数理解析研究所講究録983 巻 1997 年 86-105

86

Algorithms for -functions, restrictions, and algebraic local
cohomology groups of D-modules
Toshinori Oaku (大阿久俊則)
Department of Mathematical Sciences, Yokohama City University22-2 Seto, Kanazawa-ku, Yokohama, 236 Japan(横浜市金沢区瀬戸 22-2 横浜市立大学理学部)

1

Introduction

Let $K$ be an algebraically closed field of characteristic zero and let $X$ be a Zariski open setofwith a positive integer . We fix a coordinate system $x=$
of $X$ and write
with. We denote by
the sheaf of algebraic linear differential
operators on $X$ .
Let
be a coherent left
-module and a section of. Suppose that $f=f(x)\in K[x]$is an arbitrary non-constant polynomial of variables. If
is holonomic, then for each pointof $Y:=\{x\in X|f(x)=0\}$ , there exist a geml $P(x, \partial, s)$ of $D_{X}[s]$ at and a polynomial$b(s)\in K[s]$ of one variable so that$K^{n}$$\partial=$$(x_{1}, \ldots , x_{n})$$n$$(\partial_{1}, \ldots , \partial_{n})$$\partial_{i}:=\partial/\partial x_{i}$$\mathcal{M}$$D_{X}$$D_{X}$$u$$\mathcal{M}$$n$$\mathcal{M}$

$P$

$p$

(1.1)

$P(x, \partial, s)(f^{s+1}u)=b(s)f^{S}u$holds with an indeterminate
ative integer $m$ so that$s$(cf. [8]). More precisely, (1.1) means that there exists a nonneg-$Q:=f^{m}-s(b(S)-P(_{X}, \partial, S)f)fs\in Dx[s]$satisfies $Qu=0$ in. The monic polynomial $b(s)$ of the least degree thatsatisfies (1.1), if any, is called the (generalized) -function for and at . The bfunctionin this sense was first studied by Kashiwara [8] (cf. also [29]). Some of its applications were
given by Kashiwara-Kawai [11]. In particular, whencoincides with the sheafof regularfunctions and $u=1$ , we get the classical bfunction (or the Bemstein-Sato polynomial) of . An
algorithm for computing the Bernstein-Sato polynomial has been given in [20].
Suppose that a presentation (i.e., generators and the relations among them) of a coherent
left-moduleand a section of
are given. Then we are concerned with algorithms forsolving the following problems:$\mathcal{M}[s]:=K[s]\otimes_{K}\mathcal{M}$

$b$

$f$

$u$

$\mathcal{M}$$p$$O_{X}$$f$$D_{X}$$\mathcal{M}$$u$$\mathcal{M}$(A1) to determine whether there exists and to find, if it does, the b–function for $f$ and(A2) to obtain presentations of the algebraic local cohomology groups
left-modules (cf. [8] for the definition);$D_{X}$$u$;$\mathcal{H}_{[Y]}^{j}(\mathcal{M})(j=0,1)$

as

87

(A3) to obtain a presentation of the localization $\mathcal{M}(*Y)=\mathcal{M}[f^{-1}]$ ofmodule;$\mathcal{M}$by $f$ as a left

(A4) to obtain a presentation of the left $D_{X}[S|$ -module $\sum_{i=1}^{r}DX[S](fs\otimes u_{i})$ , whereis regarded as a section ofandare generators of$f^{s}\otimes u_{i}$$\mathcal{M}$$u_{1},$$(O_{X}[s, f^{-1}]f^{s})\otimes \mathit{0}_{x^{\mathcal{M}}}$$D_{X^{-}}$$\ldots,$$u_{r}$.

It tums out that these problems are closely related with one another not only from theoreticalneed the existence of andbut also from algorithmic point of view: Solutions to; one can solve the problem (A3) bysome information on the bfunctions for $f$ andusing a solution to (A4) by specializing the parameter to an appropriate negative integer.
left$f_{2}\in K[s]$ , we can obtain a presentation of
As an application, for two polynomials$(\mathrm{A}2)-(\mathrm{A}4)$$u_{1},$$\ldots,$$u_{r}$$s$$\mathrm{t}\tilde{\mathrm{h}}\mathrm{e}$$f_{1},$$s_{2}\in K$ .
for generic constants-moduleare holonomic if so is . In this case (moreandKashiwara [8] proved thatexist, which can be,generally, under a weaker condition that the bfunctions for andcompletely except that wedetermined algorithmically), we can solve the problems
to solve the latter part of (A1), (A3), and (A4); even if thisneed the condition
condition fails, we can obtain certain information (estimates ‘from above’) on solutions of these
problems. We solve the problem (A4) by generalizing a method developed in [21] for computing
a presentation of $D_{X}[s]fs$ .
Our algorithms for (A1) and (A2) are actually obtained as applications of algorithms for-module with $X.–K\cross X$ .
be a left coherent
more general problems as follows: Now let. We identify $X$ with the hyperplane $\{(t, x)\in\overline{X}|t=0\}$ ofbe generators ofLetalong $X$ at $p\in X$ is the monic polynomial $b(s)\in K[s]$ of the least. Then the bfunction ofdegree that satisfies$D_{X}$$D_{X}(f_{1}^{S_{1}}f^{s_{2}}2)$$s_{1},$$\mathcal{H}_{[Y]}^{j}(\mathcal{M})$$\mathcal{M}$$\mathcal{M}(*\mathrm{Y})$$f$$u_{1},$$\ldots$$u_{r}$$(\mathrm{A}1)-(\mathrm{A}4)$$\mathcal{H}_{[Y1}^{0}(\mathcal{M})=0$$\mathcal{M}$$D_{\overline{X}}$$\mathcal{M}$$u_{1},$$\ldots,$$u_{r}$$\overline{X}$$\mathcal{M}$$(b(t\mathrm{a})+tP_{i}(t, x, t\partial_{t}, \partial))ui=0$a$(i=1, \ldots , r)$is called specializable along.
at , where we writeofwith germs$X$ atif such $b(s)$ exists. On the other hand, the restriction (also called the induced system or-modules:to $X$ is the complex of left
the tangential system) of$P_{i}(t, x, t\partial_{t}, \partial)$$p$$D_{\overline{X}}$$\mathcal{M}$$:=\partial/\partial t$$p$$D_{X}$$\mathcal{M}$$\mathcal{M}_{X}^{\cdot}$:$0arrow \mathcal{M}arrow \mathcal{M}tarrow 0$.

is specializable along
It was proved by Laurent-Schapira [13] (and by Kashiwara [8]) that if$X$ (or holonomic), then the cohomology groups of-modules (holonomic
are coherent left
systems, respectively).
is given. Then we obtain-moduleAssume now that a presentation of a coherent left
a complete algorithm for solving the problem$\Lambda t$$D_{X}$$\mathcal{M}_{X}$$\mathcal{M}$$D_{\overline{X}}$(B1) to determine whether
along $X$ .

$\mathcal{M}$is specializable along $X$ and to find, if so, the bfunction of$\mathcal{M}$This algorithm is obtained by generalizing a method of Gr\"obner basis computation (the Buchberger algorithm [4] in the Weyl algebra with respect to the so-called -filtration ([9]) developedin [18], [19], [20]. We have solved (B1) for the case $r=1$ in [20]. Here we generalize an algorithmof [20] so that we can compute the bfunction as a function of the point of $X$ for arbitrary $r\geq 1$ .
is specializable along $X$ , we also get an algorithm to solve theUnder the condition thatproblem$\mathrm{V}$$)$$\mathcal{M}$88(B2) to obtain presentations of the cohomology groups ofas left$\mathcal{M}_{X}$$D_{X}$-modules.

It seems that no complete algorithm for (B2) used to be known (see $[26],[27],[19]$ for partial
algorithms). Note thatis specializable if
is holonomic ([12]). Algorithms for (A1) and (A2)are obtained by applying the algorithms for (B1) and (B2) to the module
for a given, where $\delta(t-f(x))$ denotes the modulo class of $(t-f(x))-1$ in-module$O_{\overline{X}}[(t-f(x))-1]$ . Thus we can solve (A2) under the condition thatisspecializable along $X$ , and (A1), (A3), (A4) under the additional assumption. We
can also show thatis specializable along $X$ if and only if there exists
the b–function for and each generator ofin the sense of (1.1).
, we can consider the problems explained so far withWhen
replaced by the sheaf
of analytic differential operators. Then our algorithms yield correct solutions also in thisanalytic case if the left
-module an in question is written in the form
with a coherent-modulewhose presentation is given explicitly.
We have implemented the algorithms in the present paper by using computer algebra systems$Kan[28]$ developed by Takayama of Kobe University, and $Risa/Asir[16]$ developed by Noro et al.
at Fujitsu Laboratories Limited. We use $Kan$ for Gr\"obner basis computation in Weyl algebras,
and $Risa/Asir$ for Gr\"obner basis computation, factorization, and primary decomposition inpolynomial rings.
$\mathcal{M}$$\mathcal{M}$$(D_{\overline{X}}\delta(t-f(x)))\otimes \mathit{0}_{x^{\mathcal{M}}}$$D_{X}$$\mathcal{M}$$(D\delta(\overline{\mathrm{x}}-ft(x)))\otimes_{\mathcal{O}}x\mathcal{M}$$\mathcal{H}_{[Y]}^{0}(\mathcal{M})=0$$(D_{\overline{\mathrm{x}}^{\delta(}x}t-f(x)))\otimes_{o}\mathcal{M}$$f$$\mathcal{M}$$K=\mathrm{C}$$D_{X}$$D_{X}^{\mathrm{a}\mathrm{n}}$$\mathcal{M}$$D_{X}^{\mathrm{a}\mathrm{n}}$$D_{X}$

2$\Lambda \mathrm{t}^{\mathrm{a}\mathrm{n}}=D_{X}^{\mathrm{a}\mathrm{n}}\otimes_{D_{X}}\mathcal{M}$$\mathcal{M}$V-filtration and involutory generators

Let be a Zariski open subset ofwith the coordinate system $(t, x)=(t, x_{1}, \ldots , x_{n})$ .
We denote byandthe corresponding derivations with.
$X:=\overline{X}\cap(\{0\}\cross K^{n})$$X$
Put. Then can be identified with a Zariski open subset of
. Let
$X$
and
be the sheaves of regular functions on
and onrespectively. We denote byand$X$
the sheaves of rings of algebraic linear differential operators on
and onrespectively.
Let
be the sheaf theoretic restriction of
to $X$ . Put. Then for each integer
we put$\overline{X}$a$K\cross K^{n}$$=\partial/\partial t$$\partial=(\partial_{1}, \ldots, \partial_{n})$$\partial_{i}=\partial/\partial x_{i}$$K^{n}$$\mathit{0}_{X}$$\overline{X}$$\mathcal{O}_{\overline{X}}$$D_{\overline{X}}$$D_{X}$$\overline{X}$$D_{\overline{X}}|_{X}$$J_{X}:=O_{\overline{X}}t$$D_{\overline{X}}$$k$$F_{k}(D_{\overline{X}}):=${$P\in D_{\overline{X}}|\mathrm{x}|P(Jx)^{j}\in(\mathcal{J}_{X})^{jk}-$

for any $j\geq 0$ }.

Let
be a left coherent-module. We assume thathas a presentation$N$on , where is a left-submodule of. Then let us put$\mathcal{M}$$\mathcal{M}$$D_{\overline{X}}$$\mathcal{M}=(D_{\overline{X}})^{r}/N$$\overline{X}$$(D_{\overline{X}})^{r}$$D_{\overline{X}}$$F_{k}(N):=N\cap F_{k}(D\overline{x}^{)^{r}},$$F_{k}$(A4)$:=F_{k}(D_{\overline{\mathrm{x}}^{)^{r}/F(Nl}}k$for each integer $k\in$ Z. These are called -filtrations ([9]). The graded ring and modules
associated with these filtrations are defined by$\mathrm{V}$$\mathrm{g}\mathrm{r}(D_{\overline{X}})$$:=$$\oplus F_{k}(D_{\overline{X}})/F_{k}-1(D_{X}-)$$k\in \mathrm{Z}$$\mathrm{g}\mathrm{r}(N)$$:=$$\oplus F_{k}(N)/F_{k1}-(N)$,$k\in \mathrm{Z}$$\mathrm{g}\mathrm{r}(\mathcal{M})$$:=$$\oplus F_{k}(\mathcal{M})/F_{k1}-(\mathcal{M})$$k\in \mathrm{Z}$

.

,

89

,
-module. Note thatis isomorphic to
is a coherent left
Then
that are polynomials in .
which consists of the sections of$P$
be the minimum
such that, letof
For a nonzero section$P$. Then let
be the modulo class of in$\mathrm{g}\mathrm{r}(\mathcal{M})$$t$$D_{\overline{X}}|_{X}$$k\in \mathrm{Z}$$k=\mathrm{o}\mathrm{r}\mathrm{d}_{F(P)}$$(D_{\overline{X}})^{r}|_{x}$$\hat{\sigma}(P)$$P\in F_{k}(D_{\overline{\mathrm{x}}^{)}}r$$F_{k}(D_{\overline{\mathrm{x}}^{)}}r/F_{k-1}(D\overline{x})^{r}\simeq(D_{X}[t\partial_{t}]S_{k})^{r}$

where
that

$D_{X}[t, \mathrm{a}]$$\mathrm{g}\mathrm{r}(D_{\overline{X}})$$\mathrm{g}\mathrm{r}(D_{\overline{X}})$$S_{k}:=\partial_{t}^{k}$,

if $k\geq 0$ and $S_{k}:=t^{-k}$ otherwise. Moreover, we define $\psi(P)(s)\in(D_{X}[S])r$ soholds.

$\hat{\sigma}(S_{-k}P)=\psi(P)(t\mathrm{a})$be a Zariski open subset of $X$ . A subsetof
Definition 2.1 Let$F$-involutory generators of $N$ on
ifgenerates $N|_{X}$ as a leftof-module.
as a left
generates$U$$\mathrm{G}$$U$$\hat{\sigma}(\mathrm{G}):=\{\hat{\sigma}(P)|P\in \mathrm{G}\}$$D_{\overline{X}}|_{X}$$\mathrm{g}\mathrm{r}(N)$is called a set-module on
and if$\Gamma(U,N|\mathrm{x})$$\mathrm{G}$$U$$\mathrm{g}\mathrm{r}(D_{\overline{X}})$

The following two propositions are immediate consequences of the definitions:

Proposition 2.2 Let $\mathrm{G}=\{P_{1}, \ldots, P_{m}\}\subset\Gamma(U,N|x)$ be a set of generator8 $ofN|_{X}$ on a Zariski
open set $U\subset X.$ Then
is a set of $F$-involutory generators of $N$ on if and only if for an
arbitrary nonzero element $P$ of the stalk $N_{p}ofN$ at $p\in U$ , and for an arbitrary integer , there,
so that ord
and
exist
$U$

$\mathrm{G}$$j$$Q_{1},$$\ldots$$p(Q_{i}P_{i})\leq \mathrm{o}\mathrm{r}\mathrm{d}_{F(P)}(i=1, \ldots,m)$$Q_{m}\in N_{\mathrm{p}}$$P-Q_{1}P_{1}-\ldots-Q{}_{m}P_{m}\in F_{j}(D\overline{x}^{)^{r}}p\cdot$

Proposition 2.3 Let$D_{X}[s]$-submodule of$\{\psi(P)|P\in \mathrm{G}\}$3$\mathrm{G}$be a set of -involutory generators of N. Denote by $\psi(N)$ the left
generated by $\{\psi(P)|P\in N\}$ . Then $\psi(N)$ is generated by$F$$\psi(\mathrm{G}):=$$(D_{X}[\mathit{8}])^{r}$.

Gr\"obner bases with respect to the V-filtration

The purpose of this section is to show that a set of -involutory generators of a given
submodule $N$ of
can be provided by a Gr\"obner basis in the Weyl algebra with respect
to an appropriate term ordering, which can be computed by the Buchberger algorithm ([4]) for
Gr\"obner bases of polynomial rings. The fact that the Buchberger algorithm applies to the Weyl
algebra (the ring of differential operators with polynomial coefficients) was observed by Galligo[5] (cf. also $[3],[25]$ ).
Let us denote by
and $A_{n+1}$ the Weyl algebras on the variables and on the $n+1$
variables $(t, x)$ respectively with coefficients in $K$ (cf. [1]). Let be a positive integer and put
with $\mathrm{N}:=\{0,1,2, \ldots\}$ . An element $P$ of $(A_{n+1})^{r}$ is written
in a finite sum$\mathrm{F}$$(D_{\overline{X}})^{r}$$A_{n}$

$n$

$x$

$r$

$L:=\mathrm{N}^{2+2n}=\mathrm{N}\cross \mathrm{N}\cross \mathrm{N}^{n}\cross \mathrm{N}^{n}$$P= \sum_{i=1\nu}^{r}\sum_{(\mu,,\alpha,\beta)\in L}a\nu\alpha\beta it\mu x\mu\alpha \mathrm{a}^{\nu_{\partial^{\beta}e_{i}}}$

with
for
Let

$a_{\mu\nu\alpha\beta i}\in K,$$\partial^{\beta}:=\partial 1^{\beta 1}\ldots\partial_{n}\beta n$$e_{1}:=$ $(1, 0, \ldots , 0),$ $\ldots,$ $e_{r}:=(0, \ldots , 0,1),$ $x^{\alpha}:=x_{1}\alpha_{1}\ldots\alpha x_{n}n,$$\alpha=(\alpha_{1}, \ldots, \alpha_{n}),$ $\beta=(\beta 1, \ldots, \beta_{n})\in \mathrm{N}^{n}$$\prec p$

(3.1)

.

be a total order on $L\cross\{1, \ldots, r\}$ which satisfies(O-1) $(\alpha, i)\prec p(\beta, j)$ implies $(\alpha+\gamma, i)\prec_{F}(\beta+\gamma, j)$ for any$\alpha,$$\beta,\gamma\in L$

and $i,j\in\{1, \ldots,r\}$ ;90(O-2) if $\nu-\mu<\nu’-\mu’$ , thenand any $i,j\in\{1, \ldots, r\}$ ;$(\mu, \nu, \alpha, \beta, i)\prec_{F}(\mu’, \nu’, \alpha’, \beta’,j)$$\mu,$(O-3)

for any$\alpha,$$\beta,$$\alpha’,$$\beta’\in \mathrm{N}^{n}$,$\mu’,$ $\nu’\in \mathrm{N}$$\nu,$$(\mu, \mu, \alpha, \beta, i)\succeq_{F}(0,0, \mathrm{o}, 0, i)$for any$\mu\in \mathrm{N},$$\alpha,$$\beta\in \mathrm{N}^{n},$$i\in\{1, \ldots, r\}$.

Note thatis not a well order (linear ordering). However, throughout the present paper,
every order is supposed to satisfy (O-1). Let $P$ be a nonzero element of $(A_{n+1})^{r}$ which is written
in the form (3.1). Then the leading exponent $1\exp_{F}(P)\in L\cross\{1, \ldots, r\}$ of $P$ with respect
is defined as the maximum element$\prec_{F}$$\mathrm{t}\mathrm{o}\prec_{F}$$\max\{(\mu, l^{\text{ノ}}, \alpha, \beta, i)|a_{\mu\nu\alpha\beta i}\neq 0\}$with respect to thedefined by$\mathrm{o}\mathrm{r}\mathrm{d}\mathrm{e}\mathrm{r}\prec_{F}$. The set of leading exponents $E_{F}(N)$ of a subset $N$ of $(A_{n+1})^{r}$ is$E_{F}(N):--\{\mathrm{l}\exp(P)|P\in N\backslash \{0\}\}$.

Definition 3.1 A finite setof generators of a left $A_{n+1}$ -submodule $N$ of $(A_{n+1})^{r}$ is called
an FW- Gr\"obner basis of $N$ if we have$\mathrm{G}$$E_{F}(N)=\cup(1\exp(P)+L)$ ,PEG

where we write$(\alpha, i)+L=\{(\alpha+\beta, i)|\beta\in L\}$for$\alpha\in L$

and $i\in\{1, \ldots , r\}$ .

Proposition 3.2 Let
be an $FW$-Gr\"obner basis of a lefl $A_{n+1}$ -submodule $N$ of $(A_{n+1})^{r}$ . Then$F$
is a set of -involutory generators of the left-submodule $N:=D_{\overline{X}}N$ of
on $X$ .
$\mathrm{G}$$\mathrm{G}$$D_{\overline{X}}$$(D_{\overline{X}})^{r}$

Since the order
is not a well-order, the Buchberger algorithm for computing Gr\"obnerbases does not work directly. We use the homogenization with respect to the -filtration inorder to bypass this difficulty (cf. [18], [19], [20]). The following arguments generalize those in[20], where the case with $r=1$ is treated. Since this generalization is straightforward, we omit.
the proof.
$\prec_{F}$$\mathrm{V}$Definition 3.3 For, and $i,j\in\{1, \ldots , r\}$ , anwith
is defined so that we have
if and only if one of the following conditions holds:$\lambda,$$\lambda<\lambda’$(2)$\lambda=\lambda’$$\lambda/,/\nu\mu$ $/\in \mathrm{N},$)$\alpha,$$\beta,$$\alpha’,$$\beta/\in \mathrm{N}^{n}$$\mathrm{o}\mathrm{r}\mathrm{d}\mathrm{e}\mathrm{r}\prec_{H}$on$(\lambda, \mu, \nu, \alpha, \beta, i)\prec_{H}(\lambda’, \mu^{\prime/}, \nu, \alpha\beta’/,,j)$

;

,

,

$(\mu+\ell, \iota^{\text{ノ}}, \alpha, \beta, i)\prec_{F}(\mu’+\ell/,/,/,j)\mathcal{U}\alpha\beta’$$\nu’-\mu’-\ell’$(3) ( , l ノ,$\nu,$$L_{1}:=\mathrm{N}\cross L$$L_{1}\cross\{1, \ldots, r\}$(1)$\mu,$$\lambda$$\alpha,$$\beta,$

$i$

with

$\ell,$$\ell’\in \mathrm{N}$such that $l\text{ノ}-\mu-P=$;)$=(\lambda^{\prime//}, \nu, \alpha, \beta/,j)$,$\mu<\mu’$This definition is independent of the choice of$\ell,$$\ell’$

in view of the condition (O-1).

91

For a nonzero element $P=P(x_{0})$ of $(A_{n+1}[X_{0}])^{r}$ , let us denote by $1\exp_{H}(P)\in L_{1}\cross\{1, \ldots, r\}$.
the leading exponent of $P$ with respect to$\prec_{H}$Definition 3.4 An element $P$ of $(A_{n+1}[x_{0}])^{r}$ of the form
$P= \sum_{i=1\lambda,\mu}^{r},\sum_{\nu,\alpha,\beta}a\lambda\mu\nu\alpha\beta ix_{0^{\lambda}}t^{\mu}X^{\alpha}\mathrm{a}^{\nu}\partial^{\theta}e_{i}$is said to be F-homogeneou8 of order $m$ if$a_{\lambda\mu\nu\alpha\beta i}=0$

whenever $\nu-\mu-\lambda\neq m$ .

Definition 3.5 For an element $P$ of $(A_{n+1})^{r}$ of the form (3.1), put $m:= \min\{\nu-\mu|a_{\mu\nu\alpha\beta i}\neq$, and $i\in\{1, \ldots , r\}\}$ . Then the $F$-homogenization $P^{h}\in(A_{n+1}[X_{0}])^{r}$for some
of $P$ is defined by$0$$\mu,$$\nu\in \mathrm{N},$$\alpha,$$\beta\in \mathrm{N}^{n}$$P^{h}:= \sum’\sum aX-tX^{\alpha}\mu\nu\alpha\beta i0^{\nu-\mu m\mu}\mathrm{a}^{\nu}\partial^{\mathit{0}_{e_{i}}}$$i=1\mu,\nu,\alpha,\beta$

which commutes with all the other variables and derivations.
with a parameterhomogeneous of order $m$ .

$P^{h}$$x_{0}$

is F--submodule of $(A_{n+1}[X_{0}])^{r}$ generated by F-homogeneous
Proposition 3.6 Let be a lefi
) ofoperators. Then there exists an $H$-Gr\"obner basis ($i.e$ . a Gr\"obner basis vrith respectconsisting of $F$-homogeneous operators. Moreover, such an $H$ -Gr\"obner basis can be computed by
the Buchberger algorithm.
$\overline{N}$$A_{n+1}[x\mathrm{o}]$$\overline{N}$$to\prec_{H}$$P_{d}\in(A_{n+1})^{r}$ .
Proposition 3.7 Let $N$ be a left $A_{n+1}$ -submodule of $(A_{n+1})^{r}$ generated by,.
Let us denote by
the left $A_{n+1}[x_{0}]$ -submodule of $(A_{n+1}[X_{0}])^{r}$ generated byconsisting of $F$-homogeneous operbe an $H$-Gr\"obner basis ofLet
ators. Then $\mathrm{G}(1):=\{Q_{1}(1), \ldots , Q_{k}(1)\}$ is an $FW$-Gr\"obner basis of $N$ .
$P_{1},$$N^{h}$$\ldots,$$(P_{1})^{h},$$(P_{d})^{h}$$\ldots$$N^{h}$$\mathrm{G}=\{Q_{1}(x\mathrm{o}), \ldots , Q_{k}(x\mathrm{o})\}$

These two propositions, combined with Proposition 3.2, provide us with an algorithm ofon $X$ .
computing a finite set of -involutory generators of$N=D_{\overline{X}}N$$\mathrm{F}$4

The -function of a D-module

be a left coherent
We retain the notation in the preceding section. Let$N$is given explicitly so that
We assume that a left $A_{n+1}$ -submodule of. Let $F_{k}(N),$holds with $M:=(A_{n+1})^{r}/N$ . Set
respectively defined in Section 2 and put
filtrations of $N$ and$\mathcal{M}$$(A_{n+1})^{r}$$N:=D_{\overline{X}}\otimes_{A_{n+1}}N\subset(D_{\overline{X}})^{r}$$D_{\overline{X}}$-module on$\overline{X}$.

$\mathcal{M}=D_{X}^{-}\otimes A_{n+}1M$$F_{k}(\mathcal{M})$be the V-$\mathcal{M}$In particular,[ta].
$D_{X}$$\mathrm{g}\mathrm{r}_{0}(\mathcal{M})$and$\mathrm{g}\mathrm{r}_{k}(D)\overline{x}$$:=$$F_{k}(D_{\overline{X}})/F_{k-1}(D_{\overline{X}})$$\mathrm{g}\mathrm{r}_{k}(N)$$:=$$F_{k}(N)/F_{k-1(N)}$$\mathrm{g}\mathrm{r}_{k}(\mathcal{M})$.—$F_{k}(\mathcal{M})/F_{k-1}(\mathcal{M})$$\mathrm{g}\mathrm{r}_{0}(N)$are left$\mathrm{g}\mathrm{r}_{0}(D_{\overline{x}})$

,

,

.

-modules and we can identify$\mathrm{g}\mathrm{r}_{0}(D_{\overline{x}})$

with

92

Definition 4.1 The -function $b(s,p)\in K[s]$ ofalong $X$ (with respect to the V-filtration$\{F_{k}(A4)\})$ at $P\in X$ is the monic polynomial $b(s,p)\in K[s]$ of theleast degree, if any, thatsatisfies$b$$\mathcal{M}$$b(t\partial_{tp)},\mathrm{g}\mathrm{r}\mathrm{o}(\mathcal{M})p=0$

If such $b(s,p)$ exists,
put $b(s,p)=0$ .

.

(4.1)

is called specializable along $X$ at . If$\mathcal{M}$$p$$\mathcal{M}$is not specializable at , we$p$

It is known that if
is holonomic, thenis specializable at any $p\in X([12])$ . In thesequel, we describe an algorithm for computing $b(s,p)\in K[s]$ as a function of $p\in X$ .
$\mathcal{M}$$\mathcal{M}$

Proposition 4.2 Put $J:=\psi(N)\cap(O_{X}[s])^{r}J$ which is an $O_{X}[S]$ -submodule of $(O_{X}[s])^{r}$ . Let
be the annihilator ideal for $(O_{X}[s])^{r}/J$ . Then the ideal
of $K[s]$ is generated by $b(s,p)$ for each $p\in X$ .

$\mathrm{A}\mathrm{n}\mathrm{n}((O\mathrm{x}[s])r/J)\subset O_{X}[S]$$\mathrm{A}\mathrm{n}\mathrm{n}((Ox[s])r/J)p\cap$$K[\mathit{8}]$A set of generators of $\psi(N)$ on $X$ can be computed by using Propositions 2.3, 3.6, 3,7.
Hence our first task here is to compute a set of generators of $J$ . Let
be a total order onwithwhich satisfies (O-1) with replaced byand$\prec_{D}$$L_{0}:=\mathrm{N}^{1+2n}$$L_{0}\cross\{1, \ldots , r\}$(O-4)$(\alpha, i)\succ_{D}(0, i)$(O-5)$|\beta|<|\beta’|$$\{1, \ldots, r\}$for any

implies$L$$\alpha\in L_{0}\backslash \{0\}$$L_{0}$

and $i\in\{1, \ldots, r\}$ ;$(\mu, \alpha, \beta)i)\prec_{D}(\mu’, \alpha^{J}, \beta^{J},j)$for any$\mu,$ $\mu’\in \mathrm{N},$$\alpha,$$\alpha’,$$\beta,$$\beta’\in \mathrm{N}^{n},$$i,j\in$.

Note that the$\mathrm{o}\mathrm{r}\mathrm{d}\mathrm{e}\mathrm{r}\prec_{D}$is a well-order.

Proposition 4.3 Let
be a finite subset ofwhich generates $\psi(m$ as a left $D_{X}[S]-$
Gr\"obner
module on X. Let
be abasis with respectof the submodule of $(A_{n}[s])r$
generated by. Put G3
. Then $J$ is generated by G3 on $X$ as an $O_{X}[s]-$module.
$(A_{n}[s])^{r}$$\mathrm{G}_{1}$$to\prec_{D}$$\mathrm{G}_{2}$$:=\mathrm{G}_{2}\cap K[s, x]^{r}$$\mathrm{G}_{1}$The final step will be devoted to the computation of $b(s,p)$ with a set of generators of $J$ as
an input. For $i=1,$, , put$\ldots$$r$$J^{(i)}:=${ $f=(f1,$$\ldots,$$f_{r})\in J|f_{j}=0$ if $j>i$can be regarded as an ideal of
Gr\"obner basis with respect to anThen$\mathcal{J}^{(i)}/J(i-1)$$\mathit{0}_{X}[S]$}.

whose generators can be computed via asatisfying
for any$(\alpha, i)\prec(\beta,j)$$\mathrm{o}\mathrm{r}\mathrm{d}\mathrm{e}\mathrm{r}\prec \mathrm{o}\mathrm{n}\mathrm{N}^{1+n}\cross\{1, \ldots , r\}$if $i<j$ .
So far we have used only the Buchberger algorithm, which does not require field extension,for computing Gr\"obner bases with respect to various orders. Hence we do not need to assume
that $K$ is algebraically closed from the viewpoint of algorithms. Thus, in the rest of this section,
we assume that $K$ is an arbitrary field of characteristic zero so that the inputs are defined over
$K$ . Since we will make use of
primary decomposition, which is sensitive to field extension, we
will have to pay attention to the coefficient fields.
Letbe the algebraic closure of $K$ and suppose that $X$ is a Zariski open subset. We
denote by
the sheaf of regular functions on $X$ . In particular,
is a sheaf of -algebras.
$\alpha,$$\beta\in \mathrm{N}^{1+n}$$\overline{K}$$\mathrm{o}\mathrm{f}\overline{K}^{n}$$\mathcal{O}_{X}$$O_{X}$$\overline{K}$93In general, for an idealof the ideal$Q$, let us denote by $b(s, Q,p)\in K[s]$ a generator
of $K[s, x]$ and. We may assume that $b(s, Q,p)$ is monic if it is not zero. Put$p\in\overline{K}$$K[s]\cap \mathcal{O}_{X}[s]_{\mathrm{P}}Q$$\mathrm{V}_{X}(Q):=$

Note that$\mathrm{V}_{X}(Q)$can be computed by eliminating

Lemma 4.4 In the above notation, the ideal$b(s, Q,p)$for any $f\in Q\mathrm{n}K[X]$ }.

{ $x\in X|f(x)=0$$s$

by means of a Gr\"obner basis of$O_{X}[s]_{p}Q\cap\overline{K}[s]$.

Proposition 4.5 Assume thatthe ideal $Q\cap K[s]$ of $K[s]$ .

$Q$

of

$\overline{K}[\mathit{8}]i\mathit{8}$$Q$.

also generated by

is a primary ideal of $K[s, x]$ and let $h(s, Q)$ be a generator of(1) Case $h(s, Q)\neq 0$ : In this case there exist8 an irreducible polynomial $h_{0}(s, Q)\in K[s]$ andso that $h(s, Q)=h_{0}(s, Q)^{\nu 0}$ . Put$\nu_{0}\in \mathrm{N}$$\mathrm{V}_{X}^{\nu}(Q):=${ $x\in X|f(x)=0$ for any $f\in K[x]\cap(Q$ :$h_{0}(S,$ $Q)^{\nu})$}, where: denotes the ideal quotient in $K[s,x]$ . Then we have a decreasing
sequence of algebraic setsfor each$\nu\in \mathrm{N}$$X\supset \mathrm{V}x(Q)=\mathrm{v}_{X}^{0}(Q)\supset \mathrm{V}_{X}^{1}(Q)\supset\ldots\supset \mathrm{V}_{X}^{\nu_{0}}(Q)=\emptyset$of X. If$p\in \mathrm{v}_{\mathrm{x}^{-1}}^{\nu}(Q)\backslash \mathrm{V}_{X}^{\nu}(Q)$where we put$\mathrm{V}_{X}^{-1}(Q):=X$, then we have $b(s, Q,p)=h_{0}(s, Q)^{\nu}$ for $\nu=0,$$\ldots,$$\nu_{0}$,.

(2) Case $h(s, Q)=0$ : In this case we have $b(s, Q,p)=0$ if$p\in \mathrm{V}_{X}(Q)$and $b(s, Q,p)=1$

otherwise.
Note that $h(s, Q)$ and the ideal quotientbases ([4]).

$Q$:$h_{0}(s, Q)^{\nu}$

can be computed also by Gr\"obner

Proposition 4.6 Under the above assumptions and notation, let. Letthat $O_{X}[\mathit{8}]J_{i}=J^{(i)}/J^{()}i-1$ for $i=1,$$\ldots,$$J_{i}$

be an ideal of $K[s, x]$ such$r$

$J_{i}=Q_{i,1}\cap\ldots\cap Qi,m_{[]}$

be a primary decomposition ofcommon multiple of$J_{i}$

in $K[s,x]$ . Then the -function $b(s,p)of\mathcal{M}$ at $p\in X$ is the least
where $(i,j)$ runs over the set $\{(i,j)|1\leq i\leq r, 1\leq j\leq m_{i}\}$ .
$b$$b(s, Q_{i}\mathrm{j},p))s$

Thus by combining Propositions 4.2, 4.3, 4.5 and 4.6, we have obtained an algorithm to
as a function of $p\in X$ . In particular, note that $b(s,p)$compute the $k$ function $b(s,p)$ ofbelongs to $K[s]$ for any $p\in X$ . Let us assume that $X$ is defined over $K$ , i.e., there exists an. Then the following theorem
inis the set of the zeros ofof $K[x]$ so thatidealis specializable along $X$ at every point
provides us with an algorithm to determine whetherof $p\in X$ , and to compute the set { $s\in\overline{K}|b(s,p)=0$ for some $p\in X$ }. This will be neededin order to compute the restriction and the algebraic local cohomology groups globally on $X$ inthe subsequent sections (cf. Proposition 5.2 below). Let us denote by rad $Q’$ the radical of an
ideal $Q’\subset K[x]$ .
$\mathcal{M}$$I_{X}$$I_{X}$$\overline{K}^{n}\backslash X$$\mathcal{M}$$\overline{K}$94

be as in the preceding proposition.
Theorem 4.7 Letand(1)
is specializable along $X$ at each point of $X$ if and only if the condition$Q_{ij}$$J_{i}$$\mathcal{M}$or rad $(Qij\cap K[x])\supset I_{X}$$Q_{ij}\cap K[s]\neq\{0\}$(4.2), and $j=1$ , . . . , .
holds for each $i=1,$(2) Assume thatbe a generator of $Q_{ij}\cap K[s]$ if
holds for each and . Let$(Qij\mathrm{n}K[X])\supset
I_{X}$, and put
. Let $b(s)$ be the least commonrad
if radmultiple of $b_{ij}(s)’S$ with $1\leq i\leq r$ and $1\leq j\leq m_{i}$ . Then the -function $b(s,p)$ ofdivides$b(s)$
for any $p\in X$ . $M_{oreo}ver_{J}$ for any irreducible factor $g(s)$ of $b(s)$ , there exists some $p\in X$
so that $g(s)$ divide8 $b(s,p)$ .
$(S)$ Assume
Thenis specializable along $X$ at each point of $X$ if and only if$J_{i}\cap K[s]\neq 0$
be a generator of $J_{i}\cap K[s]$ and let $b(s)$ be
for any $i=1,$ , . In this case let. Then $b(s)$ is the’sthecommon multiple ofcommon multiple ofwhere runs over $X$ .
$\ldots$$r$$m_{i}$$i$$(\mathit{4}\cdot \mathit{2})$$(Qij\mathrm{n}K[X])\not\supset I_{X}$$b_{ij}(s)$$j$$b_{ij}(\mathit{8}).--1$$b$$X=\overline{K}^{n}$$\mathcal{M}$$\ldots$$lea\mathit{8}t$$\mathcal{M}$$b_{i}(s)$$r$$b_{1}(s),$$\ldots,$$lea\mathit{8}t$$b_{r}(S)$$b(\mathit{8}, p)$$p$

The restriction of a D-module

5

We retain the notation of the preceding section. In particular, let $b(s,p)$ be the b-function
at $p\in X$ . The ( $D$ -module theoretic) restriction ofof
to $X$ is the complex$\mathcal{M}$$\mathcal{M}$:$\mathcal{M}_{X}^{\cdot}$$0arrow \mathcal{M}arrow \mathcal{M}tarrow 0$-modules, where the homomorphism denotes the one defined by $t(u)=tu$ for each$u\in$ M.
to be placed at the degree in considering the cohomology
We regard the right. Put. Thengroups ofis a $(D_{X}, D_{X}-)$ -bimodule, andisof left$D_{X}$$t$$0$$\mathcal{M}$$\mathcal{M}_{X}$$D_{Xarrow\overline{X}}.--D_{\overline{X}}/tD_{\overline{X}}$in the derived category,
isomorphic to. Let us denote by.
complex$\mathcal{M}_{X}:=\mathcal{H}^{0}(\mathcal{M}_{X})=\mathcal{M}/t\mathcal{M}$[6])$denotes the left derived functor
the O-th cohomology group of the$\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\otimes \mathrm{L}$$D_{Xarrow\overline{X}}\otimes_{D-\mathcal{M}}\mathrm{L}x$$\mathrm{o}\mathrm{f}\otimes(\mathrm{c}\mathrm{f}.

$\mathcal{M}_{X}$$D_{Xarrow\overline{X}}$$\mathcal{M}_{X}$Lemma 5.1 The homomorphism$p\in X$$t$:$\mathrm{g}\mathrm{r}_{k+1}(\mathcal{M})_{p}arrow \mathrm{g}\mathrm{r}_{k}(\mathcal{M})_{\mathrm{p}}$.

is bijective if $b(k,p)\neq 0$ forbe
Proposition 5.2 Assume thatis specializable along $X$ at each point of X. Let
integers such that the -function $b(s,p)$ of
satisfies $b(k,p)\neq 0$ for any $p\in X$ and for any$k>k_{1}$$k<k_{0}$or. Theninteger such thatis quasi-isomorphic to the complex$k_{0}\leq k_{1}$$\mathcal{M}$

$b$

$k$

$\mathcal{M}$$\mathcal{M}_{X}$$0arrow F_{k_{1}+1}(\mathcal{M})/Fk_{0}(\mathcal{M})arrow F_{k_{1}}(\mathcal{M})t/F_{k1}0-(\mathcal{M})arrow 0$of left
and$D_{X}$$k\in \mathrm{Z}$-modules on X. In particular,.

$t:\mathcal{M}arrow \mathcal{M}$is bijective if $b(k,p)\neq 0$ for any $p\in X$

The following proposition provides a sufficient condition for the -lth cohomology group
to vanish.

$\mathcal{H}^{-1}(\mathcal{M}_{X})$95

Proposition 5.3 Assume that there exists $b_{0}(s)\in K[s]$ and$[\mathfrak{v}(t\mathrm{a})\mathrm{a}m\mathrm{g}\mathrm{r}0(\mathcal{M})_{\mathrm{p}}=0$$A_{\mathit{8}}sume_{J}$moreover,) $(k)\neq 0$$\mathrm{h}$for any$m\in \mathrm{N}$so that.

Z. Then the homomorphism$k\in$$t$: Mp$arrow \mathcal{M}_{p}i_{\mathit{8}}$injective.
Now we shall give an algorithm to compute
we can write $P$ in the form$\mathcal{M}_{X}$. Let $P$ be an element of$F_{m}(D-x^{)^{r}}$. Then$P= \sum_{i=1}’\sum^{m}Pik(t\partial_{t}, X, \partial)\partial_{t}^{k}ei+Rk=0$

uniquely with $P_{ik}\in D_{X}$ [ta] and$R\in F_{-1}(D_{\overline{\mathrm{x}}^{)}}r$. Then we put$\rho(P, k_{0}):=\sum_{i=1k}’\sum_{0=k}P_{ik(x}m0,,$

for each integer$k_{0}$$\partial)\partial^{k}tei$with $0\leq k_{0}\leq m$ .

be as in Proposition 5.2.
along $X$ and let
isthat
Theorem 5.4
Redefine to be if $k_{0}<0$ . (We have $k_{0}=0$ and $k_{1}=m-1$ under the assumption of
be a finite set of $F$-involutory generators of $N$ on X. Then we have an
Proposition 5. ) Let
isomorphism$As\mathit{8}ume$$\mathit{8}pecializable$$\mathcal{M}$$k_{0},$$k_{1}$$0$$k_{0}$$S.$$\mathrm{G}$’$\mathcal{M}_{X}$of left$D_{X}$-modules, where$N_{X}$$\simeq$is the left$( \bigoplus_{i=1k}\bigoplus_{=k0}^{k_{1}}D\mathrm{x}\partial t^{k}ei)/N\mathrm{x}$$D_{X}$-module generated by a finite set$\mathrm{G}_{X}:=\{p(\partial_{t^{j}}P, k\mathrm{o})|P\in \mathrm{G}, j\in \mathrm{N}, k_{0}\leq j+\mathrm{o}\mathrm{r}\mathrm{d}F(P)\leq k_{1}\}$

In particular, we have$\mathcal{M}_{X}=0$

if $b(\nu,p)\neq 0$ for any$\nu\in \mathrm{N}$.

and $p\in X$ .

be the modulo
In order to interpret the preceding theorem more concretely, lettheorem,precedingthe. Then as is seen by the proof ofinclasses of-module.
with $k_{0}\leq k\leq k_{1}$ and $1\leq i\leq r$ as left
is generated byMoreover, for $P_{ik}\in D_{X}$ , we have$u_{1},$$e_{1},$$\ldots,$$e_{r}$$\ldots,$$u_{r}$$\mathcal{M}_{X}\simeq$$\lambda 4$$D_{X}$$1\otimes(\partial_{ti}^{k}u)$$D_{Xarrow\overline{X}}\otimes_{D-,X}\mathcal{M}$$\sum_{i=1k}^{f}\sum^{1}Pik(1\otimes\partial_{ti}^{k}u=k_{0}k)=0$

if and only if$\sum_{i=1}^{r}\sum^{k_{1}}k=k_{\mathrm{O}}P_{i}ke_{i}\in Nx$.

Our next aim is to give an algorithm for computing the structure of the kernel
has a structure of left
-module. Note thatas a left:, put-module. For two integers
which is compatible with that of left$t$$\mathcal{M}arrow \mathcal{M}$$D_{X}[t\mathrm{a}]$$\mathcal{H}^{-1}(\mathcal{M}_{X})$$D_{X}$$k_{0}\leq k_{1}$$D_{X}$$\overline{D}^{(k_{0},k)}1:=\bigoplus_{i=1k}’\bigoplus_{=k\mathrm{o}}Dx[t\partial\iota]rsk1ke_{i}$,of-module$\mathcal{H}^{-1}(\mathcal{M}_{X})$

96

where
if $k\geq 0$ , and $S_{k}:=t^{-k}$ if $k<0$ . Let $P$ be a section of
write $P$ uniquely in the form$:=\partial_{t}^{k}$$S_{k}$$F_{m}(D_{X}-)^{r}$. Then we can(5.1)$P= \sum_{i=1k}^{\tau}\sum_{\infty=-}^{m}P_{ik}(t\partial_{t}, X, \partial)s_{k}e_{i}$with$P_{ik}\in D_{X}[t\partial_{t}]$. Then we define.

$\tau(P, k_{0}):=\sum_{i=1k}\sum Pik(t’=km\mathrm{o}X\partial_{t},, \partial)ske_{i}$Proposition 5.5 Let
be a finite set of -involutory generators of $N$ on X. Then, for any
integers
we have an isomorphism$F$$\mathrm{G}$$k_{0}\leq k_{1_{l}}$$F_{k_{1}}(\mathcal{M})/F_{k0-1()}\mathcal{M}$of left $Dx[t\mathrm{a}]- modules$, where$\mathcal{G}^{(k_{0},k}1$)

is a left$\simeq$$\overline{D}^{(0,)}kk_{1}/\mathcal{G}^{(kk_{1}}0,)$$D_{X}[t\mathrm{a}]$-module generated by a finite set$\mathrm{G}^{(k_{0},k)}1:=\{\mathcal{T}(sjP, k\mathrm{o})|P\in \mathrm{G}, j\in \mathrm{Z}, k_{0}\leq j+\mathrm{o}\mathrm{r}\mathrm{d}_{F}(P)\leq k_{1}\}$Let:$\chi$$\overline{D}(k_{\mathrm{O}}+1,k1+1)arrow\overline{D}^{(k_{0},k}1)$

be a left$D_{X}$.

[ta]-module homomorphism defined by$\chi(\sum_{i=1k}^{r}\sum^{1}P_{i},k+1(t\partial_{t}, X, \partial)s=kk0k+1ei\mathrm{I}=\sum_{ki=1}^{r}\sum_{=k0}^{k_{1}}Pi,k+1(t\mathrm{a}-1, x, \partial)T_{k}ei$

with$S_{k}$$(k\leq-1)$$t\partial_{t}S_{k}$$(k\geq 0)$$T_{k}:=\{$.

Theorem 5.6 Under the same assumptions as in Proposition 5.2, we have an isomorphism$\mathcal{H}^{-1}(\mathcal{M}_{X}^{\cdot})\simeq\chi^{-1}(\mathcal{G}^{(k_{0},k_{1}}))/\mathcal{G}^{(1}k\mathrm{o}+1,k_{1}+)$

as lefl$D_{X}$[ta]-modules. $Moreover_{J}\chi^{-1}(\mathcal{G}^{(k}0,k1))/\mathcal{G}^{(+k_{1}+)}k01,1$ is a coherentA
algorithm. Put$\mathrm{p}_{\Gamma \mathrm{a}\mathrm{e}\mathrm{e}\mathrm{n}\mathrm{t}}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$of$\mathcal{H}^{-1}(\mathcal{M}_{X})$as a left coherent$D_{X}$-module can be obtained by the following$A^{(k_{0},k)}1:= \bigoplus_{i=1k}\oplus An[t\partial_{t}]t=k0k_{1}Ske_{i}$We regard$A^{()}k_{0},k1$

as a free left

$A_{n}$

left Dx-module.

.

[ta]-module of rank $k_{1}-k_{0}+1$ .

Algorithm 5.7 Input: a finite setof -involutory generators of $N$ on $X$ , andintegers
satisfying the assumption of Proposition 5.2.
$\mathrm{G}\subset(A_{n+1})^{r}$$\mathrm{F}$$k_{0},$ $k_{1}$(1) Let

$N_{1}$

be the left

$A_{n}[t\mathrm{a}, z]$-submodule of$A^{(kk_{1}}0,)[Z]:= \bigoplus_{i=1k}\oplus^{1}A’=kk_{0}n[t\mathrm{a}, z]S_{ki}e$which is generated by$\bigcup_{i=1}^{f}\cup\{(1-Z)T_{k}e_{i}\}\cup\{_{Z}P|P\in k=k0k_{1}\mathrm{G}^{(k_{0},k}1)\}$

with an indeteyminate .
$z$

97

on $L\cross\{1, \ldots, r\}$with respect to a well-order
be a Gr\"obner basis of(2) Letwhenever $\mu<\mu’$ ; here
for eliminating , i.e., satisfying$(\mu, \nu, \alpha, \beta, i)\in L\cross\{1, \ldots, r\}$ corresponds to the monomial.
with$N_{1}$$\mathrm{G}_{1}$$\prec_{z}$$(\mu, \nu, \alpha, \beta, i)\prec_{z}(\mu\nu/,/,’\alpha,\beta’,j)$$z$$z^{\mu_{S}\nu_{X^{\alpha}\partial^{\beta}e_{i}}}$(3) Each element $P$ of$\mathrm{G}_{1}\cap A^{()}k\mathrm{o},k1$$s=t\partial_{t}$can be written uniquely in the form
$P= \sum_{i=1k}’\sum Qik(t\partial t)\tau ke_{i}=k_{0}k_{1}$with$Q_{ik}(ta)\in A_{n}[t\partial_{t}]$. Then we define $\chi^{-1}(P)\in A^{(1}k_{0}+1,k_{1}+)$ by$\chi^{-1}(P):=\sum_{i=1k}^{t}\sum_{=k\mathrm{o}}Q_{ik(\partial}k1tt+1)sk+1ei$.

Put$\mathrm{G}_{2}:=\mathrm{t}x^{-1}(P)|P\in \mathrm{G}_{1}\cap A^{(k}k0,1)\}$Then$\mathrm{G}_{2}$(4) Suppose

generates the left$\mathrm{G}_{2}=\{P_{1}, \ldots, P_{d}\}$$D_{X}$[ta]-module$\chi^{-1}(\mathcal{G}^{(k,k_{1}}0))$.

.

and $\mathrm{G}^{(+k_{1}+)}k_{0}1,1=\{P_{d+1,\ldots,\ell}P\}$ and put$S:= \{(Q_{1}, \ldots, Ql)\in An[t\mathrm{a}]^{f}|\sum_{=j1}^{\ell}QjPj=0\}$.

:Compute a set of generators G3 of by means of a Gr\"obner basis. Let
be the projection to the first components. Then we have an isomorphism$S$$\pi_{d}$$A_{n}[t\mathrm{a}]^{\ell}arrow$$d$$A_{n}[t\partial_{t}]^{d}$$\chi^{-1}(\mathcal{G}^{()}k_{\mathrm{O}},k_{1})/\mathcal{G}^{(k}0+1,k_{1}+1)\simeq D\mathrm{x}[t\mathrm{a}]^{d}/(DX[t\mathrm{a}]\otimes An[t\partial_{t}1\pi d(s))$

of left$D_{X}$[ta]-modules and$D_{X}[t\mathrm{a}]\otimes_{A_{n}1^{t}\partial 1}\pi_{d()}tS$$\pi_{d}(\mathrm{G}_{3})$.

be a well-order onfor eliminating 8, where’withand $e_{1}’=(1,0, \ldots, 0),$corresponds to. At this stage,. Let G4 be a Gr\"obner basis ofwith respect to
if and only if there exists $P\in$ G4 whose leading exponent withis $(0, i)\in L_{0}\cross\{1, \ldots, d\}$ for each $i=1,$.

(5) Put $L_{0}:=\mathrm{N}^{1+2n}$ and let$L_{0}\cross\{1, \ldots , d\}$$\prec_{s}$$(\mu, \alpha, \beta, i)\in L_{0}\cross\{1, \ldots, d\}$$s=t\mathrm{a}$$s^{\mu}X^{\alpha}\partial\beta e_{i}$$(0, \ldots, 0,1)\in \mathrm{Z}^{d}$we have
respect

is generated by$\ldots,$$\pi_{d}(S)$$\prec_{s}$$\mathcal{H}^{-1}(\mathcal{M}_{X})=0$$\mathrm{t}\mathrm{o}\prec_{s}$$d$$\ldots,$(6) For an element $P$ of $(A_{n}[s])d$ of the form
$P= \sum_{i=1}^{d}\sum_{\mu,\alpha,\beta}a\alpha\beta is\mu\mu_{X^{\alpha_{\partial e_{i}}}}\beta/$,

we put{$\deg(P, S)$$:=$$\max$

lcoef $(P, s)$$:=$$\sum_{i=1}^{d}\sum a_{m\alpha\beta ii}X^{\alpha}\partial^{\beta}e’\alpha,\beta$

with $m:=\deg(P_{\mathit{8}},)$ .

$\prec_{D}’$for some$\alpha,$$\beta\in \mathrm{N}^{n},$$i\in\{1,$ $\ldots,$ $d\}$},$\in(A_{n})^{d}$for eliminating , where
be a well-order on
be a Gr\"obner, let. For eachcorresponds togenerated byof the left submodule of$\mathrm{L}\mathrm{e}\mathrm{t}\prec_{D}’$$(\alpha, \beta, i)\in \mathrm{N}^{2n}\cross\{1, \ldots, d\}$basis with respect to$\mu\in \mathrm{N}|a_{\mu\alpha\beta i}\neq 0$$\mathrm{N}^{2n}\cross\{1, \ldots, d\}$$m\in \mathrm{N}$$x^{\alpha}\partial^{\theta}e_{i}’$$(A_{n})^{d}$.

$\{\mathrm{l}\mathrm{c}\mathrm{o}\mathrm{e}\mathrm{f}(P, s)|P\in \mathrm{G}_{4}, \deg(P, S)\leq m\}$$\partial$$\mathrm{H}_{m}$$e_{d}’=$

98(7) Put$\mathrm{H}_{m0:=}\mathrm{H}_{m}\cap(K[X])^{d}$and{$p\in X|$ rank $[h(p)|h\in \mathrm{H}_{m0}]=d$ },$U_{m}:=$

where $[h(p)|h\in \mathrm{H}_{m0}]$ denotes the matrix consisting of the row vectors $h(p)$ with.
Then we have
C... and $U_{m}=X$ for some $m\in$ N. On, we have an
isomorphism$h\in \mathrm{H}_{m0}$$U_{0}\subset U_{1}\subset U_{2}$$U_{m}$$(D_{\mathrm{x}}[t\mathrm{a}])^{d}/(DX[t\mathrm{a}]\otimes_{A_{n}}[t\partial_{t}]\pi d(s))\simeq(D_{X}[t\partial t](m))^{d}/Nm$of left$D_{X}$-modules, where$(D[t \partial_{t}]^{(m}))^{d}:=\bigoplus_{i=1}^{m}d\mu=\oplus^{-}D_{X}01(t\mathrm{a})\mu/e_{i}$and$N_{m}$

is the left

$D_{X}$-submodule of$(D[t\mathrm{a}]^{(}m))^{d}$,generated by $\{P\in \mathrm{G}_{4}|\deg(P_{S},)\leq$$m-1\}$ .

6

Algebraic local cohomology groups

In this section, let $X$ be a Zariski open set ofand put $\overline{X}:=K\cross X$ . We identify $X$ withthe subset
of $K^{n+1}$ as in the preceding sections. In the sequel we consider a-moduleinstead of a-module. Let $N$ be a left-submodule ofand put $M:=(A_{n})^{r}/N$
and $\mathcal{M}:=D_{X}\otimes_{A_{n}}M$ . Then we have $\mathcal{M}=(D_{X})^{r}/N$ with $N:=D_{X}N$ .
Let $f=f(x)\in K[x]$ be a non-constant polynomial and put $Y:=\{x\in X|f(x)=0\}$ .
Then the algebraic local cohomology group
has a structure of left
-module andvanishes for $j\neq 0,1([8])$ . Our purpose is to give an algorithm of computing
as a left$F$-module. In general, for an-module , put$K^{n}$$\{\mathrm{O}\}\cross X$$D_{X}$$\mathcal{M}$$A_{n}$$D_{\overline{X}}$$(A_{n})^{r}$$D_{X}$$\mathcal{H}_{[Y]}^{j}(\mathcal{M})$$\mathcal{H}_{[Y]}^{j}(\mathcal{M})$$D_{X}$$O_{X}$$\Gamma_{[Y]}(\mathcal{F}):=${$u\in \mathcal{F}|f^{k}u=0$

for some$k\in \mathrm{N}$

}.

Then
is defined as the j-th derived functor of.
Put $Z:=\{(t, x)\in K\cross X|t-f(x)=0\}$ . Let
be a left ideal ofgenerated by $t-f(x)$ ,, and put
. We denote by $\delta(t-f)$ theresidue class ofin.
Put $\mathcal{L}:=O_{X}[f-1, s]f^{s}$ , whereis regarded as a free generator. Then
has a natural
structure of left $D_{X}[s]$ -module. As was observed by Malgrange [14],has a structure of left
-module so that$\mathcal{H}_{[Y]}^{j}(\mathcal{F})$$\Gamma_{[Y]}$$J_{Z}$$\partial_{1}+(\partial f/\partial X_{1})\partial t,$$\partial_{n}+(\partial f/\partial x_{n})\mathrm{a}$$\mathcal{B}_{[Z]}:=D_{\overline{X}}/J_{Z}$$\ldots,$$1\in D-\mathrm{x}$$D_{X}^{-}$$B_{[Z}1$$f^{s}$$\mathcal{L}$$\mathcal{L}$$D_{\overline{X}}$$t(g(s)fs)=g(s+1)f^{s+1}$ ,

a

(6.1)

$(g(s)f^{s})=-sg(S-1)f^{s-}1$

for $g(s)\in O_{X}[f-1, s]$ . This implies that there exists an injective homomorphismof left-modules such that $\iota(\delta(t-f))=f^{s}([14])$ .

$\iota$:$B_{[Z]}|_{X}arrow \mathcal{L}$$D_{\overline{X}}$

Lemma 6.1 We have an isomorphismin the derived category of left
-modules, where
denotes the right derived functor ofand [1] the translation$(B_{[]}Z)X\simeq \mathrm{R}\mathrm{r}_{[Y]}(o_{\mathrm{x})}[1]$$D_{X}$

functor ([6]).

$\mathrm{R}\Gamma_{[Y]}$$\Gamma_{[Y]f}$

99

has a
be the projection. Then the tensor productNow let :
and
to
and
be the projections of-modules. Let
structure of sheaves of left
$t\in
K$
$y\in
X$
$\pi_{2}(t,
x,
y)=y$
$X$
$\pi_{1}(t,x,y)=(t,
x)$
$x,$
.
and
Put
for
and
respectively defined byto$\overline{X}arrow X$$\pi$$\mathcal{B}_{[Z]}\otimes_{\pi O\mathrm{x}}-1\pi^{-1}\mathcal{M}$$\overline{X}\cross X$$\overline{X}$$\pi_{2}$$\pi_{1}$$D_{X}^{-}$$\Delta:=\{(t, x, y)\in\overline{X}\cross X|x=y\}$and$D_{\trianglearrow\overline{X}\mathrm{x}x}:=D_{\overline{X}\cross X^{/}}((X_{1}-y1)D+\ldots+\overline{x}_{\mathrm{X}}X(Xn-y_{n})Dx\mathrm{x}x-)$Lemma 6.2 Let

be a left$\mathcal{F}$$D_{\overline{X}}$.

-module. Then we have$\mathcal{F}\otimes_{\pi^{-1}}\mathrm{L}\mathcal{O}\mathrm{x}\pi^{-1}\mathcal{M}\simeq D_{\trianglearrow\overline{\mathrm{x}}\mathrm{x}x^{\otimes_{D-}}}\mathrm{L}X\cross X(\mathcal{F}\otimes \mathcal{M})\wedge$

with

.

$\mathcal{F}\otimes \mathcal{M}\wedge.--D_{\overline{X}\cross X^{\otimes 1}}\pi 1^{-}1D-\otimes\pi xD2-X(\pi_{1^{-11}}\mathcal{F}\otimes_{K}\pi_{2^{-}}\mathcal{M})$

Lemma 6.3 The i-th torsion group$\mathcal{T}or^{\pi \mathit{0}_{X}}i-1(g[Z]’\pi-1\mathcal{M})$vanishes for $i\neq 0$ .

Theorem 6.4 We have isomorphism8$\mathcal{H}^{j}((B1^{Z}]\otimes\pi^{-}1\mathrm{o}X).x\pi-1\mathcal{M})\simeq \mathcal{H}_{[Y}^{j+1}1(\mathcal{M})$of left$D_{X}$-modules for $j=-1,0$ .

-module . In viewfor aby
In what follows, we shall denoteof Theorems 5.6, 5.8, 5.10 and 6.3, we obtain an algorithm for computing the algebraic local
asfor $j=0,1$ if there is an algorithm for computing
cohomology groups
follows:-module. In fact, this tensor product can be computed asa left$\mathcal{F}\otimes_{\pi^{-1}}0_{X}\pi^{-}\mathcal{M}1$$\mathcal{F}$$\mathcal{F}\otimes 0_{X}\mathcal{M}$$D_{\overline{X}}$$\mathcal{H}_{[Y]}^{j}(\mathcal{M})$$B_{[z]}\otimes_{\mathcal{O}\mathrm{x}}\mathcal{M}$$D_{\overline{X}}$

Lemma 6.5 Let$J_{Z}$

be as above. Then we have an isomorphism$\mathcal{B}_{[Z]}\otimes \mathcal{M}\wedge\simeq(D_{\overline{X}\cross X})^{r}/N_{Z}$

with

.

$N_{z:=J_{Z^{\wedge}}}\otimes(D\mathrm{x})r+D_{\overline{X}^{\otimes}}N\wedge$

For $i=1,$$\ldots,$

$n$

, put
$\Delta_{i}:=${$(t,$ $x,$ $y)\in\overline{X}\cross X|x_{j}=y_{j}$for $j=1,$

, }.
$i$

$\ldots$

Then we have$B_{[z]}\otimes \mathcal{O}\mathrm{x}\mathcal{M}\simeq(\ldots((g1z1\otimes\Lambda\not\in)\triangle 1)\Delta 2\wedge\ldots)_{\triangle_{n}}$by virtue of Lemma 6.2. Since
Lemma 6.3, we can compute$\Delta_{i}$in view of the proof ofis non-characteristic forby applying Theorem 5.7 repeatedly with $k_{0}=k_{1}=0$ .
$e_{[Z}1^{\wedge}\otimes \mathcal{M}$$\mathcal{B}_{[Z:}\otimes 0_{X}\mathcal{M}$

Lemma 6.6 If$\mathcal{M}$is $holonomic_{f}$ then$B_{1^{Z}]^{\otimes_{\mathcal{O}}\mathcal{M}}}X$

is specializable along $X$ .

Thus we have obtained an algorithm for computing
under the condition that
5.7 and Algorithm 5.10 to
X. In particular, we have proved the following statement effectively:$\mathcal{H}_{[Y]}^{j}(\mathcal{M})(j=0,1)$$\mathcal{B}_{[Z]}\otimes 0_{x}\mathcal{M}$$\mathcal{B}_{[z1^{\otimes 0_{x}}}\mathcal{M}$by applying Theoremis specializable along100

Corollary 6.7 If
lefl -modules.

is specializable along$e_{[Z]}\otimes_{\mathcal{O}x}\mathcal{M}$$X_{r}$

then$\mathcal{H}_{[Z]}^{j}(\mathcal{M})(j=0,1)$

are coherent$D_{X}$Let us describe

more concretely. First note thatwith
By applying Theorem 5.7 to, we know thatis generated by the modulo classes
inwithand $1\leq i\leq r$ , and the relations among the generators
are given byof Theorem 5.7.
Actually,
with $1\leq i\leq r$ generate
and the relations among these generators canbe obtained by eliminatingwith $k<k_{1}$ .
Our next aim is to give an algorithm of computing the bfunction for a polynomial and asection of M. Put. Then we have$\mathcal{H}_{[Y]}^{1}(\mathcal{M})$$\mathcal{H}_{[Y]}^{1}(\mathcal{M})\simeq \mathcal{M}[f^{-1}]/\mathcal{M}$$\mathcal{M}[f^{-1}]:=Ox[f-1]\otimes_{\mathcal{O}}\mathcal{M}X^{\cdot}$$\mathcal{M}[f^{-1}]/\mathcal{M}$$e_{[Z1^{\otimes_{\mathcal{O}x}\mathcal{M}}}$$v_{ik}:=[f^{-k}\otimes u_{i}]$$k_{0}\leq k\leq k_{1}$$(O[f^{-1}]\otimes_{\mathcal{O}_{X}}\mathcal{M})/\mathcal{M}$$k!v_{ik}$$N_{X}$$\mathcal{M}[f^{-1}]/A4$$v_{ik_{1}}$$v_{ik}$$f$$u$$\mathcal{M}[s]:=K[s]\otimes_{K}\lambda 4$$\mathcal{L}\otimes 0_{X}[s\mathrm{J}\mathcal{M}[S]=\mathcal{L}\otimes \mathrm{o}_{X}1^{s}](O_{X}[s]\otimes 0_{X}\mathcal{M})=\mathcal{L}\otimes_{O_{X}}\mathcal{M}$Note that an arbitrary element of
some
and
.
$m\in \mathrm{N}$$\mathcal{L}\otimes_{\mathcal{O}_{X}}[s]\mathcal{M}[s]$.

can be expressed in the form$f^{s-m}\otimes u$

with$u\in \mathcal{M}[s]$Lemma 6.8 Let$f^{s-m}\otimes u=0$ in$u$be a section ofand let $m$ be a nonnegative integer. Then we haveurith some.
if and only if $f^{k}u=0$ holds in$\mathcal{M}[s]$$\mathcal{L}\otimes \mathrm{o}_{x[}s1\mathcal{M}[s]$$k\in \mathrm{N}$$\mathcal{M}[s]$be a section of
and $P$ a section of $D_{X}[s]$ . Then the identity $P(f^{s}u)=0$ means
by definition that there exists
so that $Q:=f^{m-s}Pf^{s}$ is contained in $D_{X}[s]$ and that
$Qu=0$ holds in(cf. [8]).
Let

$u$

$\mathcal{M}$$m\in \mathrm{N}$$\mathcal{M}[s]$Lemma 6.9 For$\mathcal{L}\otimes 0_{X}\mathcal{M}$and $P\in D_{X}[s]$ , we have $P(f^{s}u)=0$ if and only if $P(f^{s}\otimes u)=0$ in$u\in \mathcal{M}$.

Lemma 6.10$\mathcal{H}_{[Y]}^{0}(\mathcal{M})=0$

Lemma 6.11 Letin the form$p$

if and only if$f$:$B_{1^{Z}1x^{\mathcal{M}}}\otimes_{\mathcal{O}}arrow\beta_{[z]}\otimes \mathrm{o}_{x^{\Lambda}}t$

be a point of Y. Then any germ

$v$

of

$B_{[Z]\mathcal{O}_{X}}\otimes \mathcal{M}$atis injective.
$pi\mathit{8}$uniquely written(6.2)$v= \sum_{i=0}^{k}\mathrm{a}i\delta(t-f)\otimes u_{i}$urith$u_{i}\in \mathcal{M}_{p}$and$k\in \mathrm{N}$.

Proposition 6.12 The homomorphism$\iota\otimes 1$

is injective if and only if:$\mathcal{H}_{1^{Y}]}^{0}(\mathcal{M})=0$$e[z]\otimes_{\mathcal{O}}\mathrm{x}\Lambda \mathrm{t}arrow \mathcal{L}\otimes 0_{X}\lambda 4$.

Theorem 6.13 $A_{\mathit{8}Su}mer=1$ and let
be the residue class of $1\in D_{X}$ . Let $b_{X}(s)$ be the$X$ vrith respect toalong
the filtration-function ofandlet $b(s)$ be the -function for and defined by (1.1), both at a point of Y. Then we have thefollowing:$u\in \mathcal{M}$$b$$B_{[Z1}\otimes \mathit{0}_{X}\mathcal{M}$$b$$\{F_{k}(D_{\overline{x}^{)(\delta}}(t-f)\otimes u)\}_{kE\mathrm{Z}}$$f$(1) $b(s)$ divides $b_{X}(-s-1)$ ;

$u$

$p$

101

(2) if$\mathcal{H}_{1^{Y}]}^{0}(\mathcal{M})\mathrm{P}=0$, then we have $b(s)=\pm b_{X}(-s-1)$ ;A nonzero -function $b(s)$ for
specializable along $X$ at .

$(S)$

$b$

$f$

and

$u$

exists at $p\in X$ if and only if$e_{1^{Z}]^{\otimes_{\mathrm{o}x}\mathcal{M}}}$

is

$p$

under
Thus we have obtained an algorithm for computing the bfunction for and, which can be determined by Algorithm 5.10. Note that we dothe assumption
not need this assumption for deciding whether a nonzero b–function exists. This generalizes analgorithm of computing the Bernstein-Sato polynomial given in [19].
$u\in \mathcal{M}$$f$$\mathcal{H}_{[Y}^{0}(]D_{X}u)=0$.
in) and be the residue class of
Example 6.14 Put. On
at is 1 since $fu=0$ in
Let be a point of $Y$ . Then the $k$ function for $f$ and$X$$b_{X}(\mathit{8})=s+1$
since
fact,
. In
at is
along
the other hand, the Ofunction of$t(\delta(t-f)\otimes u)=\delta(t-f)\otimes(fu)=0$ , we know that $b_{X}(s)$ divides $s+1$ . If $b_{X}(s)=1$ , then we$\mathcal{M}=O_{X}[f-1]/\mathit{0}_{x}$$f^{-1}$$u$$\mathcal{M}:=\mathcal{H}_{[Y1^{(\mathrm{x}}}^{1}O$$\mathcal{M}$

$u$

$p$

$p$

$p$

$g_{[z1^{\otimes_{\mathcal{O}_{X}}\mathcal{M}}}$should have$\mathcal{M}=\mathcal{H}_{[1^{(}1^{Z1x}}^{0_{Y}1}\mathcal{M})\simeq \mathcal{H}-((\mathcal{B}\otimes \mathit{0}_{\mathrm{X}}\mathcal{M}).

)=0$by virtue of Proposition 5.2 and Theorem 6.4, which is a contradiction.

for algebraic set $Y$ of codimension
It is also possible (in generic cases) to compute
greater than one. For example, let $f_{1}(x),$ $f_{2(X})$ be two polynomials and put$\mathcal{H}_{[Y]}^{j}(\mathcal{M})$

Assume that$\mathcal{H}_{1^{Y_{1}}}^{j}$]$(\mathcal{M})=0$$Y_{i}$$:=$$\{x\in X|f_{i}(X)=0\}$

$Y$

$:=$

$Y_{1}\cap Y_{2}$for$j\neq j_{0}$$(i=1,2)$ ,.

. Then we can compute$\mathcal{H}_{1^{Y}1^{(\mathcal{M}}[]1Y_{1}1^{(}}^{jj})=\mathcal{H}\mathrm{o}(Y^{-}2\mathcal{H}^{j_{0}}j\mathcal{M}))$

explicitly by applying the above method first to$f_{1}$

and$\mathcal{M}$, then to$f_{2}$

and$\mathcal{H}_{1^{Y_{1}}}^{j\mathrm{o}_{1}}(\mathcal{M})$.

Example 6.15 Put $X=K^{3},$ $f_{1}:=x^{2}-y^{3},$ $f_{2}:=y^{2}-z^{3}$ , and consider the space curvefor $j\neq 2$ and$Y.–\{(x, y, z)\in X|f_{1}(X, y)z)=f2(x,y, z)=0\}$ . Then we have$\mathcal{H}_{[Y1}^{j}(O\mathrm{x})=0$,$\mathcal{H}_{[Y]}^{2}(Ox)\simeq D\mathrm{x}/\mathcal{I}$where$\mathcal{I}$is the left ideal of$D_{X}$

generated by$9x\partial_{x}+6y\partial+y+4_{Z\partial_{z}}30$

inbe the residue class of$f_{j}(x,y, z)=0\}$ . Then the bfunction forLet$f_{j}^{-1}$$u_{j}$$f_{1},$,$f_{2}$

and$9y^{2_{Z^{2}}}\partial_{x}+6Xz^{2}\partial y+4xy\partial z$$\mathcal{H}_{[Y_{j}}^{1}(1xO)=\mathcal{O}x[f_{j}^{-}1]/O_{X}$$f_{2}$

and$u_{1}$.

with $Y_{j}:=\{(x, y, z)|$is$(s+1)(s+ \frac{1}{12})(s+\frac{5}{12})(s+\frac{7}{12})(s+\frac{5}{6})(s+\frac{11}{12})(s+\frac{7}{6})$at$(0,0, \mathrm{o})$, and $s+1$ on $Y\backslash \{(0,0,0)\}$ . The kfunction for$f_{1}$

and$u_{2}$

is$(s+1)(s+ \frac{7}{18})(s+\frac{11}{18})(s+\frac{13}{18})(\mathit{8}+\frac{5}{6})(s+\frac{17}{18})(s+\frac{19}{18})(s+\frac{7}{6})(s+\frac{23}{18})$

at$(0,0, \mathrm{o})$, and $s+1$ on $Y\backslash \{(0,0,0)\}$ .

102

7

Localization of a D-module

We retain the notation of the preceding section. Our primary goal in this section is to obtainan algorithm for computing the localization
as a left-moduleunder the assumption. For this purpose, we shall first compute$\mathcal{M}[f^{-1}]:=O_{X}[f-1]\otimes_{\mathcal{O}_{X}}\Lambda 4$$D_{X}$$\mathcal{H}_{[Y]}^{0}(\mathcal{M})=0$$P:=Dx[s](fs\otimes u1)+\ldots+DX[S](f^{s}\otimes u_{r})$which is a left $D_{X}[s]$ -submodule of$\mathcal{L}\otimes 0_{X}\mathcal{M}$

Proposition 7.1 Assume
erators on $X$ of the left $D_{X}[s]$ -module$\mathcal{H}_{1^{Y}]}^{0}(\mathcal{M})=0$,, and then specialize the parameter .
$s$. Then there is an algorithm to compute a set of gen-

$Q:= \{(Q1, \ldots, Q_{r})\in(D_{X}[s])r|\sum^{r}Qi(S)(f^{s}\otimes ui)i=1=0\}$.

Now let us fix an arbitrary element
of $K$ and consider the specialization $s=s_{0}$ of
the parameter . Put $\mathcal{L}(s_{0}):=O_{X}[f-1]fs_{0}$ , whereis regarded as a free generator. Let
be the surjective homomorphism of left
-modules defined by $p(g(\mathit{8}, X)f^{s-}m)=$$g(s_{0}, x)f^{s_{0}-m}$ for $g(s, x)\in Ox[s, f-1]$ and $m\in$ N.
Then it is easy to see that induces anisomorphism
as left-modules.
Since the proof of Lemma 6.8 is also valid with specialized to an element of $K$ , we get thefollowing:$s_{0}$$s$$f^{s_{0}}$$p:\mathcal{L}arrow \mathcal{L}(s\mathrm{o})$$D_{X}$$p$$D_{X}$$\mathcal{L}(s\mathrm{o})\simeq \mathcal{L}/(s-s_{0})\mathcal{L}$$s$Lemma 7.2 Let be a section of
we have $f^{s_{0}-m}\otimes u=0$ in$u$$\mathcal{M}$and let $m$ be a nonnegative integer. Fix $s_{0}\in K.$ Thenwith.
if and only if $f^{k}u=0$ holds in$\mathcal{L}(s\mathrm{o})\otimes 0_{X}\mathcal{M}$$\mathcal{M}$$\mathit{8}omek\in \mathrm{N}$Consider the homomorphism$\rho\otimes 1$:$\mathcal{L}\otimes_{\mathcal{O}_{X}1^{s]}}\mathcal{M}[s]=\mathcal{L}\otimes_{\mathcal{O}_{X}}\mathcal{M}arrow \mathcal{L}(s_{0})\otimes 0_{X}\mathcal{M}$and put $P(s\mathrm{o}).--(\rho\otimes 1)(P)$ . Our aim is to obtain an algorithm of computing. Sinceis contained in the kernel of, there exists a surjective homomorphism $P/(s-$induced by. A sufficient condition for this homomorphism to be an
isomorphism is given as follows (cf. Proposition 6.2 of [7] for the case).
$P(s\mathrm{o})$$(s-s_{0})\mathcal{P}$$p\otimes 1$$s_{0})\mathcal{P}arrow P(s\mathrm{o})$$\rho\otimes 1$$\mathcal{M}=O_{X}$Proposition 7.3 Assume that the -function $b_{i}(s,p)$ for
at $p\in Xexi\mathit{8}ts$ for $i=$and$b_{i}(s_{0}-\nu)\neq
0$, . $AsSumeJ$ moreoverJ that$\nu=1,2,3,$, and $p\in Y$ .
for any $i=1,$$P/(s-s\mathrm{o})ParrowP(s\mathrm{o})$$isomorphi_{\mathit{8}}m$
Then the homomorphism
is a left-module. In $partiCular_{J}$$Q(s\mathrm{o}):=\{Q(s_{0})|Q(s)\inQ\}$
we have an isomorphism
with
.
$b$

$1,$

$\ldots$$f$$u_{i}$$r$$\ldots,$$r,$$\ldots$$D_{X}$$P(s\mathrm{o})\simeq(D_{X})^{r}/Q(s\mathrm{o})$Thus we have obtained an algorithm for computing
proposition. Note that it amounts to computing$P(s\mathrm{o})$$\mathcal{L}(s\mathrm{o})\otimes_{\mathcal{O}_{X}}\mathcal{M}$under the conditions of the above
as follows.

Proposition 7.4 Under the same assumptions as in the preceding $prop_{oS}ition$ , we have$\mathcal{L}(s_{0})\otimes \mathit{0}_{X}\mathcal{M}$.

$P(s\mathrm{o})=$103

Proposition 7.5
so thatinteger$A\mathit{8}\mathit{8}ume$$k_{0}$$k\geq k_{0}$$\mathcal{M}[f^{-1}]$

that
is

is specializable along X. Then there exists a-module for any integer
to $(D_{X})^{r}/Q(-k)$ as left
$po\mathit{8}itive$$\mathcal{B}_{[Z1\mathrm{x}}\otimes_{\mathcal{O}}\mathcal{M}$$i\mathit{8}omor_{\mathrm{P}}hic$.

$D_{X}$, weis specializable along $X$ and that
Thus under the condition that
More
and
7.5.
Propositions
7.1
combining
have obtained an algorithm of computing
wehave
concretely,$\mathcal{H}_{[Y1}^{0}(\mathcal{M})=0$$B_{[Z]}\otimes \mathcal{M}$$\mathcal{M}[f^{-1}]$$\mathcal{M}[f^{-1}]=\sum_{i=1}DX(f^{-}k\mathrm{o}\otimes u_{i})r$

and our algorithm computes a finite subset of$(A_{n})^{r}$,which generates the left

$D_{X}$-module$Q(-k_{0})= \{P\in D_{X}|\sum_{i=1}P_{i}(rf-k\mathrm{o}\otimes u_{i})=0\}$

with another polynomial
on $X$ . In particular, by applying the above argument tofor generic$g\in K[s]$ and a constant $s_{2}\in K$ , we obtain an algorithm for computingof$s_{2}\in K$ as follows: First, we can compute
if the Bemstein-Sato polynomial$b_{g}(s_{2}-\nu)\neq
0$(cf. [.21]). Then we havefor $\nu=1,2,3,$
satisfies$\mathcal{M}.--D_{X}g^{s_{2}}$$D_{X}(f^{S_{1}}fs_{2})$$b_{g}(s)$$Dxg^{s_{2}}$$s_{1},$$g$$\ldots$$(D_{X}f^{s_{1}})\otimes_{\mathcal{O}\mathrm{x}}(D_{X}g^{s_{2}})\simeq D_{X}(f^{\mathit{8}s_{2}}1g)$-submodule of $O_{X}[f^{-11},g^{-}]f^{s}1g^{s}2$is the leftby virtue of Lemma 7.2, where. Thus by applying the arguments in this section, we can computegenerated bysatisfies
if, in addition to the above condition, the bfunction $b_{12}(s)$ for $f$ and.
for $\nu=1,2,3,$ . Note that we always have$s_{1}=-k_{1}$and $s_{2}=-k_{2}$ satisfy the above
so thatHence by choosing positive integers
conditions, we get an algorithm to compute the localization $O_{X}[f^{-1},g^{-1}]=Ox[f^{-k_{1}},g^{-k}]2$ as-module.
as inderterminates not as constants, then it is also interesting to consider
If we regard-module $D_{X}[s_{1,2}S]fs1gs_{2}$ . An algorithm for computing this module can be
the leftobtained by generalizing a method used in [21], or also by modifying the arguments in this-module. We shall discuss thisis asection so as to be adapted to the case whereproblem elsewhere.
$D_{X}$$D_{X}(f^{s_{1}\mathit{8}2}g)$$D_{X}(f^{s_{1}}g)s_{2}$$f^{s_{1}}gs_{2}$$b_{12}(s0-\mathcal{U})\neq 0$$g^{s_{2}}$$\mathcal{H}_{[Y]}^{0}(D_{X}g^{s_{2}})=0$$\ldots$$k_{1},$ $k_{2}$$D_{X}$$s_{1},$$s_{2}$$D_{X}[S_{1}, S_{2}]$$\Lambda 4$$D_{X}[\mathit{8}_{2}]$. Put
Example 7.6 Put $X=K^{3}\ni(x, y, z)$ and write$f_{1}:=x^{2}-y^{3}$ and $f_{2}:=y^{2}-z^{3}$ . Let$s_{2}\in K$ be constants. The Bernstein-Sato polynomial$b_{\mathit{2}}(s)=(s+1)(s+\frac{5}{6})(s+\frac{7}{6})$ . We haveisat the singular pointofgenerated bywith the left ideal of$\partial_{x}:=\partial/\partial x,$ $\partial_{y}:=\partial/\partial y,$$\partial_{z}:=\partial/\partial z$$s_{1},$$D_{X}f^{s_{2}}=D_{X}/\mathcal{I}$$(0,0, \mathrm{o})$$f_{2}$$D_{X}$$\partial_{x}$,if $b_{2}(s_{2}-\nu)\neq 0$ for any $\nu=1,2,3,$$b_{12}(s)$,$3y\partial_{yz2}+2z\partial-6S$$=$$\ldots$$3z^{2}\partial_{y}+2y\partial_{z}$,$(y^{2}-z^{3})\partial_{z}+3z^{2}\mathit{8}2$. Then the bfunction for $fi$ and$f_{2}^{s_{2}}$is$s+ \frac{7}{6})(s+\frac{2}{3}s_{2}+\frac{19}{18})(\mathit{8}+\frac{2}{3}S_{2}+\frac{23}{18})$$(s+1)(s+(s+ \frac{2}{3}S_{2}+\frac{2\frac{5}{56}}{18})\{$$S+ \frac{2}{3}S_{2}+\frac{29}{18})(\mathit{8}+\frac{2}{3}s_{2}+\frac{31}{18})(s+\frac{2}{3}s_{2}+\frac{35}{18})$

104

at $(0,0,0)$ ; while at the other points we have

on $\{(0,0, z)|z\neq 0\}$ ,on $\{(x, y, z)|x^{23}-y=0, yz\neq 0\}$ ,on $\{(x, y, z)|x^{2}-y^{3}\neq 0\}$ .

$(s+1)(s+ \frac{5}{6})(s+\frac{7}{6})$$b_{12}(s)=\{$

$s+1$

1

If
satisfies $b_{12}(s_{1}-\nu)\neq 0$ for any $\nu=1,2,3,$
Under the same assumptions, we haveofgenerated by$s_{1}$$\ldots$

in addition to the above condition onwith the left ideal$D_{X}(f_{1}^{s}1f_{2}s2)=D_{X}/\mathcal{I}(s_{1}, s_{2})$$s_{2}.$.

$\mathcal{I}(s_{1}, s_{2})$$D_{X}$$9x\partial_{x}+6y\partial_{y}+4Z\partial z-6(3s_{1}+2s2)$$(y^{2}-Z^{3})\partial_{z}+3_{\mathcal{Z}}2s_{2}$$(x^{2}-y^{3})\partial x-2S1x$

,

,

,

$\{$

$9y^{2_{Z}2}\partial_{x}+6XZ^{2}\partial_{y}+4Xy\partial z$’$3y(_{X^{2}}-y^{3})\partial y+2_{Z}(_{X^{2}}-y)3\partial z+3(-2_{S}2^{X+}2(3s_{1}+2s2)y^{3})$$3z^{2}(x^{2}-y^{3})\partial+2yy(_{X}2-y^{3})\partial z+9s1y^{2}Z^{2}$,.

In particular the above assumptions are satisfied for $s_{1}=s_{2}=-1$ . Hence we have $O_{X}[f1-1, f^{-}21]\simeq$$D_{X}/\mathcal{I}(-1, -1)$ . By regarding
as indeterminates not as constants, we have also
. Then we can verify by elimination that the ideal $(\mathcal{I}(s_{1}, S2)+D_{X}[s1, s2]f1f_{2})0\cap$$K[s_{1}, s_{2}]$ of $K[s_{1}, S2]$ is generated by asingle element$D_{X}[s_{1}, S_{2}](f_{1}s1f^{s_{2}}2)=$$s_{1},$ $s_{2}$$D_{X}[S_{1}, s_{2}]/\mathcal{I}(s_{1}, s_{2})$$b(s_{1}, S_{2})$$:=$$(s_{1}+1)(6s_{1}+5)(6s_{1}+7)(s_{2}+1)(6s_{2}+5)(6S_{2}+7)(P+19)(P+23)$$(P+25)(P+29)(\ell+31)(P+35)(P+37)(\ell+41)(P+43)(\ell+47)$

with $p:=18s_{1}+12s_{2}$ . This means thatis a minimum polynomial that satisfies afunctional equation of the form $P(f_{1}^{s+1}1f^{s2+1}2)=b(s_{1}, s_{2})f^{s}11f_{2^{2}}^{S}$ with some germ $P$ ofat (cf. [22], [15]).
$b(s_{1}, s_{2})$$D_{X}[s_{1,\mathit{8}_{2}]}$

$0$

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[2] Briangon, J., Granger, M., Maisonobe, Ph., Miniconi, M., Algorithme de calcul duBernstein: cas non d\’eg\’en\’er\’e. Ann. Inst. Fourier 39 (1989), 553-610.

$\mathrm{P}^{\mathrm{o}1}\mathrm{y}\mathrm{n}\hat{\mathrm{O}}\mathrm{m}\mathrm{e}$de[3] Castro, F., Calculs effectifs pour les id\’eaux d’op\’erateurs diff\’erentiels. ’Ravaux en Cours, vol. 24, pp.
1-19, Hermann, Paris, 1987.
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[7] Kashiwara, M., $B$-functions and holonomic systems–Rationality of roots ofMath. 38 (1976), 33-53.

$f\succ \mathrm{f}\mathrm{u}\mathrm{n}\mathrm{C}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}$. Invent.

[8] Kashiwara, M., On the holonomic systems of linear differential equations, II. Invent. Math. 49(1978), 121-135.

[9] Kashiwara, M., Vanishing cycle sheaves and holonomic systems of differential equations. Lecture
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105[10] Kashiwara, M., Systems of Microdifferential Equations. Birkh\"auser, Boston, 1983.
[11] Kashiwara, M., Kawai, T., On the characteristic variety of a holonomic system with regular singularities. Advances in Math. 34 (1979), 163-184.
[12] Kashiwara, M., Kawai, T., Second microlocalization and asymptotic expansions. Lecture Notes inPhysics vol. 126, pp. 21-76, Springer, Berlin, 1980.
[13] Laurent, Y., Schapira, P., Images inverses des modules diff\’erentiels. Compositio Math. 61 (1987),

229-251.
[14] Malgrange, B., Le poly\^ome de Bernstein d’une singularit\’e isol\’ee. Lecture Notes in Math. vol. 459,

pp. 98-119, Springer, Berlin, 1975.
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