aa r X i v : . [ m a t h . A C ] F e b ON THE SUPPORT OF RELATIVE D -MODULES ROBIN VAN DER VEER
Abstract.
In this article we prove that although cyclic relative D -modules are notfinitely generated as modules over the polynomial ring, their support is open andZariski dense in the vanishing set of their annihilator. As a consequence we obtainan alternative proof of a conjecture of Budur which was recently proven by Budur,van der Veer, Wu and Zhou. Contents
1. Introduction 12. A lemma in commutative algebra, and a non-commutative corollary 33. Gr¨obner bases for D n -modules 5References 81. Introduction
The purpose of this article is to give an alternative proof of the following theorem:
Theorem A (Conjectured in [Bud15], proven in [BvVWZ21]) . Denote by Exp : C p → ( C ∗ ) p the coordinate-wise exponential map. Let F = ( f , . . . , f p ) be a tuple of polyno-mials on C n , and denote by B F the Bernstein-Sato ideal of the tuple, and by ψ F ( C C n ) the specialization complex of F . ThenExp ( Z ( B F )) = supp ( C ∗ ) p ( ψ F ( C C n )) . We refer to [Bud15], [BvVWZ21] for background on B F and ψ F . The inclusionExp( Z ( B F )) ⊃ supp ( C ∗ ) p ( ψ F ( C C n ))was proven already in [Bud15]. To analyse the reverse inclusion the following criterioncan be extracted from the proof of [Bud15, Proposition 1.7]. Proposition 1.1 ([Bud15]) . If α ∈ Z ( B F ) and (1) D n [ s , . . . , s p ] f s . . . f s p p D n [ s , . . . , s p ] f s +11 . . . f s p +1 p ⊗ C [ s ,...,s p ] C [ s , . . . , s p ] m α = 0 , then Exp ( α ) ∈ supp ( C ∗ ) p ( ψ F ( C C n )) . Mathematics Subject Classification.
Key words and phrases.
Groebner basis; Weyl algebra; Bernstein-Sato ideal; b -function. Since supp ( C ∗ ) p ( ψ F ( C C n )) is a closed subset of ( C ∗ ) p , Theorem B follows if we canprove (1) for α in an open Zariski dense subset of Z ( B F ). With this in mind thefollowing theorem is our main result. Theorem B.
Let J ⊂ D n [ s , . . . , s p ] be a left ideal, where D n is the Weyl algebra.Let p be a minimal prime divisor of J ∩ C [ s , . . . , s p ] . Then there exists a polynomial h ∈ C [ s , . . . , s p ] \ p such that for all α ∈ Z ( p ) \ Z ( h ) with maximal ideal m α , (cid:18) D n [ s , . . . , s p ] J (cid:19) ⊗ C [ s ,...,s p ] (cid:18) C [ s , . . . , s p ] m α (cid:19) = 0 . Theorem A was recently proven in [BvVWZ21] where it was deduced from thefollowing theorem.
Theorem C ([BvVWZ21]) . Let F = ( f , . . . , f p ) be a tuple of polynomials on C n . Forevery codimension irreducible component H of Z ( B F ) there is a Zariski open subset V ⊂ H such that for every α ∈ V with maximal ideal m α , D n [ s , . . . , s p ] f s . . . f s p p D n [ s , . . . , s p ] f s +11 . . . f s p +1 p ⊗ C [ s ,...,s p ] C [ s , . . . , s p ] m α = 0 . Theorem C is a special case of Theorem B, since it only deals with codimension1 components. This restriction meant that in [BvVWZ21] results from [Mai16] werenecessary to conclude A from Theorem C. More precisely, the following result from[Mai16] was used to prove Theorem A in [BvVWZ21]:
Theorem D ([Mai16]) . Every irreducible component of Z ( B F ) can be translated alongan integer vector into a codimension component of Z ( B F ) . Theorem D is a corollary of Theorem A, since it is known from [BLSW17, Theorem1.3] that supp ( C ∗ ) p ( ψ F ( C C n )) is a union of codimension 1 torsion translated subtori of( C ∗ ) p . In particular, our methods provide a new proof of Theorem D.Another advantage of Theorem B over the methods in [BvVWZ21] is that Theorem Bdoes not depend on the module D n [ s , . . . , s p ] / J being relatively holonomic, as definedin [BvVWZ21].The proof strategy for Theorem B is to specialize a suitable Gr¨obner basis for J + p toa Gr¨obner basis for ( D n [ s , . . . , s p ] / J ) ⊗ C [ s ,...,s p ] ( C [ s , . . . , s p ] / m α ). Using a result onspecializing Gr¨obner bases from [Ley01] and [Oak97] we can then conclude Theorem B.This method of proof works in every ring where Gr¨obner basis methods are available.In Section 2 we start with some technical preliminaries that allow us to conclude inLemma 2.4 that ( J + p ) ∩ C [ s , . . . , s p ] = p . This result will be used in Section 3 toconstruct the Gr¨obner basis we need and to control the specialization. Acknowledgement.
We would like to thank Nero Budur and Alexander Van Werde forthe helpful comments and suggestions.The author is supported by a PhD Fellowship of the Research Foundation - Flanders.
N THE SUPPORT OF RELATIVE D -MODULES 3 A lemma in commutative algebra, and a non-commutative corollary
We denote A = C [ s , . . . , s p ]. An ideal q ⊂ A is called primary if for all x, y ∈ A with xy ∈ q , either x ∈ q or y ∈ √ q . If q is primary, then p = √ q is a prime ideal.When we say that q is p -primary we mean that q is primary and √ q = p . Lemma 2.1.
Let q ⊂ A be a p -primary ideal. If q = p , then there exists an f ∈ p \ q such that f p ⊂ q .Proof. Let N ⊂ A/ q be the nilradical, which is non-zero since q = p . Let g , . . . , g m ∈ N be a set of generators. Let f = 1 ∈ A/ q . Using induction we define for i = 1 , . . . , m : f i +1 = f i g k i +1 − i +1 , where k i +1 ∈ Z is the smallest integer for which f i g k i +1 i +1 = 0 . Notice that such k i +1 always exists, since each g i is nilpotent and that k i +1 is alwaysat least 1 since f i is not zero, by induction. By construction we have that for each i = 1 , . . . , m , g i f m = 0 . Let f be a lift of f m to A . Since f m = 0, f q , and since f m is nilpotent, f ∈ p . Let p ∈ p . Then in A/ q we can write p + q = P mi =1 a i g i for some a i ∈ A/ q . Then f p + q = m X i =1 a i g i f m = 0 , so that f p ∈ q . (cid:3) Every ideal J ⊂ A has a primary decomposition . This means that we can write J = T mi =1 q i such that(1) every q i is primary, and(2) for all 1 ≤ j ≤ m , T mi =1 q i ( T mi =1 i = j q i , and(3) The prime ideals √ q , . . . , √ q m are pairwise distinct.The minimal (under the inclusion order) elements of the set {√ q i } are uniquely deter-mined by J , and are called the minimal prime divisors of J . Theorem 2.2.
Let J ⊂ A be an ideal with primary decomposition J = T mi =1 q i , m > .Let √ q j be a minimal prime divisor. Then there exists an h ∈ A \ q j such that h √ q j ⊂ J .Proof. We assume without loss of generality that j = 1. We claim that T mj =2 q i
6⊂ √ q .If we would have T mj =2 q i ⊂ √ q then there is some q i contained in √ q , since thelatter is prime. Hence also √ q i ⊂ √ q . Since √ q is a minimal prime, this must be anequality. However, by definition of the primary decomposition, √ q i = √ q , and thiscontradiction proves the claim. Let g ∈ T mi =2 q i \ √ q . ROBIN VAN DER VEER If √ q = q , then h = g satisfies the condition of the theorem. Namely, let q ∈ q .Then gq ∈ ( T mi =2 q i ) q ⊂ T mi =1 q i = J .If √ q = q , we get from Lemma 2.1 an f ∈ √ q \ q such that f √ q ⊂ q . Set h = f g . We claim that this h satisfies the conditions of the theorem. If h ∈ q , thensince q is primary, f ∈ q or g ∈ √ q , neither of which is possible by choice of f and g . This means that indeed h ∈ A \ q . Let p ∈ √ q . By choice of f , pf ∈ q , and thus ph = gpf ∈ ( T mi =2 q i ) q ⊂ T mi =1 q i = J , which concludes the proof. (cid:3) Corollary 2.3.
Let M be an A -module, and let p be a minimal prime divisor of theideal Ann A ( M ) . Then Ann A ( M ⊗ A ( A/ p )) = p .Proof. Let Ann A ( M ) = T mi =1 q i be a primary decomposition of Ann A ( M ) with √ q = p . From Theorem 2.2 we get an h ∈ A \ q such that h p ⊂ Ann A ( M ).Let g ∈ Ann A ( M ⊗ A ( A/ p )). This means that for every m ∈ M , gm ∈ p M . Inother words, for every m ∈ M , there exists an n ∈ M and p ∈ p such that gm = pn .We multiply this equation by h on both sides to find ghm = hpn . By construction, hp ∈ Ann A ( M ), so that ghm = 0 for all m , and thus gh ∈ Ann A ( M ). In particular, gh ∈ q , so that either h ∈ q or g ∈ √ q = p . Since h q by construction, weconclude that g ∈ p , which shows that Ann A ( M ⊗ A ( A/ p )) ⊂ p . The other inclusionis obvious, and this concludes the proof. (cid:3) Lemma 2.4.
Let J ⊂ D n [ s , . . . , s p ] be a left ideal. Let p be a minimal prime divisorof J ∩ A . Then ( J + R p ) ∩ A = p .Proof. We denote R = D n [ s , . . . , s p ]. We regard R/ J as an A -module. As such weclaim that Ann A ( R/ J ) = J ∩ A . To see this, let f ∈ Ann A ( R/ J ), so that f · (1+ J ) = f + J is zero in R/ J , which means that f ∈ J ∩ A . For the other inclusion let f ∈ J ∩ A .Then f · ( P + J ) = f P + J = P f + J , where in the second equality we use that A is contained in the center of R . Since f ∈ J , which is a left ideal, we conclude that P f ∈ J , so that f · ( P + J ) = 0 and henceindeed f ∈ Ann A ( R/ J ).We apply Corollary 2.3 to find that Ann A (( R/ J ) ⊗ A ( A/ p )) = p . The lemma followswhen we prove that Ann A (( R/ J ) ⊗ A ( A/ p )) = ( J + R p ) ∩ A , which in turn follows, asabove, when we prove that we have an isomorphism( R/ J ) ⊗ A ( A/ p ) ∼ = R/ ( J + R p ) . There is a well-defined map from the left hand side to the right hand side given by( P + J ) ⊗ ( f + p ) P f + ( J + R p ) . Note that this is well-defined because A is contained in the center of R . In the otherdirection we have the map P + ( J + R p ) ( P + J ) ⊗ (1 + p ) . N THE SUPPORT OF RELATIVE D -MODULES 5 These maps are inverse to each other, which proves the claim. (cid:3) Gr¨obner bases for D n -modules In this section we recall some facts about Gr¨obner bases for ideals in R = D n [ s , . . . , s p ]and D n and then prove Theorem B. We denote as in the previous section A = C [ s , . . . , s p ]. We will always implicitly regard A and D n as subsets of R .In D n we consider the set of standard monomials of the form x α ∂ β , α, β ∈ Z n ≥ . Onthese monomials we consider the lexicographical order with(2) ∂ > · · · > ∂ n > x n > · · · > x . Any operator in D n is a finite C -linear combination of standard monomial in a uniqueway. Writing P ∈ D n as a linear combination P = P α,β c α,β x α ∂ β with non-zero c α,β we denote by lm D n ( P ) the largest monomial x α ∂ β with respect to the lexicographicalorder. For an ideal I ⊂ D n we denote lm D n ( I ) = { lm D n ( P ) | P ∈ I } . A finitegenerating set G ⊂ I is called a Gr¨obner basis for I if the following is true: for every m ∈ lm D n ( I ) there exists a P ∈ G such that σ (lm D n ( P )) | σ ( m ) where σ denotesthe operation of taking the principal symbol. More explicitly this says that when x α ∂ β ∈ lm D n ( I ) there exists a P ∈ G with lm D n ( P ) = x α ′ ∂ β ′ and ( α ′ , β ′ ) is entry-wiseless than or equal to ( α, β ). To simply the notation we will write this divisibility notionsimply as x α ′ ∂ β ′ | x α ∂ β , but we emphasise that this does not mean that there existssome P ∈ D n for which P x α ′ ∂ β ′ = x α ∂ β . It follows immediately from the definitionthat I = D n if and only if any Gr¨obner basis for I contains a unit, i.e. an element of C . In R we consider the set of standard monomials of the form x α ∂ β s γ , α, β ∈ Z n ≥ , γ ∈ Z p ≥ . On these monomials we consider the lexicographical order with(3) ∂ > · · · > ∂ n > x n > · · · > x > s p > · · · > s . Any operator in R is a finite C -linear combination of standard monomial in a uniqueway. We denote by lm R ( P ) the leading monomial of an operator P ∈ R with respect tothe lexicographical order. For an ideal Q ⊂ R we denote lm R ( Q ) = { lm R ( P ) | P ∈ Q } .A finite generating set G ⊂ Q is called a Gr¨obner basis for Q if the following Gr¨obnerproperty is true: for every m ∈ lm R ( Q ) there exists a P ∈ G such that σ (lm R ( P )) | σ ( m ) where σ denotes the operation of taking the principal symbol. More explicitly thissays that when x α ∂ β s γ ∈ lm R ( Q ) there exists a P ∈ G with lm R ( P ) = x α ′ ∂ β ′ s γ ′ and( α ′ , β ′ , γ ′ ) is entry-wise less than or equal to ( α, β, γ ). Again we denote this divisibilityrelation simply by x α ′ ∂ β ′ s γ ′ | x α ∂ β s γ . Regarding the s , . . . , s p as parameters we willalso need the following. Any P ∈ R can be written uniquely as(4) P = X α,β h α,β ( s , . . . , s p ) x α ∂ β , ROBIN VAN DER VEER with h α,β = 0. We denote the parametric leading monomial of P by plm( P ) = x α ∂ β and the parametric leading coefficient op P by plc( P ) = h α,β , where x α ∂ β is the largestmonomial occurring in (4) with respect to the monomial order (2).In the commutative ring A consider the lexicographical monomial order with(5) s p > · · · > s . We denote by lm A ( f ) the leading monomial of f ∈ A .In all three rings D n , A and R a Gr¨obner basis for an ideal can be obtained byapplying Buchbergers algorithm to an arbitrary generating set. We note the followingrelations between the leading monomials. For f ∈ A : lm R ( f ) = lm A ( f ). For P ∈ D n ,lm D n ( P ) = lm R ( P ). For P ∈ R , lm A (plc( P )) plm( P ) = lm R ( P ). Because of theseequalities we will from now on suppress the notation of the ring and simply write lmfor leading monomials. Lemma 3.1.
Let Q ⊂ R be an ideal and let G be a Gr¨obner basis for Q with respectto the order (3) . Then G ∩ A is a Gr¨obner basis for Q ∩ A with respect to the order (5) .Proof. Let f ∈ Q ∩ A . There exists a P ∈ G such that lm( P ) | lm( f ). This meansthat lm( P ) ∈ A , and thus by definition of the order (3), P ∈ A . We conclude that forevery f ∈ Q ∩ A there exists a g ∈ G ∩ A such that lm( g ) | lm( f ). It only remains toshow that G ∩ A generates Q ∩ A . This follows from the preceding statement, since wecan reduce f to zero modulo G ∩ A by iteratively canceling leading monomials. (cid:3) Lemma 3.2.
Let Q ⊂ R be an ideal. Then there exists a Gr¨obner basis G for Q withrespect to the order (3) such that for all P ∈ G \ A , plc( P ) Q ∩ A .Proof. Let P ∈ G \ A and write P = plc( P ) plm( P ) + Q. By Lemma 3.1 G ∩ A = { f , . . . , f m } is a Gr¨obner basis for Q ∩ A . Using the divisionalgorithm [CLO15, Proposition 1.6.1] we find q , . . . , q m , r such thatplc( P ) = m X i =1 q i f i + r, where lm( r ) is not divisible by any lm( f i ). This means that P = m X i =1 q i f i + r ! plm( P ) + Q ≡ r plm( P ) + Q =: P ′ , where ≡ denotes equivalence modulo Q . If P ′ = 0 we define G ′ = G \ { P } and if P ′ = 0 then we define G ′ = G \ { P } ∪ { P ′ } .We claim that G ′ is still a Gr¨obner basis for Q . This is clear if P = P ′ so assumethat P = P ′ . We first remark that G ′ clearly still generates Q , and hence we needonly verify the Gr¨obner property. If lm( P ′ ) = lm( P ) then the Gr¨obner propery isclearly still satisfied, so assume that lm( P ′ ) = lm( P ). This assumption means that N THE SUPPORT OF RELATIVE D -MODULES 7 lm(plc( P )) was divisible by lm( f i ) for some f i . To see this, note that if this werenot the case then by the definition of the division algorithm, lm( r ) = lm(plc( P )),and hence lm( P ) = lm( P ′ ), contradicting our assumption. This means that lm( P ) =lm(plc( P )) plm( P ) is divisible by some lm( f i ). Hence any element Q ∈ Q for whichlm( P ) | lm( Q ) also satisfies lm( f i ) | lm( Q ). It follows that G ′ is still a Gr¨obner basis,since P was redundant for the Gr¨obner basis property.We keep repeating this procedure until no P ∈ G \ A is changed by applying thisreduction. This is clearly a finite process. Once we are done we thus find that for all P ∈ G \ A , lm(plc( P )) is not divisible by any lm( f i ). Since the { f , . . . , f m } form aGr¨obner basis for Q ∩ A , this means that for all P ∈ G \ A , plc( P ) Q ∩ A . (cid:3) For any α ∈ C p there is a specialization map q α : R → D n defined by q α ( P ( s , . . . , s p )) = P ( α , . . . , α p ) . Notice that for any ideal Q ⊂ R we have( R/ Q ) ⊗ A ( A/ m α ) ∼ = D n /q α ( Q ) , where m α denotes the maximal ideal corresponding to α . Theorem 3.3 ([Ley01], Lemma 2.5) . Let Q ⊂ R be an ideal, let p ⊂ Q be a secondideal, and let G ⊂ Q be a Gr¨obner basis for Q with respect to the order (3) . Let h = Q P ∈ G \ p plc( P ) . Let α ∈ Z ( p ∩ A ) \ Z ( h ) . Then q α ( G \ p ) is a Gr¨obner basis for q α ( Q ) with respect to the order (2) .Proof. Apply [Ley01] Lemma 2.5 without any y -parameters. (cid:3) We remark that a statement similar to Theorem 3.3 also appeared in [Oak97], andthat the comprehensive Gr¨obner bases from [KW91] can also be used to give a simpleproof of this result.
Proof of Theorem B.
Let G be a Gr¨obner basis for Q = J + R p with respect to theorder (3). By Lemma 3.2 we can assume that for all P ∈ G \ A ,plc( P ) Q ∩ A = p , where this equality follows from Corollary 2.4.Since Q ∩ A = p , clearly we have G ∩ A ⊂ p , and hence G ∩ p = G ∩ A , and hencealso G \ A = G \ p . Putting the preceding two statements together we conclude that for all P ∈ G \ p ,plc( P ) p . Since p is prime, this means that h = Y P ∈ G \ p plc( P ) p Since G ∩ p = G ∩ A , we have that ( G \ p ) ∩ A = ∅ , and hence for every P ∈ G \ p ,plm( P ) C . By choice of h , for all α ∈ Z ( p ) \ Z ( h ) and P ∈ G \ p , plm( P ) = lm( q α ( P )), ROBIN VAN DER VEER so that q α ( G \ p ) does not contain any units. Since this set is a Gr¨obner basis for q α ( Q )by Theorem 3.3 we conclude that D n /q α ( Q ) = 0.Now notice that D n q α ( Q ) ∼ = R Q ⊗ A A m α ∼ = (cid:18) R J ⊗ A A p (cid:19) ⊗ A A m α ∼ = R J ⊗ A A m α , which proves the claim. (cid:3) Proof of Theorem A.
We denote M = D n [ s , . . . , s p ] f s . . . f s p p D n [ s , . . . , s p ] f s +11 . . . f s p +1 p . Then B F = Ann A ( M ), and M is a cyclic D n [ s , . . . , s p ]-module. Let C ⊂ Z ( B F ) bean irreducible component. By Theorem B there is an h ∈ A which does not vanishidentically on C such that for all α ∈ C \ Z ( h ), M ⊗ A ( A/ m α ) = 0.By Proposition 1.1 we conclude thatExp( C \ Z ( h )) ⊂ supp ( C ∗ ) p ( ψ F ( C C n )) . The map Exp : C p → ( C ∗ ) p is continuous for the analytic topology. In the analytictopology we have C \ Z ( h ) = C , and henceExp( C ) ⊂ Exp( C \ Z ( h )) ⊂ supp ( C ∗ ) p ( ψ F ( C C n )) = supp ( C ∗ ) p ( ψ F ( C C n )) , which is what we wanted to prove. (cid:3) References [BLSW17] Nero Budur, Yongqiang Liu, Luis Saumell, and Botong Wang. Cohomology support lociof local systems.
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N THE SUPPORT OF RELATIVE D -MODULES 9 Robin van der Veer, KU Leuven, Department of Mathematics, Celestijnenlaan 200Bbus 2400, B-3001 Leuven, Belgium
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