Multidimensional local residues and holonomic
D-modules
新潟大学
田島慎–
(Shinichi
Tajima)
横浜市大
大阿久俊則
(Toshinori Oaku)
お茶の水女子大
中村弥生
(Yayoi
Nakamura)
Multidimensional
$1\mathit{0}$cal residues are
fundamental
objects
in complex analysis and
geom-etry. However,
if the polar divisors of
a meromorphic
differential form are not in general
position, the
actual calculation of
local residues is difficult in
many
cases. In this
paper
we
study
Grothendieck
local residue from the viewpoint of
$D$
-modules. We mainly consider
the
case
where
the
polar
divisors
are
not
in
general position.
We
propose
a
new
$\mathrm{a}\mathrm{p}\mathrm{p}\dot{\mathrm{r}}\mathrm{o}\mathrm{a}\mathrm{C}\mathrm{h}$for
calculating
multidimensional
local
residues.
In the appendix we consider the
zero-dimensional
transversal complete
intersection
case.
We
present
a simple method
$\mathrm{f}\mathrm{o}\dot{\mathrm{r}}$computing
residues for this case.
We use a
computer
algebra
system
Kan for
Gr\"obner
basis
computation
in Weyl algebra
and
a computer
algebra
system
$\mathrm{R}\mathrm{i}\mathrm{s}\mathrm{a}/\mathrm{A}\mathrm{s}\mathrm{i}\mathrm{r}$for
Gr\"obner
basis
computation,
and
primary
decomposition in
polynomial
rings.
1. Algebraic local cohomologies
Let us
recall
some basic
facts about algebraic local cohomology and holonomic
D-modules.
Let
$X$
be a complex
manifold
$\mathcal{O}_{X}$the sheaf
on
$X$
of
holomorphic
functions.
Let
$\mathrm{Y}$a
subvariety in
$X$
.
Let
$J_{\mathrm{Y}}$be the
sheaf
of
ideal of
$\mathrm{Y}$in
$X$
.
The k-th algebraic
local
cohomology group
supported
in
$\mathrm{Y}$is
defined
as
the
inductive limit of extension
groups
$\mathcal{H}_{[\mathrm{Y}]}^{kk}(\mathcal{O}x)=\lim_{\mathit{1}arrow\infty}\mathcal{E}xt_{o_{X}}(\mathcal{O}x/J_{\mathrm{Y}}^{\mathit{1}}; \mathcal{O}\mathrm{x})$
.
Note
that for a hypersurface
case,
we have
$\mathcal{H}_{[\mathrm{Y}]}^{1}(\mathcal{O}\mathrm{x})\simeq \mathcal{O}_{X}[*\mathrm{Y}]/\mathcal{O}_{X}$, where
$\mathcal{O}_{X}[*\mathrm{Y}]$stands
for
$\mathrm{t}\mathrm{h}\overline{\mathrm{e}}$sheaf
of meromorphic
functions
on
$X$
with
poles
along Y.
Let
$D_{X}$
be the sheaf of
rings
on
$X$
of linear
partial
differential
operators
with
holomorphic
coefficients.
Then
$D_{X}$
is coherent
as a
sheaf
of
rings.
It
is
easy to
see that the algebraic
local
cohomology group
$\mathcal{H}_{[\mathrm{Y}]}^{k}(\mathcal{O}_{X})$is
naturally
endowed
with a structure of left
$D_{X}$
-module.
Theorem
$(\mathrm{K}\mathrm{a}\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{W}\mathrm{a}\mathrm{r}\mathrm{a}[12], \mathrm{M}\mathrm{e}\mathrm{b}\mathrm{k}\mathrm{h}\mathrm{o}\mathrm{u}\mathrm{t}[14])$(1)
$\mathcal{H}_{[\mathrm{Y}]}^{k}(\mathcal{O}x)$is
coherent over
$D_{X}$
.
(2)
$\mathcal{H}_{[Y]}^{k}(\mathcal{O}_{X})$is
a holonomic system.
We refer to
[11], [22]
for the notion of a holonomic
system.
Recently,
one of
the
authors
(T.
Oaku
([19], [20]))
constructed an algorithm
to calculating
Gr\"obner
basis of an
algebraic
local cohomology
group.
His
algorithm
has been
implemented
in the computer
algebra
system Kan ([24]), developed by N. Takayama of
Kobe
University.
The following
computation
was
carried out
by
using
Kan.
Example Let
$f(x,y)=(x^{2}+y^{2})^{3}-4x^{2}y^{2},$
$D=\{(x, y)|f(x, y)=0\}$
.
Put
$m=$
(
$\frac{1}{(x^{2}+y)^{3}2-4xy22}$
mod
$\mathcal{O}\mathrm{x}$)
$\in \mathcal{H}_{[D}1](\mathcal{O}_{X})$.
The generator
$m$
of the
module
$\mathcal{H}_{[D]}^{1}(\mathcal{O}_{X})$satisfies
the following
holonomic
system.
$|(-15y^{22}XD_{x}+3yD.+xy_{6}3x_{2}^{3}D15yyy,243+4(_{X^{3}}Dx_{3_{XD3}}-2y^{2_{X}}Dx_{7}+2yx_{1}^{2}D_{y}-y_{2}^{3}D+(-27yx+x_{2}^{4}Dyyx_{D_{y}-}-2215y^{4}D18yx_{2D_{yx}}^{2}(-972yD2^{+}16x_{965}^{4}D-yyx-222^{+0y_{X}0’}275^{+4}6X^{-}D_{y,8}^{-}y^{4}D_{y}^{2}-8748yx(-108^{-}y_{4}^{5}D_{x_{4^{-2}}}6^{+D2}0XD_{y,3}19(-x^{6}3yx-243yx_{X^{2}-3}yx^{2}y^{2}2D24+14x^{2}D_{x_{2^{-}}}129^{+1}6yx_{312+}^{2}1468y^{3}D^{+}+y4x62-y6+92+8y_{5}xD^{+}x_{2^{-y}}+4yD_{y}+2-18y_{42}x+4X)y=yX^{+X}6yxD5D_{y}x4y)m_{15}=0_{D}208y^{2}’ D-xy386y3m_{3}0,9^{+}2y)m=0_{1}45XD-76XDx+5044yX)m=Dy.y+y4xy^{2})m=09yy^{2}xX^{2}D33yD+416’ 16^{-}y_{X^{3}}D\mathrm{o}_{y_{2}}3x\mathrm{o}_{1}x156)m=02y^{2}D_{y}2$
.
Moreover,
these operators
form
a
Gr\"obner
basis of the
annihilator
ideal of the generator
$m$
.
Example
$(\mathrm{c}\mathrm{f}. [25])$Let
$f(x, y)=x^{6}-x^{2}y^{3}-y^{5},$
$g(x, y)=y$
.
Let
$m$
be the cohomology
class
associated
to the meromorphic
function
$\frac{1}{fg}$:
$m=[ \frac{1}{fg}]\in \mathcal{H}_{[0,0]}^{2}(\mathcal{O}_{X})$
.
We have
$\{$$x^{6}m=0$
,
$ym=0$
$(_{XD_{x}+}6)m=0$
.
However,
the
$D_{X}$
-module structure of the algebraic local cohomology group supported
on
[1]
$|/$.
sml
sml
$\mathrm{K}\mathrm{a}\mathrm{n}/\mathrm{S}\mathrm{t}\mathrm{a}\mathrm{c}\mathrm{k}\mathrm{M}\mathrm{a}\mathrm{c}\dot{\mathrm{h}}$
inel
1991 April
$—$
1996.
Release
2.970417
(c)
N.
Takayama
This
software
may
be
freely distributed
as
is with
no
warranty expressed.
Please
address
bug
reports and
advices
to
$\mathrm{k}\mathrm{a}\mathrm{n}\emptyset \mathrm{m}\mathrm{a}\mathrm{t}\mathrm{h}.\mathrm{s}$.
kobe-u.
$\mathrm{a}\mathrm{c}$
.
jp
Ready
$\mathrm{s}\mathrm{m}\mathrm{l}>\mathrm{d}\mathrm{r}.\mathrm{s}\mathrm{m}\mathrm{l}$:
9/26,1995
$—$
Version
4/17,
1997.
$\mathrm{s}\mathrm{m}\mathrm{l}>\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{u}\mathrm{l}\mathrm{e}\mathrm{l}.\mathrm{s}\mathrm{m}\mathrm{l}$,
1994
$\mathrm{s}\mathrm{m}\mathrm{l}>(\mathrm{b}\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{t}.\mathrm{S}\mathrm{m}\mathrm{l})$run
;
bfrest. sml
.
. .
$\mathrm{K}\mathrm{a}\mathrm{n}/\mathrm{s}\mathrm{m}\mathrm{l}$programs
for
D-modules
Version
970623
by
T. Oaku and
N. Takayama
See
usages
by
(indicial)
usage
:
(restO)
usage
:
(rest-l)
usage
;
$\mathrm{s}\mathrm{m}\mathrm{l}>(\mathrm{b}\mathrm{S}\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{y}.\mathrm{S}\mathrm{m}\mathrm{l})$
run
$*$ $\mathrm{s}\mathrm{m}\mathrm{l}>(\mathrm{t}\mathrm{o}\mathrm{a}\mathrm{S}\mathrm{i}\mathrm{r}.\mathrm{s}\mathrm{m}\mathrm{l})$run
:
$\mathrm{s}\mathrm{m}1>(\mathrm{X}^{arrow 6}-\mathrm{x}2\wedge \mathrm{y}*3\wedge-\mathrm{y}^{\wedge}5)[(\mathrm{x})(\mathrm{y})]00$
alcl ;
$(\mathrm{x}^{arrow 6^{-}\mathrm{X}2}arrow*\mathrm{y}^{-}3-\mathrm{y}5arrow)[(\mathrm{x})(\mathrm{y})]00$alcl
:
Computing
an
$\mathrm{F}\mathrm{W}$-Groebner basis.
Completed.
$\mathrm{s}\mathrm{m}\mathrm{l}>:$
:
$[\-75*\mathrm{y}*_{\mathrm{X}}2arrow \mathrm{D}\mathrm{X}-6**_{\mathrm{X}^{-}}3*\mathrm{D}\mathrm{y}-90*\mathrm{y}^{\sim}2*\mathrm{X}*\mathrm{D}\mathrm{y}+9*\mathrm{x}^{\wedge}2*_{\mathrm{D}}\mathrm{x}-_{3\mathrm{y}2*}*\mathrm{D}\mathrm{x}\star 12*\mathrm{y}*\mathrm{x}*_{\mathrm{D}}\mathrm{y}-450*\mathrm{y}*\mathrm{X}\star 54*_{\mathrm{X}}\sim$
,
$\-9*\mathrm{x}^{\wedge}3*\mathrm{D}\mathrm{x}-15*\mathrm{y}^{\sim}2*_{\mathrm{X}*}\mathrm{D}\mathrm{x}-12*\mathrm{y}*\mathrm{x}^{\wedge}2*\mathrm{D}\mathrm{y}-18*\mathrm{y}^{-3*}\mathrm{D}\mathrm{y}-54*\mathrm{X}^{arrow 2-}90*\mathrm{y}^{\wedge}2$,
$375*\mathrm{y}^{\wedge}3*\mathrm{x}*\mathrm{D}\mathrm{X}-\perp 8*\mathrm{x}^{\wedge}4*\mathrm{D}\mathrm{y}\star 30*\mathrm{y}^{\sim}2*\mathrm{X}2\sim*\mathrm{D}\mathrm{y}arrow 450*\mathrm{y}^{\wedge}4*\mathrm{D}\mathrm{y}-54*\mathrm{y}^{\sim}2*_{\mathrm{X}}*\mathrm{D}_{\mathrm{X}^{-}543\mathrm{D}\mathrm{y}}*\mathrm{y}arrow*$
$+2250*\mathrm{y}^{-3^{-}}270*\mathrm{y}arrow 2$
,
$\-21093750*\mathrm{y}2*\mathrm{X}*\mathrm{D}\mathrm{y}*\mathrm{D}\mathrm{x}-168750\wedge \mathrm{o}*\mathrm{y}*\mathrm{X}*\mathrm{D}\mathrm{y}-2^{-2}5312500*\mathrm{y}^{\wedge}3*\mathrm{D}\mathrm{y}^{\wedge}2\star 421875*\mathrm{x}2*\mathrm{D}_{\mathrm{X}^{-}}2arrow 2\sim$ $+703125*\mathrm{y}^{-}2*\mathrm{D}\mathrm{x}^{\wedge}2+3093750*\mathrm{y}*\mathrm{X}*\mathrm{D}\mathrm{y}*\mathrm{D}\mathrm{X}+3375000*\mathrm{y}2\wedge*\mathrm{D}\mathrm{y}^{arrow}2^{-1}01953125*\mathrm{y}*\mathrm{x}*\mathrm{D}\mathrm{x}$ $-6468750*\mathrm{X}^{arrow}2*\mathrm{D}\mathrm{y}-274218750*\mathrm{y}-2*_{\mathrm{D}\mathrm{y}.\mathrm{y}7}+13500\mathrm{o}\mathrm{o}0*\mathrm{x}*\mathrm{D}\mathrm{x}\star 32625\mathrm{o}\mathrm{o}0*\mathrm{y}*\mathrm{D}-61171850*\mathrm{y}$
\star 63281250$ ,
$\-\mathrm{x}-6+\mathrm{y}-3*\mathrm{x}^{-}2+\mathrm{y}^{\wedge}5$,
$18*\mathrm{x}^{\wedge}5*_{\mathrm{D}+9*}\mathrm{y}\mathrm{y}^{\wedge}2*\mathrm{X}2-\mathrm{D}*\mathrm{x}+15*\mathrm{y}^{\wedge}4*\mathrm{D}\mathrm{x}-6*\mathrm{y}arrow 3*\mathrm{X}^{*}\mathrm{D}\mathrm{y}$,
$\-3796875\mathrm{o}\mathrm{o}*\mathrm{x}^{arrow}4*\mathrm{D}\mathrm{y}2+263671875*\mathrm{y}-3arrow*\mathrm{D}\mathrm{x}2\sim+28687500*\mathrm{y}*\mathrm{X}^{\wedge}2*\mathrm{D}\mathrm{y}arrow 2$ $\star 32906- 2500*\mathrm{y}3arrow*\mathrm{D}\mathrm{y}2-1\wedge 4501953125*\mathrm{y}^{\wedge}2*_{\mathrm{X}}*\mathrm{D}\mathrm{x}+\perp 6031250*\mathrm{x}^{\wedge}.2*\mathrm{D}\mathrm{x}2arrow$$-11250000*\mathrm{y}^{\wedge}2*\mathrm{D}\mathrm{X}2^{-}\wedge 1160156250*\mathrm{y}*\mathrm{x}^{\wedge}2*\mathrm{D}\mathrm{y}^{-}17402343750*\mathrm{y}^{\wedge}3*\mathrm{D}\mathrm{y}\star 3656250*\mathrm{y}*_{\mathrm{X}}*\mathrm{D}\mathrm{y}*\mathrm{D}\mathrm{x}$ $-236250\mathrm{o}\mathrm{o}*\mathrm{y}^{\wedge}2*_{\mathrm{D}\mathrm{y}2}-+2274609375*\mathrm{y}*\mathrm{x}*_{\mathrm{D}\mathrm{x}+1}00125000*\mathrm{x}^{\wedge}2*\mathrm{D}\mathrm{y}+50540625\mathrm{o}\mathrm{o}*\mathrm{y}2\wedge*\mathrm{D}\mathrm{y}$
$-87011718750*\mathrm{y}2\wedge+573750\mathrm{o}\mathrm{o}*\mathrm{X}*\mathrm{D}\mathrm{x}-_{203}062500*\mathrm{y}*\mathrm{D}\mathrm{y}+15925781250*\mathrm{y}-329062500$
,
$+837\mathrm{o}\mathrm{o}00*\mathrm{X}-3*\mathrm{D}\mathrm{y}3\sim 243\mathrm{o}\mathrm{o}\mathrm{o}00*\mathrm{y}\star-2*\mathrm{x}*_{\mathrm{D}\mathrm{y}^{arrow 3+}\mathrm{y}^{arrow 2*_{\mathrm{X}*}}}14501953125*\mathrm{D}\mathrm{X}2\wedge$ $-16031250*\mathrm{x}^{\wedge}2*\mathrm{D}\mathrm{X}^{\wedge}3+11250\mathrm{o}\mathrm{o}0*\mathrm{y}-2*\mathrm{D}\mathrm{x}^{-}3+17402343750*\mathrm{y}^{\sim}3*\mathrm{D}\mathrm{y}*\mathrm{D}\mathrm{x}$ $-3656250*\mathrm{y}*\mathrm{X}*_{\mathrm{D}\mathrm{D}21}\mathrm{y}*\mathrm{x}^{\sim}-763437500*_{\mathrm{X}3}*\mathrm{D}\wedge \mathrm{y}^{\wedge}2-_{1}3921875\mathrm{o}\mathrm{o}*\mathrm{y}2arrow**\mathrm{x}\mathrm{D}\mathrm{y}^{\sim}2$ $-12555000*_{\mathrm{X}^{arrow}}2*\mathrm{D}\mathrm{y}2\sim \mathrm{D}\mathrm{X}\star 27810*000*\mathrm{y}^{\wedge}2*_{\mathrm{D}\mathrm{y}\mathrm{y}*}arrow 2*\mathrm{D}\mathrm{x}-16740000*\mathrm{x}*\mathrm{D}\mathrm{y}^{\wedge}3$ $-2274609375*\mathrm{y}*\mathrm{X}*_{\mathrm{D}}\mathrm{x}^{\wedge}2+96468750*\mathrm{x}^{arrow}2*\mathrm{D}\mathrm{y}*\mathrm{D}\mathrm{X}^{-}4948593750*\mathrm{y}^{\wedge}2*\mathrm{D}\mathrm{y}*\mathrm{D}\mathrm{x}$ $+397575000*\mathrm{y}*_{\mathrm{X}}*\mathrm{D}\mathrm{y}2^{-_{1}}160156250*\mathrm{x}2arrow*_{\mathrm{D}_{\mathrm{X}+}101}513671875*\wedge \mathrm{y}2*\mathrm{D}\mathrm{X}-89437500*\mathrm{x}*\mathrm{D}\mathrm{X}^{-}2-$ $-7425\mathrm{o}\mathrm{o}\mathrm{o}\mathrm{o}\mathrm{o}\mathrm{o}*\mathrm{y}*\mathrm{x}*\mathrm{D}\mathrm{y}+216146250*\mathrm{y}*\mathrm{D}\mathrm{y}*\mathrm{D}\mathrm{X}-10881\mathrm{o}\mathrm{o}\mathrm{o}0*\mathrm{x}*\mathrm{D}\mathrm{y}^{\sim}2-18141328125*\mathrm{y}*\mathrm{D}\mathrm{x}$ $+12135375\mathrm{o}\mathrm{o}*\mathrm{x}*\mathrm{D}\mathrm{y}-6960937500*\mathrm{X}+28\mathrm{o}\mathrm{o}57500*_{\mathrm{D}\}\mathrm{x}$
,
$\-205031250000*\mathrm{y}^{\wedge}2*\mathrm{x}2*_{\mathrm{D}\mathrm{y}3}4+9887695125*\mathrm{y}3*\mathrm{D}\mathrm{x}4+1044140\sim\wedge 62500*\mathrm{y}^{arrow}3*_{\mathrm{D}2\mathrm{D}}\mathrm{y}^{\wedge}*\mathrm{x}^{\wedge}\simarrow 2$ $+674325\mathrm{o}\mathrm{o}\mathrm{o}0*\mathrm{y}*\mathrm{x}^{arrow}2*\mathrm{D}\mathrm{y}4\wedge 729\mathrm{o}\mathrm{o}\mathrm{o}\mathrm{o}000*\mathrm{y}^{arrow}+3*_{\mathrm{D}\mathrm{y}4-}-5438232421875*\mathrm{y}-2*\mathrm{x}*\mathrm{D}\mathrm{x}3arrow$$+6011718750*\mathrm{x}2arrow \mathrm{D}*\mathrm{X}^{arrow}4-4218750000*\mathrm{y}^{arrow}2*_{\mathrm{D}\mathrm{x}4^{-}}6525878906250*\sim \mathrm{y}^{arrow 3\mathrm{D}}*\mathrm{y}*_{\mathrm{D}\mathrm{X}^{-}}2$
$+1371093750*\mathrm{y}*\mathrm{x}*\mathrm{D}\mathrm{y}*\mathrm{D}\mathrm{x}^{arrow}3+32332500\mathrm{o}\mathrm{o}*\mathrm{x}2\wedge*_{\mathrm{D}\mathrm{y}00}-2*\mathrm{D}\mathrm{X}^{\wedge}2-115200\mathrm{o}\mathrm{o}0*\mathrm{y}^{\sim}2*\mathrm{D}\mathrm{y}^{arrow 2*_{\mathrm{D}\mathrm{x}^{\wedge}}}2$ $-19629843750\mathrm{o}\mathrm{o}*\mathrm{y}*\mathrm{x}^{\sim}2*_{\mathrm{D}\mathrm{y}264}arrow 3^{-}684375\mathrm{o}\mathrm{o}0*\mathrm{y}^{\sim}3*\mathrm{D}\mathrm{y}3-\wedge 437625000*\mathrm{y}*\mathrm{x}*\mathrm{D}\mathrm{y}3\sim*\mathrm{D}\mathrm{X}$ $-6331500\mathrm{o}\mathrm{o}0*\mathrm{y}^{arrow}2*\mathrm{D}\mathrm{y}^{arrow}4+852978515625*\mathrm{y}*\mathrm{x}*\mathrm{D}\mathrm{X}^{\wedge}3^{-}20250000000*\mathrm{X}2\wedge*\mathrm{D}\mathrm{y}*\mathrm{D}\mathrm{x}^{-}2$
$+182531250\mathrm{o}\mathrm{o}00*\mathrm{y}2arrow*\mathrm{D}\mathrm{y}*_{\mathrm{D}}\mathrm{x}^{-}2+67668750\mathrm{o}\mathrm{o}0*\mathrm{y}*\mathrm{x}*\mathrm{D}\mathrm{y}^{\wedge}2*_{\mathrm{D}}\mathrm{x}+37248750\mathrm{o}\mathrm{o}\mathrm{o}*\mathrm{X}^{-}2*\mathrm{D}\mathrm{y}^{\wedge}3$
$+285795000000*\mathrm{y}^{\wedge}2*_{\mathrm{D}\mathrm{y}*\mathrm{y}2}arrow 3+435058593750*_{\mathrm{X}^{\wedge}}2*_{\mathrm{D}_{\mathrm{X}}2435}\sim-05859375000\wedge*_{\mathrm{D}\mathrm{x}2}-$
$+45562500000*\mathrm{x}*\mathrm{D}_{\mathrm{X}^{\wedge}}3^{-}1560058593750*\mathrm{y}*\mathrm{X}*_{\mathrm{D}\mathrm{D}\mathrm{x}8}\mathrm{y}*-4048750000*\mathrm{y}*\mathrm{D}\mathrm{y}*_{\mathrm{D}\mathrm{X}^{\wedge}}2-38$ $349843750\mathrm{o}\mathrm{o}*\mathrm{x}^{\wedge}2*\mathrm{D}\mathrm{y}^{\wedge}2-9076640625000*\mathrm{y}2*_{\mathrm{D}\mathrm{y}0}\simarrow 2+599625\mathrm{o}\mathrm{o}0*\mathrm{x}*\mathrm{D}\mathrm{y}2*\sim \mathrm{D}\mathrm{X}$
$-81164250000*\mathrm{y}*\mathrm{D}\mathrm{y}-3+7652109375000*\mathrm{y}*_{\mathrm{D}}\mathrm{X}^{arrow}2\star 244251562500*\mathrm{X}^{*\mathrm{D}}\mathrm{y}*\mathrm{D}\mathrm{X}$
$+22572225\mathrm{o}\mathrm{o}000*\mathrm{y}*\mathrm{D}\mathrm{y}2\wedge 1\star 80\iota 75781250*\mathrm{X}*\mathrm{D}\mathrm{X}-73665000\mathrm{o}\mathrm{o}\mathrm{o}*\mathrm{D}\mathrm{X}^{arrow}2^{-}32455898437500*\mathrm{y}*_{\mathrm{D}\mathrm{y}}$ $-218907\mathrm{o}\mathrm{o}\mathrm{o}000*_{\mathrm{D}\mathrm{y}^{\wedge}241}+75150625\mathrm{o}\mathrm{o}\mathrm{o}*\mathrm{D}\mathrm{y}-1\tau 191406250\mathrm{o}\mathrm{o}0$
,
$\-2460375\mathrm{o}\mathrm{o}\mathrm{o}\mathrm{o}0*\mathrm{y}3arrow \mathrm{y}*_{\mathrm{X}^{*_{\mathrm{D}}}}5^{-98}876953125*\mathrm{y}^{\wedge}3arrow \mathrm{D}*\mathrm{x}^{-5-}949218750\mathrm{o}\mathrm{o}*\mathrm{y}^{\wedge}3*\mathrm{D}\mathrm{y}^{arrow}2*_{\mathrm{D}\mathrm{X}^{\wedge}}3$ $-4556250\mathrm{o}\mathrm{o}\mathrm{o}*\mathrm{y}^{\wedge}3*\mathrm{D}\mathrm{y}^{-4}*\mathrm{D}\mathrm{X}^{-}1730160000*_{\mathrm{X}^{\wedge}}3*_{\mathrm{D}\mathrm{y}*}arrow 5+10497600000\mathrm{y}^{\wedge}2*\mathrm{x}*\mathrm{D}\mathrm{y}^{\wedge}5$
$+5438232421875*\mathrm{y}^{\wedge}2*\mathrm{x}*\mathrm{D}\mathrm{X}^{-4^{-_{601}}}1718750*\mathrm{x}2arrow*\mathrm{D}\mathrm{x}5\sim 4+2187500\mathrm{o}\mathrm{o}*\mathrm{y}^{\wedge}2*\mathrm{D}\mathrm{x}5\wedge$
$+6525878906250*\mathrm{y}3\wedge*\mathrm{D}\mathrm{y}*\mathrm{D}\mathrm{x}^{\wedge}3-1371093750*\mathrm{y}*\mathrm{x}*\mathrm{D}\mathrm{y}*\mathrm{D}\mathrm{x}4^{-}\sim 2656125000*\mathrm{x}^{\wedge}2*_{\mathrm{D}}\mathrm{y}2\wedge \mathrm{D}*\mathrm{X}3\wedge$ $+11115000000*\mathrm{y}2*\mathrm{D}arrow \mathrm{y}^{arrow 2\mathrm{D}}*\mathrm{X}^{arrow}3+6264843750\mathrm{o}\mathrm{o}*\mathrm{y}^{-}3*\mathrm{D}\mathrm{y}3*\mathrm{D}\mathrm{x}+569250\mathrm{o}\mathrm{o}\mathrm{o}*\mathrm{y}arrow**\mathrm{X}\mathrm{D}\mathrm{y}3-*_{\mathrm{D}\mathrm{x}}2\wedge$
$+15187500\mathrm{o}\mathrm{o}*\mathrm{x}^{\wedge}3*\mathrm{D}\mathrm{y}^{arrow 4}arrow 415985625\mathrm{o}\mathrm{o}\mathrm{o}\mathrm{o}*\mathrm{y}^{\wedge}2*_{\mathrm{X}^{*_{\mathrm{D}\mathrm{y}^{arrow}*\mathrm{D}4}}}4+259524\mathrm{o}\mathrm{o}\mathrm{o}\mathrm{o}*_{\mathrm{X}2}arrow \mathrm{y}^{\wedge}*\mathrm{D}\mathrm{X}$
$+4615920\mathrm{o}\mathrm{o}0*\mathrm{y}^{arrow}2*\mathrm{D}\mathrm{y}4*\mathrm{D}_{\mathrm{X}+}3460320000*\mathrm{y}*\mathrm{x}*\mathrm{D}\mathrm{y}^{\wedge}5-85297851562arrow 5*\mathrm{y}*\mathrm{x}*\mathrm{D}\mathrm{x}^{arrow}4$
$\star 9808593750*\mathrm{x}^{\wedge}2*_{\mathrm{D}*}\mathrm{y}\mathrm{D}\mathrm{X}-3-1785761718750*\mathrm{y}\wedge 2*\mathrm{D}\mathrm{y}*\mathrm{D}\mathrm{x}^{-}3-62353125\mathrm{o}\mathrm{o}\mathrm{o}*\mathrm{y}*\mathrm{X}*\mathrm{D}\mathrm{y}^{\wedge}2*_{\mathrm{D}\mathrm{x}2}\wedge$ $+29991375000*\mathrm{x}^{arrow}2*\mathrm{D}\mathrm{y}^{-3}*_{\mathrm{D}\mathrm{x}-}238561875000*\mathrm{y}^{\sim}2*\mathrm{D}\mathrm{y}^{\sim}3*_{\mathrm{D}\mathrm{x}}+158460300\mathrm{o}\mathrm{o}\mathrm{o}*\mathrm{y}*_{\mathrm{X}*\mathrm{D}}\mathrm{y}^{\wedge}4$
$-435058593750*\mathrm{x}^{\wedge}2*\mathrm{D}\mathrm{X}^{\wedge}3+48944091796875*\mathrm{y}2*\mathrm{D}\mathrm{x}^{-}3-575859375\mathrm{o}\mathrm{o}*\mathrm{x}*\mathrm{D}\mathrm{x}^{\wedge}4-$
$\star 3039257812500*\mathrm{y}*\mathrm{X}*\mathrm{D}\mathrm{y}*\mathrm{D}\mathrm{X}2\wedge\star 81057656250*\mathrm{y}*_{\mathrm{D}\mathrm{D}}\mathrm{y}*\mathrm{x}^{\wedge}3+524742187500*\mathrm{X}^{\wedge}2*_{\mathrm{D}\mathrm{y}*}\sim 2\mathrm{D}\mathrm{X}$ $+10851679687500*\mathrm{y}-2*\mathrm{D}\mathrm{y}^{\wedge}2*\mathrm{D}\mathrm{x}-8979750000*\mathrm{X}*\mathrm{D}\mathrm{y}^{\sim}2*_{\mathrm{D}}\mathrm{x}^{\sim}2-190329750\mathrm{o}\mathrm{o}0\mathrm{o}\mathrm{o}*\mathrm{y}*\mathrm{x}*\mathrm{D}\mathrm{y}^{\wedge}3$ $+6410061\mathrm{o}\mathrm{o}\mathrm{o}\mathrm{o}*\mathrm{y}*\mathrm{D}\mathrm{y}^{arrow 3\mathrm{D}\mathrm{x}}*+29412720000*\mathrm{x}*\mathrm{D}\mathrm{y}4-8482939453125*\mathrm{y}arrow*\mathrm{D}\mathrm{x}3\wedge$
$-4715578125\mathrm{o}\mathrm{o}*\mathrm{x}*\mathrm{D}\mathrm{y}*_{\mathrm{D}}\mathrm{x}^{\wedge}2-232742250\mathrm{o}\mathrm{o}00*\mathrm{y}*\mathrm{D}\mathrm{y}^{\sim}2*\mathrm{D}\mathrm{x}+587146950000*\mathrm{X}*\mathrm{D}\mathrm{y}^{\wedge}3$ $+428906250\mathrm{o}\mathrm{o}\mathrm{o}*\mathrm{X}*\mathrm{D}\mathrm{x}2\sim+_{272925}\mathrm{o}\mathrm{o}000*\mathrm{D}\mathrm{x}^{\wedge}3+47920429687500*\mathrm{y}*\mathrm{D}\mathrm{y}*_{\mathrm{D}_{\mathrm{X}}}$
$-20893415625000*\mathrm{X}^{*}\mathrm{D}\mathrm{y}^{arrow 2}+176690790000*\mathrm{D}\mathrm{y}*25612812500*_{\mathrm{D}*_{\mathrm{D}_{\mathrm{X}}}}\mathrm{y}arrow 2\mathrm{D}_{\mathrm{X}^{-}}58$
+27365625000000*Dx$
,
$36905625000000*\mathrm{y}4*_{\mathrm{D}\mathrm{y}9}-619140625*\mathrm{y}^{-}3*\mathrm{D}\mathrm{x}\wedgearrow 61235\sim 6^{-}10678710937500*\mathrm{y}-3*_{\mathrm{D}\mathrm{y}^{\wedge}}2*_{\mathrm{D}\mathrm{X}4}\sim$
$-341718750000*\mathrm{y}3*\mathrm{D}arrow \mathrm{y}^{arrow 4*\mathrm{D}\mathrm{x}^{\wedge}}2+478224000000*\mathrm{y}*\mathrm{X}^{-}2*\mathrm{D}\mathrm{y}^{-}6-188082\mathrm{o}\mathrm{o}0\mathrm{o}\mathrm{o}\mathrm{o}\mathrm{o}*\mathrm{y}3\wedge \mathrm{D}*\mathrm{y}^{\wedge}6$
$+679779052734375*\mathrm{y}^{\wedge}2*\mathrm{x}*\mathrm{D}\mathrm{X}5arrow-751464843750*\mathrm{x}2arrow*\mathrm{D}\mathrm{x}6-52734375000\dagger \mathrm{o}*\mathrm{y}^{\wedge}2*\mathrm{D}\mathrm{x}^{arrow}6$ $+815734863281250*\mathrm{y}^{-}3*_{\mathrm{D}}\mathrm{y}*_{\mathrm{D}\mathrm{x}41713}-86718\sim 750*\mathrm{y}*\mathrm{x}*\mathrm{D}\mathrm{y}*\mathrm{D}\mathrm{x}5\wedge$
$-25987\dot{5}0\mathrm{o}\mathrm{o}\mathrm{o}00*\mathrm{x}^{\sim}2*_{\mathrm{D}}\mathrm{y}^{\sim}2*_{\mathrm{D}\mathrm{X}^{-}}4+_{133875}0000000*\mathrm{y}^{\wedge}2*\mathrm{D}\mathrm{y}-2*\mathrm{D}\mathrm{x}-4$ $+78310546875000*\mathrm{y}^{\wedge}3*\mathrm{D}\mathrm{y}^{\wedge}3*_{\mathrm{D}\mathrm{X}^{\wedge}}2+87609375\mathrm{o}\mathrm{o}0*\mathrm{y}*\mathrm{X}*\mathrm{D}\mathrm{y}-3*\mathrm{D}\mathrm{x}-3$ $+245754\mathrm{o}\mathrm{o}0\mathrm{o}\mathrm{o}0*\mathrm{X}^{\sim}2*_{\mathrm{D}\mathrm{y}0}arrow 4*\mathrm{D}\mathrm{X}^{-}2\star 39062250000*\mathrm{y}2\wedge*\mathrm{D}\mathrm{y}^{\wedge}4*\mathrm{D}\mathrm{x}arrow 2$
$-33260625\mathrm{o}\mathrm{o}000*\mathrm{y}*\mathrm{X}^{\wedge}2*_{\mathrm{D}}\mathrm{y}5+96228000\mathrm{o}\mathrm{o}0\mathrm{o}\mathrm{o}\mathrm{o}arrow \mathrm{o}*\mathrm{y}^{\sim}3*\mathrm{D}\mathrm{y}5+88560\mathrm{o}\mathrm{o}00\sim \mathrm{o}*\mathrm{y}*\mathrm{x}*\mathrm{D}\mathrm{y}^{\wedge}5*\mathrm{D}\mathrm{X}$ $-425088\mathrm{o}\mathrm{o}\mathrm{o}000*\mathrm{y}^{arrow 2\mathrm{D}}*\mathrm{y}^{\sim}6-106622314453125*\mathrm{y}*_{\mathrm{X}}*\mathrm{D}\mathrm{X}^{arrow}5-79101562500*_{\mathrm{X}^{\wedge}}2*\mathrm{D}\mathrm{y}*\mathrm{D}\mathrm{x}^{arrow}4$ $-218276367187500*\mathrm{y}^{\wedge}2*\mathrm{D}\mathrm{y}*\mathrm{D}\mathrm{x}^{\sim}4-7129687500000*\mathrm{y}*\mathrm{X}*\mathrm{D}\mathrm{y}2\wedge*\mathrm{D}\mathrm{x}^{\wedge}3$ $+3969140625000*\mathrm{x}^{\sim}2*_{\mathrm{D}3*_{\mathrm{D}\mathrm{X}}}\mathrm{y}2-\sim\wedge 23622890625\mathrm{o}\mathrm{o}0*\mathrm{y}2arrow*\mathrm{D}\mathrm{y}^{\wedge}3*\mathrm{D}\mathrm{x}^{-}2$ $+610065\mathrm{o}\mathrm{o}00000*\mathrm{y}*_{\mathrm{X}}*\mathrm{D}\mathrm{y}^{\wedge}4*_{\mathrm{D}+359110}\mathrm{X}8000000*\mathrm{x}^{\sim}2*\mathrm{D}\mathrm{y}5\sim-35450055000000*\mathrm{y}^{\sim}2*\mathrm{D}\mathrm{y}^{\wedge}5$ $-54382324218750*\mathrm{x}^{\wedge}2*_{\mathrm{D}4}\mathrm{x}\sim+6797790527343750*\mathrm{y}^{\wedge}2*_{\mathrm{D}\mathrm{X}^{\wedge}}4^{-_{87}}01171875000*\mathrm{X}*\mathrm{D}\mathrm{x}^{arrow}5$ $+564807128906250*\mathrm{y}*\mathrm{x}*\mathrm{D}\mathrm{y}*\mathrm{D}\mathrm{X}^{arrow}3+9758320312500*\mathrm{y}*\mathrm{D}\mathrm{y}*_{\mathrm{D}\mathrm{x}^{\wedge}}4$
$+803847656250\mathrm{o}\mathrm{o}*\mathrm{x}^{arrow}2*\mathrm{D}\mathrm{y}^{-}2*_{\mathrm{D}}\mathrm{X}^{arrow 2+15}78339843750000*\mathrm{y}2\wedge 2*\mathrm{D}*\mathrm{D}\mathrm{y}^{arrow}\mathrm{X}^{arrow 2}$ $-120684375\mathrm{o}\mathrm{o}00*_{\mathrm{X}}*\mathrm{D}\mathrm{y}2*_{\mathrm{D}\mathrm{x}^{arrow 3^{-_{68}}}}491406250000*\mathrm{y}*\mathrm{x}*\mathrm{D}\mathrm{y}-3*_{\mathrm{D}\mathrm{x}}arrow$
$+6137167500000*\mathrm{y}*\mathrm{D}\mathrm{y}^{\sim}3*_{\mathrm{D}2^{-}2}\mathrm{x}^{\sim}4440906250000*\mathrm{x}^{arrow}2*\mathrm{D}\mathrm{y}^{arrow 4+}786858046875\mathrm{o}\mathrm{o}00*\mathrm{y}2*\mathrm{D}\sim \mathrm{y}4\wedge$
$+726192\mathrm{o}\mathrm{o}0000*_{\mathrm{X}}*\mathrm{D}\mathrm{y}-4*\mathrm{D}\mathrm{x}-7111368\mathrm{o}\mathrm{o}\mathrm{o}000*\mathrm{y}*\mathrm{D}\mathrm{y}5-11642arrow 21191406250*\mathrm{y}*_{\mathrm{D}\mathrm{x}^{\wedge}}4$ $-925787109375\mathrm{o}\mathrm{o}*\mathrm{x}*\mathrm{D}\mathrm{y}*\mathrm{D}\mathrm{X}^{arrow}3^{-}292514062500000*\mathrm{y}*\mathrm{D}\mathrm{y}^{-}2*\mathrm{D}\mathrm{x}2\wedge\star 53763975\mathrm{o}\mathrm{o}0000*_{\mathrm{X}}*\mathrm{D}\mathrm{y}^{\sim}3*\mathrm{D}\mathrm{X}$ $-177632527500\mathrm{o}\mathrm{o}0*\mathrm{y}*\mathrm{D}\mathrm{y}-4\star 129748535156250*\mathrm{x}*\mathrm{D}_{\mathrm{X}3}arrow-_{3}887929687500*\mathrm{D}_{\mathrm{X}^{-}}4$ $+8292919921875\mathrm{o}\mathrm{o}0*\mathrm{y}*\mathrm{D}\mathrm{y}*\mathrm{D}\mathrm{x}^{\sim}2+1160648437500\mathrm{o}\mathrm{o}*\mathrm{X}*\mathrm{D}\mathrm{y}^{arrow}2*\mathrm{D}\mathrm{x}\star 1720872000\mathrm{o}\mathrm{o}\mathrm{o}\mathrm{o}*_{\mathrm{D}2}\mathrm{y}^{\wedge}*_{\mathrm{D}\mathrm{x}^{\wedge}}2$ $+22847796562500000*\mathrm{y}*\mathrm{D}\mathrm{y}^{-3}-261207720\mathrm{o}\mathrm{o}\mathrm{o}\mathrm{o}0*_{\mathrm{D}}\mathrm{y}^{\sim}4-989171718750000*_{\mathrm{D}*_{\mathrm{D}\mathrm{X}^{\wedge}}}\mathrm{y}2$
$-203966628750000*_{\mathrm{D}\mathrm{y}^{arrow 3}9}+4953515625000000*_{\mathrm{D}2}\mathrm{x}^{\wedge}+1873168593750\mathrm{o}\mathrm{o}0*\mathrm{D}\mathrm{y}2\wedge$
$]$ $\mathrm{s}\mathrm{m}\mathrm{l}>\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}$;
2.
Residue and residual duality
Let
$X$
be a
domain
in
$\mathrm{C}^{n}$.
Let
$f_{1},$ $f_{2},$$\ldots.,$$f_{n}$
be a regular sequence of holomorphic
functions
on
$X$
.
Let
$\mathcal{I}$be
the ideal
generated
by
$f_{1},$ $f_{2},$$\ldots.,$
$f_{n}$
over
$\mathcal{O}_{X}$.
Let
us
denote
by
$\in \mathcal{E}_{Xt_{\mathcal{O}}^{n}}(X\mathcal{O}\mathrm{x}/\mathcal{I}, \mathcal{O}x)$
the
Grothendieck.
residue
symbol
associated to the
meromorphic
function
$\frac{1}{f_{1}f_{2}\cdots f_{n}}$
.
Let
$i$be the canonical
map
$\mathcal{E}xt_{O}^{n_{X}}(\mathcal{O}x/I, \mathcal{O}_{X})arrow \mathcal{H}_{[A]}^{n}(\mathcal{O}x)$
,
where
$A=\{z\in X|f_{j}(z)=0, j=1,2, \ldots,n\}$
.
Set:
$m=i()\in H_{[A]}^{n}(\mathcal{O}x)$
.
We
assume
that the
common
locus
$A$
consists
of
finite
number of points
$A_{k},$$k=1,2,$
$\ldots,N$
.
Corresponding
to
the
decomposition of the algebraic local
cohomology
group
$\mathcal{H}_{[]}^{n_{A}}(\mathcal{O}x)$we have
$m=m_{1}+m_{2}+\cdots+m_{N}$
with
$m_{k}\in H_{[A]}^{n_{k}}(\mathcal{O}x)$.
Let
$\Omega_{X}$be
the sheaf on
$X$
of holomorphic
differential
$\mathrm{n}$-forms. The
canonical
pairing
$\Omega_{X}\mathrm{x}H_{[A}^{n_{k}}](\mathcal{O}_{X})arrow H_{[A]}^{n_{k}}(\Omega_{X})$
composed with
$H_{[A]}^{n_{k}}(\Omega_{X})arrow \mathrm{C}$
defines the residue
pairing at the point
$A_{k}$.
Put
${\rm Res}_{A_{k}} \langle\phi(z), m\rangle=\frac{1}{(2\pi i)^{n}}\oint_{A_{k}}\phi(_{Z})mdz$
.
We regard
${\rm Res}_{A_{k}}$$\langle \cdot, m\rangle$as a linear map
$\Omega_{X}\ni\phi(z)dZarrow{\rm Res}_{A_{k}}\langle\phi(_{Z}), m\rangle\in \mathrm{C}$
.
Note There exists
$m_{k}\in \mathrm{N}$and complex constants
$c_{\beta,k}(0\leq|\beta|\leq m_{k})$
such that for every
$\phi dz\in\Omega x$
,
${\rm Res}_{A_{k}} \langle\phi(_{Z}), m\rangle=\sum_{0\leq|\beta|\leq m_{k}}C_{\beta,k}\cdot((\frac{\partial}{\partial z})^{\beta}\phi)(A_{k})$
.
3.
Main Theorems
Let
$X$
be
a
domain in
$\mathrm{c}^{n}$:
Let
$f_{1},$ $f_{2},$ $\ldots.,$$f_{n}$
be
a regular sequence of holomorphic
functions
on
$X$
.
Let
$A=\{z\in X|f_{j}(z)=0, j=1,2, \ldots, n\}$
.
Let
us denote by
$m$
the
residue class
$m=i()\in H_{[}^{n_{A}}(\mathrm{J}\mathcal{O}_{X}).$
.
We assume that
the
common
locus
$A$
consists of finite number of points
$A_{k},$$k=1,2,$
$\ldots,$
$N$
.
We
have
$m=m_{1}+m_{2}+\cdots+m_{N}$
with
$m_{k}\in H_{[A]}^{n_{k}}(\mathcal{O}x)$.
The following theorem asserts that the
cohomology
class
$m$
can
be
characterized as a
solution
of linear
partial differential
equations up to
constant factor.
Theorem
A
Let
$J=\{P\in D_{X}|Pm=0\}$
be the annihilator ideal
of
$m$
.
Then at
each
point
$A_{k}$,
we
have
$\{u| Pu=0, u\in H_{[A]}^{n_{k}}(\mathcal{O}_{X}), P\in J\}=\{cm_{k}|c\in \mathrm{C}\}$
.
Proof
Put
$\mathcal{M}_{k}=\mathcal{H}_{[]}^{n_{A_{k}}}(\mathcal{O}_{X})$.
We
have
$m_{k}\in \mathcal{M}_{k}$. Since
the
algebraic local cohomology group
Hence
we have
$\mathcal{H}om_{\mathcal{D}_{X}}(Dx/J, \mathcal{M}_{k})$
$=\mathcal{H}om_{\mathcal{D}_{X}}(Dxm,\mathcal{M}_{k})$
$=\mathcal{H}omv_{X}(Dxm_{k}, \mathcal{M}_{k})$
$=\mathcal{H}om_{\mathcal{D}_{X}}(\mathcal{M}k, \mathcal{M}_{k})$
.
The claim
follows from
th.e
fact that
$\mathcal{H}om_{\mathcal{D}_{X}}(\mathcal{M}_{k}, \mathcal{M}_{k})=\mathrm{c}|_{A_{k}}$
.
q.e.d.
Let
us recall the fact that
$\Omega_{X}$is
naturally
endowed with
a structure of a right
$D_{X}$
-module.
The
right action of
$P= \sum a_{\alpha}(z)(\frac{\partial}{\partial z})^{\alpha}$is described explicitly as follows.
$(\phi(z)dZ)P=(P^{*}\phi)(Z)d_{Z}$
,
where
$P^{*}= \sum(-\frac{\partial}{\partial z})^{\alpha}a_{\alpha}(z)$is the formal
adjoint
of
$P\in D_{X}$
.
If
$P\in J$
,
then we have
${\rm Res}_{A_{k}}\langle(P^{*}\psi(z))d_{Z,m\rangle}={\rm Res}_{A_{k}}\langle(\psi(z)dz,Pm\rangle=0$
.
Furthermore,
we
$\mathrm{h}\mathrm{a}.\mathrm{v}\mathrm{e}$the
$\mathrm{f}\mathrm{o}\mathrm{l}\mathrm{i}_{0}\mathrm{w}\mathrm{i}\mathrm{n}\mathrm{g}$
Theorem,
which is a
direct
consequence of a result
of
Kashiwara
[11].
Theorem
$\mathrm{B}$Let
$J$
be
the annihilator ideal
of
$m$
.
Then, we
have
$\{\phi(z)d_{Z}\in\Omega_{X}|{\rm Res}_{A_{k}}\langle\phi(z)dz,m\rangle=0,\forall k=1,2, \ldots, N\}$
$=\{(P*\psi)(z)dZ|P\in J, \psi(\mathcal{Z})dz\in\Omega \mathrm{x}\}$
.
Note
that,
for
the
case
of
one
variable,
Theorem
$\mathrm{B}$provides
a new theoretical foundation
of the
Horowitz-Ostrogradski algorithm
([10])
for
$\dot{\mathrm{t}}$he
integration of rational functions.
4.
Examples
Example Let
$X=\{(x, y)|x,y\in \mathrm{C}\},$
$f(\dot{x}, y)=y^{2},$
$g(x,y)=y-x^{2}$
.
The
$\tilde{\mathrm{m}}$ultiplicity
of
intersection
of these two curves at the
origin
is equal to 4. The
coho-mology class
satisfies
the
following
system
of
linear
partial
differential equations.
$\{$
$y^{2}m=0$
,
$(y-x^{2})m=0$
,
$(xD_{x}+2yD_{y}+6)m=0$
.
It
is
easy to see
that the
annihilator ideal
$J$
of
$m$
is
generated
by
these three
operators:
$J=\langle y^{2}, y-x^{2}, xD_{x}+2yD_{y}+6\rangle$
.
Put
$m= \sum a_{\alpha,\beta}[\frac{1}{x^{\alpha}y^{\beta}}]$.
Since
$(xD_{x}+2yD_{y}+6) \frac{1}{x^{\alpha}y^{\beta}}=(-\alpha-2\beta+6)\frac{1}{x^{\alpha}y^{\beta}}$
,
we
have
$m=[ \frac{a_{2,2}}{x^{2}y^{2}}+\frac{a_{4,1}}{x^{4}y}]$.
The second equation
$(y-x^{2})m=0$
implies
$m= \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}[\frac{1}{x^{2}y^{2}}+\frac{1}{x^{4}y}]$
.
Let
$\mathcal{I}=\langle y^{2}, y-X^{2}\rangle$be the
ideal
generated by
$y^{2}$and
$y-x^{2}$
over
the ring
$\mathcal{O}_{X}$.
Then the
quotient
space
$\Omega_{X}/\Omega_{X}\mathcal{I}$is
a
4-.dimensional
vector
space.
Put
$K=$
{
$\phi(x,$
$y)dx$
A
$dy|{\rm Res}_{[]}0,0(\phi(x,$
$y)dX$
A
$dy,$
$m)=0$
}.
Obviously
we have
$\Omega \mathcal{I}\subset K$.
Since
$P^{*}=-xD_{x}-2yD_{y}+3$
,
we have
$P^{*}1=3,$
$P^{*}x=2x,P^{*}x^{2}=x^{2},$
$P^{*}x^{3}=0$
.
Therefore,
the differential
forms
$dx$
A
$dy,$
$xd_{X}$
A
$dy$
and
$x^{2}dx$
A
$dy$
belong
to
$K$
and the
differential
form
$x^{3}dx$
A
$dy$
gives a
representative of a
non-trivial
element of
$\Omega_{X}/IC$
.
Example
Take
$f(x, y)=(x^{2}+y^{2})^{2}+3x^{2}y-y^{3},$
$g(x, y)=y-x^{2}$
.
Let
$A=\{(x, y)|f(x, y)=g(x, y)=0\}$
.
Then
$A=\{(0,0)\}\cup\{(x, y)|y-x^{2}=0, y^{2}+y+4=0\}$
.
Put
$m=[ \frac{1}{fg}]\in \mathcal{H}_{[A]}2(\mathcal{O}X)$
.
Let
$J\subset D_{X}$
be the
annihilating
ideal
of
$m$
.
Then
$\{Q_{1}, Q_{2}, P_{1}, P_{2}, P_{3}\}$
is an
involutory
base of
the ideal
$J$
,
where
$Q_{1}$
$=$
$-x^{2}+y$
,
$Q_{2}$
$=$
$-y^{4}-y^{\mathrm{s}_{-4y}2}$
,
$P_{1}$
$=x(y^{2}+y+4)D_{x}+2y(y^{2}+y+4)D_{y}+10y^{2}+8y+24$
,
$P_{2}$$=$
$y(y^{2}+y+4)D_{x}+2xy(y^{2}+y+4)D_{y}+2x(4y^{2}+3y+8)$
,
Since
$P_{1}$
$=$
$(xD_{x}+2yD_{y}+6)(y^{2}+y+4)$
,
$P_{2}$
$=$
$(yD_{x}+2xyD_{y}+4X)(y^{2}+y+4)$
,
$P_{3}$
$=$
$(-yD_{x}^{2}+6yD_{y}+12)(y^{2}+y+4)$
,
hold,
the annihilator ideal
$J$
of
the cohomology
class
$m$
is
generated
by
$y-x^{2},$
$y^{2}+y+4$
over
$D_{X}$
at
$\{(x, y)|y-x^{2}$
.
$=0, y^{2}+y+4=0\}$
.
5.
Appendix
Let
$f_{1},$ $f_{2},$$\ldots,$
$f_{n}\in \mathrm{C}[z_{1}, Z_{2}, \ldots, z_{n}]$
be
a regular
sequence
of
polynomials.
Let
$A=\{z\in$
$\mathrm{C}^{n}|f_{1}(z)=f_{2}(z)=\cdots=f_{n}(z)=0\}$
be the
common locus of
$f_{1},f_{2},$$\ldots,$
$f_{n}$
.
We assume
that
$f_{1},$$f_{2},$$\ldots,f_{n}$are
in general position,
i.e.,
the
Jacobian determinant
$Jac= \frac{\partial(f_{1},f_{2},\ldots,fn)}{\partial(z_{1},z2,\ldots,Zn)}$
does
not
vanish
at any point
$A_{k}\in A$
.
Then
we
have
${\rm Res}_{A_{k}} \langle\phi(_{Z),m\rangle}=\frac{\phi(A_{j})}{Jac(A_{k})}$
.
By
rewriting
the above
relation:
we
get
$Jac(A_{k})\cdot ResAk\langle\phi(z), m\rangle-\phi(Aj)$
.
Let us introduce a new indeterminant
$t$.
We see that the residues of
$\frac{\phi(z)d_{Z}}{f_{1}f_{2}\cdots f_{n}}$should
satisfy
$\{$
$JaC(\mathcal{Z})t-\phi(z)=0$
,
$f_{1}(z)=f2(Z)=\cdots=f_{n}(\mathcal{Z})=0$
.
We arrive
at
the
following method for computing the residues of
$[ \frac{\phi(z).d_{Z}}{f_{1}f_{2}\cdot\cdot f_{n}}]$.
$\bullet$
Set
$I=\langle f_{1}(z), f2(_{Z)}, \ldots, f_{n}(z), JaC(_{\mathcal{Z})-}t\phi(Z)\rangle\subset \mathrm{C}[z_{1,2,n}z\ldots, Z,t]$
.
$\bullet$
Compute
the
Gr\"obner
basis of the
$\mathrm{i}\mathrm{d}\mathrm{e}\mathrm{a}\dot{\mathrm{l}}$$I$
with respect to pure.lexicographic order
$z\succ t$
and then perform the primary decomposition of the
polynomial ideal
$I$
.
Note that
the above method
is a natural
generalization
of
$\mathrm{T}\mathrm{r}\mathrm{a}\mathrm{g}\mathrm{e}\mathrm{r}- \mathrm{L}\mathrm{a}\mathrm{Z}\mathrm{a}\mathrm{r}\mathrm{d}- \mathrm{R}\mathrm{i}\mathrm{o}\mathrm{b}_{0}\mathrm{o}$and
Example Let $f(x, y)=y-x2,$
$g(x, y)=y-X-2,$
$\phi(x, y)=1$
.
Put
$h(x, y, t)=Jac(x, y)\cdot t-1$
,
where
$Jac(X, y)=-2x+1$
is the jacobian
determinant
of
$f,g$
.
Let
$I=\langle f,g, h\rangle\subset IC[x, y, t]$
.
Then the
Gr\"obner
base
of
$I$
with respect to the pure
lexicographic
$\mathrm{o}\mathrm{r}\mathrm{d}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g}x\succ y\succ t$is
$\{-9t^{2}+1,2y+9t-5,2X+9t+1\}$
.
The
primary decomposition
of this
ideal is given
by
$\langle 3t+1, y-4,x-2\rangle,$
$\langle 3t-1,y-1, x+1\rangle$
.
We thus get
${\rm Res}_{(2,4)}\langle 1,$ $[ \frac{1}{fg}])=-\frac{1}{3}$