The Gröbner fan of an An-module
Abstract
Let I be a non-zero left ideal of the Weyl algebra An of order n over a field k and let L:R2n→R be a linear form defined by L(α,β)=∑i=1neiαi+∑i=1nfiβi. If ei+fi≥0, then L defines a filtration F•L on An. Let grL(I) be the graded ideal associated with the filtration induced by F•L on I. Let finally U denote the set of all linear form L for which ei+fi≥0 for all 1≤i≤n. The aim of this paper is to study, by using the theory of Gröbner bases, the stability of grL(I) when L varies in U. In a previous paper, we obtained finiteness results for some particular linear forms (used in order to study the regularity of a D-module along a smooth hypersurface). Here we generalize these results by adapting the theory of Gröbner fan of Mora-Robbiano to the D-module case. Our main tool is the homogenization technique initiated in our previous paper, and recently clarified in a work by F. Castro-Jiménez and L. Narváez-Macarro.
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The Grobner fan of an A n -mo dule A. Assi , F.J. Castro-Jimenez y and M. Granger z Abstract Let I b e a non-zero left ideal of the Weyl algebra A n of order n over a eld k and let L : R 2 n ?! R b e a linear form dened by L ( ; ) = P n i =1 e i i + P n i =1 f i i . If e i + f i 0, then L denes a ltration F L on A n . Let gr L ( I ) b e the graded ideal asso ciated to the ltration induced by F L on I . Let nally U denote the set of all linear forms L for which e i + f i 0 for all 1 i n . The aim of this pap er is to study, by using the theory of Grobner bases, the stability of gr L ( I ) when L varies in U . In a previous pap er, we obtained niteness results for some particular linear forms (used in order to study the regularity of a D -mo dule along a smo oth hyp ersurface). Here we generalize these results by adapting the theory of Grobner fan of Mora-Robbiano to the D -mo dule case. Our main to ol is the homogenization technique initiated in our previous pap er, and recently claried in a work of F. Castro-Jimenez and L. Narvaez-Macarro. 1991 Math. Sub j. Class: Primary 35A27, Secondary 13P10, 68Q40 1 Intro duction Let A n ( k ) denote the Weyl algebra of order n over a eld k : A n ( k ) ( A n for short) is the central k -algebra generated by x i ; D i ; i = 1 ;::: ;n with relations [ x i ; x j ] = [ D i ; D j ] = 0 and [ D i ; x j ] = ij . Let P = P ; p ; x D b e a non-zero element of A n and denote by N ( P ) the Newton diagram of P , namely N ( P ) = f ( ; ) 2 N 2 n ; p ; 6 = 0 g : If L : R 2 n ?! R is the linear form dened by L ( ; ) = P n i =1 e i i + P n i =1 f i i , then the L - order ord L ( P ) of P is dened to b e the maximal element in the set of L ( ; ) ; ( ; ) 2 N ( P ). If furthermore e i + f i 0 for all 1 i n , then ord L ( P :Q ) = ord L ( P ) + ord L ( Q ) for all non-zero elements P ; Q 2 A n , in particular L denes a ltration on A n (where for all k 2 R ; F L k ( A n ) = f P 2 A n ; ord L ( P ) k g ). If e i + f i > 0 for all i = 1 ;::: ;l (resp. e i + f i = 0 for all i = l + 1 ;::: ;n ), then the asso ciated graded algebra is: Universite d'Angers y Universidad de Sevilla. Partially supp orted by the DGICYT PB94-1435 z Universite d'Angers 1
gr L ( A n ) ' k [ x 1 ;::: ;x n ; 1 ;::: ; l ][ D l +1 ;::: ;D n ] with relations: x i x j = x j x i ; x i m = m x i ; x i D p = D p x i ? ip ; m D p = D p m for all 1 i; j n , 1 m l and l + 1 p n . The principal symb ol of P is the element of gr L ( A n ), L ( P ) = X L ( ; )=ord L ( P ) p x 1 1 l l D l +1 l +1 D n n Let us p oint out that in the commutative case, there is no condition of the typ e e i + f i 0. Here, since ? x i D i + D i x i = 1, we must require ord L (1) ord L ( x i ) + ord L ( D i ). Let I b e a non-zero left ideal of A n and let gr L ( I ) b e the graded ideal asso ciated with the ltration induced by F L on I . Let nally U denote the set of all linear form L for which e i + f i 0 for all 1 i n . The aim of this pap er is to study, by using the theory of standard and Grobner bases, the stability of gr L ( I ) when L varies in U . Let Y b e the hyp ersurface of k n dened by x 1 = 0. Given two non negative reals p; q , we dene the linear form L p;q on R 2 n by L p;q ( ; ) = p: ( P n i =1 i ) + q : ( 1 ? 1 ): this is an interp olation b etween the ltration F by the order of op erators ( q = 0) and the V -ltration of MalgrangeKashiwara ( p = 0). In [10], Y. Laurent proved, using 2-micro dierential op erators, that the radical ideal p gr L p;q ( I ) is not a ( F ; V )- homogeneous ideal for only a nite set of rational numb ers r = p=q (eventually empty), when p; q vary in R 2 + (in [11] and [13], an analytic interpretation of these numb ers is given). C. Sabbah and F. Castro proved in [15 ] the same result by using a lo cal attener. In [2] we obtained, using the theory of standard bases, a constructive pro of of this result. This allowed us to give an algorithm for the calculation of these numb ers. So, it was natural to think ab out general niteness results when L varies in U . Recall that the theory of Grobner bases (cf [5]) works very well in the Weyl algebra A n (cf [6],[7] and [8]). However, when the co ecients of the linear form L 2 U are negative, the division pro cess in A n can b e innite. In [2], in order to avoid this diculty, we worked in A n [ t ] by homogenizing with resp ect to the total order in a way inspired by [12]. However, the non commutativity of A n [ t ] causes some diculty, since divisions by homogeneous elements do not pro duce necessarily homogeneous remainders. Although our algorithm (which consists in rehomogenizing the remainders and iterating division) allows us the calculation of standard bases with resp ect to any L 2 U , it do es not seem to b e adapted to the question we are working on. In [9], this diculty is avoided in the following natural way: consider the graded k -algebra B , generated by x i ; D i ; i = 1 ;::: ;n and t with homogeneous relations: [ t; x i ] = [ t; D i ] = [ x i ; x j ] = [ D i ; D j ] = 0 ; [ D i ; x j ] = ij t 2 : This k -algebra coincides with the Rees algebra asso ciated with the Bernstein ltration on A n . The homogenization pro cess b etween A n and B verify the same prop erties as in the commutative case, in particular the notion of the homogenized ideal h ( I ) of I is well dened. On the other 2
hand, we can get the dierent graded ideals gr L ( I ) from calculations of the graded ideals of h ( I ). Since the notion of reduced standard bases exists for ideals in B , then the natural way in order to study our question is in adapting to the D {mo dule case the theory of Grobner fan develop ed by T. Mora and L. Robbiano in [14]. Let us summarize the structure of the pap er: in Section 2 we recall some of the results of [9] related to the homogenization problem. We also prove that a standard basis w.r.t. the Bernstein ltration of an ideal I in A n gives us a generating system of h ( I ) in B = A n [ t ]. In section 3, paragraph 3.2., we obtain niteness results for the set of graded ideals gr L ( J ) where J is an homogeneous ideal of A n [ t ]. The main to ol we use here is the Hilb ert function of J . This notion has also b een used by one of us [1] in order to prove similar results in the commutative case. The results of paragraph 3.2. are then applied in order to prove that the set of graded ideals gr L ( I ) ; L 2 U is nite (paragraph 3.3.). Finally, in section 4, we study the repartition of the graded ideals gr L ( h ( I )), where L varies in U . We dene rst the notion of privileged exp onent (or stairs) Exp L ( h ( I )) of an ideal asso ciated with a xed well ordering on N 1+2 n (see 3.3.). Our main result is then the following, which generalizes the results in the commutative case as found in [1], [14] and [17]: Theorem 1.1 There exists a partition E of U into convex rational polyhedral cones, such that for al l element 2 E , gr L ( h ( I )) and Exp L ( h ( I )) do not depend on L 2 (and the same is true for gr L ( I ) ). Some results of this article has b een used in [16]. 2 Homogenization We shall use here the results of [9]. Let A n [ t ] denote the algebra A n [ t ] = k [ t; x ][ D ] = k [ t; x 1 ;:::;x n ][ D 1 ;:::;D n ] with relations: [ t; x i ] = [ t; D i ] = [ x i ; x j ] = [ D i ; D j ] = 0 ; [ D i ; x j ] = ij t 2 : The algebra A n [ t ] is a graded algebra, the degree of the monomial t k x D b eing k + j j + j j : In fact, the k {algebra A n [ t ] is isomorphic to the Rees algebra asso ciated with the Bernstein ltration on A n . The algebra k [ t ] is central in A n [ t ], and the quotient algebra A n [ t ] = h t ? 1 i is isomorphic to A n . Let P = P ; p ; x D b e a non-zero op erator of A n . We denote by N ( P ) the Newton diagram of P , N ( P ) = f ( ; ) 2 N 2 n ; p ; 6 = 0 g ; then we denote by ord T ( P ) the total order of P 3
ord T ( P ) = max fj j + j j ; p ; 2 N ( P ) g : The dierential op erator h ( P ) = X ; p ; t ord T ( P ) ?j j?j j x D 2 A n [ t ] is called the homogenization of P . If H = P k ;; h k ;; t k x D is an element of A n [ t ], we denote by H j t =1 the op erator of A n H j t =1 = X k ;; h k ;; x D : With the notations ab ove, for all P ; Q 2 A n and for all homogeneous element H 2 A n [ t ], 1. h ( P Q ) = h ( P ) h ( Q ). 2. There exists k ; l ; m 2 N such that t k h ( P + Q ) = t l h ( P ) + t m h ( Q ). 3. There exists k 2 N such that t k h ( H j t =1 ) = H . Let < b e a total ordering on N 2 n (not necessarily a well ordering), compatible with sums. We recall that the extension of < , denoted by < h , is the total well ordering on N 1+2 n (compatible with sums) dened by: ( k ; ; ) < h ( k 0 ; 0 ; 0 ) () 8 < : k + j j + j j < k 0 + j 0 j + j 0 j or k + j j + j j = k 0 + j 0 j + j 0 j and ( ; ) < ( 0 ; 0 ) Since < h is a total well ordering compatible with sums, we have for all non-zero element G = P a;; g ( a;; ) t a x D the notion of privileged exp onent of G w.r.t. < h , which we denote by exp < h ( G ): If N ( G ) = f ( a; ; ); g ( a;; ) 6 = 0 g denote the Newton diagram of G , then exp < h ( G ) = max < h N ( G ). Also we have for all non-zero ideal J of A n [ t ], the notion of Grobner (or standard) basis of J , namely, if we denote by Exp < h ( J ) = f exp < h ( P ) j P 2 J g ; then f P 1 ;::: ;P r g J is a standard basis of J if Exp < h ( J ) = r [ i =1 (exp < h ( P i ) + N 1+2 n ) : We have nally a division theorem in A n [ t ], analogous to that in the ring of p olynomials or in the Weyl algebra A n . For more details, see [9]. Let : N 1+2 n = N N 2 n ! N 2 n denote the natural pro jection, then we have: 1. If P 2 A n , then (exp < h ( h ( P ))) = exp < ( P ). 4
2. More generally, if H is an homogeneous element of A n [ t ], then (exp < h ( H )) = (exp < h ( h ( H j t =1 ))) = exp < ( H j t =1 ) : Let I b e a left ideal of A n . We denote by h ( I ) the homogeneous ideal of A n [ t ], generated by f h ( P ) j P 2 I g . We call h ( I ) the homogenized ideal of I . With these notations we have the following (see [9]): 1. (Exp < h ( h ( I ))) = Exp < ( I ) : 2. Let f P 1 ;::: ;P m g b e a generating system of I and let e I b e the ideal generated by f h ( P 1 ) ;::: ;h ( P m ) g in A n [ t ]. Then (Exp < h ( e I )) = Exp < ( I ). Let B ( A n ) denote the Bernstein ltration on A n (that is the case with e i = f i = 1 for all i = 1 ;::: ;n ). If P is a dierential op erator in A n , then we denote by B ( P ) the principal symb ol of P w.r.t. the Bernstein ltration. If I is an ideal of A n , then we denote by gr B ( I ) the graded ideal asso ciated with the induced Bernstein ltration on I . A standard basis w.r.t. the Bernstein ltration has the following interesting prop erty: Lemma 2.1 Let I be a non-zero left ideal of A n and let f P 1 ;::: ;P m g be a family of dierential operators of I . The fol lowing assertions are equivalent: i) h ( I ) = ( h ( P 1 ) ;::: ;h ( P m )) : ii) gr B ( I ) = ( B ( P 1 ) ; : : : ; B ( P m )) : Pro of. The pro of is classical and uses the structure of graded algebra of A n [ t ] (see for details [3]). Remark that a standard basis with resp ect to the Bernstein ltration satises ii), but the converse is in general false. 3 Finiteness results Let L 2 U (see 1) and consider the extension of L to R R 2 n (by abuse of notation we continue to write L and U in R R 2 n ), L : R R 2 n ! R , such that L ( a; ; ) = P n i =1 e i i + P n i =1 f i i . Recall in particular that e i + f i 0 for all 1 i n . Let P b e a non-zero dierential op erator of A n [ t ]. We dene the L {order of P in the usual way (we denote this element by ord L ( P )). If P ; Q 2 A n [ t ], then ord L ( P Q ) = ord L ( P ) + ord L ( Q ), consequently the L {order denes a ltration on A n [ t ], which we shall call the L -ltration and we shall denote by F L ( A n [ t ]). We denote by L ( P ) the principal symb ol of P w.r.t. the L -order, precisely, if P = P p ( t ) x D , then L ( P ) = P L ( ; )=ord L ( P ) p ( t ) x 1 1 l l D l +1 l +1 D n n with l is as dened in the intro duction. If J is a non-zero homogeneous ideal of A n [ t ], we denote by gr L ( J ) the graded ideal asso ciated with the induced L {ltration on J (i.e. gr L ( J ) is the ideal of gr L ( A n [ t ]) generated by f L ( P ) j P 2 J g ). In this section we shall prove that, if the co ecients e i ; f i vary in R , then the set of gr L ( J ) is nite. We shall use in the pro of the Hilb ert function, therefore we shall start by recalling some of its prop erties. 5
3.1 Hilb ert function Let E N 1+2 n such that E + N 1+2 n = E . We dene the Hilb ert function of E (and we denote it by H E ) to b e the map H E : N 7?! N : H E ( k ) = ] f ( a; ; ) 2 N 1+2 n n E ; a + j j + j j = k g ; 8 k 2 N : Let J b e an homogeneous ideal of A n [ t ] = k 2 N A n [ t ] k , where A n [ t ] k is the k {vector space generated by the monomials t a x D of total degree a + j j + j j = k . We set J k = A n [ t ] k \ J . Let b e a total well ordering on N 1+2 n compatible with sums, and let E = Exp ( J ). Lemma 3.1 For al l k 2 N , we have: dim k ( A n [ t ] k =J k ) = ] f ( a; ; ) 2 N 1+2 n n E ; a + j j + j j = k g = H E ( k ) Pro of. Let f P 1 ;::: ;P m g b e a family of homogeneous op erators of J such that: E = m [ i =1 (exp ( P i ) + N 1+2 n ) : If we denote by k i = ord T ( P i ), then for all P 2 A n [ t ] k , there exists a family of homogeneous elements Q 1 ;::: ;Q m ; R of A n [ t ] such that: 1. P = P m i =1 Q i P i + R . 2. ord T ( Q i ) = k ? k i ; ord T ( R ) = k . 3. If R 6 = 0, then the Newton diagram N ( R ) N 2 n +1 n E . Thus P 2 J k () R = 0. In particular, P + J k = R + J k . This proves that the classes, mo dulo J k , of the monomials t a x D , with a + j j + j j = k , ( a; ; ) 62 E form a basis for A n [ t ] k =J k over k . This proves our assertion. Let, for all k 2 N , H J ( k ) = dim k ( A n [ t ] k =J k ). This denes a map H J : N ! N which we call the Hilb ert function of J . By Lemma 3.1, H J = H E do es not dep end on . 3.2 Finiteness Theorems for homogeneous ideals Let O ( N 1+2 n ) denote the set of total well ordering on N 1+2 n compatibles with sums (for such an order, 0 is the smallest element, this implies in particular that exp ( P Q ) = exp ( P ) + exp ( Q ))). Theorem 3.2 Let J be a non-zero homogeneous ideal of A n [ t ] . Then f Exp ( J ) j 2 O ( N 1+2 n ) g is a nite set. 6
Pro of. By Lemma 3.1, it suces to prove that the set of subsets E N 1+2 n such that: 1. E + N 1+2 n = E . 2. H E = H J . is nite. Denote this set by E and assume that E is innite. Given an element E of E and an integer k 2 N , we set E ( k ) = f 2 E ; j j k g : Let k 0 2 N b e the smallest integer for which H J ( k ) < dim k ( A n [ t ] k ) (such an integer exists b ecause J 6 = (0)). Since N 1+2 n ( k 0 ) is a nite set, one of the p ossible choices of E ( k 0 ) o ccurs for all E in an innite subset E 1 = f E i g i 1 of E . Thus, there are elements i 2 N 1+2 n ( k 0 ) ; 1 i r such that E i; ( k 0 ) = ( [ r i =1 ( i + N 1+2 n )) ( k 0 ) for all i 1 : Assume, without loss of generality, that E 1 = E and set S 0 = [ r i =1 ( i + N 1+2 n ) : Clearly S 0 E i for all i 1, on the other hand E i 6 = E j for all i 6 = j . In particular H J 6 = H S 0 . Let consequently k 1 > k 0 b e the smallest integer for which H J ( k 1 ) < H S 0 ( k 1 ). For all j 2, there exists j 2 E j n S 0 such that j j j = k 1 . The set N 1+2 n ( k 1 ) b eing nite, there is an innite subset E 2 E and elements r + i ; 1 i r + r 1 , in ( N 1+2 n n S 0 ) ( k 1 ) such that: E j; ( k 1 ) = ( [ r + r 1 i =1 ( j + N 1+2 n ) ( k 1 ) for all E j 2 E 2 : Let S 1 = [ r + r 1 i =1 ( j + N 1+2 n ) ; then S 0 S 1 . Now rep eat the same argument with E 2 and S 1 ,... We construct this way an innite sequence S 0 S 1 ::: of subsets of N 1+2 n with S i + N 1+2 n = S i for all i 0. This is imp ossible. As a consequence of Theorem 3.2. we get the following result: Theorem 3.3 Let J be a non-zero homogeneous ideal of A n [ t ] . Then f gr L ( J ); L 2 U g is a nite set. 7
Pro of. Fix 2 O ( N 1+2 n ), then for any L 2 U , denote by L the total ordering on N 1+2 n such that: ( k ; ; ) L ( k 0 ; 0 ; 0 ) () 8 > > > > > > > > < > > > > > > > > : k + j j + j j < k 0 + j 0 j + j 0 j or k + j j + j j = = k 0 + j 0 j + j 0 j and 8 > > < > > : L ( k ; ; ) < L ( k 0 ; 0 ; 0 ) or L ( k ; ; ) = L ( k 0 ; 0 ; 0 ) and ( k ; ; ) ( k 0 ; 0 ; 0 ) (Where we recall that L ( k ; ; ) = P n i =1 e i i + P n i =1 f i i ). Clearly L 2 O ( N 1+2 n ). On the other hand, by 3.2, f Exp L ( J ) j L 2 U g is a nite set. Consequently we have only to prove that, if E N 1+2 n with E + N 1+2 n = E , then f gr L ( J ) j Exp L ( J ) = E ; L 2 U g is a nite set. Fix to this end E and let L 2 U b e such that E = Exp L ( J ). Then consider a reduced standard basis B = f Q 1 ; : : : ; Q m g of J w.r.t. L (i.e. [ m i =1 (exp L ( Q i ) + N 1+2 n ) = E and N ( Q i ) n f exp L ( Q i ) g N 1+2 n n E , for all 1 i m , where N ( Q i ) is the Newton diagram of Q i ). Clearly B is also a reduced standard basis of J w.r.t. L 0 , for all L 0 2 U such that Exp L 0 ( J ) = E (indeed, if exp L 0 ( Q i ) 6 = exp L ( Q i ), we would have exp L 0 ( Q i ) = 2 E ). In particular, as proved in [2], Lemma 1.3.3., f L 0 ( Q 1 ) ;::: ; L 0 ( Q m ) g generates gr L 0 ( J ) for all L 0 2 U such that Exp L 0 ( J ) = E . Every N ( Q i ) b eing nite, we have only a nite numb er of p ossibilities. This proves our assertion. We shall nally give a b ound for the cardinality of O ( J ) = f Exp ( J ); 2 O ( N 1+2 n ) g . Let for all E 2 O ( J ), J E = ( y 1 ;::: ;y s ) k [ y 1 ;::: ;y 2 n +1 ], where y 1 ;::: ;y 2 n +1 are indeterminates and f 1 ;::: ; s g is the minimal b oundary of E , that is E = [ s i =1 ( i + N 1+2 n ) and for all k = 1 ;::: ;s , k = 2 [ i 6 = k ( i + N 1+2 n ). Clearly H J E = H E , then we have: ] O ( J ) = ] f J E ; E 2 O ( J ) g ] f M k [ y 1 ;::: ;y 2 n +1 ] monomial ideal ; H M = H J g Let d ( J ) denote the maximal degree of the elements arising in the minimal b oundaries of f Exp ( J ) ; 2 O ( N 2 n +1 ) g . If ( d 1 ; d 2 ;::: ) denote the values of the Hilb ert function of J , then we have: Prop osition 3.4 ] O ( J ) d ( J ) Y k =1 C a k a k ? d k ; where a k = dim k A n [ t ] k = C 2 n + k k and C a b is the binomial coecient. 8
Pro of. The numb er of p oints in E which are exp onents of monomials of degree k is exactly a k ? d k . This proves our assertion. 3.3 Finiteness Theorems for ideals in A n Let I b e non-zero left ideal of A n . The aim of this paragraph it to give for I analogous results to those of 3.2. Let to this end < b e a total well ordering on N 2 n , compatible with sums, and denote, for all L 2 U , by < L the total ordering on N 2 n such that: ( ; ) < L ( 0 ; 0 ) , 8 < : L ( ; ) < L ( 0 ; 0 ) or L ( ; ) = L ( 0 ; 0 ) and ( ; ) < ( 0 ; 0 ) Let P 2 A n b e a non-zero dierential op erator. We denote by exp < L ( P ) the privileged exp onent of P w.r.t. < L , i.e. exp < L ( P ) = max < L N ( P ) (See [2] for the main prop erties of the privileged exp onent of an op erator). We also set Exp < L ( I ) = f exp < L ( P ) j P 2 I n f 0 gg : Clearly Exp < L ( I ) + N 2 n = Exp < L ( I ). Theorem 3.5 For a given total wel l ordering < on N 2 n , compatible with sums, f Exp < L ( I ) j L 2 U g is a nite set. Pro of. This results follows from Theorem 3.2 as follows: rstly we remark that, with the notations of section 2, L = < h L , for the following choice of , ( k ; ; ) ( k 0 ; 0 ; 0 ) , 8 < : ( ; ) < ( 0 ; 0 ) or ( ; ) = ( 0 ; 0 ) et k < k 0 Now apply (Exp < h ( h ( I ))) = Exp < ( I ), to the order < = < L . Theorem 3.6 f gr L ( I ) j L 2 U g is a nite set. Pro of. Let h ( I ) b e the homogenized ideal of I in A n [ t ]. The asso ciated graded ideal gr L ( h ( I )) is an ideal of the ring gr L ( A n [ t ]) ' (gr L ( A n ))[ t ] (where [ x i ; i ] = 0 if e i + f i > 0 and [ D i ; x i ] = t 2 if e i + f i = 0). Let : A n [ t ] 7?! A n ; ( H ) = H j t =1 denote the deshomogenization morphism. If L 2 U , gives rise to a morphism L : gr L ( A n [ t ]) 7?! gr L ( A n ) ' gr L ( A n [ t ]) = ( t ? 1) : 9