Intuitive Representation of Local Cohomology Groups Within This Framework

Home , Local cohomology

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H(ˆ F ( )) = F (W ) / , (1.2) W W⊕∈W R where is a C-vector space so that two sections which coincide on their non- emptyR common domain give the same element in H(ˆ F ( )). We shall show the equivalence of local cohomology groupsW and their intuitive representation by constructing the isomorphisms concretely. In this course, we can construct the boundary value morphism bW from their intuitive representation to local cohomology groups in a functorial way due to Schapira’s idea [KS]. The inverse map ρ of the boundary value morphism is, however, not obvious. Therefore by introducing an particular resolution which depends on the choices of a stratification χ , we obtain the expression of local cohomology groups depending on χ. Then∈ T we realize the inverse map ρ thanks to this expression. As an application, for example, we can construct a framework which realizes intuitive representation of Laplace hyperfunctions. The theory of Laplace hyperfunctions in one variable was established by H. Komatsu as a framework of operational calculus, and is extended to the one in several variables by K. Umeta and N. Honda [HU]. In particular, they established a vanishing theorem of cohomology groups on a pseudoconvex open subset for holomorphic functions with exponential growth at infinity. Their result is the extension of Oka-Cartan’s vanishing theorem to DCn , and is a crucial key for constructing a framework in n the application. Here DCn is the radial compactification of C . In addition, we establish a theorem of Grauert type which guarantees the existence of a good open neighborhood in our sense, from which we can conclude the existence of the framework for Laplace hyperfunctions.

At the end of the introduction, we would like to show our greatest apprecia- tion to Professor Naofumi Honda for the valuable advice and generous support in Hokkaido University.

2 Intuitive Representation

In this section, we introduce several deﬁnitions which are needed for constructing a framework. Then we deﬁne intuitive representation of local cohomology groups within this framework.

2.1 Preparation Let X be a topological manifold and M a closed submanifold of X. Note that these manifolds are allowed to have their boundaries. Let F be a sheaf on X, ZX the constant sheaf on X having stalk Z and m a non negative integer. We

2 m F denote by HM (X, ) the m-th local cohomology group of X supported by M. We also denote by Dm the open unit ball in Rm with the center at the origin and set Sm−1 = ∂Dm. Then we deﬁne X as follows.

X = M Dm. (2.3) ×e We assume that there existse a homeomorphism ι : X X which satisfies ι(M)= M 0 . Hereafter we identify X with X by ι. → ×{ } As the first step of constructing framework, we introducee an appropriate partition of Sm−1. e Definition 2.1. Let k = 0, 1, ,m 1. A simplex σ is said to be a linear k-cell in Sm−1 if σ is oriented and··· it is− written in the form

σ = ϕ( H ) Sm−1. (2.4) ∩ i ∩ k+1 m Here ϕ : R R is a linear injective map and Hi is a finite family of → k+1 { } open half spaces in R with 0 ∂Hi whose intersection is non-empty. For convenience, we regard a empty set∈ as a linear cell. ∅ Let us recall the definition of a stratification of a set U. A partition U = Uλ is called a stratification of U, if it is locally finite, and it satisfies for all λ⊔∈Λ the pairs (Uλ,Uτ ) U U = U U , λ ∩ τ 6 ∅ ⇒ τ ⊂ λ and we call each Uλ a stratum. Let σ denote the realization, i.e., just the set in Sm−1 forgetting its orientation, of| the| simplex σ. Let χ be a stratification of Sm−1 such that each stratum is a linear cell of m−1 S . We denote by ∆(χ) and ∆k(χ) the set of all linear cells of χ and the set of all linear k-cells of χ, respectively. We also denote by ∆ (χ) the set of realizations σ of cells σ ∆(χ), that is, | | | | ∈ ∆ (χ)= σ ; σ ∆(χ) . | | {| | ∈ } The set ∆ (χ) is also defined in the same way. | |k Definition 2.2. For two stratifications χ and χ′ of Sm−1, χ′ is finer than χ if and only if for any σ′ ∆(χ′) there exists σ ∆(χ) with σ′ σ, which is denoted by χ χ′. ∈ ∈ ⊂ ≺ Definition 2.3. For a cell σ ∆(χ) we define a star open set Stχ( σ ) as follows. ∈ | | Stχ( σ )= τ . (2.5) | | σ⊔⊂τ | | τ∈∆(χ)

m−1 As a special case, we set Stχ( ) = S . We write St( σ ) instead of St ( σ ) if there is no confusion. |∅| | | χ | |

3 Definition 2.4. Let χ be a stratification of Sm−1 whose stratum is a linear cell and let σ, τ ∆(χ). We write σ τ = δ if and only if there exists a linear cell δ such that σ∈ τ = δ holds. ∧ ∩ Remark 2.5. δ is allowed to be . ∅ Definition 2.6. For a subset K in Sm−1 we define the subset M K in X by ∗ M K = (x,ty) M Dm ; x M,y K, 0

4 n Deﬁnition 2.7. Let W be an open set in C and Γ an R+-conic open subset in Rn. We say that W is an inﬁnitisimal wedge of type M √ 1Γ if and only if the following conditions hold. × − 1. W M √ 1Γ. ⊂ × − ′ 2. For any R+-conic open proper subset Γ of Γ, there exists an open neighborhood U X of M such that ⊂ M √ 1Γ′ U W. (2.8) × − ∩ ⊂

Example 2.8. Let F be the sheaf OX of holomorphic functions on X. We set , and as follows. T U W = χ ; χ is associated with partitioning by the finite family of T { hyperplanes in Rn passing through the origin. , } = all the Stein open neighborhoods of M in X , U { } = all the infinitisimal wedges of type M √ 1Γ ; W { × − Γ runs through all the connected open cones. . } Example 2.9. Let F = O . We set = χ , where χ is the stratification X T { } which is associated by the set of hyperplanes Hi = y = (y1,y2, ,yn) Rn { ··· + ∈ ; yi =0 (i =1, 2, ,n), and set = X . Moreover we define Hi and − } ··· U { } Hi by H± = y = (y ,y , ,y ); y > 0 , i { 1 2 ··· n ± i } respectively. Then we set

∗ = W ; Each W is either X itself or the finite intersection of Hi with =+ or . . W { ∗(2.9) − } Now we are ready to define intuitive representation of local cohomology groups. Definition 2.10. We define intuitive representation H(ˆ F ( )) of local cohomology groups as follows. W

H(ˆ F ( )) = F (W ) / . (2.10) W W⊕∈W R i Here is the C-vector space generated by the following elements. R f ( f ) (f F (W ), W , W with W W ). (2.11) ⊕ − |W2 ∈ 1 1 2 ∈W 2 ⊂ 1 Remark 2.11. We can consider a restriction map of intuitive representation. Let X be a topological manifold, M a closed submanifold of X and F a sheaf on X. We assume that there exists ( , , ) satisfying the conditions (T), T U W

5 (U) and (W). Moreover let M ′ be an open subset in M and a homeomorphism ι′ : X′ ∼ M ′ Dm with the following commutative diagram. −→ × ∼ X′ M ′ Dm −−−−→ι′ × (2.12)  ∼  X M Dm. y −−−−→ι ×y We set

′ = , (2.13) T T ′ = X′ = U X′ ; U , (2.14) U U ∩ { ∩ ∈ U} ′ = X′ = W X′ ; W . (2.15) W W ∩ { ∩ ∈ W} Then ( ′, ′, ′) satisfy the conditions (T) and (W). Hence by assuming the conditionT (U-1)U W for ′ and ′, we can obtain intuitive representation of T U m ′ F ′ HM ′ (X , ) orM ′/X′ (M ) Z ′ M′⊗(M ) by using ( ′, ′, ′). Furthermore the restriction map of intuitive representation is inducedT U fromW the one F (W ) F (W X′) of the sheaf F . −→ ∩ 3 The equivalence of local cohomology groups and their intuitive representation

Through this section, we shall prove the following main theorem. This theorem guarantees the equivalence of local cohomology groups and their intuitive representation.

Theorem 3.1. The boundary value morphism bW deﬁned below gives an isomorphism from intuitive representation to local cohomology groups. ˆ F ∼ m F bW : H( ( )) HM (X, ) orM/X (M). (3.16) W −→ ZM⊗(M)

3.1 Construction of the boundary value morphism bW

First of all, we construct the boundary value morphism bW from H(ˆ F ( )) to local cohomology groups thanks to Schapira’s idea (p.497[KS]). W Let G be a sheaf on X. We deﬁne the dual complex D(G ) of G by G H G C D( ) = R omCX ( , X ). Fix W . By the condition (W-2), we can ﬁnd U , χ and σ ∆(χ) satisfying∈ W (M St( σ )) U W . Hence we obtain∈ U the restriction∈ T map∈ ∗ | | ∩ ⊂ F (W ) F ((M St( σ )) U). (3.17) → ∗ | | ∩

6 We also get the restriction map of sheaves as M St( σ ) M: ∗ | | ⊃ C C . M∗St(|σ|) −→ M

By applying D( ) and CU to the above morphism, we have • C⊗X

C D(C ). (M∗St(|σ|))∩U ←− M Remark 3.2. We note the following well-known facts.

D(CM ) CM orM/X [ m], (3.18) ≃ Z⊗M −

Hm (X, Z ) Γ(X, or ), (3.19) M X ≃ M/X H m Z where orM/X = M ( X ) is the relative orientation sheaf on M. F Thus by applying RHomCX ( , ) to the above and taking the 0-th cohomology, we obtain •

F m F ((M St( σ )) U) HM (X, ) orM/X (M). (3.20) ∗ | | ∩ → ZM⊗(M)

We can get the morphism bW by composing (3.17) and (3.20); F F m F (W ) ((M St( σ )) U) HM (X, ) orM/X (M). (3.21) → ∗ | | ∩ → ZM⊗(M)

Then bW satisﬁes the proposition below.

Proposition 3.3. The map bW does not depend on the choices of U, χ and σ.

Proof. The fact that bW does not depend on the choices of U is shown by the same argument as the following one. Hence we only show∈ U that it is independent of the choices of χ and σ. Let χ1 and χ2 be stratiﬁcations of m−1 S which belong to , let σ1 ∆(χ1) and σ2 ∆(χ2). We can ﬁnd an ′ T ∈ ∈ open neighborhood U , χ and τ1,...,τℓ ∆m−1(χ) which satisfy the condition (W-3). Furthermore,∈ U ∈ byT (U-2), we may∈ assume U U ′. Then we obtain ⊃ M St( τ ) M St( τ τ ) (i = k, k + 1). ∗ | i| ⊂ ∗ | k ∧ k+1| Therefore we obtain the following commutative diagram for i = k, k + 1.

C / C M∗St(|τk∧τk+1|) M∗St(|τi|) PPP PPP PPP PPP ( CM .

7 By applying RHom(D( ) CU ′ , F ) to the above diagram, we obtain the following commutative diagram• ⊗ for i = k, k + 1.

F F ′ (W ) ❘ / ((M Stχ( τk τk+1 )) U ) ❘❘❘ ∗ | ∧ ❱|❱❱∩❱ ❘❘❘ ❱❱❱❱ ❘❘❘ ❱❱❱❱❱ ❘❘❘ ❱❱❱❱ ❘❘❘ ❱❱❱❱ ❘) ❱+ F ′ m F ((M Stχ( τi )) U ) / HM (X, ) orM/X (M). ∗ | | ∩ ZM⊗(M)

By the repeated applications of the same argument for k = 1, 2,...,ℓ 1, we have −

′ F (W ) / F ((M Stχ( τ1 )) U ) PP ∗ | | ❯∩❯❯ PPP ❯❯❯❯ PPP ❯❯❯❯ PPP ❯❯❯❯ PP ❯❯❯❯ P( ❯* F ′ m F ((M Stχ( τℓ )) U ) / HM (X, ) orM/X (M). ∗ | | ∩ ZM⊗(M)

Because of τ1 σ1 and τℓ σ2, we ﬁnally get the following commutative diagram. ⊂ ⊂

′ F (W ) / F ((M Stχ( σ1 )) U ) ◗◗ ∗ | | ❯∩❯❯ ◗◗◗ ❯❯❯❯ ◗◗◗ ❯❯❯❯ ◗◗◗ ❯❯❯❯ ◗◗◗ ❯❯❯❯ ( ❯* F ′ m F ((M Stχ( σ2 )) U ) / HM (X, ) orM/X (M). ∗ | | ∩ ZM⊗(M)

Hence bW does not depend on the choices of χ and σ. The next corollary follows from Proposition 3.3 immediately. Corollary 3.4. For any W and W in with W W , and for any f 1 2 W 2 ⊂ 1 ∈ F (W1), we have b (f)= b (f ). (3.22) W1 W2 |W2 We ﬁnally extend bW to bW . We get bW by assigning fW F (W ) W⊕∈W ∈W ⊕∈W to bW (fW ). WX∈W F m F bW : (W ) HM (X, ) orM/X (M). (3.23) W⊕∈W −→ ZM⊗(M)

It follows from Corollary 3.4 that bW (f) = 0 holds for f . Hence bW passes through H(ˆ F ( )) and we obtain ∈ R W ˆ F m F bW : H( ( )) HM (X, ) orM/X (M). (3.24) W −→ ZM⊗(M)

8 3.2 The interpretation of the boundary value map bW

Our next purpose is to compute the boundary value map bW concretely. Remember that all the cells are oriented. Let 1 denote a section which generates or over Z . Each σ ∆ (χ) induces a section of or (M), M/X M ∈ m−1 M/X which is denoted by 1σ. Let σ ∆(χ) and τ ∆(χ′) for χ,χ′ . We define < , > which reflects the orientation∈ of two cells∈ under the following∈ T situations: • • 1. σ and τ are cells of the same dimension and σ τ or τ σ holds. ⊂ ⊂ 2. χ = χ′, σ ∆ (χ) and τ ∆ (χ) with σ τ. ∈ k ∈ k+1 ⊂ Definition 3.5. Let σ and τ be cells satisfying the above condition either 1 or 2. We define <σ,τ > by

1 (the orientation of σ or the induced orientation from σ has <σ,τ >=  the same orientation of τ)  1 (otherwise). − Furthermore, we introduce a mapping whose image is in locally constant functions on M a1( ) : ∆ (χ) Z (M) • m−1 −→ M deﬁned by the formula

1 = a1(σ) 1 (σ ∆ (χ)). σ · ∈ m−1 We construct an exact sequence which is needed when we compute the concrete expression of b . Let χ be in . W T L k−m+1 Deﬁnition 3.6. Let k = 0, 1, ,m 1. We deﬁne the sheaf χ as follows. ··· −

L k−m+1 C χ = σ, (3.25) σ∈∆⊕k(χ) C C C C where σ = M∗St(|σ|). More precisely, σ is the sheaf such that σ(U) consists of pairs (σ, s) with s C (U) for an open set U X. ∈ M∗St(|σ|) ⊂ For convenience, we set L −m = C . We also note that, for σ τ, the image χ X ⊂ of cσ Cσ by the canonical map Cσ Cτ is denoted by cσ τ for simplicity. Then∈ we have the following sequence.→ |

d−m d−1 d0 L −m χ χ L 0 χ C 0 χ i∗orM/X χ i∗orM/X M 0, (3.26) −→ Z⊗X −→ · · · −→ Z⊗X −→ −→ where the map i is the embedding i : M X. Note that, in the above L −m −→ sequence, the leftmost term χ i∗orM/X is located at degree m and the Z⊗X −

9 L 0 term χ i∗orM/X is at degree 0. Here, for any σ ∆k(χ) (k =0, 1, ,m 2) Z⊗X ∈ ··· − and for any c C , we deﬁne dk−m+1 as follows. σ ∈ σ χ k−m+1 1 1 dχ (cσ ℓ )= (<σ,τ >cσ τ ℓ ), ⊗ σ⊕⊂τ | ⊗ τ∈∆k+1(χ) where ℓ1 i or with ℓ Z (M). We also deﬁne d−m and d0 by ∈ ∗ M/X ∈ M χ χ −m 1 1 dχ (c ℓ ) = c σ ℓ , ⊗ σ∈∆⊕0(χ) | ⊗ 0 d (c ℓ1) = ℓa1(σ)c , χ σ ⊗ σ 1 where c CX , cσ Cσ with σ ∆m−1(χ) and ℓ i∗orM/X with ℓ ZM (M). ∈ 0∈ ∈ ∈ 1 ∈ We remark that dχ does not depend on the choices of . k L k k L k We write d and x instead of dχ and ( χ )x, respectively if there is no risk of confusion. Lemma 3.7. We have dk+1 dk =0 for k = m, m +1, , 1. ◦ − − ··· − 0 −1 Proof. Firstly we prove d d = 0. Let cσ Cσ with σ ∆m−2(χ) and 1 ◦ ∈ ∈ i∗orM/X . Here we remark that, there exists just two cells whose closure contains∈ σ. Let τ and τ ′ be mutually distinct cells whose closure contains σ. Then we obtain

d0 d−1(c 1) ◦ σ ⊗ 0 ′ = d (<σ,τ >cσ τ 1 < σ, τ >cσ τ ′ 1) | ⊗ ⊕ ′ |′ ⊗ = <σ,τ >a1(τ)c + < σ, τ >a1(τ )c ′ σ |τ σ|τ =0.

Here the last equation follows from the fact

′ ′ <σ,τ >a1(τ)= < σ, τ >a1(τ ). − Next we prove dk+1 dk = 0 for k = m, m +1, , 2. We remark ◦ − − ··· − that for any σ ∆k(χ) and τ ∆k+1(χ) we get < σ,τ > by comparing the orientation induced∈ by τ with the∈ orientation of σ. Here the orientation of σ is determined so that the outward pointing vector of τ followed by a positive frame of σ from that of τ. Note that, for any δ ∆k+2(χ) and σ ∆k(χ) which satisfy σ δ, there exist just two (k + 1)-cells∈ which are contained∈ in the closure of δ and⊂ whose closure contain σ. We denote by them τ and τ ′. By the

10 above observation we have

k+1 k k+1 d d (cσ)= d ( (<σ,τ >cσ τ 1)) ◦ σ⊕⊂τ | ⊗ τ∈∆k+1(χ)

= (<σ,τ ><τ,δ>cσ δ 1) τ⊕⊂δ σ⊕⊂τ | ⊗ δ∈∆k+2(χ) τ∈∆k+1(χ) ′ ′ = (<σ,τ ><τ,δ> + < σ, τ >< τ ,δ>)cσ δ 1 σ⊕⊂δ | ⊗ δ∈∆k+2(χ) =0, where c C and 1 i or . σ ∈ σ ∈ ∗ M/X By this lemma we see that the sequence (3.26) is a complex. Proposition 3.8. The complex (3.26) is exact. Before starting the proof of Proposition 3.8, we introduce some definitions and lemmas. Definition 3.9. The set ∆ (χ, x) is defined as follows. If x M, we define k ∈ ∆k(χ, x) = ∆k(χ), and if x / M, we define e∈ e ∆k(χ, x)= τ ∆k(χ) τ St( σx ) , (3.27) { ∈ | ⊂ | | } where σ ∆(χ) is the unique cell with x M σ . x ∈ e ∈ ∗ x Then the following lemma holds. Lemma 3.10. We have the isomorphism αk

k ∼ k−m+1 α : HomC(∆ (χ, x), C) L . (3.28) k −→ x The proof follows immediatelye from correspondence below.

αk : HomC(∆ (χ, x), C) L k−m+1 k −→ x ∈ ∈ (3.29) ϕe ϕ(σ)1σ . e 7−→σ∈∆ ⊕k (χ,x)

Here 1 denotes the image of 1 by C (X) C (X). σ X X −→ σ Lemma 3.11. Two sequences

0 ∆ (χ, x) ∂ ... ∂ ∆ (χ, x) 0, (3.30) ←− 0 ←− ←− m−1 ←− e e d−m+1 d−1 0 L −m+1 χ χ L 0 0 (3.31) −→ x −→ · · · −→ x −→ are dual to each other. Here ∂ is the boundary operator of the simplicial complex.

11 Proof. By applying the exact functor HomC( , C) to (3.30), we obtain • ∂∗ ∂∗ 0 HomC(∆ (χ, x), C) ... HomC(∆ (χ, x), C) 0, (3.32) −→ 0 −→ −→ m−1 −→ ∗ C where ∂ = HomeC(∂, ). Due to Lemma 3.10 wee just prove the commutativity of the diagram below.

k−m+1 L k−m+1 d L k−m+2 x −−−−−→ x αk αk+1 (3.33) x x HomC(∆k(χ, x), C) HomC(∆k+1 (χ, x), C).  −−−−→∂∗  Here we recall thate e ∂∗(ϕ)= < σ, τ > ϕ(σ). σ⊕⊂τ Then we have

k−m+1 k (d α )(ϕ)= < σ, τ > ϕ(σ)1τ . ◦ τ∈∆k⊕+1(χ,x) σ⊂τ On the other hand, we have

k+1 ∗ (α ∂ )(ϕ)= < σ, τ > ϕ(σ)1τ . ◦ σ∈∆⊕k(χ,x) σ⊂τ Note that both the indices sets in the above two computations coincide because

(σ, τ) σ τ, σ ∆ (χ, x) = (σ, τ) σ τ, σ ∆ (χ), σ St( δ ) { | ⊂ ∈ k } { | ⊂ ∈ k ⊂ | x| } = (σ, τ) σ τ, σ τ St( δ ) = (σ, τ) σ τ, τ ∆ (χ), τ St( δ ) { | ⊂ ⊂ ⊂ | x| } { | ⊂ ∈ k+1 ⊂ | x| } = (σ, τ) σ τ, τ ∆ (χ, x) , { | ⊂ ∈ k+1 } where the cell δx is the unique cell satisfying x M δx. Hence (3.33) becomes a commutative diagram. ∈ ∗ Now we are ready to prove Proposition 3.8. Proof. It is enough to prove that the following sequence is exact.

−m −1 0 −m d d 0 d 0 L (i∗orM/X )x L (i∗orM/X )x (CM )x 0. −→ x ⊗Z −→ · · · −→ x ⊗Z −→ −→

By Lemma 3.11, the claim follows from the homology groups of St( δx ) if x / M and Sm−1 if x M. | | ∈ ∈ 1. The case of x / M. ∈ Since St( δx ) is contractible to a point in δx , its homology groups vanish except for| the| 0-th group, that is | |

H (St( δ ), C)=0 (k = 0). k | x| 6

12 Hence the complex (3.31) is concentrated in degree m + 1 and its ( m + 1)-th cohomology group is C, and we have the exact− sequence: −

−m −1 −m d d 0 0 L (i∗orM/X )x L (i∗orM/X )x 0. −→ x ⊗Z −→ · · · −→ x ⊗Z −→

2. The case of x M. ∈ Except for the 0-th and the (m 1)-th homology groups, the homology groups of the (m 1)-dimensional− spherical surface vanish. Its 0-th homology group is isomorphic− to C, and the (m 1)-th homology gives the − orientation of the surface which is isomorphic to C (orM/X )x. Hence we ⊗Z obtain the exact sequence;

−m −1 0 −m d d 0 d 0 L (i∗orM/X )x L (i∗orM/X )x (CM )x 0. −→ x ⊗Z −→ · · · −→ x ⊗Z −→ −→

The proof of Proposition 3.8 has been completed. By Proposition 3.8 we have the quasi-isomorphism.

L • C χ i∗orM/X M . (3.34) Z⊗X ≃ L • Here χ i∗orM/X designates the complex Z⊗X

L −m L −m+1 L 0 0 χ i∗orM/X χ i∗orM/X χ i∗orM/X 0. −→ Z⊗X −→ Z⊗X −→ · · · −→ Z⊗X −→ (3.35) Let U be in . For convenience we set U F (σ, U) = F (M St( σ ) U) (σ ∆(χ)), (3.36) ∗ | | ∩ ∈ Fk(χ,U) = F (σ, U) (k =1, 2, ,m). (3.37) σ∈∆⊕k−1(χ) ···

Moreover we set

F (σ) = lim F (σ, U), (3.38) U−→∈U F (χ) = lim F (χ,U) (k =1, 2, ,m). (3.39) k k ··· U−→∈U

Lemma 3.12. We have the isomorphism βχ as follows. F m(χ) ∼ m F orM/X (M) HM (X, ) orM/X (M). Im(F (χ) F (χ)) Z ⊗(M) −→ Z ⊗(M) m−1 → m M M (3.40)

13 Proof. By applying D( ) to (3.34) we obtain • L • ∼ C D( χ i∗orM/X ) D( M ). Z⊗X ←−

Because the support of D(C ) is contained in M we obtain, for U , M ∈U

D(CM )= D(CM ) CU . Z⊗X ∼ L • C D( χ i∗orM/X ) U . (3.41) −→ Z⊗X Z⊗X

Moreover, since each M St( σ ) is cohomologically trivial we get ∗ | | L • C N • D( χ i∗orM/X ) U χ i∗orM/X . Z⊗X Z⊗X ≃ Z⊗X N • Here the χ designates the complex

0 N 0 N 1 N m−1 0, −→ χ −→ χ −→ · · · −→ χ −→ where N k C χ = σ,U , σ∈∆m⊕−k−1(χ)

Cσ,U = CM∗St(|σ|)∩U , N m−1 C F for k =0, 1, ,m 2 and χ = U . By applying RHomCX ( , ) to (3.41) we obtain ··· − • N • F ∼ C F RHomCX ( χ i∗orM/X , ) RHomCX (D( M ), ). (3.42) Z⊗X −→

Since U satisﬁes the condition (U-1), we obtain N • F N • F RHomCX ( χ i∗orM/X , ) = HomCX ( χ i∗orM/X , ). Z⊗X Z⊗X By taking the 0-th cohomology, we get F m(χ,U) ∼ m F orM/X (M) HM (X, ) orM/X (M). Im(F (χ,U) F (χ,U)) Z ⊗(M) −→ Z ⊗(M) m−1 → m M M Finally we take the inductive limit in the above isomorphism with respect to U , and then, the claim of the lemma follows. ∈U Hereafter we identify the objects in both sides of (3.40) by βχ. For any σ ∆ (χ) the following diagram commutes. ∈ m−1 L 0 C χ i∗orM/X M Z⊗X −−−−→

εχ,σ (3.43) C x ∼ C x σ i∗orM/X M∗St( |σ|). Z⊗X  −−−−→ηχ,σ 

14 Here εχ,σ is the embedding map and ηχ,σ is deﬁned by

c 1 a1(σ) c . (3.44) σ⊗ 7→ · σ −1 Note that ηχ,σ is given by c a1(σ)c 1, (3.45) 7→ ⊗ where c C . By (3.43) the diagram of complexes below commutes. ∈ M∗St(|σ|) L • ∼ C χ i∗orM/X M Z⊗X −−−−→

εχ,σ

C x ∼ C x σ i∗orM/X M∗St( |σ|). Z⊗X  −−−−→ηχ,σ  C F By applying RHomCX (D( ) U , ) to the above commutative diagram, taking the 0-th cohomology• groups⊗ and taking the inductive limit lim of these U−→∈U cohomology groups, we obtain the commutative diagram. F m(χ) ∼ m F orM/X (M) HM (X, ) orM/X (M) Im(F (χ) F (χ)) Z ⊗(M) −−−−→βχ Z ⊗(M) m−1 → m M M 0 b εχ,σ W|σ| (3.46) F x ∼ F x (σ) orM/X (M) lim (M St( σ ) U). Z ⊗(M) −−−−→η0 ∗ | | ∩ M  χ,σ U−→∈U  Here vertical map on the right side is b = lim b with W = M W|σ| W|σ|,U |σ|,U ∗ U−→∈U St ( σ ) U and χ | | ∩ 0 0 ε = lim H (RHomC (D(ε ) C , F )). χ,σ X χ,σ ⊗ U U−→∈U

Moreover the upper horizontal map is βχ and 0 0 η = lim H (RHomC (D(η ) C , F )). χ,σ X χ,σ ⊗ U U−→∈U By taking a direct sum with respect to σ ∆ (χ), we obtain ∈ m−1 F m(χ) ∼ m F orM/X (M) HM (X, ) orM/X (M) Im(F (χ) F (χ)) Z ⊗(M) −−−−→βχ Z ⊗(M) m−1 → m M M ⊕ b ε0 W|σ| χ |σ|∈|∆|m−1(χ) (3.47) x x  ∼  F (σ)  orM/X (M) lim F(M St( σ ) U) , σ∈∆ ⊕ (χ) Z ⊗(M) −−−−→η0 |σ|∈|∆⊕| (χ) ∗ | | ∩ m−1 M χ m−1 U−→∈U ! 0 0 0 0 0 where ηχ = ηχ,σ and εχ = εχ,σ. We note that εχ is surjec- σ∈∆m⊕−1(χ) σ∈∆m⊕−1(χ) tive. 0 −1 0 On account of (3.45), tracing the map (ηχ) and εχ in (3.46) we get the proposition below.

15 Proposition 3.13. The boundary value morphism bW ˆ F m F H( ( )) HM (X, ) orM/X (M) (3.48) W −→ ZM⊗(M) is given by, for W and f F (W ) ∈W ∈ f fa1(σ) 1 F (σ, U) or (M) F (χ) or (M), (3.49) 7→ ⊗ ∈ ⊗ M/X ⊂ m ⊗ M/X 1 through the identiﬁcation by βχ. Here is a section of orM/X (M) which generates orM/X over ZM , σ ∆m−1(χ) and U are taken such that M St ( σ ) U W holds. ∈ ∈ U ∗ χ | | ∩ ⊂ Note that a1(σ) 1 does not depend on the choices of the generator 1 because another choice⊗ of the generator is, if M is connected, 1, and hence, we get on each connected component −

a 1(σ) 1 = a1(σ) 1. − ⊗− ⊗

3.3 The inverse map ρ of bW

The aim of this subsection is to construct the inverse map ρ of bW and complete the proof for Theorem 3.1. Firstly we prove the commutativity of diagram [1],[2] and [3].

F β m(χ) χ m F orM/X (M) / HM (X, ) orM/X (M) Im(F (χ) F (χ)) Z ⊗(M) Z ⊗(M) m−1 → m M ❘ M O ❘❘ ρχ b ❧❧6 O O ❘❘❘ W❧❧ ❘❘❘ [3] ❧❧❧ ❘❘❘ ❧❧❧ ❘) ❧❧ ⊕ b ε0 H(ˆ (F )) [1] W|σ| χ ◗ |σ|∈|∆|m−1(χ) W h ◗◗◗ ◗◗◗ ◗◗̟◗ [2] ◗◗◗

F F (σ) orM/X (M) 0 / lim (M St( σ ) U) . σ∈∆ ⊕ (χ) Z ⊗(M) η |σ|∈|∆⊕| (χ) ∗ | | ∩ m−1 M χ m−1 U−→∈U ! (3.50) Here ̟ is a canonical morphism and ρχ is deﬁned as follows.

Deﬁnition 3.14. The map ρχ F m(χ) ˆ ρχ : orM/X (M) H(F ( )) (3.51) Im(F (χ) F (χ)) Z ⊗(M) −→ W m−1 → m M is deﬁned by, for f F (σ), σ ∈

fσ 1 a1(σ) fσ, ⊗Z 7→ · 1 where is a section of orM/X (M) which generates orM/X over ZM .

16 We note that ρχ is well-deﬁned. As a matter of fact, the set Im(Fm−1(χ) Fm(χ)) is generated by elements → + 1 − 1 <τ,σ >f σ+ + < τ,σ >f σ− , | ⊗Z | ⊗Z

∗ ∗ for f Fm−2(τ), τ ∆m−2(χ) and σ ∆m−1(χ) with τ σ . Here = + or and∈ σ+ = σ−. Hence∈ we obtain ∈ ⊂ ∗ − 6 + 1 − 1 ρχ(< τ,σ >f σ+ + <τ,σ >f σ− ) | ⊗Z | ⊗Z + + − − =(a1(σ ) < τ,σ > + a1(σ ) < τ,σ >)f =0.

The commutativity of [1] follows from Corollary 3.4. And the diagram [2] also commutes by Definition 3.14 and the definition of ̟. We prove the com- 0 mutativity of the diagram [3]. Since εχ is surjective, it suffice to prove that

β ε0 = b ρ ε0 , χ ◦ χ W ◦ χ ◦ χ and this commutativity follows from (3.47) and the commutativity of [1] and [2]. Then we get the following Theorem 3.15. The map −1 m F ˆ F ρχ βχ : HM (X, ) orM/X (M) H( ( )) ◦ ZM⊗(M) −→ W does not depend on the choice of χ. Moreover the map ρ = ρ β−1 is the χ ◦ χ inverse map of bW . In the sequel, we shall prove Theorem 3.15. To see the first claim in Theorem 3.15, as satisfies the condition (T-2), it suffices to show the claim only for the pair χ Tχ′ in . We construct the map ≺ T 0 L 0 L 0 Θ : χ i∗orM/X χ′ i∗orM/X Z⊗X −→ Z⊗X as follows. Let us take a choice function ψ

ψ : ∆ (χ) ∆ (χ′) m−1 −→ m−1 such that ψ(σ) σ for any σ ∆ (χ). Then we deﬁne the Θ0 by | | ⊂ | | ∈ m−1 Θ0 : c 1 < σ, ψ(σ) >c 1, σ ⊗ 7→ σ|ψ(σ) ⊗ where c C and 1 i or . We have the following commutative diagram. σ ∈ σ ∈ ∗ M/X d0 L 0 χ C χ i∗orM/X M Z⊗X −−−−→

Θ0 id L 0  C ′ i∗orM/X M . χ Z 0 ⊗X y −−−−→dχ′ y

17 As a matter of fact the commutativity of diagram above follows from

(d0 Θ0)(c 1)= d0(< σ, ψ(σ) >c 1) ◦ σ ⊗ σ|ψ(σ) ⊗ 0 = c < σ, ψ(σ) >a1(ψ(σ)) = c a1(σ)= d (c 1), σ|ψ(σ) σ σ ⊗ where σ ∆m−1(χ). ∈ L • We can extend this commutative diagram to that of the complexes χ and L • χ′ by the following proposition. Proposition 3.16. Let Θ0 be the map deﬁned above. There exist Θk−m(k = 0, 1, ,m 1) such that the following diagram commutes. ··· − d−m d−1 d0 L −m χ χ L 0 χ C χ i∗orM/X χ i∗orM/X M 0 Z⊗X −−−−→ · · · −−−−→ Z⊗X −−−−→ −−−−→

Θ−m Θ0 id

 d−m d−1  d0  L −m  χ′ χ′ L 0  χ′ C χ′ yi∗orM/X χ′ yi∗orM/X yM 0. Z⊗X −−−−→ · · · −−−−→ Z⊗X −−−−→ −−−−→

Since X = M Dm, it suffices to show Proposition 3.16 when M is a point. Hence, in what follows,× we may assume that M is contractible until the end of the proof of the proposition. To show the proposition, we prepare for several lemmas. Let Hχ be the family H of open half spaces H in Rm passing through the origin which { i}i∈Λχ i defines the stratification χ, that is, Hχ is the minimal set satisfying that, for any σ ∆m−1(χ), σ is given by the intersection of some Hi’s belonging to Hχ and Sm∈−1. The most important property of the family H is that; for any σ ∆(χ) χ ∈ and for any H Hχ, σ H = implies σ H, and also σ ∂H = implies σ ∂H. In subsequent∈ arguments,∩ 6 ∅ this fact⊂ is constantly used.∩ 6 ∅ ⊂ Lemma 3.17. For any χ and σ ∆(χ), the star open set Stχ( σ ) has the following expression. ∈ T ∈ | |

m−1 Stχ( σ )= Hi S . (3.52) | | σ⊂H∩i∈Hχ ∩

m−1 Proof. Firstly we prove Stχ( σ ) Hi S . Let x Stχ( σ ). | | ⊂ σ⊂H∩i ∈Hχ ∩ ∈ | | By the deﬁnition of Stχ( σ ) there exists a cell τ such that x τ and σ τ. By considering the condition| | (T-1), there exist only two cases∈ of the inclusion⊂ between τ and each Hi. 1. τ (H )c. Here we denote by ( )c the complement of ( ). ⊂ i • • 2. τ H . ⊂ i

18 c However the first case never occurs. As a matter of fact, we have τ (Hi) and σ τ, that is, σ (H )c = (H )c, which contradicts σ H . Therefore⊂ we can ⊂ ⊂ i i ⊂ i m−1 ignore the first case and we obtain Stχ( σ ) Hi S . | | ⊂ σ⊂H∩i∈Hχ ∩ m−1 Next we prove Stχ( σ ) Hi S . It is sufficient to show the | | ⊃ σ⊂H∩i∈Hχ ∩ m−1 implication; x / Stχ( σ ) x / Hi S . For any x / Stχ( σ ), ∈ | | ⇒ ∈ σ⊂H∩i∈Hχ ∩ ∈ | | there exists a cell δ with x δ and δ / Stχ( σ ), that is, σ δ holds. For such δ ∈ ∈ | | 6⊂ c and σ we can find an open half space H Hχ satisfying σ H and δ (H) . In fact, this can be shown by the induction∈ on m in the following⊂ way; When⊂ m = 0, the claim clearly holds. Then we assume that the claim holds for m<ℓ (ℓ 1). ≥ If δ ∆ℓ−1(χ), it follows from the definition of χ that H = δ holds, where ∈ H∈∩Hχ,δ Hχ,δ denotes the set H Hχ ; δ H . In particular, we have H = δ. { ∈ ⊂ } H∈∩Hχ,δ Then we can find H H with σ (H)c because otherwise we get σ H = ∈ χ,δ ⊂ ∩ 6 ∅ for any H Hχ,δ, which implies σ H = δ. This contradicts the fact ∈ ⊂H∈ ∩Hχ,δ δ / St ( σ ). ∈ χ | | Now we assume δ ∆k(χ) with k<ℓ 1. Then we can find L Hχ,δ with δ ∂L. If σ ∂L∈= , then clearly L −or L gives the desired half∈ space H. Therefore⊂ we may∩ assume∅ σ ∂L also. Then,− by applying the induction ⊂ hypothesis to δ and σ in ∂L, we can find H Hχ,δ such that δ ∂L H and σ ∂L (H)c. Hence we have obtained the∈ claim. ⊂ ∩ ⊂Therefore∩ the converse implication follows and this completes the proof. The following lemma is the crucial key for the proof of Proposition 3.16.

Lemma 3.18. Assume that M is contractible. Then M Stχ′ ( τ ) M Stχ( σ ) is contractible for any σ ∆(χ) and τ ∆(χ′). ∗ | | \ ∗ | | ∈ ∈ Remark 3.19. The assumption χ χ′ is essential, otherwise the claim does not hold. ≺

Proof. We set U = M Stχ( σ ) and V = M Stχ′ ( τ ). We also set Hχ,σ = H H ; σ H . By∗ Lemma| | 3.17 we have∗ | | { i ∈ χ ⊂ i} m−1 V U = M Stχ′ ( τ ) ( Hi) S \ ∗ | | \ Hi∈∩Hχ,σ ∩ m−1 = M (Stχ′ ( τ ) Hi S ). Hi∈∪Hχ,σ ∗ | | \ ∩ m−1 Therefore we just prove that M (Stχ′ ( τ ) Hi S ) is contractible. Hi∈∪Hχ,σ ∗ | | \ ∩ By considering χ χ′ there are three cases of the inclusion between τ and each ≺ Hi containing σ. 1. The case of τ H Sm−1. ⊂ i ∩ ′ Since χ is ﬁner than χ we have Stχ′ ( τ ) Hi, that is, M (Stχ′ ( τ ) H Sm−1)= and we can ignore this| case.| ⊂ ∗ | | \ i ∩ ∅

19 2. The case of τ H Sm−1. ⊂− i ∩ m−1 We have M (Stχ′ ( τ ) Hi S ) = M Stχ′ ( τ ) = V . Therefore if such an H exists,∗ then| | V\ U∩= V , and hence∗ it becomes| | contractible. i \ 3. The case of τ ∂H Sm−1. ⊂ i ∩ By the observations in the cases 1 and 2, we may assume that all the Hi con- c m−1 taining σ satisfy the case 3. We ﬁrst show the claim that (Hi) S Hi∈∩Hχ,σ ∩ has an interior point. We take points x σ and y τ. Let z be a point in the line passing through x and y so that∈ the points x,y,z∈ are situated in this order. Then we can easily see, by noticing σ = Hi and y Hi (Hi Hi∈∩Hχ,σ ∈ ∈ c m−1 Hχ,σ), that z becomes an interior point of (Hi) S . By choos- Hi∈∩Hχ,σ ∩ ing z in close enough to the point y, z belongs to Stχ′ ( τ ) and hence we have c m−1 | | z Stχ′ ( τ ) (Hi) S . Finally, using z, we prove the con- ∈ | | ∩ {Hi∈ ∩Hχ,σ ∩ } tractibleness of V U and complete the proof. We have \ c m−1 Stχ′ ( τ ) (Hi) S | | ∩ {Hi∈ ∩Hχ,σ ∩ } m−1 = Stχ′ ( τ ) (Hi S ). Hi∈∩Hχ,σ | | \ ∩

m−1 Because each Stχ′ ( τ ) Hi S is a convex set, M (Stχ′ ( τ ) | | \ ∩ Hi∈∪Hχ,σ ∗ | | \ m−1 m−1 Hi S ) is contractible to the point z St( τ )χ′ Hi S . This ∩ ∈Hi∈ ∩Hχ,σ | | \ ∩ complete the proof. Lemma 3.20. Assume χ χ′ and M to be contractible. Then for any σ ∆(χ) and for any τ ∆(χ′) we≺ have ∈ ∈ Extℓ(C , C )=0 (ℓ = 0). (3.53) σ τ 6

Proof. We set U = M St ( σ ) and V = M St ′ ( τ ). Then we obtain ∗ χ | | ∗ χ | | Extℓ(C , C ) = Hℓ(RHom(C , C )) Hℓ (V, C ). (3.54) σ τ U V ≃ U∩V V Here we have the following long exact sequence.

0 H0 (V, C ) H0(V, C ) H0(V U, C ) −−−−→ U∩V V −−−−→ V −−−−→ \ V (3.55) H1 (V, C ) H1(V, C ) . −−−−→ U∩V V −−−−→ V −−−−→ · · · By Lemma 3.17, V is contractible and we obtain

C (k = 0), Hk(V, C ) V ≃ 0 (k = 0). ( 6

20 Moreover, by Lemma 3.18 we have

C (k = 0), Hk(V U, C ) \ V ≃ 0 (k = 0). ( 6 By (3.55), Hk (V, C ) = 0 holds for any k = 0, 1. Moreover we have the U∩V V exact sequence 6

0 H0 (V, C ) C C H1 (V, C ) 0. → U∩V V → → → U∩V V → From which we obtain H0 (V, C )=0 and H1 (V, C ) = 0. The proof of U∩V V U∩V V Lemma 3.20 has been ﬁnished. Now we are ready to prove Proposition 3.16. Proof. We will show existence of Θk under the assumption of that of Θk+1. Then by the induction we obtain all the Θk (k = m, m+1, , 1). Assume that we construct Θk+1. By (3.34), we obtain the− following− exact··· − sequence for k = m, m +1, , 1. − − ··· − dk k id L k χ k 0 Ker dχ′ χ′ i∗orM/X Im dχ′ 0. (3.56) −→ −→ Z⊗X −→ −→ The following long exact sequence is induced from (3.56).

k k k k k k 0 Hom(R , Ker d ′ ) Hom(R , R ′ ) Hom(R , Im d ′ ) −−−−→ χ χ −−−−→ χ χ −−−−→ χ χ (3.57)

1 k k 1 k k Ext (R , Ker d ′ ) Ext (R , R ′ ) , −−−−→ χ χ −−−−→ χ χ −−−−→ · · · Rk L k 1 Rk k where χ = χ i∗orM/X . Hence it is enough to show Ext ( χ, Ker dχ′ )=0 Z⊗X k k k for the existence of Θ Hom(R , R ′ ). By the deﬁnition of R we have k ∈ χ χ χ 1 Rk k 1 C k Ext ( χ, Im dχ′ ) =Ext σ i∗orM/X , Ker dχ′ σ∈∆ ⊕ (χ) Z⊗X k+m−1 1 k Ext (Cσ i∗orM/X , Ker d ′ ) Z χ ≃σ∈∆k+ ⊕m−1(χ) ⊗X 1 k Ext (Cσ, Ker d ′ ) i∗orM/X . χ Z ≃σ∈∆k+ ⊕m−1(χ) ⊗X Therefore it is suﬃcient to prove the following equation for any k = m, m + 1, , 1 and for any σ ∆(χ). − − ··· − ∈ 1 C k Ext ( σ, Ker dχ′ )=0. (3.58) By taking the short exact sequence below into account

dk−1 k−1 id k−1 χ′ k−1 k 0 Ker d ′ R ′ Im d ′ = Ker d ′ 0, −→ χ −→ χ −→ χ χ −→

21 we can obtain the following long exact sequence.

k−1 k−1 k 0 Hom(C , Ker d ′ ) Hom(C , R ′ ) Hom(C , Ker d ′ ) −−−−→ σ χ −−−−→ σ χ −−−−→ σ χ (3.59)

1 k−1 1 k−1 Ext (C , Ker d ′ ) Ext (C , R ′ ) . −−−−→ σ χ −−−−→ σ χ −−−−→ · · · Let us prove the following equation for any ℓ> 0. ℓ C Rk−1 Ext ( σ, χ′ )=0. (3.60) Rk l C Rk By remembering the deﬁnition of χ′ , we can calculate Ext ( M∗St(|σ|), χ′ ) as follows.

ℓ C Rk ℓ C C Ext ( σ, χ′ ) =Ext σ, τ i∗orM/X τ∈∆ ⊕ (χ′) Z⊗X m+k−1 ℓ Ext (Cσ, Cτ ) i∗orM/X ′ Z ≃τ∈∆m+ ⊕k−1(χ ) ⊗X =0. The last equation follows from Lemma 3.18. By (3.59) and (3.60) we obtain

ℓ k ℓ+1 k−1 Ext (C , Ker d ′ ) Ext (C , Ker d ′ ). σ χ ≃ σ χ This means that ℓ k ℓ+k+m −m Ext (C , Ker d ′ ) Ext (C , Ker d ′ )=0 σ χ ≃ σ χ −m because of Ker dχ′ = 0. Hence we have shown (3.58) and the proof of Propo- sition 3.16 has been ﬁnished. By Proposition 3.16, the following diagram of complexes commutes for χ χ′. ≺ L • ∼ C χ i∗orM/X M Z⊗X −−−−→

Θ• (3.61) L •  ∼ C χ′ i∗orM/X M . Z⊗X y −−−−→ C F By applying RHomCX (D( ) U , ) to the above diagram, taking the 0-th cohomology groups and taking• ⊗ inductive limit lim of these cohomology groups, U−→⊃M we obtain F m ∼ m(χ) H (X, F ) orM/X (M) orM/X (M) M −1 F F ZM⊗(M) −−−−→β Im( (χ) (χ)) ZM⊗(M) χ m−1 → m A0(Θ) F ′  m ∼ m(χ ) H (X, F ) orM/X (M)  orM/X (M), M −1 F ′ F y ′ ZM⊗(M) −−−−→β Im( m−1(χ ) m(χ )) ZM⊗(M) χ′ →

22 where A0(Θ) = lim H0(RHomC (D(Θ•) C , F )). Moreover by the calcula- X ⊗ U U−→∈U tion

0 ρ ′ A (Θ)(f 1)=ρ ′ (< σ, ψ(σ) >f 1) χ ◦ σ ⊗ χ σ|ψ(σ) ⊗ = < σ, ψ(σ) >a1(ψ(σ))f σ |ψ(σ) =a1(σ)f = a1(σ)f = ρ (f 1), σ|ψ(σ) σ χ σ ⊗ we have the following commutative diagram. F m(χ) ∼ ˆ orM/X (M) H(F ( )) Im(F (χ) F (χ)) Z ⊗(M) −−−−→ρχ W m−1 → m M A0(Θ) (3.62) F ′  m(χ )  ∼ ˆ F ′ y ′ orM/X (M) H( ( )). Im(F (χ ) F (χ )) Z ⊗(M) −−−−→ρχ′ W m−1 → m M Hence we get

−1 0 −1 −1 ρ β = ρ ′ A (Θ) β = ρ ′ β ′ , χ ◦ χ χ ◦ ◦ χ χ ◦ χ and we have seen that ρ = ρ β−1 does not depend on the choices of χ. χ ◦ χ We finally show bW ρ = id and ρ bW = id. The formula bW ρ = id follows from the commutativity◦ of (3.50).◦ ◦ To show ρ bW = id, we may fix χ and W , and take f F (W ). By Proposition◦ 3.13 and Definition 3.14∈ we T have ∈ W ∈

−1 2 ρ b (f)= ρ β b (f 1 )= ρ (fa1(σ) 1 ) = (a1(σ)) f = f. ◦ W χ ◦ χ ◦ W ⊗ σ χ ⊗ σ (3.63) These complete the proof of Theorem 3.1.

4 Application to Laplace hyperfunctions

In the last section, we shall introduce intuitive representation of Laplace hyperfunctions as an application of Theorem 3.1. First we recall the sheaf of Laplace hyperfunctions. Let n be a natural number and S2n−1 the unit sphere in Cn R2n. ≃ n Definition 4.1. We define the radial compactification DCn of C as follows.

n 2n−1 DCn = C S . (4.64) ⊔ ∞ In the same way we deﬁne

n n−1 DRn = R S . (4.65) ⊔ ∞ A family of fundamental neighborhoods of z Cn consists of 0 ∈ B (z )= z Cn ; z z <ε (4.66) ε 0 { ∈ | − 0| }

23 for ε> 0, and that of ζ S2n−1 consists of an open cone 0∞ ∈ ∞ z G (Γ) = z Cn ; z > r, Γ Γ , (4.67) r ∈ | | z ∈ ∪ ∞ | | 2n−1 where r> 0 and Γ runs through open neighborhoods of ζ0 in S . We deﬁne the set of holomorphic functions of exponential type on DCn .

exp DCn O Deﬁnition 4.2. Let U be an open subset of . We deﬁne the set DCn (U) of holomorphic functions of exponential type on U as follows. A holomorphic function f(z) on U Cn belongs to Oexp (U) if, for any compact set K U, ∩ DCn ⊂ there exist positive constants CK and HK such that

f(z) C eHK |z| (z K Cn). (4.68) | |≤ K ∈ ∩ Oexp Moreover we denote by DCn the sheaf of holomorphic functions of exponential type on DCn . To construct intuitive representation of Laplace hyperfunctions, we need the vanishing theorem of the global cohomology groups on a Stein open set with Oexp coefficients in DCn . However, by Umeta and Honda ([HU]), we cannot expect such a vanishing theorem generally. To overcome this difficulty, they introduce a new property called regular at for a subset in DCn . D ∞ 1 2n−1 Let U be a subset in Cn . We define the closed set clos∞(U) S as follows. A point z S2n−1 belongs to clos1 (U) if there exists⊂ a point∞ ∈ ∞ ∞ sequence z N in U Cn such that { k}k∈ ∩

zk+1 z z and | | 1 in DCn (k ). (4.69) k → z → → ∞ | k| Set N 1 (U)= S2n−1 clos1 (Cn U). (4.70) ∞ ∞\ ∞ \ D 1 2n−1 Deﬁnition 4.3 ([HU]). Let U be an open set in Cn . If N∞(U)= U S holds, we say that U is regular at . ∩ ∞ ∞ Then we have the following theorem.

Theorem 4.4 ([HU]). Let U be an open set in DCn . Assume that U is regular at and U Cn is a Stein open set in Cn. Then we obtain ∞ ∩ k exp H (U, OD )=0 (k = 0). (4.71) Cn 6 2n−1 As a similar notion, we deﬁne N (U) S for a subset U in DCn as ∞ ⊂ ∞ N (U)= S2n−1 (Cn U), (4.72) ∞ ∞\ \ 1 where the closure ( ) is taken in DCn . Note that N (U) N (U) holds. • ∞ ⊂ ∞

24 Definition 4.5. For an open set A Cn, we define ⊂ A = A N (A). (4.73) ∪ ∞ n Remark that A is the largest open set in DCn such that A C = A holds. b n ∩ We also define the similar set B DRn for a set B R . Sometimes we write ⊂ ⊂ instead of . b b • • b exp B DRn Definition 4.6 ([HU]). The sheaf DRn of Laplace hyperfunctions on is bdefined as follows.b exp n exp B = HD (O ) orD n D n , (4.74) DRn Rn DCn ⊗ R / C H n Z where orDRn /DCn is the orientation sheaf DRn ( DCn ). Let Ω be an open cone in Rn with vertex a Rn. Note that Ω is not assumed to be convex. We define an open set X in Cn∈as follows. X = z = x + √ 1y Cn; x Ω, y 2 < x a 2 +1 . (4.75) { − ∈ ∈ | | | − | } Note that Ω= X DRn . ∩ exp We define intuitive representation of Laplace hyperfunctions B n (Ω) on Ω. DR b b First we define the family of open sets in X. LetΓbean R+-conic connected open set in Rn. W b b Definition 4.7. Let W be an open set in Xb. We say that W is an infinitisimal wedge of type Ω √ 1Γ if and only if the following conditions hold. × − 1. W (Ω √ 1Γ). b ⊂ × − 2. For any open proper subcone Γ′ of Γ, there exists an open neighborhood U Xb of Ω such that ⊂ (Ω √ 1Γ′) U W. (4.76) b b × − ∩ ⊂ Let (√ 1Γ) be a family of all infinitisimal wedges of type Ω √ 1Γ in W − b × − X, and we set = (√ 1Γ). (4.77) W W − b Γ [ n Here Γ runs through all the R+-conic connected open set in R . Theorem 4.8. For aforementioned , there exists ( , , ) which satisfies the conditions (T), (U) and (W), andW we have intuitive represT U entationW of Laplace exp hyperfunctions B n (Ω) by DR

ˆ Obexp Oexp H( DCn ( )) = DCn (W ) / , (4.78) W W⊕∈W R where is a C-vector space generated by the following elements. R exp f ( f ) (f OD (W )). (4.79) ⊕ − |W2 ∈ Cn 1 Here W and W are open sets in with W W . 1 2 W 2 ⊂ 1

25 To prove this theorem, we construct ( , , ) satisfying the condition (T), T U W (U) and (W). First we show the theorem of Grauert type on DCn .

n Theorem 4.9. Let Ω be an R+-conic open set in R and let V be an open neighborhood of Ω in DCn . Then we can ﬁnd an open set U in DCn satisfying the following three conditions. b 1. Ω U V . ⊂ ⊂ 2. U is regular at . b ∞ 3. U Cn is a Stein open set. ∩ Proof. Retaking V smaller we can assume the following conditions. (a) V Rn = Ω. ∩ (b) V Cn z = x + √ 1y Cn ; y max 1, x /2 . ∩ ⊂{ − ∈ | |≤ { | | }} (c) V\Cn = V . ∩ We introduce some new functions. Let r(x) be a function on Rn such that

r(x) = dist(x, (Cn V ) ( x √ 1Rn)) (x Rn). (4.80) \ ∩ { }× − ∈ We also deﬁne the function r∗(ω) on Sn−1 by

r(tω) r∗(ω) = inf (ω Sn−1). (4.81) t≥1 t ∈ Here r∗ has the following properties. (i) r∗(ω) is a lower semi-continuous function on Sn−1. (ii) r∗(ω) > 0 for (ω Ω Sn−1) and r∗(ω) = 0 otherwise. ∈ ∩ For x∗ Rn and for c > 0, we deﬁne an analytic polyhedron U(x∗,c) as follows. ∈

U(x∗,c)= z = x + √ 1y Cn ; y 2 < x x∗ 2 + c2 . (4.82) { − ∈ | | | − | } n Moreover we deﬁne an open set U1 in C as the interior of the following set. x∗ r∗(x∗/ x∗ ) W x∗, | | | | . (4.83) ∗ Rn ∗ 2 x ∈ \,x 6=0

Then U1 has the following properties.

(1) It is a R+-conic set. (2) It is a Stein open set. (3) Ω U and U z = x + √ 1y Cn ; x 1 V hold. ⊂ 1 1 ∩{ − ∈ | |≥ }⊂

26 These properties are shown in the following way: ∗ n We first show (1). Let z = x + √ 1y U1. For any x R 0 , ∗ ∗ ∗ ∗ − ∈ ∈ \{ } ∗ x r (x / x ) z U ∗ := U x , | | | | holds. Hence for any t R and for any ∈ x 2 ∈ + ∗ n x R 0 it is sufficient to prove tz Ux∗ . Since z Ux∗ holds for any x∗ ∈ Rn \{0 }, by the definition we get ∈ ∈ ∈ \{ } x∗ r∗(x∗/ x∗ ) 2 y 2 < x x∗ 2 + | | | | . (4.84) | | | − | 2 By multiplying the both sides by t2

tx∗ r∗(tx∗/ tx∗ ) 2 ty 2 < tx tx∗ 2 + | | | | . (4.85) | | | − | 2 Furthermore, for any t R and for any x∗ Ω, tx∗ Ω holds. Hence we have ∈ + ∈ ∈ tz U1 and we have shown (1). ∈Next we give the proof of (2). Because x∗ r∗(x∗/ x∗ ) U x∗, | | | | 2 x∗ r∗(x∗/ x∗ ) 2 = z Cn ; exp (z x∗)2 | | | | < 1 , (4.86) ( ∈ − − − 2 ! )

U1 is a Stein open set. The proof of (2) has been completed. Finally we show (3). Ω U1 is obvious from the construction of U1. More- ⊂ n over it is easy to see U1 z = x + √ 1y C ; x 1 V by the deﬁnition ∗ ∩{ − ∈ | |≥ }⊂ of r (ω). Hence (3) follows. Let U1 be the interior of x∗ r∗(x∗/ x∗ ) Uf x∗, | | | | . ∗ Rn ∗ 2 x ∈ \,x ≥1 Let us deﬁne U2 by the interior of

∗ ∗ U2 = U (x , r(x )/2) . (4.87) ∗ Rn ∗ x ∈ \, |x |≤1 Since there exists R> 0 such that

U = U in z = x + √ 1y ; x > R 1 1 { − | | } we see that U1 is regularf at and a Stein open set. Then U = (U1 U2) satisﬁes the conditions of the theorem.∞ ∩ f f Set M = Ω. We deﬁne a homeomorphism ι : X Cn M Dn bas follows. ∩ → × y b ι(x + √ 1y)= x, b . (4.88) − x 2 +1! | | p 27 By extending the morphism to X continuously, we obtain the homeomorphism between X and Ω Dn = M Dn. × × b Let bbe a familyb of all the open subsets U in X which satisfy U 1. U is a neighborhood of M. b 2. U is regular at . ∞ 3. U Cn is a Stein open set. ∩ Such an open subset surely exists thanks to Theorem 4.9.

Let be a family of all the stratifications of Sn−1 which are associated with partitioningT by a finite family of hyperplanes passing through the origin in Rn. It is clear that satisfies the condition (T). T On the aforementioned preparations, we can easily prove that ( , , ) satisfies the conditions (T), (U) and (W) by applying Theorem 4.4 andTTheoremU W 4.9, and thus intuitive representation of Laplace hyperfunctions is justified.

References

[HU] Honda, N. and Umeta, K., On the Sheaf of Laplace Hyperfunctions with Holomorphic Parameters. J. Math. Sci. Univ. Tokyo 19 (2012), 559–586. [Kn] Kaneko, A., Introduction to hyperfunctions. KTK Scientiﬁc Publishers, Kluwer Academic Publishers (1988). [Ks] Kashiwara, M., Kawai, T. and Kimura, T., Foundation of algebraic analysis. Kinokuniya math book series, 18 (1980).

[KS] Kashiwara, M. and Schapira. P., Sheaves on Manifold. Grundlehren der mathematischen Wissenschaften, 292, Springer-Verlag, (1990). [M] Morimoto, M., An Introduction to Sato’s Hyperfunctions. In Translations of Mathematical Monographs, American Mathematical Society, (1993). [S] Sato, M., Theory of hyperfunctions. I and II, J. Frac. Sci. Univ. Tokyo Sect. IA 8 (1959/1960).

[SKK] Sato, M., Kawai, T. and Kashiwara, M., Microfunctions and pseudo- diﬀerential equations. In Hyperfunctions and Pseudo-Diﬀerential Equations (Proc. Conf., Katata, 1971; dedicated to the memory of Andre Martineau). Springer, Berlin, 265–529. Lecture Notes in Math., 287, (1973).