The Weyl algebra Modules over the Weyl algebra

The Weyl algebra p. The Weyl algebra Modules over the Weyl algebra Francisco J. Castro Jiménez 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. I thank very much the participants for their comments and suggestions.

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 II Castro-Jiménez F.J., Modules Over the Weyl Algebra, in Le Dũng 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[x 1,...,x n ] ring of polynomials. (informal definition) The Weyl algebra A n (C) of order n over the complex numbers C is the set

The Weyl algebra p. The Weyl algebra C[x] = C[x 1,...,x n ] ring of polynomials. (informal definition) The Weyl algebra A n (C) of order n over the complex numbers C is the set p α (x) α=(α 1,...,α n ) N n ( x 1 ) α1 ( ) αn p α (x) C[x] x n A n (C) is a ring (and a C-algebra). The addition + is the natural one.

The Weyl algebra p. The Weyl algebra C[x] x i C[x] is a C linear map (i.e. xi End C (C[x]))

The Weyl algebra p. The Weyl algebra C[x] x i C[x] is a C linear map (i.e. The multiplication by f C[x] xi End C (C[x])) C[x] φ f C[x] (i.e. φ f (g) = fg for all g C[x]) is also a C linear map (i.e. φ f End C (C[x]))

The Weyl algebra p. The Weyl algebra End C (C[x]) the ring of endomorphisms of the C-vector space C[x]. The product in this ring is the composition of endomorphisms. End C (C[x]) is a noncommutative ring (if the number of variables n 1). The unit of this ring is the identity map (i.e. φ 1 : C[x] C[x]) (Formal definition) A n (C) is the subring (the subalgebra) of End C (C[x]) generated by the endomorphisms φ x1,...,φ xn, 1,..., n.

The Weyl algebra p. The Weyl algebra Exercise.- Prove the following equalities in A n : i φ xi = φ xi i +1 i φ xj = φ xj i if i j i j = j i for all pairs (i,j). φ xi φ xj = φ xj φ xi for all pairs (i,j). A n is a non-commutative ring (for n 1). Proposition.- The map C[x] A n defined by f φ f is an injective morphism of rings (and of C-algebras).

The Weyl algebra p. 1 The product in A n a) f C[x] and β N n. Then ( β β f = σ σ β ) σ (f) β σ ( where ) σ β stands for σ i β i for i = 1,...,n, β β! = σ σ!(β σ)! and β! = β 1! β n!

The Weyl algebra p. 1 The product in A n a) f C[x] and β N n. Then ( β β f = σ σ β ) σ (f) β σ ( where ) σ β stands for σ i β i for i = 1,...,n, β β! = σ σ!(β σ)! and β! = β 1! β n! For n = 1 the equality j j 1 f = j k k=0 k t (f) j k 1 can be proved by induction on j. The general case follows from the previous one and the distributivity of the product in A n with respect to the sum.

The Weyl algebra p. 1 The product in A n b) β,γ N n then β (x γ ) = β! ( γ β ) x γ β where ( γ β ) = 0 if the relation β γ doesn t hold.

The Weyl algebra p. 1 The product in A n b) β,γ N n then β (x γ ) = β! ( γ β ) x γ β where ( γ β ) = 0 if the relation β γ doesn t hold. c) α,α,β,β N n then σ β,σ α,σ 0 x α β x α β = x α+α β+β + ( )( ) β α σ! x α+α σ β+β σ. σ σ

The Weyl algebra p. 1 The Weyl algebra Proposition.- : The set of monomials B = {x α β α,β N n } is a basis of the C-vector space A n.

The Weyl algebra p. 1 The Weyl algebra Proposition.- : The set of monomials B = {x α β α,β N n } is a basis of the C-vector space A n. Each nonzero element P in A n can be written in an unique way as a finite sum P = α,β p αβ x α β for some nonzero complex numbers p αβ.

The Weyl algebra p. 1 A n (K) K be a field of characteristic zero. Polynomial ring K[x] = K[x 1,...,x n ].

The Weyl algebra p. 1 A n (K) K be a field of characteristic zero. Polynomial ring K[x] = K[x 1,...,x n ]. Definition.- Let n 1 be an integer number. The n-th Weyl algebra over K, denoted by A n (K), is the subalgebra of End K (K[x]) generated by the endomorphisms φ x1,...,φ xn, 1,..., n. All the results given so far for A n (C) are also valid for A n (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. 1 Modules over A n C[x] is a left A n module.

The Weyl algebra p. 1 Modules over A n C[x] is a left A n module. P = β p β(x) β A n, f C[x], P f = P(f) = β p β (x) β 1+ +β n (f) x β 1 1 xβ n n

The Weyl algebra p. 1 Modules over A n C[x] is a left A n module. P = β p β(x) β A n, f C[x], P f = P(f) = β p β (x) β 1+ +β n (f) x β 1 1 xβ n n A n f = {P(f) P A n } C[x]. C[x] = A n 1.

The Weyl algebra p. 1 Modules over A n C[x] is a left A n module. P = β p β(x) β A n, f C[x], P f = P(f) = β p β (x) β 1+ +β n (f) x β 1 1 xβ n n A n f = {P(f) P A n } C[x]. C[x] = A n 1. C[x] A n Ann A n(1) Ann An (1) = {P A n P(1) = 0} = A n ( 1,..., n )

The Weyl algebra p. 1 Systems of LPDE P 1 (u) = 0 S.. P l (u) = 0 (0.0.0) where P i is a linear differential operator P i A n = A n (C).

The Weyl algebra p. 1 Systems of LPDE P 1 (u) = 0 S.. P l (u) = 0 (0.0.0) where P i is a linear differential operator P i A n = A n (C). Assume u(x) C[x] is a solution of S.

The Weyl algebra p. 1 Systems of LPDE P 1 (u) = 0 S.. P l (u) = 0 (0.0.0) where P i is a linear differential operator P i A n = A n (C). Assume u(x) C[x] is a solution of S. Then ( i Q ip i )(u(x)) = 0 for all Q i A n. One says then that u(x) is a solution of the left ideal A n (P 1,...,P l ) generated by the P i.

The Weyl algebra p. 1 Systems LPDE A n Modules To the system S we associate the left A n module M = M(S) = A n A n (P 1,...,P l )

The Weyl algebra p. 1 Systems LPDE A n Modules To the system S we associate the left A n module M = M(S) = A n A n (P 1,...,P l ) Example.- 1 variable x = x 1. To the equation (x 2 x + x )(u) = 0 (here x = d dx ) we associate the (left) A 1 module M = A 1 A 1 P.

The Weyl algebra p. 1 Solutions P 1 (u) = 0 S.. P l (u) = 0 (0.0.0)

The Weyl algebra p. 1 Solutions P 1 (u) = 0 S.. P l (u) = 0 (0.0.0) Sol(S; C[x]) (vector space) polynomial solutions of the system S.

The Weyl algebra p. 1 Solutions P 1 (u) = 0 S.. P l (u) = 0 (0.0.0) Sol(S; C[x]) (vector space) polynomial solutions of the system S. Any solution u C[x] of S induces a morphism ψ u : M = given by ψ u (P) = P(u). A n A n (P 1,...,P l ) C[x]

The Weyl algebra p. 1 Solutions Any morphism ψ : M = A n A n (P 1,...,P l ) C[x] induces a solution of S (just by considering u ψ = ψ(1)).

The Weyl algebra p. 1 Solutions Any morphism ψ : M = A n A n (P 1,...,P l ) C[x] induces a solution of S (just by considering u ψ = ψ(1)). Exercise.- The mappings and u Sol(S;C[x]) ψ u Hom An (M,C[x]) ψ Hom An (M,C[x]) u ψ Sol(S;C[x]) are both C linear maps and moreover they are inverse to each other.

The Weyl algebra p. 1 Solutions The example before justifies the following

The Weyl algebra p. 1 Solutions The example before justifies the following Definition.- Let F be a left A n module and M be a finitely generated A n module. The solution space of M in F is the C vector space Hom An (M,F) There are several algorithms computing a basis of the vector space Hom An (A n /I,C[x]) [for A n /I holonomic]. These algorithms use Groebner basis in the ring A n. T. Oaku-N. Takayama (2000) and H. Tsai-U. Walther (2000)

The Weyl algebra p. 2 Order and total order of a LDO Denote β = i β i = β 1 + +β n for β N n.

The Weyl algebra p. 2 Order and total order of a LDO Denote β = i β i = β 1 + +β n for β N n. Definition.- [order and total order] P = αβ p αβ x α β = β p β (x) β A n ord(p) = max{ β s.t. p β (x) 0}

The Weyl algebra p. 2 Order and total order of a LDO Denote β = i β i = β 1 + +β n for β N n. Definition.- [order and total order] P = αβ p αβ x α β = β p β (x) β A n ord(p) = max{ β s.t. p β (x) 0} ord T (P) = max{ α + β s.t. p αβ 0} Convention: ord(0) = ord T (0) =.

The Weyl algebra p. 2 Order and total order of a LDO Example.- (n=2) P = x 2 1 x 2 3 1 5x3 2 2 1 2 + 2 1 2 2 1

The Weyl algebra p. 2 Order and total order of a LDO Example.- (n=2) P = x 2 1 x 2 3 1 5x3 2 2 1 2 + 2 1 2 2 1 ord(p) = 4

The Weyl algebra p. 2 Order and total order of a LDO Example.- (n=2) P = x 2 1 x 2 3 1 5x3 2 2 1 2 + 2 1 2 2 1 ord(p) = 4 ord T (P) = 6

The Weyl algebra p. 2 Symbols Definition.- [principal symbol] P = αβ p αβ x α β = β p β (x) β σ(p) = β =ord(p) p β (x)ξ β C[x][ξ] = C[x][ξ 1,...,ξ n ].

The Weyl algebra p. 2 Symbols Definition.- [principal symbol] P = αβ p αβ x α β = β p β (x) β σ(p) = β =ord(p) p β (x)ξ β C[x][ξ] = C[x][ξ 1,...,ξ n ]. Definition.- [principal total symbol] σ T (P) = p αβ x α ξ β C[x][ξ]. α+β =ord T (P)

The Weyl algebra p. 2 Symbols Example.- (n=2) P = x 2 1 x 2 3 1 +x3 2 2 1 2 + 2 1 2 2

The Weyl algebra p. 2 Symbols Example.- (n=2) P = x 2 1 x 2 3 1 +x3 2 2 1 2 + 2 1 2 2 σ(p) = ξ 2 1ξ 2 2

The Weyl algebra p. 2 Symbols Example.- (n=2) P = x 2 1 x 2 3 1 +x3 2 2 1 2 + 2 1 2 2 σ(p) = ξ 2 1ξ 2 2 σ T (P) = x 2 1x 2 ξ 3 1 +x 3 2ξ 2 1ξ 2

The Weyl algebra p. 2 Symbols σ(p) homogeneous polynomial of degree ord(p) in the ξ-variables

The Weyl algebra p. 2 Symbols σ(p) homogeneous polynomial of degree ord(p) in the ξ-variables σ T (P) homogeneous polynomial of degree ord T (P) in the variables (x, ξ).

The Weyl algebra p. 2 Symbols σ(p) homogeneous polynomial of degree ord(p) in the ξ-variables σ T (P) homogeneous polynomial of degree ord T (P) in the variables (x, ξ). In general σ(p) σ T (P). In general we do not have neither σ(p +Q) = σ(p)+σ(q) nor σ T (P +Q) = σ T (P)+σ T (Q). e.g. P = 1 +x 1 +1, Q = 1 x 1. )σ(p) = ξ 1 ξ 1 +x 1 = σ T (P). Moreover )1 = σ(p +Q) 0 = σ(p)+σ(q) )1 = σ T (P +Q) 0 = σ T (P)+σ T (Q).

The Weyl algebra p. 2 Properties ofσ andσ T Exercise.- For P,Q A n one has ord(pq) = ord(p)+ord(q) and σ(pq) = σ(p)σ(q).

The Weyl algebra p. 2 Properties ofσ andσ T Exercise.- For P,Q A n one has ord(pq) = ord(p)+ord(q) and σ(pq) = σ(p)σ(q). ord T (PQ) = ord T (P)+ord T (Q) and σ T (PQ) = σ T (P)σ T (Q).

The Weyl algebra p. 2 Properties ofσ andσ T Exercise.- For P,Q A n one has ord(pq) = ord(p)+ord(q) and σ(pq) = σ(p)σ(q). ord T (PQ) = ord T (P)+ord T (Q) and σ T (PQ) = σ T (P)σ T (Q). ord(pq QP) ord(p)+ord(q) 1 and ord T (PQ QP) ord T (P)+ord T (Q) 2. ord(p +Q) max{ord(p),ord(q)} (and similarly for ord T ). If ord(p) = ord(q) and σ(p)+σ(q) 0 then σ(p +Q) = σ(p)+σ(q) (and similarly for ord T and σ T ).

The Weyl algebra p. 2 A n is an integral domain. Proposition.- A n is an integral domain.

The Weyl algebra p. 2 A n is an integral domain. Proposition.- A n is an integral domain. Proof.- Assume P,Q A n are both non-zero. Then ord(pq) = ord(p)+ord(q) 0. Then PQ is non-zero.

The Weyl algebra p. 2 Graded ideals Definition.- For each left (or right) ideal I A n we call the graded ideal associated with I gr(i) = C[x][ξ]{σ(P) P I}.

The Weyl algebra p. 2 Graded ideals Definition.- For each left (or right) ideal I A n 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 gr T (I) of C[x][ξ] generated by the family of principal total symbols of elements in I gr T (I) = C[x][ξ]{σ T (P) P I}.

The Weyl algebra p. 2 Graded ideals I = A n P a principal left ideal.

The Weyl algebra p. 2 Graded ideals I = A n P a principal left ideal. Exercise.- gr(i) = C[x, ξ]σ(p)

The Weyl algebra p. 2 Graded ideals I = A n P 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. 2 Graded ideals I = A n P 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 A n. Then σ(q) = σ(r)σ(p) C[x,ξ]σ(P).

The Weyl algebra p. 2 Graded ideals Problem.- Input: P 1,...,P l A n

The Weyl algebra p. 2 Graded ideals Problem.- Input: P 1,...,P l A n Output.- A finite system of generators of gr(i) (similarly for gr T (I))

The Weyl algebra p. 2 Graded ideals Problem.- Input: P 1,...,P l A n Output.- A finite system of generators of gr(i) (similarly for gr T (I)) If l = 1 then Problem is solved (previous exercise). The general case can be solved using Groebner basis theory in A n. (see, Castro F. Théorème de division pour les opérateurs différentiels et calcul des multiplicités, Thèse de 3 eme 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).)

A n is Noetherian The Weyl algebra p. 3

The Weyl algebra p. 3 A n 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[y 1,...,y m ] is Noetherian (for all m 0).

The Weyl algebra p. 3 A n is a simple ring A n is a simple ring (i.e. the only two-sided ideals in A n are (0) and A n )

The Weyl algebra p. 3 Filtrations ona n [Order filtration] F k (A n ) = {P A n ord(p) k} for k Z. We will simply write F k = F k (A n )

The Weyl algebra p. 3 Filtrations ona n [Order filtration] F k (A n ) = {P A n ord(p) k} for k Z. We will simply write F k = F k (A n ) 1. F k = {0} for k 1. 2. F k F k+1 for k Z. 3. F k F l F k+l for k,l Z. 4. A n = k F k. 5. 1 F 0 = C[x]. 6. Each F k is a free C[x]-module with rank ( n+k k ).

The Weyl algebra p. 3 Filtrations ona n [Total order filtration] B k (A n ) = {P A n ord T (P) k} for k Z. We will simply write B k = B k (A n ).

The Weyl algebra p. 3 Filtrations ona n [Total order filtration] B k (A n ) = {P A n ord T (P) k} for k Z. We will simply write B k = B k (A n ). 1. B k = {0} for k 1. 2. B k B k+1 for k Z. 3. B k B l B k+l for k,l Z. 4. A n = k B k. 5. 1 B 0 = C. 6. dim C (B k ) = ( 2n+k k ).

The Weyl algebra p. 3 Classical characteristic vectors P(u) = ( β p β(x) β )(u) = 0 with p β (x) R[x]

The Weyl algebra p. 3 Classical characteristic vectors P(u) = ( β p β(x) β )(u) = 0 with p β (x) R[x] ξ 0 R n is characteristic for P at x 0 R n if σ(p)(x 0,ξ 0 ) = 0.

The Weyl algebra p. 3 Classical characteristic vectors P(u) = ( β p β(x) β )(u) = 0 with p β (x) R[x] ξ 0 R n is characteristic for P at x 0 R n if σ(p)(x 0,ξ 0 ) = 0. The classical characteristic variety of the operator P is by definition the set Char(P) = {(x 0,ξ 0 ) R n R n σ(p)(x 0,ξ 0 ) = 0}. Assume ord(p) 1. P is elliptic if Char(P) R n {0}.

The Weyl algebra p. 3 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 A 1 (R) (x = x 1 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 R 2. 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. 3 Characteristic variety To define the analog to the classical characteristic variety for a general system of LPDE

The Weyl algebra p. 3 Characteristic variety To define the analog to the classical characteristic variety for a general system of LPDE P 11 (u 1 ) + + P 1m (u m ) = 0 S.... P l1 (u 1 ) + + P lm (u m ) = 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. 3 Characteristic variety To the system S we associate the A n module M(S) = A m n A n (P 1,...,P l )

The Weyl algebra p. 3 Characteristic variety To the system S we associate the A n module M(S) = A m n A n (P 1,...,P l ) where P i = (P i1,...,p im ) and A n (P 1,...,P l ) denote the submodule of A m n generated by P 1,...,P l. If m = 1 (only one unknown) M(S) = A n /A n (P 1,...,P l ). Definition.- I A n left ideal. The characteristic variety of the left A n module A n /I is defined as Char(A n /I) = V C (gr(i)) C 2n.

The Weyl algebra p. 3 Characteristic variety V C (gr(i)) C 2n.

The Weyl algebra p. 3 Characteristic variety V C (gr(i)) C 2n. V C (gr(i)) = {(x 0,ξ 0 ) C 2n σ(p)(x 0,ξ 0 ) = 0, P I}.

The Weyl algebra p. 3 Characteristic variety Example.- I = A n P principal ideal.

The Weyl algebra p. 3 Characteristic variety Example.- I = A n P principal ideal. gr(i) = C[x,ξ]σ(P) Char(A n /I) = V C (gr(i)) = V C (σ(p)) = Char(P)

The Weyl algebra p. 3 Characteristic variety Example.- I = A n P principal ideal. gr(i) = C[x,ξ]σ(P) When Char(A n /I) = V C (gr(i)) = V C (σ(p)) = Char(P) A n (P 1,...,P l ) is not principal (or we have more than 1 unknowns the computation is more involved. We need filtrations on A n -modules.

The Weyl algebra p. 4 Graded rings The graded ring gr B (A n ) Exercise.- B k /B k 1 is a C vector space with dimension ( 2n+k 1 2n 1 ).

The Weyl algebra p. 4 Graded rings The graded ring gr B (A n ) Exercise.- B k /B k 1 is a C vector space with dimension ( 2n+k 1 2n 1 ). Quick answer.- The residue classes x α β = x α β +B k 1 with α+β = k generate the quotient vector space B k /B k 1 and, moreover, they are linearly independent since a linear combination λ αβ x α β α+β =k belongs to B k 1 if and only if all λ αβ are 0.

The Weyl algebra p. 4 Graded rings Proposition.- The Abelian group gr B (A n ) := k Z B k 1 B k has a natural structure of commutative ring with unit.

The Weyl algebra p. 4 Graded rings Proposition.- The Abelian group gr B (A n ) := k Z B k 1 B k has a natural structure of commutative ring with unit. Proof.- k,l Z, defined by µ kl : for P B k,q B l. B k B k 1 B l B l 1 B k+l B l+k 1 µ kl (P,Q) = PQ

The Weyl algebra p. 4 Graded rings Proof continued.- The map µ kl is well defined. Since PQ QP B k+l 1 we have PQ = QP.

The Weyl algebra p. 4 Graded rings Proof continued.- The map µ kl is well defined. Since PQ QP B k+l 1 we have PQ = QP. We define by bilinearity: µ : gr B (A n ) gr B (A n ) gr B (A n ) µ ( k P k, l Q l ) = k,l P k Q l where P k B k and Q l B l for all k,l.

The Weyl algebra p. 4 Graded rings Proposition.- The graded ring gr B (A n ) is isomorphic to the polynomial ring C[x,ξ] := C[x 1,...,x n,ξ 1,...,ξ n ] endowed with the natural grading.

The Weyl algebra p. 4 Graded rings Proposition.- The graded ring gr B (A n ) is isomorphic to the polynomial ring C[x,ξ] := C[x 1,...,x n,ξ 1,...,ξ n ] endowed with the natural grading. Proof.- k N, η k : B k /B k 1 C[x,ξ] k defined η k α+β k p αβ x α β +B k 1 = α+β =k p αβ x α ξ β.

The Weyl algebra p. 4 Graded rings Proof continued.- C[x,ξ] k homogeneous polynomials of degree k.

The Weyl algebra p. 4 Graded rings Proof continued.- C[x,ξ] k homogeneous polynomials of degree k. The family η k yields by bilinearity a natural isomorphism of graded rings. η : gr B (A n ) C[x,ξ]

The Weyl algebra p. 4 Graded rings We can prove similar results for gr F (A n ) := k Z F k 1 F k

The Weyl algebra p. 4 Graded rings We can prove similar results for gr F (A n ) := k Z F k 1 F k gr F (A n ) has a natural structure of commutative ring with unit.

The Weyl algebra p. 4 Graded rings We can prove similar results for gr F (A n ) := k Z F k 1 F k gr F (A n ) has a natural structure of commutative ring with unit. gr F (A n ) is isomorphic to the polynomial ring C[x,ξ] endowed with the ξ grading.

The Weyl algebra p. 4 Filtrations ona n modules AnF filtration on a A n module M is a family Γ = (M k ) k N of finitely generated C[x] submodules of M such that:

The Weyl algebra p. 4 Filtrations ona n modules AnF filtration on a A n module M is a family Γ = (M k ) k N of finitely generated C[x] submodules of M such that: (i) M k M k+1 for all k N.

The Weyl algebra p. 4 Filtrations ona n modules AnF filtration on a A n module M is a family Γ = (M k ) k N of finitely generated C[x] submodules of M such that: (i) M k M k+1 for all k N. (ii) k M k = M.

The Weyl algebra p. 4 Filtrations ona n modules 1.- (F k (A n )) k is an F filtration on A n.

The Weyl algebra p. 4 Filtrations ona n modules 1.- (F k (A n )) k is an F filtration on A n. 2.- k N, F k (C[x]) = C[x] for k 0. The family (F k (C[x])) k is an F filtration on C[x].

The Weyl algebra p. 4 Filtrations ona n modules 1.- (F k (A n )) k is an F filtration on A n. 2.- k N, F k (C[x]) = C[x] for k 0. The family (F k (C[x])) k is an F filtration on C[x]. 3.- Let I A n be an ideal and denote F k (I) = F k (A n ) I for k N. The family (F k (I)) k is an F filtration on I. 4.- Let I A n be an ideal and denote F k ( An I ) = F k(a n )+I I for k N. The family (F k (A n /I)) k is an F filtration on A n /I. It will be called the induced F filtration on A n /I

The Weyl algebra p. 4 Graded modules M, an A n module. Γ = (M k ) k an F filtration on M.

The Weyl algebra p. 4 Graded modules M, an A n module. Γ = (M k ) k an F filtration on M. gr Γ (M) := k M k 1 M k is also an Abelian group (and a vector space).

The Weyl algebra p. 4 Graded modules M, an A n module. Γ = (M k ) k an F filtration on M. gr Γ (M) := k M k 1 M k is also an Abelian group (and a vector space). An element in gr Γ (M) is a finite sum k m k where each m k belongs to M k.

The Weyl algebra p. 4 Graded modules Proposition.- The Abelian group gr Γ (M) has a natural structure of gr F (A n ) module. [The graded module associated with Γ on M]

The Weyl algebra p. 4 Graded modules Proposition.- The Abelian group gr Γ (M) has a natural structure of gr F (A n ) module. [The graded module associated with Γ on M] Proof.- ν : gr F (A n ) gr Γ (M) gr Γ (M) defined by bilinearity from the maps defined by B k B k 1 M l M l 1 M k+l M k+l 1 P k m l = P k m l.

The Weyl algebra p. 5 gr Γ (M) f.g. M f.g. Proposition.- M an A n module. Γ = (M k ) k a filtration on M. If gr Γ (M) if finitely generated over C[x,ξ] then M is finitely generated.

The Weyl algebra p. 5 gr Γ (M) f.g. M f.g. Proposition.- M an A n module. Γ = (M k ) k a filtration on M. If gr Γ (M) if finitely generated over C[x,ξ] then M is finitely generated. Idea of the Proof.- Assume {m 1,...m l } is a homogeneous generating system of gr Γ (M). Then (using induction on the Γ order of the elements in M one proves that) M is generated by {m 1,...,m l }

The Weyl algebra p. 5 Good filtrations (I) Exercise.- I A n left ideal. Prove that: (i) The induced F filtration on I is a good filtration. (ii) The induced F filtration on A n /I is a good filtration. (iii) Any finitely generated A n module M admits a good F filtration. (Hint for (iii). If m 1,...,m r is a generating system for M, define M k := j F km j for k N. The family (M k ) k is a good filtration on M)

The Weyl algebra p. 5 Good filtrations (II) Proposition.- M a (left) A n module. Γ = (M k ) k a F filtration on M. The following conditions are equivalents

The Weyl algebra p. 5 Good filtrations (II) Proposition.- M a (left) A n module. Γ = (M k ) k a F filtration on M. The following conditions are equivalents i) Γ is a good F filtration on M.

The Weyl algebra p. 5 Good filtrations (II) Proposition.- M a (left) A n module. Γ = (M k ) k a F filtration on M. The following conditions are equivalents i) Γ is a good F filtration on M. ii) There exists k 0 N s.t. M k+l = F l M k for all l and all k k 0.

The Weyl algebra p. 5 Good filtrations (III) Proposition.- M a (left) A n module. Γ = (M k ) k, and Γ = (M k ) k two F filtrations on M. Then we have:

The Weyl algebra p. 5 Good filtrations (III) Proposition.- M a (left) A n module. Γ = (M k ) k, and Γ = (M k ) k two F filtrations on M. Then we have: i) If Γ is a good F filtration then there exists k 1 N s.t. M k M k+k 1 for all k N. ii) If Γ and Γ are both good filtrations then there exists k 2 N s. t. M k k 2 M k M k+k 2 for all k N.

The Weyl algebra p. 5 Good filtrations (IV) Proposition.- M a finitely generated A n module. Γ = (M k ) k, and Γ = (M k ) k two good F filtrations on M. Then we have:

The Weyl algebra p. 5 Good filtrations (IV) Proposition.- M a finitely generated A n module. Γ = (M k ) k, and Γ = (M k ) k two good F filtrations on M. Then we have: Ann grf (A n )(gr Γ (M)) = Ann grf (A n )(gr Γ (M))

The Weyl algebra p. 5 Characteristic variety Definition.- M finitely generated A n module. The characteristic variety of M is defined as Char(M) = V(Ann grf (A n )(gr Γ (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 = A n /I. Characteristic varieties are not easy to compute. There are algorithms computing characteristic varieties that use Groebner basis in the Weyl algebra A n. These algorithms have been implemented: Macaulay2, Risa/Asir, Singular...

The Weyl algebra p. 5 Dimension We associate to any f.g. A n module M its dimension which is a positive integer number (unless for M = (0); we write dim((0)) = 1. Definition.- M finitely generated A n module. The dimension of M (denoted dim(m)) is the Krull dimension of Char(M) the characteristic variety of M. M = A n /I for I a left ideal in A n. dim(m) is the Krull dimension of the quotient ring C[x, ξ]/gr(i).

The Weyl algebra p. 5 Dimension Example.- I = A n P, for P A n.

The Weyl algebra p. 5 Dimension Example.- I = A n P, for P A n. if P is a nonzero constant then A n /I = (0) and its dimension is -1.

The Weyl algebra p. 5 Dimension Example.- I = A n P, for P A n. if P is a nonzero constant then A n /I = (0) and its dimension is -1. If P = 0 then Char(A n /I) = Char(A n ) = C n C n and then its dimension is 2n. If P A n \C then σ(p) C[x][ξ] is a non-constant polynomial and the Krull dimension of V C (σ(p)) is 2n 1. So in this case dim(a n /I) = 2n 1.

The Weyl algebra p. 5 Dimension. Bernstein inequality [Bernstein s inequality] Let M be a nonzero finitely generated A n -module. Then 2n dim(m) n.

The Weyl algebra p. 5 Dimension. Bernstein inequality [Bernstein s inequality] Let M be a nonzero finitely generated A n -module. Then 2n dim(m) n. A finitely generated A n module M is said to be holonomic if either M = (0) or dim(m) = n.

The Weyl algebra p. 5 Dimension. Hilbert polynomials M finitely generated A n module. Γ = (M k ) k a B filtration on M.

The Weyl algebra p. 5 Dimension. Hilbert polynomials M finitely generated A n module. Γ = (M k ) k a B filtration on M. Definition.- Let M be an A n -module. A B filtration on M is a family Γ = (M k ) k N of finitely dimensional C vector subspaces of M such that: (i) M k M k+1 for all k N. (ii) k M k = M. (iii) B k M l M k+l for all (k,l). Denote by HF M,Γ : N N the map

The Weyl algebra p. 6 Dimension. Hilbert polynomials M finitely generated A n module. Γ = (M k ) k a good B filtration on M.

The Weyl algebra p. 6 Dimension. Hilbert polynomials M finitely generated A n module. Γ = (M k ) k a good B filtration on M. Proposition.- There exists a unique polynomial HP M,Γ (t) Q[t] such that for ν big enough. HF M,Γ (ν) = HP M,Γ (ν) HP M,Γ (t) is called the Hilbert polynomial associated with the B filtration Γ on M.

The Weyl algebra p. 6 Dimension. Hilbert polynomials (J. Bernstein, 1971) M finitely generated A n module. Γ = (M k ) k a good B filtration on M.

The Weyl algebra p. 6 Dimension. Hilbert polynomials (J. Bernstein, 1971) M finitely generated A n module. Γ = (M k ) k a good B filtration on M. dim(m) = deg(hp M,Γ (t))

The Weyl algebra p. 6 Multiplicities. Hilbert polynomials M finitely generated A n module.

The Weyl algebra p. 6 Multiplicities. Hilbert polynomials M finitely generated A n module. Definition.- The multiplicity of M is e(m) := a d d!.

The Weyl algebra p. 6 Multiplicities. Hilbert polynomials M finitely generated A n module. Definition.- The multiplicity of M is e(m) := a d d!. where a d t d is the leading term of the Hilbert polynomial HP M,Γ (t) for a (or any) good B filtration Γ on M.