Residue Current Approach to Ehrenpreis-Malgrange Type

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Theorem 1.1 ([17, Theorem 4.2], [22, Theorem 7.6.12, 7.6.13], [38, IV, 5,5◦]). § Let Ω be a convex subset of Rn and let E denote the C[∂]-module consisting of exponential polynomials. One has the following properties:

(1) C∞(Ω) is an injective C[∂]-module, i.e., for any ﬁnitely generated C[∂]- module M and any positive integer i, we have an identity

i ExtC[∂](M,C∞(Ω)) = 0. (1.1)

r0 1 r1 1 (2) For any matrix P(∂) M(r1,r0; C[∂]), Ker(P(∂): E × E × ) is ∈ r0 1 r1 1 → dense in Ker(P(∂): C∞(Ω) × C∞(Ω) × ). → We call the theorem above Ehrenpreis-Malgrange type theorem. In the following, we explain the meaning of (1) of Theorem 1.1. To this end, let us consider a more general setting. Let R be a subring of the ring of distributions with compact supports n ′(R ) and F be a vector space of functions of a suitable kind, say F = E n C∞(R ) for simplicity. Then, R acts on F by

def T f = T f (T R, f F ). (1.2) · ∗ ∈ ∈ Here, is the usual convolution product. Note that usual partial diﬀerential operator∗ ∂ can be realized as a convolution operator by ∂xi

∂f ∂ = δ f, (1.3) ∂x ∂x ∗ i i where δ is Dirac measure with support at the origin. For any r1 r0 matrix t r1 1 × P with entries in R and for any f = (f , , f ) F × , we consider a 1 ··· r1 ∈ system of equations Pu = f, (1.4)

r0 1 where u F × is an unknown vector of functions. Now, suppose this system of equations∈ (1.4) has a solution u. Then, we can easily ﬁnd constraints on f:

For any r r matrix Q with entries in R, QP = 0 implies Qf =0. (1.5) 2 × 1

2 The condition (1.5) is called the compatibility condition of the system (1.4). One may ask if this condition is suﬃcient for the solvability of the system (1.4). After a simple discussion on homological algebra, the problem is reduced to proving the identity

1 1 r0 1 r1 ExtR(R × /R × P, F )=0. (1.6) Thus, by (1) of Theorem 1.1, the system (1.4) is solvable if and only if (1.5) is satisﬁed when R = C[∂]. Since the formulation above is quite general, it is natural to ask if one can extend Ehrenpreis-Malgrange type theorem to more general convolution equations, i.e., if one can replace C[∂] by a larger ring R in Theorem 1.1. For example, consider a system

P(∂, σ)u = f, (1.7) where P(∂, σ) is a r r matrix with entries in C[∂, σ] = C[∂ , , ∂ , σ]. 1 × 0 1 ··· n Here, σ is a diﬀerence operator to the ﬁrst coordinate which can be realized as a convolution operator by

σf = f(x +1, x , , x )= δ(x +1, x , , x ) f. (1.8) 1 2 ··· n 1 2 ··· n ∗ When we regard the first coordinate as a time variable, this system of difference-differential equations is called linear partial differential equations with constant coefficients and commensurate time lags. For brevity, we call such a system D∆-equations in this paper. This system has attracted at- tentions of many mathematicians, especially in connection with ring theory ([15], [19]). One might hope that Ehrenpreis-Malgrange type theorem holds for R = C[∂, σ]. However, this optimistic conjecture turns out to be false even if n = 1 as was pointed out by H.Glürsing-Lürßen. We will see a counterexample in 2. She also showed that one must consider a non-Noetherian extension of the§ ring C[∂, σ] in order to obtain an Ehrenpreis-Malgrange type theorem for D∆-equations. Our interest, therefore, is to introduce a suitable ring extension of C[∂, σ] and prove Ehrenpreis-Malgrange type theorem for this ring. OurH main result is the following: Theorem 1.2. n Let F be either C∞, , or and Ω be a convex subset of R (when F = C∞, n O B ) or of C (when F = ) such that Ω+ Re1 = Ω. Here, is the sheaf Bof hyperfunctions. Let Odenote a ring of generalized D∆-operatorsB which will be introduced in 3.H Finally, let E denote an -module consisting of § H exponential polynomials. For any matrix P M(r1,r0; ), we put M = 1 p 1 q ∈ H × / × P. One has the following properties: H H 3 (1) For any positive integer i, one has

Exti (M, F (Ω)) = 0. (1.9) H

r0 1 r1 1 (2) If F is either C∞ or , then Ker(P(∂, σ): E × E × ) is dense in r0 O1 r1 1 → Ker(P(∂, σ): F (Ω) × F (Ω) × ). → Although Theorem 1.1 and Theorem 1.2 concern ring theoretic study of convolution equations, the technology of topological vector spaces enables us to reduce the proofs to solving a problem in complex harmonic analysis. Namely, Ehrenpreis-Malgrange type theorem follows if, for any r r matrix 1 × 0 P with entries in C[∂] (resp. ), one can show the identity H

n 1 r1 n 1 r0 n 1 r1 (C ) × P (C ) × = (C ) × P, (1.10) O ∩ O p O p where stands for the Fourierb transform (of convolutionb operators) and (Cn) (Cn) is a set of holomorphic functions with a suitable growth O p ⊂ O encodedb from the Fourier transform. This identity which is called “division with bounds”, was developed by many mathematicians around 60’s and gave rise to fruitful theorems in the theory of differential equations with constant coefficients. However, the classical proof of Theorem 1.1 requires a refined version of Noether’s normalization lemma as well as Cechˇ cohomology with bounds, neither of which can be adapted for the proof of Theorem 1.2. The crucial step of the proof of Theorem 1.2 is to combine our ring theoretic study of with integral representation technique developed by M. Andersson: the residueH current method ([5]). For the readers’ convenience, we review a brief history of Ehrenpreis- Malgrange type theorem. Soon after the celebrated work of L. Ehrenpreis ([17]), there naturally appeared some people investigating a generalization of Theorem 1.1 for more general convolution equations. Their interests were mainly about generalizing (2) of Theorem 1.1. However, D. I. Gurevich gave the following counterexample against the density of exponential polynomial solutions for general convolution equations.

Proposition 1.1 ([20]). There are two convolution operators µ1 and µ2 in ′ so that the exponential E n polynomial solutions are not dense in the solution space f C∞(R ) µ f = µ f =0 . { ∈ | 1 ∗ 2 ∗ } Due to this example, generalizations of (2) of Theorem 1.1 have been limited to convolution equations whose Fourier transforms deﬁne complete intersection varieties and for a certain period of time, this constraint remained to

4 be necessary. (In the case of discrete varieties, this constraint is significantly relaxed to the “locally slowly decreasing condition” in [9]. See [9, Defini- tion 2.1].) The main technique of their proofs was based on explicit integral formulae in the spirit of multidimensional residue theory: residue currents (cf. [8], or [10]). In this direction, the most general version of such currents was finally introduced in [5]. The authors of [5] gave a new proof of (2) of Theorem 1.1, which is a prototype of that of Theorem 1.2. On the other hand, if we restrict our attention to D∆-equations, ring theoretic study comes into play. It enables us to prove a variant of (1) of Theorem 1.2 for smaller function module such as F (Cn) = (Cn; exp) O (entire functions of exponential growth). See [15, Theorem 7.4]. Though ring theoretic approach cannnot solve division with bounds in general, the observation of H. Glüsing-Lürßen in [19] is of significant importance: in the theory of (ordinary) D∆-equations, one must consider a system of equations as a module over a non-Noetherian ring , a univariable version of . She H1 H also conducted an extensive study of ring theoretic properties of 1, which serves as a bridge between analysis of systems and homological algeHbra in this paper. This paper makes full use of these progresses in complex analysis and non-Noetherian ring theory and therefore, is positioned at an intersection of two different points of views. Let us summarize the content of this paper. In Section 2 we introduce the ring defined by Glüsing-Lürßen and prove the integral representation of hyperfunctionH solutions as a superposition of exponential polynomial solutions in dimension 1. Section 2 is independent of other sections, but it would help the reader to understand the latter part of this paper. In Section 3 we study a generalization of which only contains difference operators to one H direction, and examine some of its cohomological properties. In Section 4 we recall the notion of the residue currents and give some proofs of necessary results. In Section 5 we combine the techniques of Sections 3 and 4 to obtain our version of the division with bounds. On the way, we need an estimate of residue currents for exponential polynomials which was established in [12]. At the beginning of Section 5, however, we will observe that division with bounds is impossible if there are more than two independent difference di- rections suggesting that our result is optimal. In Section 6 we prove the Theorem 1.2 and, as a corollary, a description of cohomology groups with values in solution sheaves of systems The integral representation of Section 2 is similar to the result of Y. Okada (cf. [36]), but the way of proving it is different. I also owe the idea of the proof of the integral representation of hyperfunction solutions to [28] and [37].

5 2 D∆-equations in dimension 1

In this section, we introduce a certain ring extension of the ring of D∆- operators when the number of independent variable is 1. We will also observe that various function spaces enjoy better cohomological properties over this ring than the usual ring of D∆-operators C[∂, σ]. In concordance with the notation of H. Glüsing-Lürßen, let us denote the standard coordinate of C by z and consider the polynomial ring C[z, σ] of 2 variables. We define an action of C[z, σ] on C∞(R) by

df (z f)(x)= (x), (2.1) · dx and (σ f)(x)= f(x + 1). (2.2) · As we remarked in the introduction, C∞(R) is not injective over C[z, σ]. The following simple counterexample is due to H. Gl¨using-L¨urßen. Example 2.1. Take two matrices P =t (σ 1, z), Q = (z, 1 σ) and consider an exact sequence − − Q 1 2 P C[z, σ] × C[z, σ] × × C[z, σ](exact). → → Applying Hom ( ,C∞(R)), it yields a complex C[z,σ] −

P 2 1 Q C∞(R) · C∞(R) × · C∞(R)(not exact). → → In fact, we have t(0, 1) Ker(Q ) Im(P ). ∈ · \ · This example suggests that we need to introduce another appropriate ring of operators to recover the classical vanishing theorem of higher extension groups. The above counterexample indicates that it should necessarily be a non-flat extension of the polynomial ring C[z, σ]. Following H. Glüsing- Lürßen, we introduce the ring of D∆-operators . For any element q of H the fraction field C(z, σ), we define the meromorphic function q∗ by q∗(z)= q(z, ez).

Deﬁnition 2.1. We deﬁne a ring extensions of C[z, σ] by

1 1 = q = pφ− p C[z, σ, σ− ],φ C[z], q∗ (C ) . (2.3) H { | ∈ ∈ ∈ O z } Definition 2.2. A commutative domain R is called a Bézout domain if any finitely generated ideal of R is principal.

6 Theorem 2.1 ([19, Theorem 3.2.1]). (1) is a non-Noetherian ring but a B´ezout domain. H p q (2) is an elementary divisor domain, i.e., for any P × one can ﬁnd H ∈ H two matrices V and W such that

V PW = diag(d1, ..., dr, 0, ..., 0), (2.4) where V and W are products of elementary matrices and d 0 , i ∈H\{ } d d . i| i+1 Roughly speaking, this theorem means that any system of D∆-equations can be transformed into a direct sum of single equations in a certain sense as we explain below. Notice that function spaces C∞(R), (D), ′(R), and (R) are all -modules. Here, D is a horizontal strip DO = zD C a < ImB z < b (aH < b, a, b [ , ]). The action of the ring { ∈on these| } ∈ −∞ ∞ 1 H function spaces is deﬁned as follows: when q = p(z, σ)φ(z)− and f(x) ∈ H is a function of one of four kinds above, say ∞(R), we take g ∞(R) so C ∈ C that φ(z) g(x) = f(x). We deﬁne q f = p(z, σ) g(x). Since q∗ is entire, this action· is independent of the choice· of g. Hence,· dealing with the ring amounts to discussing generalized D∆- operators. H Theorem 2.2.

(1) (real case) Let F be either C∞(R), ′(R), or (R). For any finitely presented -module M and for anyD positive integerB i, one has the fol- H lowing vanishing result Exti (M, F )=0. (2.5) H (2) (holomorphic case) Let D be a subset of C defined by D = z C a< { ∈ | Im z < b , where a < b, a, b [ , ]). For any finitely presented - module M} and for any positive∈ integer−∞ ∞i, one has the following vanishingH result Exti (M, (D))=0. (2.6) H O

Corollary 2.1 (integral representation formula). Take any q , and f (R) and assume q f =0. If (αk, mk) k 1 (αk ∈ H ∈ B · { } ≥ ∈ C, q∗(αk)=0, mk = ordαk q∗) are zeros of q∗ with its multiplicities, then f has a representation

α x f = P (x)e k· (P (X) C[X],degP < m ). (2.7) k k ∈ k k k 1 X≥ Here the sum is convergent in the sense of hyperfunctions.

7 Proof of Theorem 2.2 We first prove (2). By the definition of a finitely 1 p 1 q q p presented module, M is of the form M = × / × P for some P × . H H · ∈ H By Theorem 2.1, we can assume P = diag(d1, ..., dr, 0, ..., 0). Therefore, it is reduced to showing that for any q 0 , q : (D) (D) is surjective. We can assume q = p(z, σ) σl(l ∈H\{Z),p =} M Op (z)σ→j. OThe surjectivity of · ∈ j=0 j q now follows from Theorem 1 of [32] (see also the remark after Theorem 1). The proof of (1) is parallel to that of (2).P One can find the corresponding surjectivity results in [7, Theorem 6.1.23.] (see also Proposition 3.1.31. of the same reference) for F = C∞(R) or in [16, Theorem 1] for F = ′(R). The surjectivity result for F = (R) follows from the commutative diagramD B in the proof of Corollary 2.1.

Proof of Corollary 2.1 First, observe that q : (R √ 1(R 0)) (R √ 1(R 0)) is surjective from Theorem 2.2O (2). Now× − let us\ consider→ theO commutative× − \ diagram

0 / Ker q / Ker q / Ker q

0 / (C) / (R √ 1(R 0)) / (R) / 0 O O × − \ B q q q 0 / (C) / (R √ 1(R 0)) / (R) / 0 O O × − \ B

0 / 0 / 0

. The ﬁrst two columns are exact and the two middle rows are also exact. By the snake lemma, we obtain an identity

Ker(q : (R √ 1(R 0)) (R √ 1(R 0))) Ker(q : (R) (R))= O × − \ → O × − \ , B → B Ker(q : (C) (C) O → O (2.8) and the third column is exact. Applying Theorem 2.3 below to

Ker(q : (R √ 1(R 0)) (R √ 1(R 0))), (2.9) O × − \ → O × − \ one gets the existence of such series representation.

8 Theorem 2.3 ([7, Proposition 6.4.17], see also Remark 6.2.11). Take any µ ′(C), and let K C be a convex carrier of µ and Ω C be ∈ O ⊂ ⊂ a convex open set. We assume µ is regularly decreasing. Denoting the zeros of µ with their multiplicities by (αk, mk) k 1 (αk C, µ(αk)=0, mk = { } ≥ ∈ ord q∗), there is an increasing sequence 1= k

b αk x f = P (x)e · (P (X) C[X],degP < m ). (2.11) k k ∈ k k k 1 X≥

We do not define the terminology ”slowly decreasing” in this paper since the notion is not necessary in the following sections. We only note that any element of is ”slowly decreasing” in the sense of [7, Definition 2.2.13]. H 3 Ring Hn We introduce a ring of generalized partial D∆-operators in the spirit of H. Glüsing-Lürßen and investigate the basic properties of this ring. The reader should be aware that most of the following propositions do not hold if we consider the smaller ring C[z , , z , ez1 ]. 1 ··· n Definition 3.1. We define the ring n of generalized partial D∆-operators inductively as follows: H For n = 1, we put 1 = . For n 2, we put H = H [z , , z ]. ≥ Hn H1 2 ··· n Remark.

(1) From now on, we use the symbol for n when there is no risk of confusion. H H

(2) We embed into (Cn) by replacing σ by ez1 . H O (3) As we shall see below, the coordinates z correspond to partial deriva- tives and σ corresponds to a diﬀerence operator to the direction of the ﬁrst coordinate. Therefore, the ring is a ring of generalized D∆- Hn operators whose frequencies generate a rank one additive subgroup of Cn.

9 In order to state basic properties of the ring , we need some lemmas from algebra. The most fundamental property of His the so called coherency. H Deﬁnition 3.2. Let R be a commutative ring.

(1) An R module M is said to be coherent if there is an exact sequence of the form Rk Rl M 0 where k and l are positive integers. → → → (2) The ring R is said to be coherent if for any positive integers k and l, and for any morphism f : Rk Rl, Ker f is finitely generated. → The following proposition is taken from [18] and it ensures the coherency of . H Proposition 3.1 ([18, Corollary 7.3.4]). Let R be a semihereditary ring and let x1,...,xn be indeterminates. Then R[x1,...,xn] is a coherent ring. Recall that a commutative ring R is semihereditary if every finitely generated ideal of R is a projective R-module. For example, a Bézout domain is a typical example of a semihereditary ring. Proposition 3.2.

(1) is a coherent ring. H (2) (C) is a flat ring extension. H1 ⊂ O (3) (Cn) is a flat ring extension. H ⊂ O Proof. (1) is immediate from Proposition 3.1 and Theorem 2.1 (1). (2) goes as follows: First, we prove that for any finitely generated ideal I of 1 1, Tor1H ( 1/I, (C)) = 0. Since 1 is Bézout, we can assume I =(q) for Hsome q H. ConsiderO the exact sequenceH ∈ H1 q 0 × /(q) 0. (3.1) → H1 → H1 → H1 → By tensoring (C), we have a complex O q 0 (C) × (C) (C)/(q) 0. (3.2) → O → O → O → This is exact since (C) is a domain. Now by a standard argument of homological algebra,O one has the flatness since for any ideal I of , /I H1 H1 is an inductive limit of 1-modules of the form 1/J where J is a finitely generated ideal of . H H H1 10 (3) We can observe that = 1[z2,...,zn] (C)[z2,...,zn] is a flat ring extension by (2). Therefore,H itH is enough to⊂ prove O that (C)[z , , z ] O 2 ··· n ⊂ (Cn) is flat, namely O

(C)[z2,...,zn] n TorO ( (C)[z ,...,z ]/I, (C )) = 0 (3.3) 1 O 2 n O for all finitely generated ideals I of (C)[z ,...,z ]. Since (C) is a Bézout O 2 n O domain, (C)[z ,...,z ] is a coherent ring by Proposition 3.1. This implies O 2 n that for any finitely generated ideal I of (C)[z2,...,zn], (C)[z2,...,zn]/I has a finite free resolution. Now it is enoughO to prove thatO for a given exact sequence L M N of coherent (C)[z ,...,z ] modules, → → O 2 n (Cn) L (Cn) M (Cn) N (3.4) O (C)[z⊗2,...,zn] → O (C)[z⊗2,...,zn] → O (C)[z⊗2,...,zn] O O O is again exact. The last complex is exact if and only if

n z0 L n z0 M n z0 N (3.5) O (C)[z⊗2,...,zn] → O (C)[z⊗2,...,zn] → O (C)[z⊗2,...,zn] O O O for any z0 Cn since Cn is a Stein open set. See [15, Result 6.1.7]. Here, we ∈ can replace n 0 by n 0 0 [z ,...,zn] , z z 1 z1 2 O (C)[z⊗2,...,zn]− O 1 z0 [z⊗2,...,zn] O (C)[z⊗2,...,zn]− O O 1 O 0 0 0 n where z = (z1 ,...,zn) C . Without loss of generality, we can assume 0 ∈ z = 0. Since (C) 1 0 is a flat ring extension by [15, Result 6.1.7], we only need to considerO ⊂ theO ring extension [z ,...,z ] and prove its 1O0 2 n ⊂ nO0 flatness. We consider a triple [z ,...,z ] C[[z ,...,z ]]. If we denote 1O0 2 n ⊂ nO0 ⊂ 1 n the maximal ideal of C[[z ,...,z ]] by m, then we can observe that m 1 n ∩ nO0 and m 1 0[z2,...,zn] are maximal ideals of n 0 and 1 0[z2,...,zn], respec- tively.∩ ByO taking completions of these rings withO respectO to these maximal ideals, we see that C[[z ,...,z ]] is faithfully flat and [z ,...,z ] nO0 ⊂ 1 n 1O0 2 n ⊂ C[[z1,...,zn]] is flat. Therefore, we can conclude that 1 0[z2,...,zn] n 0 is flat. O ⊂ O

Notice that one can actually show that the ring extension 1 (C) is a faithfully flat extension. See [15, Theorem 7.7]. We do not knowH ⊂ whet O her the ring extension (Cn) is a faithfully flat ring extension. If it is, we can get a refined estimateH ⊂ O of proj.dimM for coherent -modules M. H

The second property of the ring is the following identity which shows H that this ring is actually identical with the one discussed in Section 7 of [15].

11 Proposition 3.3. One always has the following identity:

= (Cn) C(ez1 , z , , z ). (3.6) H O ∩ 1 ··· n Proof. Take any f = P (Cn) 0 where P, Q C[ez1 , z , , z ]. By Q ∈ O \{ } ∈ 1 ··· n the generalization of the theorem of Ritt (cf. [6, MAIN THEOREM]), we p θj ,z can assume f = , for some p = a (z)eh i, a (z) C[z ,...,z ], θ q j j j ∈ 1 n j ∈ Cn, q C[z ,...,z ]. Furthermore, a careful reading of the proof of [6, ∈ 1 n P MAIN THEOREM] tells us that θj is obtained as a Z-linear combination of frequencies of P and Q and their complex conjugates, meaning that we may assume p C[z ,...,z , ez1 ]. ∈ 1 n We can assume n = 2 since the proof of the proposition for n 3 is essentially the same. We want to show ∂q = 0. Assume the opposite,≥ that ∂z2 is, assume ∂q = 0. In this case, we should have ∂p = 0 since the fraction p ∂z2 ∂z2 q is entire. 6 6 Now we factorize p as p = p (z , z )p (z , z , ez1 ), p C[z , z ], p 1 1 2 2 1 2 1 ∈ 1 2 2 ∈ C[z1, z2,X] so that ∂p p = pm1 pml , 2j =0, p are irreducible in C[z , z ,X], 2 21 ··· 2l ∂X 6 2j 1 2

n1 nk and q and p1 are relatively prime. We also factorize q as q = q1 qk , ∂q1 ··· qi C[z1, z2], and qi are irreducible. We further assume = 0. Let ∈ ∂z2 6 ∆1(z1) C[z1] 0 be the discriminant of q1 with respect to z2. We can 0 ∈ \{ } take z1 C so that ∈ ∆ (z0) =0. 1 1 6 Take z0 C so that ∂q1 (z0, z0) = 0 and q (z0, z0) = 0. We can solve the 2 ∂z2 1 2 1 1 2 ∈ 6 0 0 0 algebraic equation q1(z1, z2) = 0 on a neighbourhood of z = (z1 , z2 ) with respect to z , i.e., z = z (z ) around z0 = (z0, z0) on the zero set q = 0 2 2 2 1 1 2 { 1 } of q1. Since p1 and q1 are relatively prime, we have p1(z1, z2(z1)) 0. Thus z1 0 6≡ we have p2(z1, z2(z1), e )=0on q1 =0 around z . In particular, there is z1 { } 0 j so that p2j(z1, z2(z1), e )=0on q1 =0 around z . Here we need the following elementary{ lemma.} Lemma 3.1. The function z2(z1) can be analytically continued as a single valued function to a simply connected domain D of C which is obtained by subtracting ﬁnitely many rays from the whole plane. Furthermore, z2(z1) has polynomial growth at inﬁnity, i.e., there are positive real numbers C, N such that z2(z1) C(1 + z )N . | | ≤ | 1|

12 z1 Now the identity p2j(z1, z2(z1), e ) = 0 is still valid on the extended do- M j main D. If we expand p2j as p2j = βj(z1, z2)X , this identity is expressed j=0 M X jz1 as βj(z1, z2(z1))e = 0. We put αj(z1) = βj(z1, z2(z1)), and claim that j=0 thereX is at least one j 1 such that α (z ) 0. Suppose, conversely, that ≥ j 1 6≡ α 0 for all j 1, then one necessarily has α 0. This implies that for all j ≡ ≥ 0 ≡ j, q1 βj(z1, z2). This contradicts the assumption that p2j is irreducible. C[z1|,z2] Thus, the claim was conﬁrmed. Now let 1 J M be the largest number such that αJ 0. Since M ≤ ≤ J 6≡ jz1 αj(z1) (j J)z1 αj(z1)e = 0, we have e − = 0. When Re z1 tends to αJ (z ) j=0 j=0 1 +X , we get 0=1. This is a contradiction.X ∞ For a commutative ring R, let us denote by gl.dimR the global dimension of R (cf. [?] Deﬁnitions 4.1.1). Amongst all homological properties of , the most important one is the following. H Theorem 3.1. One always has the following inequality:

gl.dim n +1. (3.7) Hn ≤

In order to prove Theorem 3.1 above, it is enough to estimate gl.dim 1 since we have the following general estimate. H

Theorem 3.2 ([18, Theorem 1.3.16]). If R is a commutative ring, and T is an indeterminate, then

gl.dimR[T ] gl.dimR +1. (3.8) ≤

Furthermore, it is enough to estimate projective dimensions of 1/I for all ideals I of (see [?] Theorem 4.1.2). Note that the structure ofH finitely H1 generated ideals are completely understood since 1 is Bézout. The key point of the proof is that the structure of nonfinitelyH generated ideals was also deeply investigated by H. Glüsing-Lürßen (cf.[19]). In general, for any commutative ring R the notation a b means that an element b of R is divisible R| by another element a of R.

13 Proposition 3.4 ([19, Theorem 3.4.10]). For any ideal 0 = I 1, one can ﬁnd a polynomial p C[z, σ] 0 , and def6 ⊂ H ∈ \{ } a set M Dp = φ C[z] φ : monic and φ p such that ⊂ { ∈ | |1 } H (1) 1 M, ∈ (2) For any φ M, and for any ψ M, one has LCM(φ, ψ) M, ∈ ∈ ∈ (3) For any φ M, and for any ψ C[z] 0 such that ψ φ, one has ∈ ∈ \{ } C|[z] ψ M, ∈ def and one has the identity I = p = h p h ,φ M . hh ii(M) { φ | ∈ H1 ∈ } Proof of Theorem 3.1. In view of Theorem 3.2, it is enough to prove the inequality gl.dim 2. Let I be a non-zero ideal. By Proposition H1 ≤ ⊂ H1 3.4, we can assume I = p (M) for some p C[z, σ] 0 and M with properties (1), (2), and (3).hh ii ∈ \{ } We consider an exact sequence

1 M × /I 0(exact). (3.9) H1 → H1 → H1 → p Here the ﬁrst morphism is given by hφeφ hφ , where eφ φ M 7→ φ { } ∈ φ M φ M 1 M X∈ X∈ 1 M stands for a free basis of × . We put K = Ker( × ) and for a H1 H1 → H1 ﬁnite subset F of M, we also put φˆ = ψ. We can observe ψ=φ ψY6∈F p p h =0 ( h φˆ) =0 h φˆ =0. (3.10) φ φ ⇐⇒ φ ⇐⇒ φ φ F φ F ψ φ F ∈ ∈ ∈ X X ψ F X Y∈

1 F F, pol We deﬁne a submodule of C[z] × by K = h e h φˆ =0, h C[z] . φ φ| φ φ ∈ (φ F φ F ) Now we claim that the following identity is valid:X∈ X∈

F, pol F def 1 F K = K = h e × h φˆ =0, h . ⊗C[z] H1 φ φ ∈ H1 | φ φ ∈ H1 (φ F φ F ) ∈ ∈ X X (3.11)

14 In fact, putting F = φ ,...,φ , we have an exact sequence { 1 m} ˆ  φ1  .  .  × .      φˆ  F, pol 1 F m 1 1 0 K C[z] × C[z] × (exact). (3.12) → → −−−−−→ Taking into account that C[z] is a flat ring extension, and tensoring ⊂ H1 1 to this sequence, we obtain the claim. −C ⊗[z] H Let us go on with the proof of the theorem. Now the totality of finite subsets of M forms a cofinal partially ordered set with respect to inclusion. So we have inductive systems KF and KF, pol where morphisms are natural { } { } inclusions KF1 ֒ KF2 (resp. KF1, pol ֒ KF2, pol) for a pair of finite subsets F F of M. Here→ we have the sequence→ of identities 1 ⊂ 2

K = KF = lim (KF, pol )= lim KF, pol . ⊗C[z]H1 ⊗C[z]H1 F M; finite F ⊂M−→; finite F ⊂M−→; finite ! ⊂ [ (3.13) If we put Kpol = lim KF, pol, since gl.dimC[z] = 1, we have a projective F ⊂M−→; finite resolution: 1 Λ pol 0 P C[z] × K 0(exact), (3.14) → → → → where P is a projective C[z]-module. By tensoring 1, we have a projective resolution for K: H

1 Λ 0 P × K 0(exact), (3.15) → ⊗C[z] H1 → H1 → → Attaching this resulting sequence to (3.9), we have a projective resolution of /I of length 2: H1 1 Λ 0 P × /I 0(exact). (3.16) → ⊗C[z] H1 → H1 → H1 → H1 →

Combining Theorem 3.1 with Theorem 3.4 below we get the following result. Theorem 3.3. Any coherent -module has a ﬁnite free resolution of length n +1, that is, if M is a coherentH -module, we have a free resolution ≤ H

1 rN PN 1 r1 P1 1 r0 0 × × × × × M 0(exact), (3.17) → H −−→· · · → H −−→H → → where N n +1 and r are positive integers. ≤ j 15 Theorem 3.4 ([29, Theorem b]). If R is a B´ezout domain, any ﬁnitely generated projective R[X1,...,Xn]- module is free.

We are now constructing the Hefer forms which admit a certain estimate of Paley-Wiener type. We can even determine their explicit forms as well so that the required estimate is satisﬁed in a trivial manner. Proposition 3.5. For any given complex

1 rN PN 1 r1 P1 1 r0 0 × × × × × M 0, (3.18) → H −−→· · · → H −−→H → → where Pj M(rj,rj 1; ), there exist matrix-valued forms ∈ − H Hk(ζ, z)= A dζI with A M(r ,r ; ˜) such that l I I ∈ k l H I =k l | X| − k Hl =0(k

α q(ζ1) q(z1) α α k k 1 h = − z , h = q(ζ ) ζ − z − . (3.22) 1 ζ z 2 2 1 2 2 1 1 k ! − X=1 Let k>l + 1 and consider general n. Since for each i,

˜/((ζ z ), , (ζ z )) A[ζ , ,ζ , z , , z ] (3.23) H n − n ··· i+1 − i+1 ≃ 2 ··· n 2 ··· i

16 is a domain, (ζn zn), , (ζ1 z1) is a regular sequence. By a general result on Koszul{ complex− ··· we have− an exact} sequence:

n δ δ δ δ 0 ˜dζ dζ ζ−z ˜dζI ζ−z ζ−z ˜dζ ζ−z ˜ (exact). → H 1 ∧···∧ n → H → ··· → H i → H I =n 1 i=1 | |X− X (3.24) Since the right hand side of the equation (3.21) is δζ z closed, we can take − k I the desired matrix valued form Hl (ζ, z)= AI dζ . I =k l | X| − 4 Residue currents and integral formulae

In the proof of our main theorem, we make full use of the theory of residue currents. To begin with, we need to review the concrete construction of residue currents following [5]. From now on, X always denotes a connected complex manifold of dimension n. Let E and Q be two holomorphic hermitian vector bundles on X. We denote by ( , ) the hermitian metric on E. Take any morphism f Hom(E, Q). Let σ· :· Q E be the minimal inverse of ∈ → f, i. e., σξ is the minimal solution η of fη = ξ if ξ Im f and σξ = 0 if ξ is orthogonal to Im f. The minimal inverse σ is divergent∈ when the rank of f degenerates. The order of divergence can be described in terms of the determinant of f. Let us remember that the optimal rank ρ = sup rank fx of x X 1 ∈ ρ ρ f is well-defined. We can also define the canonical section F = ρ! f = det f ρ ρ ρ ∧ of E∗ Q. We put Z = x X fx =0 . Since Z is a locally common zero∧ set⊗∧ of finitely many holomorphic{ ∈ |∧ functions,} it is a proper analytic subset of X and σ is smooth outside Z. Recall that for a given section s of a hermitian vector bundle E, one can define a section s∗ of E∗ by s∗ = ( ,s); · it is called the dual section of s. The following lemma is the key tool for obtaining a good estimate of residue currents as we will see in Section 5.

Lemma 4.1 ([3, Lemma 4.1]). Let s be a global section of Hom(Q, E) E Q∗ such that f s Im f = 2 ≃ ⊗ ◦ | F IdIm f and s (Im f)⊥ = 0 with pointwise minimum norms on X Z, and | | | ρ ρ 2 \ let S be a global section of E Q∗ such that FS = F with point wise ∧ ⊗ ∧ | | minimum norms on X Z. Then one has the identity \ s = F 2σ (4.1) | | on X Z and both s and S are smooth across Z. \

17 Though we do not include the proof of Lemma 4.1, we recall the explicit formula for s which can be found in [3]. Take any smooth local frame r ǫj j=1 (r = rank Q) of Q. With the aid of this frame, we can write f locally as{ } ρ f = (f j) ǫ , (4.2) ⊗ j j=1 X j j where f E∗. If we denote by δf (resp. δǫj ) the interior multiplication by j ∈ f (resp. ǫj), the formula of the section s is given by ρ 1 r − s = δ j δ S/ρ!. (4.3) f ⊗ ǫj j=1 ! X r

Notice that the operator δ j δ does not depend on a particular choice f ⊗ ǫj j=1 X r of the local representation f = f j ǫ even if local frame ǫ of Q is ⊗ j { j}j j=1 X non-holomorphic. We also remark that S is nothing but the dual section F ∗ of F . Before discussing the residue currents, we introduce a convention on composition of vector bundle-valued forms. Let us denote by ′ the sheaf of Dk k-currents on X. For any given three vector bundles E1, E2, E3, for any currents ω ′ , η ′, and for any morphisms f Hom(E , E ) and ∈ Dk ∈ Dl ∈ 1 2 g Hom(E , E ), we take the following composition rule: ∈ 2 3 (ω f) (η g)= η ω f g. ⊗ · ⊗ ∧ ⊗ ◦ With these preparations, we can give the construction of residue currents following M. Andersson and E. Wulcan. In [5] they constructed the residue current by using the terminologies of super-connection (cf. [40]). We do not need to employ this formalism in this paper. Instead, we only give the explicit way of constructing it. Let us consider a generically exact complex of hermitian vector bundles f f 0 E N E 1 E . (4.4) → N →···→ 1 → 0 Take the minimal inverse σk of each morphism fk : Ek Ek 1, and let Z be → − the set of points of X where some fk do not have optimal rank. We deﬁne a l smooth Hom(El, Ek)-valued (0,k l 1)-form uk for (k l +1) on X Z by the following formula: − − ≥ \ ul =(∂σ¯ ) (∂σ¯ )σ . (4.5) k k ··· l+2 l+1 With this notation, we have

18 Proposition 4.1 ([5]). For any local holomorphic function F X such that Z F = 0 holds 2λ l ¯ 2λ l ∈ O ⊂ { } locally, F uk and ∂ F uk can be continued analytically to a current across Z| with| a holomorphic| | ∧ parameter in Re λ> ε for some small ε> 0. The value at λ =0 of these currents are independent− of the particular choice of F . Furthermore, if we put

U l = F 2λul for k l +1, (4.6) k | | k|λ=0 ≥ and Rl = ∂¯ F 2λ ul for k l +1, (4.7) k | | ∧ k|λ=0 ≥ if the holomorphic function F above can be factorized as F = F˜1 F˜p, if (tλ) t1λ ··· tpλ t1, , tp are positive real numbers, and if F˜ ∗ denotes F˜1 F˜p , ··· (tλ) l (tλ) l | | | | ···| | then F˜ ∗ u and ∂¯ F˜ ∗ u can be continued analytically to a distribution | | k | | ∧ k across Z with a holomorphic parameter in Re λ > ε for some ε > 0 and one has the formulae −

l (tλ) l U = F˜ ∗ u for k l + 1 (4.8) k | | k|λ=0 ≥ and l (tλ) l R = ∂¯( F˜ ∗ ) u for k l +1. (4.9) k | | ∧ k|λ=0 ≥

The formulae (4.8) and (4.9) are not explicitly written in [5]. However, a l l careful reading of the constructions of Uk and Rk in [5] tells us that such formulae are valid. We put

l U = Uk (4.10) l 0 k l+1 X≥ X≥ and l R = Rk, (4.11) l 0 k l+1 X≥ X≥ and call them associated currents to the complex (4.4). In particular, the current R is called the residue current of the complex. The following result states that R measures the exactness (hence non-exactness) of the complex (4.4). Theorem 4.1 ([5, Theorem 1.1]). l The complex (4.4) is exact on X if and only if Rk =0 for all positive integers k >l. Furthermore, if the complex (4.4) is exact, a holomorphic section φ E0 is in Im f1 if and only if φ is generically in Im f1 and for all k > 0, ∈ 0 one has Rkφ =0.

19 Thus, the residue current R actually corresponds to the notion of Noethe- rian operators deﬁned originally by L. Ehrenpreis (see [38], and [17] ). This theorem gives us an abstract criterion of membership, but we need to know explicitly what φ in Theorem 4.1 is in view of applications to the theory of equations. To this end, we use the integral representation technique that was introduced by B. Berndtsson and M. Andersson and further developed by M. Andersson (cf. [1], [4]).

Let D Cn be a domain and we regard z D as a parameter. We put ⊂ ∈ m = ′ , (4.12) L D(k,k+m) k 0 M≥ ζ z = δζ z ∂.¯ (4.13) ∇ − − − Here, δζ z is the operator introduced in Proposition 3.5. Note that any − element ω m can be decomposed as ω = ω + ω + , where ∈ L 0,m 1,m+1 ··· ω ′ . k,k+m ∈D(k,k+m) Proposition 4.2 ([4, Proposition 2.1]). Let z D be a ﬁxed point and g = g + + g 0(D) be a current ∈ 0,0 ··· n,n ∈ L with compact support in D such that ζ zg = 0, g is smooth around z, and − g (z)=1. In this setting, for any holomorphic∇ function φ (D), one has 0,0 ∈ O an integral representation formula

φ(z)= gφ = gn,nφ. (4.14) ZD ZD Any g 0(D) with properties in the proposition above is called a weight with respect∈ L to z. In [1] and [4] it was explained that one can obtain various classical integral formulae in a unified manner thanks to the proposition above. Let us now introduce yet another tool to get an explicit solution of the membership problem. We consider a generically exact complex. f f 0 E N E 1 E 0. (4.15) → N →···→ 1 → 0 → The readers should be aware that this complex is different from the complex (4.4). We recall the following general existence theorem of Hefer forms. Proposition 4.3 ([4, Proposition 5.3]). Assume D is Stein and consider either (4.4) or (4.15). Then, for any integers k k and l, one can find Hl (ζ, z) (k l,0)(Hom(Ek, El))(Dζ) such that ∈E − k Hl (ζ, z) is holomorphic both in ζ and z. k l Hl (ζ, z)=0(k

k l k l HU = Hl+1U = Hl+1Uk, Hl+1Uk (′k l 1,k l 1)(Hom(El, El+1)) ∈D − − − − l k l+1 X X≥ k l k l HR = HlR = Hl Rk, Hl Rk (′k l,k l)(End(El)) ∈D − − l k l+1 X X≥ f = fj, g′(ζ, z)= f(z)HU + HUf + HR. j 1 X≥

Now we can check by a direct computation that ζ zg′ = 0. Furthermore, ∇ − for any z D Z, one has the identity g′ (z, z) = Id , where Z is the ∈ \ 0,0 Ej j 0 singular locus of currents U and R. With this notationX≥ Proposition 4.2 and Proposition 4.3 lead to

Proposition 4.4 ([4, Proposition 5.4]). Let D be a Stein open subset of Cn and consider the generically exact complex (4.15).

(1) If g 0(D ) is a smooth weight with respect to z / Z, then for any ∈ L ζ ∈ holomorphic section φ (D , E ), one has an integral representation ∈ O ζ l

φ(z)= f (z) H Uφ g + H Uf φ g + H Rφ g, (4.16) l+1 l+1 ∧ l l ∧ l ∧ ZDζ ZDζ ZDζ

0 (2) If g(ζ, z) (Dζ ) is a smooth weight with respect to z D, g is holomorphic∈ Lin z, and if for all z D, supp g( , z) is compact∈ in D, ∈ { · } then the (4.16) is valid across Z.

Proof. Part (1) follows immediately from Proposition 4.2 since g′ g is a weight. To see Part (2) notice that in this case, both sides of the eq∧uation (4.16) are holomorphic so we have the equation even across Z by the identity theorem.

Remember that we assumed f1 is generically surjective in the argument above. However, this assumption is not necessary. We shall see that we anyway can get a representation of φ as soon as it belongs to the image of f1. To this end, we again consider the complex (4.4). We can extend this to another generically exact complex in a neighbourhood of each point by the following proposition.

21 Proposition 4.5. Let X be a Stein manifold, and K be a holomorphically convex compact subset of X. If is a coherent X -module on a neighbourhood of K, it can be embeddedM into a genericallyO exact complex of the folowing type in a neighbourhood of K:

1 r0 1 rM−1 0 × × 0, (4.17) →M→OX →···→OX →E→ where is a locally free sheaf in a neighbourhood of K. E Proof. We put ∗ = om( , X ) for any X -module . By Theorem 7.2.1 in [22], oneN can takeH theN followingO resolutionO of inN a neighbourhood of K: M 1 rM 1 r0 × × ∗ 0 (exact). (4.18) OX →···→OX →M → Here, we put M = dim X. By Hilbert’s syzygy theorem, we can conclude 1 rM 1 rM−1 = Ker( × × ) is locally free. By putting = ∗, taking K OX → OX E K ( )∗ of the above complex, and composing the canonical map ∗∗, we− have a complex M→M

1 r0 1 rM−1 0 × × 0. (4.19) →M→OX →···→OX →E→ Since this complex is exact where is locally free, this is the desired com- M plex. With the help of this proposition, we can prove

Proposition 4.6. f f Let D be a Stein open subset of Cn and 0 E N E 1 E be a → N → ···→ 1 → 0 generically exact complex on D. 0 (1) If g (Dζ ) is a smooth weight with respect to z / Z, then, for any holomophic∈ L section φ (D , Im f ), one has an integral∈ representation ∈ O ζ 1 φ(z)= f (z) H Uφ g. (4.20) 1 1 ∧ ZDζ (2) If g(ζ, z) 0(D ) is a smooth weight with respect to z D, g is holo- ∈ L ζ ∈ morphic in z, and if for all z D, supp g( , z) is compact in D, then the formula of (1) is valid across Z∈. { · } Proof. We ﬁrst, embed the complex (4.4) into a generically exact sequence

fN f1 f0 0 EN E1 E0 E n 0 (4.21) → −→··· → −→ −→··· → − → on a neighbourhood of supp g by Proposition 4.5. We can now prolong the k k Hefer forms taken above to Hl n l k N , where Hl (k l,0)(Hom(Ek, El)) { }− ≤ ≤ ≤ ∈E − satisfy the relations of Proposition 4.3. Applying Proposition 4.5 to this complex and φ (D , Im f ), we obtain the proposition. ∈ O ζ 1 22 5 Division with bounds

In this section, we solve a certain kind of the division with bounds which naturally arises in the theory of D∆-equations. Before starting a discussion, we would like to note that the division with bounds is no longer possible if there are more than two independent frequencies. Example 5.1. n For any positive integer n, let us denote by p(C ) the space of entire func- n O tions on C for which there exist constants C > 0, M > 0, and k Z>0 such that for every ζ Cn one has the estimate ∈ ∈ f(ζ) C(1 + ζ )k exp(M Im ζ ). (5.1) | | ≤ | | | | 2 sin ζ1 Consider now an ideal I of p(C ) generated by three elements , sin ζ2, O ζ1 and ζ2 αζ1, where α R Q¯ is a Liouville number. We can easily check − 2 ∈ \ 2 that 1 is in ( (C ) I) p(C ) since the zero locus of I is empty. On the other hand,O 1 can· never∩ O belong to the ideal I. Otherwise we have a representation 1 = f (ζ ,ζ ) sin ζ1 + f (ζ ,ζ ) sin ζ + f (ζ ,ζ )(ζ αζ ) for 1 1 2 ζ1 2 1 2 2 3 1 2 2 1 2 − some f1, f2, f3 p(C ). Substituting ζ2 = αζ1, we can conclude that 1 ∈ O sin ζ1 belongs to the ideal I′ of p(C) generated by and sin αζ , where the ζ1 1 O 2 definition of p(C) is similar to that of p(C ). However, L. Ehrenpreis O O 2 2 showed that I′ does not contain 1, meaning that I ( ( (C ) I) (C ) O · ∩ Op (cf. [17, pp 319-320]). This example suggests that the division with bounds is impossible when there are more than two independent frequencies. In spite of this sort of example, the division with bounds is possible for D∆-equations with one independent frequency as we shall see in this section. We begin with some elementary estimates related to the Hefer forms. Lemma 5.1. q(ζ) q(z) Take any q(z) and put p(ζ, z) = − . Assume that q does not ∈ H1 ζ z contain any negative power of ez. Then one has− the following estimates: (1) Regard q C(z)[ez]. If N denotes the degree of q as a polynomial of ez with∈ coefficients in C(z), then there are a non-negative integer M Z 0 and a positive constant C > 0 such that for all z C, ∈ ≥ ∈ M N Re z q(z) C(1 + z ) e | |. (5.2) | | ≤ | | (2) With the same notation as above, there are a non-negative integer M ∈ Z 0 and a constant C > 0 so that the following inequality is valid for ≥ all ζ, z C: ∈ M M N Reζ N Re z p(ζ, z) C(1 + ζ ) (1 + z ) e | |e | |. (5.3) | | ≤ | | | | 23 (3) For any non-negative integer k Z 0, one can find a non-negative ∈ ≥ integer M Z 0 and a constant C > 0 so that the following inequality ∈ ≥ is true for all ζ, z C: ∈ k ∂ M M N Reζ N Re z p(ζ, z) C(1 + ζ ) (1 + z ) e | |e | |. (5.4) ∂ζk ≤ | | | |

Proof. We omit the proof of (1) since it is straightforward. To see (2) ﬁrst recall that we have a formula

1 q(ζ) q(z)= q′(z + t(ζ z))dt (ζ z) (5.5) − − − Z0 by integration by parts. Note that q′(z) satisﬁes the estimate of (5.2). There- fore, we have

M N Re z+t Re(ζ z) q′(z + t(ζ z)) C(1 + z + t(ζ z) ) e | − | (5.6) | − | ≤ | − | M N (1 t) Re z+t Re ζ C(1+(1 t) z + t ζ ) e | − | (5.7) ≤ − | | | | M M N Re ζ N Re z C(1 + ζ ) (1 + z ) e | |e | |. (5.8) ≤ | | | | As for (3), if we prove the estimate when k = 1, we inductively have the estimate for all k. When k = 1, the inequality is immediately veriﬁed by means of Cauchy’s integral formula as follows:

∂ M N Re z M N Re ξ p(ζ, z) C(1 + z ) e | | sup (1 + ξ ) e | | (5.9) ∂ζ ≤ | | ξ ∂∆(ζ;1) | | ∈ M M N Re ζ N Re z C˜(1 + ζ ) (1 + z ) e | |e | |. (5.10) ≤ | | | |

Proposition 5.1. Suppose a complex of free -modules, H

1 rN PN 1 r1 P1 1 r0 0 × × × × × (5.11) → H −−→· · · → H −−→H

1 rj is given. Put Ej = n × and consider the associated complex obtained by tensoring to the aboveO complex2: nO f f 0 E N E 1 E . (5.12) → N −→··· → 1 −→ 0 2Recall that (Cn) is an exact functor by Proposition 3.2 (3). −⊗H O

24 Assume (5.12) is generically exact and equip these trivial bundles with standard Hermitian metrics and consider the associated currents U and R. De- compose U and R as l l U = Uk, R = Rk (5.13) k>l k>l X X and l l ¯I l l ¯I Uk = Uk,I dζ , Rk = Rk,I dζ , (5.14) I =k l 1 I =k l | | X− − | X| − where ζ is the standard coordinate of Cn. Under the assumptions and notation above, there is a non-zero polynomial (ζ ) such that (ζ )U l and (ζ )Rl are distributions with Paley-Wiener R 1 R 1 k,I R 1 k,I growth, i.e., there are non-negative integers M, N, k, and k′ Z 0 and a ≥ positive constant C > 0 so that for any compactly supported smooth∈ function n φ C∞(C ) the following inequalities hold: ∈ c

l α β M N Re ζ1 (ζ1)Uk,I ,φ C sup (∂ζ ∂ζ¯ φ(ζ))(1 + ζ ) e | | , (5.15) hR i ≤ ζ supp φ | | |α|≤∈k,|β|≤k′

l α β M N Re ζ1 (ζ1)Rk,I ,φ C sup (∂ζ ∂ζ¯ φ(ζ))(1 + ζ ) e | | . (5.16) hR i ≤ ζ supp φ | | |α|≤∈k,|β|≤k′

Proof. We use the same symbols as in Section 4. Let E and Q be trivial n bundles on C . We take global holomorphic frames ǫ1,...,ǫr (r = rank Q), e ,...,e (s = rank E), and equip E and Q with hermitian{ } metrics so that { 1 s} these frames are orthonormal basis at each point of Cn. Notice that the dual section ei∗ of ei is a holomorphic section of E∗. Take f Hom(E, Q) n ∈ and express it as f = f e∗ ǫ , f (C ). If the optimal rank of ij i ⊗ j ij ∈ O i,j 1Xρ f is ρ, we have F = f = F e∗ ǫ , where each coefficient ρ! ∧ I,J I ⊗ J I = J =ρ n | |X| | FI,J (C ) is a product of finitely many fi,j. Taking into account that ∈ O j the standard dual ǫ of ǫj is equal to the hermitian dual ǫj∗ of it, we have s r ρ 1 S = F¯I,J eI ǫ∗ . Since s =( δf e∗ δǫ ) − S/ρ!, if we represent ⊗ J ij i ⊗ j I = J =ρ i=1 j=1 | |X| | X X s as s = s e ǫ∗, each s is a product of finitely many f and f¯ . Now ij i ⊗ j ij ij ij i, j X we take f = fk, E = Ek, and Q = Ek 1. We are going to estimate ∂σ¯ for − s pI,J ζ1 σ = 2 . Writing FI,J = , φ C[ζ ], pI,J C[ζ ,...,ζn, e ], letting φ F φI,J 1 1 | | ∈ ∈

25 F˜ be the least common multiplier of φ , F = I,J , F˜ C[ζ ,...,ζ , eζ1 ] I,J I,J φ I,J ∈ 1 n and putting F˜ = F˜ e∗ ǫ , we have an identity I,J I ⊗ J I,J X s s = φ 2 . (5.17) F 2 | | F˜ 2 | | | | By Lemma 5.1, each fij satisﬁes the estimate

M N Reζ1 f (ζ) C(1 + ζ ) e | |. (5.18) | ij | ≤ | | Therefore we can write s˜ ∂σ¯ = , s˜ = s˜ijei ǫ∗, (5.19) ˜ 4 ⊗ j F i, j | | X so that eachs ˜ij satisﬁes the estimate of the type (5.18). In the followings, we construct the associated current U. Under the notation analogous to the argument above, we have a representation

l 1 (k) (l) u = gije e ∗, (5.20) k,I ˜ 4 ˜ 4 i ⊗ j Fk Fl+1 i,j | | ···| | X (k) (l) where ei and ej are global holomorphic frames of Ek and El and gij is a linear combination{ } { of} products of holomorphic and antiholomorphic functions satisfying the estimates of the type (5.18). We rewrite the denominator as a sum F˜ 2 F˜ 2 = f 2 + + f 2 = f 2, (5.21) | k| ···| l+1| | 1| ··· | p| || || ζ1 where each fj C[ζ1,...,ζn, e ] is a product of some entries of F˜l+1, , F˜k. In view of Proposition∈ 4.1, if t , , t are positive numbers, the current··· U l 1 ··· p k,I is equal to (tλ) f ∗ (k) (l) | | g e e ∗ . (5.22) f 4 ij i ⊗ j |λ=0 || || A priori, this current is holomorphicX at λ = 0. Therefore, the resulting f ∗(tλ) current is a combination of the constant term of the Taylor expansion of | |f 4 multiplied by some products of holomorphic and anti-holomorphic functions|| || of Paley-Wiener growth. Now we need the following lemma. Lemma 5.2. n n Let T ′(C ) be a distribution with Paley-Wiener growth. If g (C ) is a holomorphic∈D function which satisﬁes the estimate ∈ O

M N Re ζ1 g(ζ) C(1 + ζ ) e | | (5.23) | | ≤ | | 26 n n for any ζ C , then two distributions gT and gT¯ ′(C ) both have Paley- Wiener growth.∈ ∈D Proof. We prove the lemma for gT since the argument forgT ¯ is similar. n Take any φ ∞(C ). By deﬁnition, ∈ Cc

α β M N Re ζ1 gT,φ = T,gφ C sup (∂ζ (g(ζ)∂ζ¯ φ(ζ)))(1 + ζ ) e | | . |h i| |h i| ≤ ζ suppφ | | |α|≤∈k,|β|≤k′

(5.24) It can be veriﬁed from Cauchy’s integral representation theorem that deriva- tives of g satisfy the estimates of the type (5.23). Combining this observation with the inequality above, we have the lemma. End of proof of Proposition 5.1 Thanks to this lemma, the problem is f ∗(tλ) reduced to showing that the constant term of the Taylor expansion of | |f 4 around the origin is of Paley-Wiener growth. The Theorem 5.1 below| sa| ys that the constant term is indeed of Paley-Wiener growth if it is multiplied by a non-zero polynomial (ζ1). The proof is completed for U. For R, we R l l (tλ) l can observe that an identity ∂U¯ = R +( f ∗ ∂u¯ ) holds. Using this k k | | k |λ=0 identity, one can show that there is a univariate polynomial (ζ1) so that (ζ )Rl is of Paley-Wiener growth in the same manner as weR proved it for R 1 k U. Theorem 5.1 ([12, Proposition 3.2]). Let f , , f C[ζ , ,ζ , eζ1 ], where p is a positive integer, then, for any 1 ··· p ∈ 1 ··· n t (0, 1)p outside a countable union of algebraic hypersurfaces and any m ∈ ∈ Z>0, there is a polynomial (ζ1) and constants C > 0, M, N, k, k′ Z>0 n R ∈ such that if a ′(C ) denote the coeﬃcients of the Taylor expansion j ∈D (tλ) f ∗ ∞ | | = a λj, (5.25) f 2m j j=0 k k X (tλ) t1λ tpλ 2 2 2 where f ∗ = f1 fp and f = f1 + + fp , then for | | n | | ···| | k k | | ··· | | φ C∞(C ), ∈ c

α β M N Re ζ1 (ζ1)a0,φ C sup ∂ζ ∂ζ¯ φ(ζ) (1 + ζ ) e | | . (5.26) |hR i| ≤ ζ supp φ | | |α|≤∈k,|β|≤k′

Example 5.2. ζ We consider a univariate function e 1 (Cζ) and the associated generically exact complex of trivial vector bundles− ∈ O

eζ 1 0 − 0. (5.27) → O → O→ 27 By deﬁnition, the current U associated to the complex (5.27) is the principal 1 value current v.p. eζ 1 . A direct computation leads to the following formula: −

1 ζ ∂ ζ 2 v.p. = e− log e 1 (5.28) eζ 1 ∂ζ | − | − ζ ∂ ζ ∂ ζ ζ 2 = e− e− (e 1) log e 1 . (5.29) ∂ζ ∂ζ − | − | Since (eζ 1) log eζ 1 2 is locally bounded and its growth order at inﬁnity − | − | 1 is exponential 1, we can conﬁrm that v.p. eζ 1 is of Paley-Wiener growth. We can also compute the residue current R explicitly:−

R = π δ(ζ 2π√ 1m)dζ.¯ (5.30) − − m Z X∈ This formula also tells us that R is indeed of Paley-Wiener growth.

Now we can ﬁnally prove the division with bounds. The essential point is the construction of the suitable weight in view of Theorem 4.2. We ﬁrst work on the growth condition which arises from Fourier-Borel transform of analytic functionals. For any compact convex subset K of Cn with smooth boundary, and n 1 r0 h (C ) × , we put ∈ O

HK (ζ) n h K = sup e− h(ζ) , HK(ζ) = sup Re z,ζ , ζ C , (5.31) | | ζ Cn | | z K h i ∈ ∈ ∈ where , is the standard Euclidian product. H is smooth except at the h· ·i K origin so we smoothen out HK(ζ) around ζ = 0 so that the resulting function is convex. This function is denoted by ρK . We put

∂ρK ∂ρK ρ′ (ζ)= 2 (ζ),..., 2 (ζ) , (5.32) K ∂ζ ∂ζ 1 n and

√ 1 ¯ ∂ρK gK(ζ, z) = exp ρK′ (ζ),ζ z − ∂∂ρK = exp ζ z( − ) . (5.33) {−h − i− π } {∇ − π√ 1 } −

It can easily be seen that (gK)0,0(z, z) = 1 and ζ zgK = 0. Furthermore, ∇ − since ρK is convex, we have an inequality

n exp ρ′ (ζ),ζ z exp ρ (z) ρ (ζ) (ζ, z C ). (5.34) | {−h K − i}| ≤ { K − K } ∈

28 def def In the same manner, we putρ ˜(ζ) =ρ ˜(ξ ) = ξ , where ζ = ξ + √ 1η. 1 | 1| − We smoothen outρ ˜ around ξ1 = 0 so that the resulting function is convex and this function is still denoted byρ ˜. Under the similar notation as above, we have identities (˜g)0,0(z, z) = 1, ζ zg˜ =0, and an inequality ∇ − n exp ρ˜′(ζ),ζ z exp ρ˜(z) ρ˜(ζ) , (ζ, z C ). (5.35) | {−h − i}| ≤ { − } ∈ We have now reached one of principal results of this paper: the division n 1 r0 with bounds. We denote by (C )p× the set of r0 vectors h with entries in (Cn) such that h < forO some compact convex subset K Ω. O | |K ∞ ⊂ Theorem 5.2. n Let Ω be an open convex subset of C such that Ze1 +Ω=Ω. Then for any matrix P M(r1,r0; ) of generalized D∆-operators, one has the fundamental identity:∈ H

n 1 r0 n 1 r1 n 1 r1 (C ) × ( (C ) × P)= (C ) × P. (5.36) O p ∩ O · O p · For the proof, we need a simple algebraic lemma which will reduce the problem from the division with bounds for a submodule to that for an ideal. Lemma 5.3. If for any finitely generated ideal I of , one has the identity H ( (Cn) I) (Cn) = (Cn) I, (5.37) O · ∩ O p O p · then for any positive integer r Z>0, and for any finitely generated submod- 1 r ∈ ule N of × , one has the identity H n n 1 r n ( (C ) N) (C ) × = (C ) N. (5.38) O · ∩ O p O p · Proof. First of all, notice that it is equivalent to proving the following claim to prove this lemma: (Claim) For any coherent module M, the complex H n n 0 (C )p M (C ) M (5.39) → O ⊗H → O ⊗H is exact. 1 r n In fact, if we put M = × /N in the claim, we have an inclusion (C ) n 1 r n H O · N (C )p× (C )p N. Since the other inclusion is obvious, we get the lemma.∩ O ⊂ O · Now we prove the claim by the induction on the number of generators of M. When M is cyclic, we can assume there is a finitely generated ideal I of such that M /I so the claim follows by the assumption of lemma. H ≃ H 29 Assume that the claim is valid when the number of generators of M is less than n. Let M be a coherent -module generated by n elements. We can find H an exact sequence 0 M ′ M M/M ′ 0, where M ′ is generated by → → → → at most n 1 elements and M/M ′ is cyclic. Consider the following diagram. − 0 0

n / n / n / (C ) M ′ (C ) M (C ) M/M ′ 0 O p ⊗ O p ⊗ O p ⊗ H H H

/ n / n / n / 0 (C ) M ′ (C ) M (C ) M/M ′ 0. O ⊗ O ⊗ O ⊗ H H H The ﬁrst and the second row are exact by the exactness of tensor and Propo- sition 3.2 (3), and the ﬁrst and the third column are exact by the inductive assumption. We can now apply the snake lemma to conclude that the middle vertical map is injective.

Proof of Theorem 5.2 We basically follow the argument of [12]. The crucial difference is that we make use of a more general division formula than in [12]. By the previous lemma, we can assume r0 = 1 and we write t n n 1 r1 P = (P1, ,Pr1 ). Let h (C )p ( (C ) × P) such that h K < for a compact··· convex set K. We∈ O can assume∩ O ∂K is smooth· since there| | is always∞ a smooth convex function ϕ on Ω such that ϕ < a is relatively compact { } in Ω and K ϕ < 0 . The existence of such smooth convex function is verified in the⊂ same { manner} as that of plurisubharmonic function ψ which exhausts a given pseudoconvex domain D and satisfies that L ψ< 0 for given holomorphically convex compact subset L of D. See Theorem⊂{ 5.1.6} in [22]. By Theorem 3.3 we can take a finite free resolution of finite length

1 rN PN 1 r1 P1=P 0 × × × × (exact). (5.40) → H −−→· · · → H −−−−→H We can also take Hefer forms for this exact sequence as in Proposition 3.5, and choose a polynomial (z1) as in Proposition 5.1. We denote its distinct roots by α , ,α , and letR µ , ,µ be their respective multiplicities. 1 ··· k 1 ··· k We can expand each P as a power series of z α whose coeﬃcients are j 1 − l polynomials in z′ =(z2, , zn). If we truncate this series at the term (z1 µl ··· − αl) , we obtain a polynomial Pj,l. By the classical Ehrenpreis fundamental principle (or division with bounds, see [17, Theorem 4.2] or [38, IV, 5, 5◦ The fundamental theorem]), possibly replacing K by its compact convex§

30 neighbourhood, we have a representation

r1 h = G P +(z α )µl G , G < . (5.41) j,l j,l 1 − l r1+1,l | j,l|K ∞ j=1 X

Now for each ﬁxed z′ we can construct a polynomial Gj(z1, z′) in z1, by means of Lagrange interpolation formula such that for each l,

µ G (z , z′) G (z , z′)= O((z α ) l ). (5.42) j 1 − j,l 1 1 − l The explicit interpolation formula implies that G < . | j|K ∞ Now by means of these functions Gj, we have

r1 r1 r1 h G P = h G P + (G G )P = O((z α )µl ). (5.43) − j j − j,l j j,l − j j 1 − l j j j X X X We can conclude that there is a representation

r1 h = G P + (z )G . (5.44) j j R 1 r1+1 j=1 X

Note here that we have an inequality (z1)Gr1+1 K < again by possibly replacing K by its compact convex neighbourhood.|R | This∞ implies G < | r1+1|K in view of Pólya-Ehrenpreis-Malgrange division lemma. (This is nothing∞ but the division with bounds for principal ideal.) We can confirm that (z )G is in the ideal generated by P , ,P so we want to give bounds R 1 r1+1 1 ··· r1 for coefficients of each Pj by means of the residue current. Choose a function χ C∞(R, R) so that ∈ χ(t) = 1 (if t < 1), χ(t) = 0 (if t > 2). (5.45) | | | | Putting ζ χ (ζ)= χ(| |), (5.46) k k we introduce a weight b gk = χk ∂χk , (5.47) − ∧ ζ zb ∇ − where b is a (1,0)-form defined by the formula

1 ∂ ζ z 2 b = | − | . (5.48) 2π√ 1 ζ z 2 − | − | 31 b Here, b is M. Andersson’s notation in [4], which reads ∇ζ−z

b n 1 = b + b ∂b¯ + + b (∂b¯ ) − . (5.49) ζ zb ∧ ··· ∧ ∇ − Note that g is a smooth weight when z

It can be veriﬁed that g satisﬁes g0,0(z, z) = 1 and ζ zg = 0. µ ν ∇ − Now consider a current gk gK g g˜ for non-negative integers µ and ν. This is a weight with respect∧ to∧z when∧ z < k. By Proposition 4.6, we have | |

(z )G (z)= H1U (ζ )G (ζ) g g gµ g˜ν P(z), z k. R 1 r1+1 R 1 r1+1 ∧ k ∧ K ∧ ∧ · | | ≤ Zζ (5.51) 1 Let us observe that g = o((1 + ζ )− ) for each ﬁxed z. If we take suitably | | large µ, ν, since (z )U is of Paley-Wiener growth, each derivative of g and R 1 g˜ satisﬁes an estimate of Paley-Wiener type, and gk 1 and since ∂g¯ k 0 n → → in C∞(C ) as k , we can observe that the integral →∞ H1U (ζ )G (ζ) g gµ g˜ν (5.52) R 1 r1+1 ∧ K ∧ ∧ Zζ converges so that we have

(z )G (z)= H1U (ζ )G (ζ) g gµ g˜ν P(z), (5.53) R 1 r1+1 R 1 r1+1 ∧ K ∧ ∧ · Zζ and

′ H1U (ζ )G (ζ) g gµ g˜ν C(1 + z )M eρ(z)eνρ˜(z). (5.54) R 1 r1+1 ∧ K ∧ ∧ ≤ | | Zζ

Since for the convex hull K˜ = c.h.(K νe ) Ω, we have ρ(z)+ νρ˜(z) ± 1 ⊂ ≤ HK˜ (z) as long as z is outside a small neighbourhood of the origin and (1 + ′ M ˜ ε z n 1 r1 z ) Ce | | for any ǫ > 0, we can conclude that h (C )p× P holds.| | ≤ ∈ O ·

In the same manner, we can solve the division with bounds where growth conditions arise from the Fourier transform. For a convex compact subset

32 n K of R , a non-negative integer M Z 0, and a vector of holomorphic ≥ n 1 r0 ∈ functions h (C ) × , we put ∈ O M HK (ζ) h M,K = sup (1 + ζ )− e− h(ζ) (5.55) | | ζ Cn | | | | ∈ and n HK(ζ) = sup x, η , ζ = ξ + √ 1η C , (5.56) x Kh i − ∈ ∈ where , is the standard Euclidian product. h· ·i We use a diﬀerent embedding (Cn) from that in Section 3 by H ⊂ O ζ z1 √ 1ζ1 putting zi √ 1ζi, e e− − . Note that under this embedding, all 7→ − − 7→ n 1 r0 the results in Section 3 still hold. We denote by (C )p×′ the set of vectors h with entries in (Cn) such that h < forO some positive integer M O | |M,K ∞ and a compact convex subset K Ω. ⊂ Theorem 5.3. n Let Ω be an open convex subset of R such that Ze1 +Ω=Ω. Then for any matrix P M(r1,r0; ) of generalized D∆-operators, one always has the identity ∈ H

n 1 r0 n 1 r1 n 1 r1 (C ) ×′ ( (C ) × P)= (C ) ×′ P. (5.57) O p ∩ O · O p · Proof. Since the proof is parallel to that of Theorem 5.2, we only give deﬁnitions of the weights. Let us take any compact convex subset K of Ω and deﬁne ρK (ζ)= ρ(η) to be the smoothened version of the convex support function HK (ζ). We put

∂ρK gK(ζ, z) = exp ζ z( ) . (5.58) {∇ − −π√ 1 } − def def Similarly, we putρ ˜(ζ) =ρ ˜(η1) = η1 and again smoothen this out around | | ∂ρ˜ the origin. Deﬁningg ˜ byg ˜ = exp ζ z( ) , we can mimic the proof of − π√ 1 Theorem 5.2. {∇ − − }

6 Ehrenpreis-Malgrange type theorems for generalized D∆ equations

We can ﬁnally establish Ehrenpreis-Malgrange type theorems for D∆-equations. Firstly, we give the simplest version of our main theorem which does not re- quire any topological consideration.

33 Let us begin with defining the action of on holomorphic functions. Let Ω Cn be an open set which satisfies Ω+ReH = Ω, that is, an open set which ⊂ 1 is an inverse image of another open set with respect to the natural projection n n n 1 π : C Y = C /Re1 √ 1R C − . Each connected component Ωi of Ω again→ satisfies Ω + R≃e =− Ω . On× each Ω , we can define the action of i 1 i i H on (Ωi) as in Section 2. OLet E be the subring of (Cn) generated by C[z] and eαz (α C). We O ∈ can decompose E as a C-vector space:

E = C[z]eαz. (6.1) α C M∈ We put E = C[z]eαz. We can easily see that E is an submodule of α α H (Cn). O Proposition 6.1. For any positive integer i and any coherent -module M one has the following vanishing result: H Exti (M, E) = 0 (6.2) H Proof. By the coherency of , it is enough to prove that for any short exact sequence H 1 r Q(z,σ) 1 s P(z,σ) 1 t × × × × × , (6.3) H −−−−→H −−−−→H the associated complex obtained by applying Hom ( , E) H −

t 1 P(∂,σ) s 1 Q(∂,σ) r 1 E × · E × · E × (6.4) −−−−→ −−−−→ is exact. Furthermore, it is enough to prove the statement above replacing E by E0 = C[z]. We consider a perfect pairing

, : C[z] C[[z]] C (6.5) h· ·i × → deﬁned by

∞ p, f = n!p f , p = p zn C[z], f = f zn C[[z]]. (6.6) h i n n n ∈ n ∈ n=0 n n X X X One can easily show that for any p C[z], any f C[[z]], and any positive ∈ ∈ integer i,

∂ z1 p, f = p, zif and σp,f = p, e f . (6.7) h∂zi i h i h i h i

34 Therefore, the exactness of (6.4) is equivalent to that of

z z 1 r Q(z,e 1 ) 1 s P(z,e 1 ) 1 t C[[z]] × × C[[z]] × × C[[z]] × . (6.8) −−−−−→ −−−−−→ The last complex is actually exact since (Cn) C[[z]] is a tower of H ⊂ O ⊂ ﬂat extensions by Proposition 3.2 (3) and [15, Result 6.1.7].

Next, we proceed to our main result. We use some basic terminologies of topological vector spaces. See e.g., [26] or [38, V, 1]. First of all, remember § that we can deﬁne the Fourier transform of any distribution with compact n √ 1ξx support T ′(R ) by the formula ( T )(ξ) = T (x), e− − . In the ∈ E F h n i same manner, for any analytic functional T ′(C ), we can deﬁne the ∈ O ζz Fourier-Borel transform of T by the formula ( BT )(ζ) = T (z), e . The classical Paley-Wiener-Schwartz theorem and Ehrenpreis-MartinF h eau theoremi state that these are linear topological isomorphisms.

Theorem 6.1 ([38, V, 3, Proposition 2], [33, Theorem 6.4.5]). For any open convex subset§ Ω of Rn, the Fourier transform gives rise to F a linear topological isomorphism

F n ′(Ω) (C ) ′ , (6.9) E →O p n where (C ) ′ is equipped with the standard (DFS) topology. O p f Similarly, for any convex open subset Ω of Cn, the Fourier-Borel transform gives rise to a linear topological isomorphism

B F n ′(Ω) (C ) , (6.10) O →O p n where (C )p is equipped with the standard (DFS) topology. O f In the following argument, we assume that Ω Cn is convex. Consider ⊂ any short exact sequence

1 r Q(z,σ) 1 s P(z,σ) 1 t × × × × × . (6.11) H −−−−→H −−−−→H Since (Cn) is exact by Proposition 3.2 (3) , we have an exact sequence −⊗H O ζ ζ n 1 r Q(ζ,e 1 ) n 1 s P(ζ,e 1 ) n 1 t (C ) × × (C ) × × (C ) × . (6.12) O −−−−−→O −−−−−→O In view of Fourier-Borel transform, one can show that the complex

ζ ζ n 1 r Q(ζ,e 1 ) n 1 s P(ζ,e 1 ) n 1 t (C ) × × (C ) × × (C ) × (6.13) O p −−−−−→O p −−−−−→O p 35 be exact and P(ζ, eζ1 ) have a closed image implies the exactness of

t 1 P(∂,σ) s 1 Q(∂,σ) r 1 (Ω) × · (Ω) × · (Ω) × . (6.14) O −−−−→O −−−−→O The proof of the claim above can be found in, for example, [27, Theorem VII.1.3]. See also [38, V, 1, Proposition 8]. The exactness of the former § complex is satisﬁed by Theorem 5.2. To prove the closed range property, n 1 s ζ1 n 1 t take any sequence fj j (C )p× such that fjP(ζ, e ) j (C )p× { } ⊂ O { }n 1⊂s O ζ1 converges. Then, we can see that the limit belongs to (C ) × P(ζ, e ) n 1 t n 1 s ζ1 n 1 t O ∩ (C )p× since (C ) × P(ζ, e ) (C ) × is closed. See [15, Result O O ⊂n O1 s ζ1 6.1.8]. Thus the limit belongs to (C ) × P(ζ, e ) by Theorem5.2. O p Since is a coherent ring, we have proved H Theorem 6.2. n For any open convex subset Ω of C such that Re1 +Ω=Ω, for any coherent -module M, and for any positive integer i, one has the following vanishing Hresult: Exti (M, (Ω)) = 0. (6.15) H O

We can deﬁne the action of on C∞ functions in the same way as we deﬁned one on , and we can proveH the following theorem in view of Fourier O transform. Theorem 6.3. n For any open convex subset Ω of R such that Re1 +Ω=Ω, for any coherent -module M, and for any positive integer i, one has the following vanishing Hresult: i Ext (M,C∞(Ω)) = 0. (6.16) H We can further prove the injectivity result for hyperfunctions by means of spectral sequence due to H. Komatsu and T. Oshima (cf. [37]). Theorem 6.4. n For any open convex subset Ω of R such that Re1 +Ω=Ω, for any coherent -module M, and for any positive integer i, one has the following vanishing Hresult: Exti (M, (Ω)) = 0. (6.17) H B Proof. Put U = Ω √ 1Rn, U = z U Im z =0 , = U, U , U , × − i { ∈ | i 6 } U { 1 ··· n} and ′ = U , U . By the Leray spectral sequence, one has U { 1 ··· n} p p H ( , ′, )= H (U, ). (6.18) U U O Ω O

36 Now take a ﬁnite free resolution

1 rN PN 1 r1 P1 1 r0 0 × × × × × M 0(exact) (6.19) → H −−→· · · → H −−→H → → of M. We put

p,q p rq 1 K = C ( , ′, × ), (6.20) U U O p rq 1 p+1 rq 1 δ = d′ : C ( , ′, × ) C ( , ′, × ), (6.21) U U O → U U O p rq 1 p rq+1 1 P = d′′ : C ( , ′, × ) C ( , ′, × ). (6.22) q U U O → U U O By Theorem 6.2 and the purity of relative cohomology ([24] Chap. 2.),

p r0 1 × p,q q p, C ( , ′, P1 ) (p n, q = 0) ′E1 = ′′H (K ·)= U U O ≤ (0 (otherwise)

rq 1 p,q p ,q (Ω) × (p = n) ′′E1 = ′H (K· )= B (0 (otherwise)

p r0 1 × p,q HΩ( , ′, P1 ) (p n, q = 0) ′E2 = U U O ≤ (0 (otherwise) rq 1 rq+1 1 Ker(P : (Ω) × (Ω) × ) q B → B (p = n) p,q rq−1 1 rq 1 ′′E2 = Im(Pq 1 : (Ω) × (Ω) × )  − B → B 0 (otherwise).

p r0 1 p r0 1 p r1 1 Here, C ( , ′,  × ) = Ker(d′′ : C ( , ′, × ) C ( , ′, × )) . This U U OP1 U U O → U U O shows that this spectral sequence degenerates at E2 terms so that

n,q n,q q 0= ′E2 = ′′E2 = Ext (M, (Ω))=0 (6.23) H B if q > 0.

We give another application of the division with bounds which is known as the problem of spectral synthesis (analysis). We follow the argument of L. H¨ormander. The following lemma is taken from [22].

Lemma 6.1 ([22, Lemma 6.3.7]). ∞ Let ξ ,ξ , be complex variables, L Cξ (j = 1, 2, ), and b = 1 2 ··· j ∈ i ··· i=1 M ∞ bj j 1 C. We put ξ = ξi i. In these settings, an inﬁnitely many lin- { } ≥ ∈ { } j=1 Y ear equations Lj(ξ)= bj has a solution if and only if the following condition

37 is satisﬁed:

For any ﬁnite subset F 1, 2, , and for any complex numbers ⊂{ ···} c C (j F ) with c L =0, one has c b =0. j ∈ ∈ j j j j j F j F X∈ X∈ Theorem 6.5. n Let F be either C∞ or and Ω be a convex subset of R (when F = C∞) n O or of C (when F = ) such that Ω+ Re1. For any matrix of generalized O r0 1 r1 1 D∆-operators P M(r1,r0; ), Ker(P(∂, σ): E × E × ) is dense in ∈ r0 1 H r1 1 → Ker(P(∂, σ): F (Ω) × F (Ω) × ). → Proof. We prove the theorem only for F = C∞ since the arguments are similar. First, note that we have the identity

r 1 r 1 r0 1 Ker(P(∂, σ): E 0 E 1 )= Ker L L (C∞(Ω) × )′, L ↾ =0 × → × { | ∈ V } \ (6.24) by the Hahn-Banach theorem, where the bar stands for the closure and

r0 1 r1 1 V = Ker(P(∂, σ): E × E × ). (6.25) →

t r0 1 Let L be a linear functional L = (f , , f ) (C∞(Ω) × )′ with L = 0. 1 ··· r0 ∈ |V For this L, there exists a compact convex subset K of Ω, a positive real number C > 0, and a non-negative integer M such that, for all ζ Cn, ∈ L(ζ) C(1 + ζ )M eHK (ζ), (6.26) | | ≤ | | where L is the Fourier transformb of L. r1 1 Now suppose we could ﬁnd an element g (C∞(Ω) × )′ such that t 1 ∈r0 1 L = P(b ∂, σ− )g. Then, for any u C∞(Ω) × , − ∈ t 1 L, u = P( ∂, σ− )g, u = g, P(∂, σ)u . (6.27) h i h − i h i This implies L, u = 0 if P(∂, σ)u = 0 in a neighbourhood of supp g, hence we can get theh theorem.i Now the proof of this theorem is reduced to solving the equation

L = Q(ζ)gˆ(ζ) (6.28)

n r1 1 with gˆ (C ) × via the Fourier transform, where Q(ζ) is a matrix of ∈ O p b t √ 1ζ1 holomorphic functions deﬁned by Q(ζ) = P( √ 1ζ, e− − ). The proof − − falls naturally into two parts. Step1 Construction of the formal solution.

38 n For any ﬁxed ζ0 C , we are going to construct a formal series solution gˆ = G C[[ζ ζ ∈]] of (6.28). Note that the equation (6.28) in this case ζ0 ∈ − 0 can be written as an inﬁnite system of equations

α α n ∂ζ L(ζ) ↾ζ=ζ0 = ∂ζ (Q(ζ)Gζ0 (ζ)) ↾ζ=ζ0 , (α Z 0). (6.29) ∈ ≥ By Lemma 6.1, itb is enough to show the following: Take a polynomial vector 1 r0 q C[∂] × . If the equation ∈ q(∂ ) (Q(ζ)h(ζ)) ↾ = 0 (6.30) ζ · ζ=ζ0

r1 1 holds for all formal series h(ζ) C[[ζ ζ ]] × , then one has ∈ − 0 q(∂ ) L(ζ) ↾ =0. (6.31) ζ · ζ=ζ0 1 r0 Take any q =(q1, , qr0 ) C[∂] ×b which satisﬁes the equation (6.30) and ··· √∈ 1zζ0 t put u (z)= q ( √ 1z)e− − , u = (u (z), ,u (z)). We have k k − − 1 ··· r0 t √ 1zζ P(∂ , σ ) u = P(∂ , σ ) (q (∂ ), , q (∂ )) e− − ↾ (6.32) z z · z z · 1 ζ ··· r0 ζ · ζ=ζ0 t t √ 1ζ1 √ 1zζ = (q (∂ ), , q (∂ )) P( √ 1ζ, e− − )e− − ↾ 1 ζ ··· r0 ζ · − − ζ=ζ0 n o(6.33) =0. (6.34)

Thus, u(z) is an exponential polynomial solution. Since L ↾V = 0, we have

√ 1zζ 0= L, u = L, (q (∂ ), , q (∂ ))e− − ) ↾ = q(∂ ) L(ζ) ↾ , h i h 1 ζ ··· r0 ζ iz ζ=ζ0 ζ · ζ=ζ0 (6.35) b and hence we get the formal solution Gζ0 . Step2 Construction of the solution with bounds Since Ω is Stein, global section functor Γ(Ω, ) is exact for complexes of − n r1 1 coherent sheaves. We get from Step1 that L Q(ζ) (C ) × . Now we can ∈ ·O n r1 1 apply Theorem 5.3 to L to obtain the desired solution gˆ (C )p × . b ∈ O Lastly, we give a descriptionb of cohomology groups of the solution sheaves. n n 1 Let π : R R − be the projection which truncates the ﬁrst coordinate. → For any given matrix of D∆-operators P M(r1,r0; ), we can deﬁne the r0 1 ∈ H subsheaf olP,F of π F × for F = C∞, by the formula S ∗ B

r0 1 r1 1 olP,F = Ker P : π F × π F × . (6.36) S ∗ → ∗

39 Theorem 6.6. 1 r0 1 r1 n 1 We put M = × / × P. For any open set Ω′ R − , there is a H H · ⊂ canonical cohomology isomorphism i i H (Ω′, olP,F ) Ext (M, π F (Ω′)), (6.37) S ≃ H ∗ where F = C∞, and i is any integer. B Proof. We ﬁrst prove the theorem for F = C∞. Consider a ﬁnite free resolution

1 rN PN 1 r1 P1= P 1 r0 0 × × × × × × M 0(exact) (6.38) → H −−→· · · → H −−−−−→H → → of M. Applying the functor om ( , π F ) to this sequence, we have a H ∗ complex H −

r0 1 P1 r1 1 PN rN 1 0 olP,F π F × · π F × · π F × 0. (6.39) → S → ∗ −→ ∗ ··· −−→ ∗ → In view of Theorem 6.3 and the fact that open convex subsets are fundamental n 1 system of neighbourhoods in R − , we can conclude that the complex above is exact. Furthermore, since for any i> 0, we have i H (Ω′, π F )=0, (6.40) ∗ the exact sequence above is a Γ(Ω′, )-injective resolution of olP,F . There- fore, by the standard theory of sheaf− cohomology, we obtain S

F ri 1 F ri+1 1 i Ker(Pi : π (Ω′) × π (Ω′) × ) H (Ω′, olP,F ) · ∗ → ∗ . (6.41) ri 1 ri 1 S ≃ Im (Pi 1 : π F (Ω′) −1× π F (Ω′) × ) − · ∗ → ∗ On the other hand, if one applies the functor Hom ( , π F (Ω′)) to the ﬁnite H ∗ free resolution (6.38) of M, one has −

F ri 1 F ri+1 1 i Ker(Pi : π (Ω′) × π (Ω′) × ) Ext (M, π F (Ω′)) · ∗ → ∗ . (6.42) F ri−1 1 F ri 1 H ∗ ≃ Im (Pi 1 : π (Ω′) × π (Ω′) × ) − · ∗ → ∗ As for F = , the argument is similar since B i H (Ω′, π F ) = 0 (6.43) ∗ for i> 0 follows from the ﬂabbiness of . B Acknowledgement

The author would like to thank H. Sakai and T. Oshima for their constant en- couragement and uncountably many suggestions, to Y. Nozaki and S. Wakat- suki for various comments, and to anonymous referees whose comments im- proved the exposition of this paper.

40 References

[1] M. Andersson, Integral representation with weights I, Math. Ann. 326 (2003) 1–18

[2] M. Andersson, Residue currents and ideals of holomorphic functions, Bull. Sci. math. 128 (2004) 481–512

[3] M. Andersson, Residue currents of holomorphic morphisms, J. reine angew. Math. 596 (2006) 215–234

[4] M. Andersson, Integral representation with weights II, division and interpolation, Math. Z. (2006) 315–332

[5] M. Andersson, E. Wulcan, Residue currents with prescribed annihila- tor ideals, Sci. Ecole´ Norm. Sup. 40 (2007), 985–1007

[6] G. A. Berenstein, and M. A. Dostal, The Ritt theorem in several variables, Arkiv for Matematik , Volume 12, Issue 1-2, (1974) 267– 280

[7] C. A. Berenstein, and R. Gay, Complex Analysis And Special Topics in Harmonic Analysis, Springer (1995)

[8] C. A. Berenstein, R. Gay, A. Vidras, A. Yger, Residue currents and Bezout identities, Progress in Mathematics Volume 114, Birkh¨auser, (1993)

[9] C. A. Berenstein, T. Kawai, D. Struppa, Interpolating varieties and the Fabry-Ehrenpreis-Kawai gap theorem, Advances in Math 122 (1996), 280–310.

[10] C. A. Berenstein and B.A.Taylor, Interpolation problems in Cn with applications to harmonic analysis, J. Analyse Math. 38 (1980), 188– 254.

[11] C. A. Berenstein, A. Yger, Ideals generated by exponential polynomials, Advances in Math 60 (1986), 1–80

[12] C. A. Berenstein, A. Yger, Exponential polynomials and -modules, Compositio Mathematica 95 (1995), 131–181 D

[13] C. A. Berenstein, A. Yger, The use of -modules to study exponential D polynomials, topics in complex analysis Banach center publications, volume 31 (1995)

41 [14] B. Berndtsson and M. Passare, Integral Formulas and an Explicit Version of the Fundamental Principle, Journal of functional analysis 84, (1989), 358–372

[15] H. Bourlès and U. Oberst, Elimination, fundamental principle and duality for analytic linear systems of partial differential-difference equations with constant coefficients. Math. Control Signals Systems 24 no.4 (2012), 351–402

[16] L. Ehrenpreis, Solution of some problems of division. Part IV. Invert- ible and Elliptic Operators, American Journal of Mathematics, Vol82, No3,(1960), 522–588

[17] L. Ehrenpreis, Fourier Analysis in Several Complex Variables, Dover, (2006)

[18] S. Glaz, Commutative coherent rings Lecture Notes in Mathematics, Vol. 1371(1989)

[19] H. Glüsing-Lürßen, Linear Delay-Differential Systems with Com- mensurate Delays: An Algebraic Approach, Springer-Verlag, Berlin, (2001)

[20] D. I. Gurevich, A counter example to the problem of L.Schwartz, Funktsional. Anal. i Prilozhen., Volume 9, Issue 2, (1975), 29–35

[21] H. Hironaka, On resolution of singularities (characteristic zero). 1963 Proc. Internat. Congr. Mathematicians (Stockholm, 1962) 507–521

[22] L. H¨ormander, An Introduction to Complex Analysis in Several Vari- ables, 3rd ed. North-Holland,(1990)

[23] T. Kawai, On the theory of Fourier hyperfunctions and its applications to partial diﬀerential equations with constant coeﬃcients, J. Fac. Sci. Univ. Tokyo (Sec. 1A Math.), 17 (1970), 467–517

[24] T. Kawai, M. Kashiwara, T. Kimura, Foundations of algebraic analysis, Princeton University Press (1986)

[25] A. G. Khovanskii, Newton polyhedra and toroidal varieties, Funct. Anal. Appl. 11, (1978), 289–295

[26] H. Komatsu, Projective and injective limits of weakly compact se- quences of locally convex spaces, J. Math. Soc. Japan, Vol. 19, No. 3, (1967), 366-383

42 [27] H. Komatsu, Sato’s hyperfunction and linear partial diﬀerential equations, Seminary note of the faculty of mathematics of the university of Tokyo, 22, (1968) in Japanese

[28] H. Komatsu, Relative cohomology sheaves of solutions of diﬀerential equations. Lecture Notes in Math., 287, Springer, Berlin,(1973),192– 261

[29] Y. Lequain and A. Simis, Projective modules over R[X1,...,Xn], R a Prufer domain, Journal of Pure Applied Algebra 18, (1980), 165–171

[30] O. Lezama, V. Cifuentes, W. Fajardo, J. Monta˜no, M. Pinto, A. Pulido, M. Reyes, Quillen-Suslin Rings, Extracta Mathematica, Vol. 24, N´um. 1, (2009), 55–97

[31] B. Malgrange, Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution, Ann. Inst. Fourier 6, (1956) 271–355

[32] Meril, A. and Struppa, D. C., Convolutors in spaces of holomorphic functions, Lecture Notes in Math., 1276, Springer, Berlin, (1987), 253–275

[33] M. Morimoto, An Introduction to Sato’s Hyperfunctions, Translations of Mathematical Monographs, American Math. Society (1993),

[34] U. Oberst, Duality theory for Grothendieck categories and linearly compact rings. J. Algebra 15, (1970), 473–542

[35] U. Oberst, Multidimensional constant linear systems. Acta Appl. Math. 20 no. 1-2,(1990), 1–175

[36] Y. Okada, On a class of convolution systems in one variable. Mi- crolocal analysis and PDE in the complex domain (Japanese), Su- urikaisekikenkyuusho Koukyuuroku No. 1159, (2000), 127–134

[37] T. Oshima, A proof of Ehrenpreis’ fundamental principle in hyperfunctions, Proc. Japan Acad. Volume 50, Number 1,(1974),16–18

[38] V. P. Palamodov, Linear Diﬀerential Operators with Constant Coef- ﬁcients, Springer,(translated by A.A.Brown in 1970)

[39] M. Passare, A. Tsikh, A. Yger, Residue currents of the Bochner- Martinelli type, Publications Mathem`atiques, Vol 44, (2000) 85–117

43 [40] D. Quillen, Superconnections and Chern characters Topology Vol. 24. No. 1, (1985) 89–95.

[41] L. Schwartz, Theorie generale des fonctions moyenne-priodiques. Ann. of Math. 48 , (1947), 857–929

[42] A. N. Varchenko, Newton polyhedra and estimation of oscillating in- tegrals, Funct. Anal. Appl. 10, (1976), 175–196

[43] A. Yger, A conjecture by Leon Ehrenpreis about zeros of exponential polynomials, From Fourier analysis and number theory to radon transforms and geometry, Dev. Math., 28, Springer, New York, (2013), 517–535.