }k∈N.
The Weyl algebra – p. 68 Rational functions f nonzero polynomial in C[x]
C[x]f rational functions with poles along f = 0, or the localization of C[x] with respect to the multiplicative set k {f }k∈N.
C[x]f is a left An–module.
−k −k −k−1 ∂i(gf ) = ∂i(g)f +(−k)∂i(f)f g.
The Weyl algebra – p. 68 Rational functions f nonzero polynomial in C[x]
C[x]f rational functions with poles along f = 0, or the localization of C[x] with respect to the multiplicative set k {f }k∈N.
C[x]f is a left An–module.
−k −k −k−1 ∂i(gf ) = ∂i(g)f +(−k)∂i(f)f g.
In general, C[x]f is not finitely generated as C[x]–module.
The Weyl algebra – p. 68 C[x]f is holonomic
−k −k−1 n = 1, f = x1. The equality ∂1(x1 )=(−k)x1 (for all ) implies C 1 . k ≥ 1 [x1]x1 = A1 • x1
Theorem.- [J. Bernstein] For any non-zero f ∈ C[x], C[x]f is holonomic (in particular it is finitely generated as An-module).
The Weyl algebra – p. 69 C[x]f is holonomic
−k −k−1 n = 1, f = x1. The equality ∂1(x1 )=(−k)x1 (for all ) implies C 1 . k ≥ 1 [x1]x1 = A1 • x1
Theorem.- [J. Bernstein] For any non-zero f ∈ C[x], C[x]f is holonomic (in particular it is finitely generated as An-module).
Idea of the Proof.- Put N = C[x]f and deg(f) = d ≥ 0. k ∈ N g N = { ∈ N | deg(g) ≤ (d + 1)k}. k f k
Γ=(Nk)k is a B–filtration on N.
The Weyl algebra – p. 69 C[x]f is holonomic
−k −k−1 n = 1, f = x1. The equality ∂1(x1 )=(−k)x1 (for all ) implies C 1 . k ≥ 1 [x1]x1 = A1 • x1
Theorem.- [J. Bernstein] For any non-zero f ∈ C[x], C[x]f is holonomic (in particular it is finitely generated as An-module).
Idea of the Proof.- Put N = C[x]f and deg(f) = d ≥ 0. k ∈ N g N = { ∈ N | deg(g) ≤ (d + 1)k}. k f k
Γ=(Nk)k is a B–filtration on N.
The Weyl algebra – p. 69 C[x]f is holonomic
Proof continued.- dimC Nk is bounded by the number of monomials xα in C[x] with degree |α|≤ (d + 1)k.
The Weyl algebra – p. 70 C[x]f is holonomic
Proof continued.- dimC Nk is bounded by the number of monomials xα in C[x] with degree |α|≤ (d + 1)k. This number is
(d + 1)k + n 1 = (d + 1)nkn + p(k) n ! n! where p(k) is a polynomial in k with rational coefficients and degree less or equal that n − 1.
The Weyl algebra – p. 70 C[x]f is holonomic
Proof continued.- dimC Nk is bounded by the number of monomials xα in C[x] with degree |α|≤ (d + 1)k. This number is
(d + 1)k + n 1 = (d + 1)nkn + p(k) n ! n! where p(k) is a polynomial in k with rational coefficients and degree less or equal that n − 1.
[Technical result] With this kind of behavior for dimC Nk one proves that • N if f.g. • dim(N) ≤ n. By Bernstein’s inequality dim N = n and N is holonomic.
The Weyl algebra – p. 70 Mf is holonomic
Assume M is an An–module. Denote Mf = C[x]f ⊗C[x] M which is a C[x]-module.
The Weyl algebra – p. 71 Mf is holonomic
Assume M is an An–module. Denote Mf = C[x]f ⊗C[x] M which is a C[x]-module.
Mf has a natural structure of (left) An–module:
−k −k−1 −k ∂i(f m) = −kf ∂i(f)m + f ∂i(m)
The Weyl algebra – p. 71 Mf is holonomic
Assume M is an An–module. Denote Mf = C[x]f ⊗C[x] M which is a C[x]-module.
Mf has a natural structure of (left) An–module:
−k −k−1 −k ∂i(f m) = −kf ∂i(f)m + f ∂i(m)
Proposition.- 0 =6 f ∈ C[x]. If M is holonomic then Mf is holonomic.
The Weyl algebra – p. 71 Mf is holonomic
Assume M is an An–module. Denote Mf = C[x]f ⊗C[x] M which is a C[x]-module.
Mf has a natural structure of (left) An–module:
−k −k−1 −k ∂i(f m) = −kf ∂i(f)m + f ∂i(m)
Proposition.- 0 =6 f ∈ C[x]. If M is holonomic then Mf is holonomic. The proof is similar to the one for M = C[x].
The Weyl algebra – p. 71 Bernstein polynomial f nonzero polynomial in C[x].
The Weyl algebra – p. 72 Bernstein polynomial f nonzero polynomial in C[x]. s new variable. C(s) field of rational functions in s. An(s) Weyl algebra over C(s).
The Weyl algebra – p. 72 Bernstein polynomial f nonzero polynomial in C[x]. s new variable. C(s) field of rational functions in s. An(s) Weyl algebra over C(s).
An[s] = An ⊗C C[s].
The Weyl algebra – p. 72 Bernstein polynomial
C(s)[x]f a ring of rational functions. Notice that “numerators" in this ring are polynomials in C(s)[x].
The Weyl algebra – p. 73 Bernstein polynomial
C(s)[x]f a ring of rational functions. Notice that “numerators" in this ring are polynomials in C(s)[x]. s s C(s)[x]f f free module over C(s)[x]f . Here f is just a symbol.
The Weyl algebra – p. 73 Bernstein polynomial
C(s)[x]f a ring of rational functions. Notice that “numerators" in this ring are polynomials in C(s)[x]. s s C(s)[x]f f free module over C(s)[x]f . Here f is just a symbol. s C(s)[x]f f is a left An(s)–module:
s∂ (f) ∂ (f s) = i f s i f
The Weyl algebra – p. 73 Bernstein polynomial
s+1 s We have ∂1(x1 )=(s + 1)x1
The Weyl algebra – p. 74 Bernstein polynomial
s+1 s We have ∂1(x1 )=(s + 1)x1 Theorem.- Let 0 =6 f ∈ C[x]. There exists a nonzero polynomial b(s) ∈ C[s] and a differential operator P (s) ∈ An[s] such that the equality
P (s)ff s = b(s)f s
s holds in C(s)[x]f f .
The Weyl algebra – p. 74 Bernstein polynomial
s+1 s We have ∂1(x1 )=(s + 1)x1 Theorem.- Let 0 =6 f ∈ C[x]. There exists a nonzero polynomial b(s) ∈ C[s] and a differential operator P (s) ∈ An[s] such that the equality
P (s)ff s = b(s)f s
s holds in C(s)[x]f f . s Proof.- We first prove that N = C(s)[x]f f is An(s)– holonomic (and hence finitely generated over An(s)). Proof is similar to the one of “C[x]f is An–holonomic."
The Weyl algebra – p. 74 Bernstein polynomial
s+1 s We have ∂1(x1 )=(s + 1)x1 Theorem.- Let 0 =6 f ∈ C[x]. There exists a nonzero polynomial b(s) ∈ C[s] and a differential operator P (s) ∈ An[s] such that the equality
P (s)ff s = b(s)f s
s holds in C(s)[x]f f . s Proof.- We first prove that N = C(s)[x]f f is An(s)– holonomic (and hence finitely generated over An(s)). Proof is similar to the one of “C[x]f is An–holonomic." s An(s)f ⊂ N is also holonomic. Then it is Artinian over An(s). [We use here multiplicities]
The Weyl algebra – p. 74 Bernstein polynomial
Proof continued.- Then the descending chain
s s k s An(s)f ⊃ An(s)ff ⊃···⊃ An(s)f f ⊃··· must be stationary.
The Weyl algebra – p. 75 Bernstein polynomial
Proof continued.- Then the descending chain
s s k s An(s)f ⊃ An(s)ff ⊃···⊃ An(s)f f ⊃··· must be stationary. Thus there exists ℓ > 0 such that
ℓ s ℓ+1 s f f ∈ An(s)f f and so there exists an operator Q(s) ∈ An(s) such that
f ℓf s = Q(s)f ℓ+1f s.
The Weyl algebra – p. 75 Bernstein polynomial
Proof continued.- Then the descending chain
s s k s An(s)f ⊃ An(s)ff ⊃···⊃ An(s)f f ⊃··· must be stationary. Thus there exists ℓ > 0 such that
ℓ s ℓ+1 s f f ∈ An(s)f f and so there exists an operator Q(s) ∈ An(s) such that
f ℓf s = Q(s)f ℓ+1f s. c(s) common denominator for the coefficients of Q(s).
c(s)f ℓf s = R(s)f ℓ+1f s
The Weyl algebra – p. 75 for some R(s) ∈ An[s]. Bernstein polynomial
Proof continued.- Here c(s) is a nonzero polynomial. We also have b(s)f s = P (s)ff s just by considering b(s) = c(s − ℓ) and P (s) = R(s − ℓ).
The Weyl algebra – p. 76 Bernstein polynomial
Proof continued.- Here c(s) is a nonzero polynomial. We also have b(s)f s = P (s)ff s just by considering b(s) = c(s − ℓ) and P (s) = R(s − ℓ). Given f ∈ C[x], b(s) and P (s) are not easy to compute.
The Weyl algebra – p. 76 Bernstein polynomial
Proof continued.- Here c(s) is a nonzero polynomial. We also have b(s)f s = P (s)ff s just by considering b(s) = c(s − ℓ) and P (s) = R(s − ℓ). Given f ∈ C[x], b(s) and P (s) are not easy to compute. Assume f(0) = 0. Then P (−1)f 0 = b(−1)f −1. This implies b(−1) = 0.
The Weyl algebra – p. 76 Bernstein polynomial
Proof continued.- Here c(s) is a nonzero polynomial. We also have b(s)f s = P (s)ff s just by considering b(s) = c(s − ℓ) and P (s) = R(s − ℓ). Given f ∈ C[x], b(s) and P (s) are not easy to compute. Assume f(0) = 0. Then P (−1)f 0 = b(−1)f −1. This implies b(−1) = 0.
Let −k0 be the smallest integer root of b(s). Then C 1 [x]f = An • f k0 . So, C[x]f is even a cyclic An-module.
The Weyl algebra – p. 76 Bernstein polynomial
The family of polynomials b(s) satisfying the equation
b(s)f s = P (s)ff s for some P (s) ∈ An[s] is a nonzero ideal in C[s].
The Weyl algebra – p. 77 Bernstein polynomial
The family of polynomials b(s) satisfying the equation
b(s)f s = P (s)ff s for some P (s) ∈ An[s] is a nonzero ideal in C[s]. C[s] is a principal domain. Previous ideal has a unique monic generator that we will denote by bf (s).
The Weyl algebra – p. 77 Bernstein polynomial
The family of polynomials b(s) satisfying the equation
b(s)f s = P (s)ff s for some P (s) ∈ An[s] is a nonzero ideal in C[s]. C[s] is a principal domain. Previous ideal has a unique monic generator that we will denote by bf (s).
Definition.- The polynomial bf (s) is called the Bernstein polynomial (or the Bernstein-Sato polynomial) associated with f.
The Weyl algebra – p. 77 Computation of bf (s)
There are algorithms (using again Groebner basis in An) computing bf (s) for f ∈ C[x]. Works of T. Oaku and T.Oaku-N. Takayama.
The Weyl algebra – p. 78 Computation of bf (s)
There are algorithms (using again Groebner basis in An) computing bf (s) for f ∈ C[x]. Works of T. Oaku and T.Oaku-N. Takayama. These computations are very hard. With the known algorithms they are, sometimes, even impossible in “human time".
The Weyl algebra – p. 78 Computation of bf (s)
There are algorithms (using again Groebner basis in An) computing bf (s) for f ∈ C[x]. Works of T. Oaku and T.Oaku-N. Takayama. These computations are very hard. With the known algorithms they are, sometimes, even impossible in “human time". We need new ideas.
The Weyl algebra – p. 78 Computation of bf (s)
s s • ∂1(x1x1)=(s + 1)x1
The Weyl algebra – p. 79 Computation of bf (s)
s s • ∂1(x1x1)=(s + 1)x1 2 2 2 2 s+1 2 s • f = x1 + x2. Then (∂1 + ∂2)(f )=4(s + 1) f .
The Weyl algebra – p. 79 Computation of bf (s)
s s • ∂1(x1x1)=(s + 1)x1 2 2 2 2 s+1 2 s • f = x1 + x2. Then (∂1 + ∂2)(f )=4(s + 1) f . 2 3 • f = x1 − x2. Then 5 7 18x ∂2∂ − 8∂3 + 54∂2s + 81∂ (f s+1)=(s+1)(s+ )(s+ )f s 2 1 2 2 1 2 6 6
The Weyl algebra – p. 79 Computer Algebra in An
The theory of Groebner bases in the polynomial ring C[x] can be extended to the Weyl algebra An (and to the rings D and D)
b
The Weyl algebra – p. 80 Computer Algebra in An
The theory of Groebner bases in the polynomial ring C[x] can be extended to the Weyl algebra An (and to the rings D and D)
Groebner bases in An are used: b • to compute a generating system of SyzAn (P1,...,Pm), the An–module of syzygies of a given family P1,...,Pm of elements in An.
The Weyl algebra – p. 80 Computer Algebra in An
The theory of Groebner bases in the polynomial ring C[x] can be extended to the Weyl algebra An (and to the rings D and D)
Groebner bases in An are used: b • to compute a generating system of SyzAn (P1,...,Pm), the An–module of syzygies of a given family P1,...,Pm of elements in An. r (or in a free module An (r ≥ 1)
The Weyl algebra – p. 80 Computer Algebra in An
• to compute dim(M) for
Am M = n An(P1,...,Pℓ)
(so, to decide if a finitely presented An–module is holonomic)
The Weyl algebra – p. 81 Computer Algebra in An
• to compute dim(M) for
Am M = n An(P1,...,Pℓ)
(so, to decide if a finitely presented An–module is holonomic) • to compute polynomial equations defining Char(M)
The Weyl algebra – p. 81 Computer Algebra in An
• to compute dim(M) for
Am M = n An(P1,...,Pℓ)
(so, to decide if a finitely presented An–module is holonomic) • to compute polynomial equations defining Char(M) • to construct a finite free resolution of a given finitely presented An–module.
The Weyl algebra – p. 81 Division theorem (DTh) in An
DTh in An generalizes Euclidean division in C[t], 1 variable.
The Weyl algebra – p. 82 Division theorem (DTh) in An
DTh in An generalizes Euclidean division in C[t], 1 variable.
DTh is used in Groebner bases theory in An
The Weyl algebra – p. 82 Division theorem (DTh) in An
DTh in An generalizes Euclidean division in C[t], 1 variable.
DTh is used in Groebner bases theory in An
The Weyl algebra – p. 82 Division theorem in An
[Division in An] Let (P1,...,Pm) be an m–tuple of nonzero elements of An. Then, for any P in An, there exists an unique (m + 1)–tuple (Q1,...,Qm,R) of elements in An, such that:
The Weyl algebra – p. 83 Division theorem in An
[Division in An] Let (P1,...,Pm) be an m–tuple of nonzero elements of An. Then, for any P in An, there exists an unique (m + 1)–tuple (Q1,...,Qm,R) of elements in An, such that:
P = Q1P1 + · · · + QmPm + R.
The Weyl algebra – p. 83 Division theorem in An
[Division in An] Let (P1,...,Pm) be an m–tuple of nonzero elements of An. Then, for any P in An, there exists an unique (m + 1)–tuple (Q1,...,Qm,R) of elements in An, such that:
P = Q1P1 + · · · + QmPm + R.
The set of monomials of the “quotients" Qi and of the “remainder" R satisfy some combinatorial conditions
The Weyl algebra – p. 83 Division theorem in An
[Division in An] Let (P1,...,Pm) be an m–tuple of nonzero elements of An. Then, for any P in An, there exists an unique (m + 1)–tuple (Q1,...,Qm,R) of elements in An, such that:
P = Q1P1 + · · · + QmPm + R.
The set of monomials of the “quotients" Qi and of the “remainder" R satisfy some combinatorial conditions
Castro F. Théorème de division pour les opérateurs différentiels et calcul des multiplicités, Thèse de 3eme cycle, Univ. of Paris VII, (1984); see also Castro-Jiménez F. J. and Granger, M. Explicit calculations in rings of differential operators. Éléments de la théorie des systèmes différentiels géométriques, 89–128, Sémin. Congr., 8, Soc. Math. France, Paris, 2004.
The Weyl algebra – p. 83