Publicacions
Ma emá iques,
Vol
36
(1992),
693-714
.
A
bs ac
QUANTUM
SECTIONS
ANDGAUGE
ALGEBRAS
L
.
LE
BRUYN
*
AND
F
.
VAN
OYSTAEYEN
To
he
memo y
o
Pe e
Menal
Using
quan um
sec ions
o
il e ed
ings
and
he
associa ed
Rees
ings
one
can
li
he
scheme
s uc u e
en P oj
o
he
associa ed
g aded
ing
e
he
P oj
o
he
Rees
ing
.
The
algeb as
o
in e es
he e
a e
posi i ely
il e ed
ings
ha ing
a
non-commu a i e
egula
quad a ic algeb a
o
he
associa ed
g aded
ing
;
hese
a e he
so-
called
gauge
algeb as
ob aining
hei
name
om
special
examples
appea ing
in
E
.
Wi en's
gauge
heo ies
.
The
pape
su eys
basic
de ini ions
and
p ope ies
bu
concen a es
en
he
de elopmen
o
se e al
conc e e
examples
.
0
.
In oduc ion
_
Speci ic
p oblems
in
de ining
a
"scheme"
s uc u e
on P oj(W), whe e
W
is
he
4-dimensional
quan um
space
o
he
Rees
ing
o
he
Wi en
gauge
algeb as,
may
be
ackled
by
i s
in oducing
such
a
scheme
s uc-
u e
on
P oj(G(W))
_whe e
G(W)
is
he
associa ed
g aded
ing
o
W
and
hen
ying
o
li
his
s uc u e
o
a
scheme
s uc u e
on
P oj(W)
.
In
[LVW]
his
li ing
p oblem
is
sol ed
by
using
quan um
sec ions
in-
oduced
by
he
second au ho
in
[VOS],
[RVO]
and
his
explains
why
he
de elopmen
o
he heo y
o
gauge
algeb as
he e goes
hand
in
hand
wi h
ha o
quan um
sec ions
.
In
ac
Noe he ian
gauge
algeb as
a e
pa icula
Za iski
ings
in
he
sense
o
[LVO,
1, 2,
. .
.]
.
Now
quan um
sec ions
a ise
in
he
shea
o
il a ion
deg ee
ze o
o
a
mic os uc u e
shea
o a Za iski
ing o e
he
p ojec i e
scheme
associa ed
o
he
as-
socia ed
g aded
ing
ha
is
supposed
o
be
commu a i e
in
[VOS]
.
The
commu a i i y
o
he
associa ed
g aded
ing
is
nowhe e
essen ial in
he
s uc u e
heo y
o
he
ings
o
sec ions
o
hose
shea es,
excep
o
cou se
*This
au ho
is
suppo ed
by an
NFWO-g an
694

L
.
LE
BRUYN,
F
.
VAN
OYSTAEYEN
in
he
de ini ion
o
he
opological
space
and
he
scheme
s uc u e
o
P oj
.
Howe e ,
M
.
A in
has
ecen ly
in oduced
in
[A ]
he
quan um
p ojec i e
space
o
a
quan um
n-space
gi en
by
i s
g aded
quad a ic
al-
geb a
Q
as P oj
(Q)
=
Q-g /
.F,
whe e
F
is
he
ull
subca ego y
o
g aded
ini e
leng h
modules,
oge he
wi h
a
sui able
shi -ope a o
.
F om
his
poin
o
iew
i is
na u al
o
y
o
combine
he
echniques o
Za iski
il e ed
ings
inhe en
in
he
s udy
o
quan um
sec ions
wi h
he
heo y
o
p ojec i e
quan um
spaces
.
This
is
done by
in oducing
he
class
o
posi i ely
il e ed
ings
ha ing
anon-commu a i e
egula
in
he
sense
o
A in
and
Schel e
[AS]
quad a ic algeb a
in
he
sense
o
Manin,
[Man],
o
he associa ed
g aded
ing
.
The
es ic ion o
posi i e
il a ions
is
no
essen ial
because
he
de ini ion
o
he
unde lying
p ojec i e
scheme
may
easily
be
modi ied
o
deal
wi h
his
;
ne e heless
we
do
es ic
o
he
posi i e
case he e
.
The
algeb as
oughly
de ined
abo e
a e
called
gauge
algeb as
.
Since
he
ing edien e
o
he
heo y
lay
sp ead ou
o e
se e al
sou ces
no
all
equally
a ailable,
we
ha e
concei ed
his
pape
as
a
su ey
pape
in oducing
necessa y
basic
de ini ions
and
p ope ies
as
well
as
expanding
a
ew
conc e e
examples
.
A
mo e
ex ensi e
s udy
o
gauge
algeb as
is
unde aken
by
he
i s
au ho
in
[LB]
;
o
ecen
esul e
on
o mal
quan um
sec ions
o e
o mal
schemms
we
e e
o
[RVO]
.
1
.
Fil e ed
ings
and
associa ed
g aded
ings
All
ings
a e
associa i e
wi h
uni
.
A
il a ion
FR
on a
ing
R
is
gi en
by an
ascending
chain
o
addi i e
subgoups
F
n
R,
n
E
7L,
sa is ying
1
E
F
O
R,
F
n
RF
n
R
C
Fn+mR
o
m,
n
E
7L
.
We
always
assume
ha
he
il a ions
conside ed
a e
exhaus i e,
ha
is
UnE7[FnR
=
R, and
sepa a ed,
ha
is
nnEzFnR
=
0
.
The
ob ious
ope a ions
induced
on
he
abelian
addi i e
g oup
G(R)
=
T
nE
aF
n
R/F
n
_
1
R
make
G(R)
in o
a
g aded
ing
wi h
G(R)
n
=
FnR/F
n
_
1
R, n
E
7L
.
The
p incipal
sym-
bol
mapa
:
R
-3
G(R)
is
de ined
by
pu ing
u(x)
=
x
mod
F
n
_
1
R
whe e
n
is
such
ha
x
E
F
n
R
-
F,-,R
.
The
Rees
ing
R
=
®n,EZFnR
may
be
iden i ied
wi h
he
sub ing
EnEa
F
n
,RXn
o
he
polynomial
ing
R[X,
X
-1
]
.
The
no ion
o
he
Rees
ing
o
a
Z- il a ion
ex ends
in
a
na u al
way
he
no ion
o
he
blow-up
ing
o
an
I-adic
il a ion
used
in
singula i y
heo y
and
commu a i e
Za iski
ings
.
A
il a ion
FR
is
comple e
when
Cauchy-sequences
con e ge
in
R,
o
equi alen ly
R
=
limR/FnR
;
when
F,,,R
=
0
o
n<
0
hen
we
say
ha
FR
is
posi-
n
i e o
R
is
posi i ely
il e ed
and
i is
clea
ha
posi i e
il a ions
a e
comple e
.
Comple e
il e ed
ings
ha ing
a
Noe he ian
associa ed
g aded
ing
a e
an
impo an
clase
o
Za iski
ings
in
he
sense
o
[LVO1],
in
pa icula
such
a
ing
R
has
a
Noe he ian
Rees
ing
R
.
QUANTUM
SECTIONS
AND
GAUGE
ALGEBRAS

69
5
Example
1
.
The
n h
Weyl
algeb a
A
n
(O)
is
he
algeb a gene a ed
by 2n
inde e mina es
x,
....
,
x
n
and
yi,
. . . ,
yn
sa is ying
he
commu a-
o ela ions
:
[xi,
x
j
]
=
[yi,
y
j
]
=
0,
[xi,
y
j
]
=
bij
.
His o ically,
An(O)
has
been
in oduced
as
he
ope a o
algeb a
gene a ed
by
he
compo-
nen s
xi
o
he
posi ion
ec o
and
he
componen s
p
j
=
ihy
j
o
he
momen um
ec o o
a
quan um
pa icle
in
n-dimensional space
.
The
non- anishing
o
he
commu a o
[x
j
,
p
j
]
=
ice
exp esses
ha
one canno
ha e
simul aneous
knowledge
o
posi ion
and
momen um
o
a
pa icle
in
he
quan um
case
.
The
associa ed
g aded
ing
G(AnP)
is
he
poly-
nomial
ing
C[xi,
.
.
.,
xn,
Y,
. .
.
,
yn]
ha
is
he
ope a o
algeb a
o
he
classical
(Le
.
non-quan um)
si ua ion
.
So
in
a
sense
he
il e ed
da a
may
be
iewed
as
quan iza ions
o
he
associa ed
g aded
da a
.
The
Rees
ing
An(O)
-
is
he
posi i ely
g aded
algeb a
gene a ed
by
he
deg ee
one
elemen
X
and
xiX,
yiX,
1
<
i
<_
n
sa is ying
he
commu a i e
ela ions
:
[xi,
xj
]Xz
=
[yi,
yj
]Xz
=
0
and
[xi,
yj
]Xz
=
Ói7X2
.
Since
X
is
cen al
in
AnP_
we
may
subs i u e
new
inde e mina es
X
=
X,Xá
=
xiX,
Yi
=
yiX
sa is ying
homogeneous
ela ions
A1
X
i
]
_
[Y¡,
Y
j
]
=
0
and
[X,,
YI
]
=
Sij
X
2
.
The e o e
we
may
iew
A
n
(C)
i
as a
quad a ic
ex ension
o
he
en eloping
algeb a
o
he
.Heisenbe g
algeb a
(see
below)
.
No e
ha
we
may
specialize
X
o
1
and
we
ob ain
An(C)
asa
specializa ion
o
i s
Rees
ing
;
mo eo e
specializing
X
o
0
yields
G(An,(O))
as
a
specializa ion
.
1
.1
.
Lemma
.
we
le
X
s and
o
he
canonical
homogeneous
cen-
al
egula
elemen
o
deg ee
one
in
R
hen
a
.
R
_
/(1
-
X)R
=R
b
.
R/XR
=
G(R)
.
The
obse a ions
in
h_e
lemma
exp ess
ha
R
is
a
"de o ma ion"
o
G(R)
ia
he
Rees
ing
R
.
No e
ha
o
any
g aded
ing
wi h
a
cen al
egula
homogeneous
elemen
he
cons uc ion
in
a in
he
lemma
yields
he
dehomogenized
il a ion
co esponding
o
a
g ada ion,
c
.
[LV02]
.
A
special
case
o his
dehomogeniza ion
p inciple
is
well-known
in
p o-
jec i e
algeb aic
geome y
(a ine
models)
and
i
is
also
e iden
in
he
ela ion
be ween
de e minen al
ings
and
Schube
cycles
.
Example
2
.
Le
g be
a
ini e
dimensional
Lie
algeb a,
say
g
=
Cxl
+
. .
+Cx
n
wi h
de ining
ela ions
[xi,
xj]
=
1
:
a
~xk
sa is ying
he Jacobi
iden i y
.
By
de ini ion
all
commu a o s
d op
in
il a ion
deg ee
o
he
usual
il a ion
de ined
on
he
uni e sal
en eloping
algeb a
U(g)
.
One
easily
checks
ha
G(U(g))
=
C[xl,
.
.
. ,
xn] (Poinca é,
Bi kho ,
Wi )
.
The
Rees
ing
U(g)-
is
gene a ed
by X,
X,
=
x1X,
. . .
,
X
n
=
x,,,X,
696

L
.
LE
BRUYN,
F
.
VAN
OYSTAEYEN
sa is ying
=
[Xi,
Xj]
=
[xi,
xj]X
2
=Ek
a
1
3
-
XkX
.

Le
us
p o ide
a
mo e
conc e e
example
and
conside
he
2-dimensional
non-Abelian
Lie
algeb a
g
=Cx
+
Cy
sa is ying
[x,
y]
=
x
.
Then
U(g)
-
is
he
egula
algeb a
in
he
sense
o
A in-Schel e
[AS]
o
[ATV1-2]
gene a ed
by
X
=
xZ,
Y
=
yZ
and
Z
(now
Z
plays
he
ole
o
he
cen al
elemen
X
be o e) sa is ying
he
quad a ic
ela ions
:
X
Z
-
ZX
=
0,
YZ
-
ZY=
0,
XY
-
YX
-
X
Z=
0
.
We
poin
ou
ha
in
he
classi ica ion
o
[ATV1]
his
algeb a
is
o
ype
SI, in
pa icula
i is
no
o gene ic,
i
.e
.
ellip ic,
ype
.
The
la e
p ope y
is
one
ha
U(g)-
will
sha e
wi h
all
o he
gauge
algeb as
de ined
la e in his
pape
.
Simila ly,
when
g
=
s12,
s1
2
=
Cx
+
Cy
+
Cz
wi h
[x,
y]
=
2y,
[x, z]
=
-2z,
[y,
z]
=
x
hen
he
Rees
algeb as
is
he
4-dimensional
quad a ic algeb a
(o
quan um
space)
gene a ed
by
X
=
xT,
Y
=
yT,
Z
=
2T
and
T
(now
playing
he
ole
o
he
cen al
elemen
o
deg ee
one),
sa is ying
:
XT
-
TX
=
0,
YT-TY
=
0,
ZT-TZ
=
0,
XY-YX-2YT
=
0,
XZ-ZX+2ZT
=
0,
YZ-ZY-
XT
=
0
.
Bu
again
u(s12)_
is
no
a
Sklyanin
algeb a
in
he
sense
o [SS]
.
Mos
o
he
algeb as
we
shall
conside
in his
pape
will
be
posi i ely
il e ed
howe e
some
localiza ions o
hese
will
be
o in e es
oo
and
so
non-posi i e
il a ions
will
appea
na u ally
.
Ex eme
amongs
he
non-
posi i e
il a ions
a e he
so-called
s ongly
il e ed
ings
.
The
il a ion
FR
is
said
o
be s ong
i
F
n
RF
n
,R
=
F
n+

,,R
holds
o
e e y
n,
m
E
7L
.
I
is
easy
o
check
ha
FR
is
s ong
i
and
only
i
G(R)
is
a
s ongly
g aded
ing,
i
.e
.
G(R)nG(R),n
=
G(R)
n+n
,,
o
n,
m
E
7L
;
o
i
and
only
i
R
is
a
s ongly
g aded
ing
.
We
say
ha
G(R)
is
d-s ongly
g aded
i
G(R)
n
dG(R)
m
d
=
G(R)(n+m)d
o
n,
m
E
7L,
and d and
is
minimal
as
such
.
Fo
a
commu a i e
posi i ely
g aded
ing
A,
P oj(A)
is
locally
s ongly
g aded
in
he
sense
ha
o
any
Za iski
open
UC
P oj(A)
he
g aded
ing
o
sec ions
is
d-s ongly
g aded
o
some
d
(depending
on
U)
;
when
A
is
gene a ed
o e
Ao
by
A
l
as
a
ing
hen
i is
e en
locally
s ongly
g aded
because
e e y
g aded
ing
o
sec ions
will
con ain
a uni
o deg ee
one
.
Kashiwa a's
ing
o
ge ms
o
mic o-di e en ial
ope a o s
on
holonomic
unc ions
p o ides
an
in e es ing
example
o
a
s ongly
il e ed
ing
.
Example
3
.
Le
z
=(z,,
...
,
z
n
)
be
coo dina es
in
Cn
and
1
=
. . . ,
in)
he
coo dina es
o
co angen
ec o s
.
Pu
Tó
(Cn)
=
{(z,
~),
0}, his
is
an open
subse
o
C2n
and
zl,
.
. .
,
zn,
~i,
.
. .
,
~n
a e
holo
mo phic
unc ions
on Tó
(en)
.
Take p
=
(z*,
~*)
ETó(e
n
)
.
Le
O
p
be
he
local
ing
o
ge ms
o
holomo phic
unc ions
(isomo phic o
he
local
ing
o
con e gen
powe
se ies
in
2n
a iables
.
Following
J .E
.
Bjd k
(p
.
136, [B])
we
le
O
p
(m)
be
he
~-homogeneous
elemen s
o
o de
m
.
QUANTUM
SECTIONS
AND
GAUGE
ALGEBRAS

69
7
I
U
is
an open
se
in
Tó
(en)
hen
O(m)(U)
is
he
se
o
holomo phic
unc ions
in
U
which
a e
~-homogeneous
o
o de
m
.
The
ing
E
p is
consis ing
o
he
E
,
(z,
~)
such
ha
E 0( )(U)
o
some
open
neighbou hood
U
o
p
and
sa is ying
he
condi ions
.
i
.
=
0
o
all
>
w,
o
a
ce ain in ege
w
.
ii
.
The e
exis
cons an s
A
and
,
such
ha
lu
=sup{i (z,~),(z,~)
E
U}
<
A(1 i!)Kwl
o
all
.
I
F
= E
,
G=
Eg
N
,
E
E
p
hen
FG
=
E(a!)
á
aa
in
mul i-index
no a ion
.
Now,
i
F=
E
,
E
Ep,
hen
he
unique
la ges
w
E
Z
such
ha

,
:7~
0
is
called
he
o de
o
F
;
his
de ines
he
il a ion
o
E
p
and
u(F)
=
w
is
hen
he
p incipal
symbol
.
Wi h
espec
o his
il a ion
G(E
p
)
=
OZ,,_1[T,T-1]
is
a
s ongly
g aded
ing
which
is
mo eo e
a
egula
Noe he ian
ing
o
pu e dimension
2n
.
The
ac
ha
E
p
is
a
Za iski
ing
(see
Sec ion
4) en ails
all
he
desi ed
p ope ies
o
E
p
.
Al hough
we
a e
mainly
conce ned wi h
ings
he e
i is
use ul
o
es-
ablish
he
co esponding
module
heo y as
well
.
An
R-module
M
is
il e ed
i
he e
is
an
ascending
chain
o
addi i e
subg oups
F
n
M,n
E
7L,
sa is ying
F
n
RF
n
M
C
Fn+n,M
o
n,m
E
7L
.
The
ca ego y
R- il
is
ob ained
by
aking
he
il e ed
R-modules
and
he
R-linea
maps
p e-
se ing
il a ion
deg ee
o
he
objec s
and
mo phisms
.
We
w i e
FM
o
he
il a ion
o
M
and
G(M)
=
®,EZFnM/Fn_1M
o
he
associ-
a ed
g aded
G(R)-module
.
The
G o hendieck
ca ego y
o
g aded
G(R)-
modules
_
will
be
deno ed
_
by G(R)-g
.
Simila ly,
we
may
de ine
he
Rees
module
M
o
FM
by
M
=
®nEaFnM
and
iden i y
i
wi h
a
submod-
ule
o
M[X,
X
-1
]
.
Again
we
always
assume
ha
FM
is
exhaus i e,
M
=
U
n
M_and_sepa a ed
Le
.
n
n
F
n
M
=
0
.
We
m_ay
ex end
Lemma
1 .1
.
o
:
M/XM
=
G(M),
M/(1-
X)M
--
M,
ú(x)
=
M[X,X-1]
whe e
(-)(X)

_
s ands
o
he objec
localized
a
he
cen al
mul iplica-
i e se o
homogeneous
elemen s
{1
,X,
X2,
.
.
.
}
.
In
R-g
we
ha e
a
ull
subca ego y
FX
consis ing
o
he
X- o sion ee
g aded
R-modules
.
1 .2
.
Lemma
.
The
unc o
-
:
R- il
---->
R-g
de ines
an
equi alen e
o
ca ego ies
be ween
R- il
and
.FX
.
The
il e ed
mo phism
in
R- il
co esponding
o
he
mo phisms
in
.FX
a e
he
s ic
mo phisms
( ecall
ha
a
il e ed
mo phism
:
M
-->
N
is
s ic
i
FNN
nIm
=
(F
n
M))
.
The
unc o
G
:
R- il
,
G(R)-g
is
no
eally
exac
bu
á is
exac
o_n
s ic
mo phims
an_d
sequences_o
s ic
mo phisms
;
he
unc o
D
:
R-
g
-+
R- il
M
H
M/(X
-
1)M,
is
exac
.
Wemay
de ine
a
p incipal

69
8

L
.
LE
BRUYN,
F
.
VAN
OYSTAEYEN
symbol
map
o
m
:
M
-->
G(M)
by
pu ing
o
m
(m)
=
m
mod
F,,
M
when
m
E
F
n
M
-
F
n
_1M
.
Fo
E
R
such
ha
QR(a)UM(m)
qÉ
0
we
ha e
QR( )om(m)
=
om( m)
.
In
pa icula ,
when G(R)
is
a
domain
hen
a
=
R
is
mul iplica i e
.
A
il a ion
FM
is
said
o
be
a
good
il a ion
i
he e
exis
m,,
...
M
and
d,
...
d
EZ
such ha
o
all
n
E
7L,
F
n
M
=
~i-1
Fñ_d
¡
Rm
i
.
The
u ili y
o
he
Rees
objec s
is
ha
p ope ies
con-
ce ning
he
il a ion
FM_
a e
ansla ed
o
p ope ies
in
.FX
conce ning
he
X-adic
il a ion
on
M
.
1
.3
.
P oposi ion
.
Wi h
no a ion
as be o e
a
.
FM
is
sepa a ed
i
and
onl_y
i
M
is
X-adically
sepa a ed
.
b
.
FM
is
good
i
and
only
i
M
is
ini ely
gene a ed
.
c
.
F_1R
C
J(FoR)
i
and
only
i
X
E
J
9(R),
whe e
J
9
(-)
s ands
o
he
g aded
Jacobson
adical
c
.
[NVO]
.
d
.
FM
is
comple e
i
and
only
i
M_is
X-adically
comple e
e
.
FM
is
p ojec i e
i
and
only
i
M
is
p ojec i e (simila
o
la -
ness)
.
.
Amap
:
M

+
N
is
s ic
i
and
only
i
Coke
E
.F
X
.
2
.
Quan um
sec ions
as
de o ma ions
o
localiza ions
In his
sec ion
we
es ic
a en ion o
posi i ely
il e ed
ings
R
ha ing
a
commu a i e
Noe he ian
domain
o
he
associa ed
g aded
ing
.
The
essen ial
pa
o his
sec ion
can
and
will
be
conside ably gene alized
in
a
u he sec ion
( o
Za iskian
il a ions)
.
Conside
a
mul iplica i ely
closed
se
S
in
R,
1
E
S,
0
1
S,
such
ha
u(S)
is
mul iplica i ely
closed
in
G(R)
( his
holds
au oma ically
when
G(R)
is
a
domain
because
u
is
mul iplica i e
in
ha case)
.
Since
Q(S)
consis s o
homogeneous
elemen s,
(S)
-1
G(R)
is
a
g aded
ing
.
In gen-
e al
S
is
no
an
O e
se
and
so
one
canno
necessa ily
o m
S`R
.
This
d aw-back
may
be o e come by
in oducing he
algeb aic
mic olocaliza-
ion
o
R
a
o,(S),
we
ollow
he
ideas
o
[AVV]
.
To
he
se
S_we
associa e
he
mul iplica i ely
closed
se
S
in
R,
S
__
{s,
s
_
=
sX'
E
R
n
o
s
E
S
such
ha
s
E
F,,
R-F_
n
_1R}
.
Clea ly
1
E
S,
0
1
S
and
S
consis s
o
homogeneous
elemen s_o
R
._
Fo
n_
E
N_ he e
is
a
canonical
_
epimo phism
o
_
g ade
_
d
ings
:
in,
R/XnR
->
R/XR
--
G(R)
.
Le
S(n)
be
he
image
o
S
in
R/XnR
.
Now
ke ~i
n
i
_
s
nilpo en
o
index
n
and
~!n(S(n))
=
o,(S)
is
an
O e
se
in
G(R),
hence
S(n)
is
an
O e
se
in
QUANTUM
SECTIONS
AND
GAUGE
ALGEBRAS

69
9
R/X'R
.
The e o e
we
can
de ine
:
B
=
Q'
(R)
=
1lim
95(n)-1(R/XnR),
n
whe e
lim
s ands
o
he
g aded
in e se
limi
( he
di ec
sum
o
he
9
in e se
limi s
o
he
sys ems
ob ained
in
each
deg ee
o
he
g ada ion)
.
2
.1
.

Lemma
.
The
i

map
js
=
-
,ú
---_>
B
is
a
g aded
ing
mo phism
.
We
ha e
B
E
.FX
and
also
B/js(R)
E
.FX,
mo eo e
B/XB
=
o,(S)-1G(R),
c
.
[AVV]
.
A e
his
lemma
we
may
ake
he
dehomogeniza ion
o
B,
Q
S
'(R)
_
B/(1
-
X)B
and
we
ob ain
a
il e ed
ing
Qs(R)
such
ha js
:
R
Ql(R)
is
a
s ic
il e ed
mo phism,
ha
is
he
il a ion
o
R
is
induced
by
he
il a ion
o
B=
Q"
(R),
and
G
(B)
--
a(S)
-1
G(R)
.
Tha
Q
5
`
(R)
is
comple e
(bu
no
posi i ely
il e ed)
is
easily
e y ied
.
In
ac
Q
S
'(R)
is
no hing
bu
he
mic o-localiza ion
a
u(S)
as
de ined
by
T
.
Sp inge
in
[Sp ]
.
This
ollows
om
he
ac
ha
B
has
he
uni e sal
p ope y
men ioned
in
he
ollowing
.
2
.2
.
Lemma
.
Fo
s
E
S,
js(s)
is
in e ible
in
B
and
i
B'
is
ano he
il e ed
ing
such
ha
FB'
is
comple e,
js
:
R
---
~
B'
is
a
s ic
inclusion
and
o
e e y
s
E
S
wi h
o,(s)
E
G(R)
n
we
ha e
s
-1
E
B'
wi h
u(s
-1
)
E
G(B')_
n
,
hen
he e
exis s
a
s ic
ac o iza ion
h
:
B
-> B' such
ha
hjs
=
js
,
(c
.
[AVV])
.
Since
Q
S
'(R)
is
comple e
and
G(Q'(R))
=
u(S)
-1
G(R)
is
a
commu a-
i e
Noe he ian
domain,
his
ing
will
be
in
he
class
o
ings
we
conside
( hough
no
posi i ely
il e ed)
.
We
may
iew
Q
S
'(R)
as
a
"de o ma ion"
o
o,(S)-1G(R)
ia he
co esponding
Rees
ing
.
When
conside ing
he s uc u e
o
P oj(G(R))
we
ha e
o
es ic
a -
en ion
o
he
pa
o
deg ee
ze o
o
o,(S)-1G(R),
say
G(R)(a(s))
=
(Q
(S)-1G(R))o
.
In
pa icula
we ha e
:
FOQs(R)/F_1QS(R)
=
G(R)Q(s))
.
We
de ine
he
quan um
sec ions
o
R
a
S
o
he
ing
FoQ1(R)
=
Qs(R)0
equipad
wi h
he
induced
il a ion
.
We
deno e
his
ing
by
R(s
) .
The
F-sa u a ion
o
S
is
S
=
{
E
R,
u( )
E
u(S)}
.
I
is
clea
ha
S
is
mul iplica i ely
closed, 1
E
S,
01
S,
and
a(S)
=
o,(S)
.
2
.3
.
P oposi ion
.
Wi h
no a ion
and
con en ions
as be o e
we
ha e
Q"
(R)
=
Q"
(R)
.
Mo eo e
;
S
is
an O e
se in
R
and
o
he
localizad
il a ion
on
(S)
-1
R
(being
he
one
induced
on
i
om
Q'
(R))
we
ha e
ha
Q
S
'(R)
=
((S)
-1
R)
^
,
i
.e
.
he
mic olocaliza ion
can
always
be
ob-
ained
as a
classical
localiza ion
ollowed by
a
comple ion
.
700

L
. .
LE
BRUYN,
F
.
VAN
OYSTAEYEN
Combining
P oposi ion
2
.3
.
wi h
o egoing p ope ies
and
P o_pos_i ion
1
.3 .d
.
we know
o
a
sa u a ed
S,
i
.e
.
S
=
S_

_

Qs(R)
-
:
Qs(R)
is
he
g aded X-adic
comple ion
o

The
de ini ion
o
he g aded
comple ion
yields
ha
Q'(R)o
=
1S)-1RIXn
.(S)-1R)o)
n
=1~((S)-1R)o/(Xn(S)-1R)o)
n
=lS)-1R)o/Xn((S)-1R)-n)
n
llimF
o
S
-1
R/F-
n
S
-1
R
=
(FoS_1R)n
n
whe e
A
s ands
o
he
comple ion
wi h
espec
o
he
il a ion
induced
F5
-1
R
(o
by
FQ"(R))
in
FOS
-1
R
.
This
p o ides
a
way
o
calcula e
quan um
sec ions
e ec i ely
by
i s
calcula ing
(S
-1
R)o
and
hen
a1-
lowing
he
sui able
comple ion
.
Example
4
.
[RVO]
Conside
he
i s
Wey]
.
algeb a
R
=
A,
(C)
.
F om
he
o egoing
sec ion
we know
ha
G(A
1
(C))
=
C
[X,
y]
and
A1
(C)
^'
C
[X,
Y,
Z]
/
(XY
-
YX
-
X2,
YZ
-
ZY,
X
Z-
ZX
),
whe e
we
lla e
pu
X
=
xZ,
Y=
yZ,
Z
being
he
egula
cen al
homogenous
elem
_
en
o
deg ee
one
.
Conside
S=
{1, x,
x2
. . .
}
in
C
[x, y]
.
The
sa u a ion
S
o
S
consis s
o
all
elemen s
such
ha a( )
=Xn
o
some
n
and
his
is
an
O e
se
.
No e
ha
all
elemen s
o _ he
o m
A
+
x,
A
E
C,
a e
con ained
in
S
.
The
homogeneous
O e
se
is
S=
{ Zn,
=
xn+Ek+ <n
CUMxkya}
and
we
may
w i e
Zn
as
a
homogeneous
o m
o deg ee
n
in
he
new
a iable
:
X
=
xZ,
Y
=yZ
and
Z,
e
.g
.
x2
+y+1
is
li ed
o
X
2
+YZ+Z
2
.
In
(S
--1
R)o
we
ind
he
elemen s
YX
-1
,
X
-1
Y,
ZX
-1
,
X
-1
Z
and
hese
sa is y
he
ela ions
:
YX
-1
-
X
-1
Y
=
(ZX
-1
)
2
,
ZX
-1
-
X
-1
Z =
0
.
So
we
ha e
o
comple e
he
commu ing
ela ions o
he
gene a o s
p
=
YX
-1
and
q
=
ZX
-1
.
Now
qp
=
ZX
-1
YX
-1
=
Z(YX
-1
-
Z
2
X
-2
)X
-1
=
YZX
-2
-Z
3
X
-3
=
XY
-1
ZX
-1
-Z
3
X
-3
=
pq
-
q3
.
Vence
we
a i e a
[p,
q]
=
q
3
.
No e
ha
Z
E
J
9
(R)
hence
Z
C-
J
9
(9
-
1
,ú)
and
he e o e
e e y
elemen
o
he
o m
Xn
+ZF
n
-
1
(X,
Y,
Z),
whe e
F
n
-1
(X,
Y,
Z)
is
homogeneous
o
deg ee
n
-
1,
has
o
yield
an
in e ible
1
+
ZX
-1
(X
1-
nF
n
_
1
(X,Y_
_ Z))
in
he
comple ion
o
_(S
-1
R)o
because
ZX
-1
E
J
9
(S
-
1R)o,
X
1-n
Fn_1(X,
Y,
Z)
E
(S
-1
R)o
.
The e o e
we
may
Conside
he
algeb a
C
(p,
q)/(pq
-
qp
=
q3
)
as
de e mining
he
quan um
sec ions
up
o
comple ion
.
QUANTUM
SECTIONS
AND
GAUGE
ALGEBRAS

70
1
Example
5
.
Conside
he
wo-dimensional
Lie
algeb a
g
=
Cx+Cy
wi h
[x,
y]
=
x
.
Then
U(g)
-
is
he
quad a ic
algeb a
gene a ed
by
X,
Y,
Z
sa is ying
:
XZ-ZX
=
0,
YZ-ZY
=
0,
XY-YX
=
XZ
.
Le
u(S)
be
{1, y, y2,
. . .
}
in
C
[x,
y]
=
G(U(g))
.

The
sa u a ed
O e
se in
U(g)
is
S=
{
=
yn
+
Ek+ <
.
.
aklxky'}
and
his
li s
o
a
homogeneous
O e
se in
U(g)
-
,_S
=
{
Z'
2
,
E
S
ha ing
Q( )
=
yn}
.
Some
ela ions
in
deg ee
ze o
o
S
-1
U
(g)
-
de i e
om
he
de ining
ela ions
abo e
ZY
-1
=
Y
-
'Z,
Y
-1
X
-
XY
-1
=Y
-1
XZY
-1
.
Conside
he
canoncial
gene a o s
p
=
XY
-1
and q
=ZY
-1
,
hen
we
calcula e
qp
=
ZY-1XY-1
=
Z(XY
-1
+Y
-1
XZY
-1
)Y
-1
=
XZY
-2
+
ZY-1XY-1ZY-1
=
pq
+
qpq
.
Yielding
he
a he
odd
ela ion
[p, q]
=
-qpq
.
Howe e ,
since
Y-Z
E
S
we
mus
in e 1
-
q
in
he
quan um
sec ions
so
we
may
ew i e
he
commu a ion
ela ion
as
qp
=
p
11
4
and
so
we
may
look
a
he
skew
polynomial
ing
C
[[q]]
[p,
-y]
whe e
,y
is
he
au omo phism
de ined
by
p
H
1
q
q
and
see
ha
C
[[q]]
[p,
-y]
de e mines
he
quan um
sec ions
o
U(g)
(no e
ha
as in
Example
4,
i
su iced
o
in e
one
elemen , he e
Y,
in
o de
o
ind
up
o
comple ion
he
quan um-sec ions)
.
One
should
no
conclude
om
he
examples
4
.
and
5
.
ha
quan um-sec ions
o
some
Q(S)
o
he
ype
{1,
a,
a
2
. . . .
}
a e
always
ha
easy
o ob ain
.
Example
6
.
Le
g
be
he
Lie
algeb a
812
and
change
he
812
-
basis
such
ha
[Y,
Z]
=
X,
[Z,
X]
=Y
and
[X,
Y]
=Z
.
Conside
he
mul iplica i e
se
Q
(S)
=
{1,
X,
X2
,.. .
}
.
The
eade
may
check
ha
he
commu a ion
o mulas
de e mining
he
quan um
sec ions
o
U(sl2)
a
Q(S)
may
be
gi en
as
[A,
B]
=
(A2
+
B2
+
1)C
C
2
C2
[A,C]=AC
1+CZ
+B
1
+C
2
C
z
C
Z
[B,C]=BC
1+C
2-A
1+C
2
whe e
A=
YX
-1
,
B=
ZX
-1
and
C=
TX-1
.
Again
i is
use ul
o
in oduce
he
quan um
sec ion
o
il e ed
mod-
ules
.
Fi s ,
in
a
way
o mally
simila
o
he
way
Q'(R)
had
been
con-
s uc ed
we
may
de ine
Q'(M)
o
any
sepa a ed
il e ed
R-module
M
70
8

L
.
LEBRUYN,
F
.
VAN
OYSTAEYEN
good
il a ion
.
ii
.
E e y
good
il a ion
is
sepa a ed
.
I
u ns
ou
h_a
i
and
ii
a e
equi alen
o
he
ollowing
:
R
is
Noe he ian
and
X
E J
9
(R)
( o
se e al
equi alen
s a emen s
we
e e
o
[LVO1],
and
in his
case
he
il a ion
FR
is
said
o
be
Za iskian
.
E e y
comple e
il e ed
ing
R
such
ha
G(R)
is
Noe he ian
is
a
Za iskian
ing
and
in
pa icula
he
posi i e
case
wi h
Noe he ian
associa ed
g aded
ing
is
also a
pa icula
case
.
The
esul s o
Sec ion
1
and
Sec ion
2
emain
alid
o
Za iski
ings
in
gene al
;
e en
o
Sec ion
3 one can do a
lo
bu
one has
o
de ine
a
sui able opological
space
i s
.
No e
ha
we
will
assume
he
condi ions
i
and
ii
o
bo h
le
and
igh
modules,
so
he
Za iski
ings
men ioned
he e
a e
le
and
igh
Za iski
ings,
as
in
[LVO,
1,2]
.
We
men ion
some
undamen al
esul s
s emming
om
[LVO,
1, 2,
.
.
.] .
4
.1
.

Theo em
.
Le
R
be
a
Za iski ing
.
I
G(R)
has
ini e
global
dimension
hen
i .
gldim
R
=
1
+
gldim
G(R)
ii
.
gldim
R
<_
g gldim
G
(R)
=
g gldim
R
-
1,
whe e
g gldim
s ands
o
he_gldim
in he
g aded
ca ego y
.
iii
.
gldim
R=
gldim
R
4
.2
.
Theo em
.
Le
R
be a
Za iski ing
.
I
_
G(R)
i
.s
a
egula
Noe he-
ian
(in
he
sense
o
Auslande )
hen
R
and
R
a e
egula
Noe he ian
.
Recall
ha
a
non-commu a i e
ing
is
egula
in
he
sense
o
Auslan-
de
i
i
has
ini e global
dimension
and
e e y
ini ely
gene a ed
le
o
igh
module
sa is ies
he
Auslande
condi ion
;
ecall
ha
a
ini ely
gene a ed
R-module
M
sa is ies
he
Auslande
condi ion
i
o
e e y
0
<_
k
<
lc
=
gldim
R
and any
nonze o
R-submodule
N
o
Ex
k
(M, R)
we
ha e
jR(N)
>
k,
whe e
jR(-)
s ands o
he
g ade
numbe ,
i .e
.
he
smalles
na u al
numbe
j
such
ha
Ex j(-, R)
=,A
0
.
4
.3
.

Co olla y
.
I
9
is
a
Noe he ian
gauge
algeb a
hen
9
and
1
a e
Noe he ian
egula
algeb as
.
I
_
G(9)
is
an
n-dimensional
quan um
space
hen
:
gldim

=
1
+
n,
gldim
9
<n
.
Mo eo e ,
G
is
an n
+
1-dim
quan umspace
.
The
"scheme"- heo e ic
ea men
o
gauge
algeb as
has
i s
oo s
in
ying
o
unde s and
he
geome y
o
so-called
innocen
quan um
spaces,
i .e
.
quan um
spaces ha
a e
Noe he ian
and
posessing
a
cen al
elemen
o
deg ee
one
.
The
innocen
quan um
space co esponding
o
a gauge

QUANTUM
SECTIONS
AND
GAUGE
ALGEBRAS

70
9
algeb a
g
is
i s
Rees
ing
9
.
In
[A ]
he
quan um
p ojec i e
space
o
a
quan um
n-space
Q
is
de ined
o
be
P oj(Q)
=
Q-g
.F,
whe e
*F
is
he
ull
subca ego y
o
ini e
leng h
modules,
oge he
wi h
a
shi
ope a ion
[A , De ini ion
1
.2
.]
.
Fo
a
commu a i e
ing,
o
one
ha
is
a
ini e
module
o e
i s
cen e ,
one
may
eco e
he
unde lying
scheme
s uc-
u e
om
Se e's
heo em
.
In
gene al
howe e
he
"scheme"
s uc u e
o
P oj(Q)
as
de ined
abo e
is
a
om
being
unde s ood
.
A
i s
eeling
o
he
unde lying
p oblems
can be
ob ained
by
conside ing
pa icula
modules,
i.e
.
he
poin -
and
line-modules,
c
.
[ATV2],
and
he
a
poin
modules
in oduced
in
[A ]
.
I
we
es ic
a en ion
o
he
geome y
o
innocen
quan um
spaces
we
can
use
he
gauge
algeb a
and
i s
quan-
um
sec ions
o
pu
a
"scheme"
s uc u e
on
P oj(~)
which
educes
he
s udy
o
( a )
poin -modules
o
ha
o
ini e
dimensional
ep esen a-
ions o
algebas,
c
.
[LB]
.
In
ac ,
we
may
iew
P oj(U)
as
an
a ine
piece
co esponding
o
he
gauge
algeb a
g
and
a
piece
a
in ini y
iden i-
ied
o
P oj(~/ ~)
=
P oj(G(Cg))
.
Tha
is
a
lowe dimensional
p ojec i e
quan um
space
.
Assume
by
induc ion
ha
we
ha e been
able
o
pu
a
scheme
s uc u e
on
P oj(G(g))
wi h
a ine
open
se s
co esponding
o
some
g aded
O e
se s S,,
. . .
,
Sk and
i
is
no
es ic i e
o
assume
ha
each
o
hese
O e
se s
may
be
gene a ed
by a
single
elemen
.
Thenwe
may
co e
P oj(~)
by
open
se s
co esponding
o
he
quan um
sec ions
wi h
espec
o
he
Si,¡
=
l,
. . . ,
k
plus
he
app op ia e
glueing
mo -
phisms
.
No e
ha
hese
quan um
sec ions
and
hei
"glues"
wi h
Gmay
be
iewed
as
a scheme
s uc u e
on
P oj(1)
.
Le
us
p o ide
some
easy
examples
he e
.
Example
11
.
Reconside
he
i s
Weyl
algeb a
A
l
(O)
.
We
use
he
calcula ions
made
abo e
o
ex end he schema ic
pic u e
in
Example
7
by
glueing
o
he
open
se
co esponding
o
Y(Z)
i.e
.
he
a ine
piece
co esponding
o
(Aj(C)
.
We
ob ain he
ollowing
diag am
o
glueing
da a
C{XZ
-1
,YZ
-1
,ZX
-1
}
~-
Aj(C)=C{XZ-1,YZ-1}
~
C{XZ
-1
,YZ
-1
,
ZY
-1
}
[XZ
-1
,YZ
-1
]
=
1
C{YX
-1
,
ZX
-1
}

C{XY
-1
,
ZY
-1
}
[YX
-1
,
ZX
-1
]
=
(
ZX
-1
)
3

[XY
-1
,
ZY
-1
]
=
(ZY-1)3
1
e{ZX
-1
,YX
-1
,
XY
-1
}
To
ind
i s
poin
modules
we
ha e
o
s udy
he one-dimensional
ep e-
710

L
.
LEBRUYN,
F
.
VAN
OYSTAEYEN
sen a ions
and
hei
glueing
da a
.
V((ZX_
1)
3)

V((ZY_
1
)3)
V((ZX
-1
)
3 )
n
V((ZY
-1
) 3
)
co esponding
o
he
ac
ha
he
associa ed
deg ee
3
di iso
o
he
quan um
3-space
A1(C)-
is
Z
3
.
This
may
be pic u ed
in
he
usual
p2
.
y(y)-poin s
z=0
whe e
he
ci cle
means
ha
he
in e sec ion
poin
is
missing
and
we
ha e
d awn
a
double
copy
o
he
same
z
=
0
locus
.
Example
12
.
Reconside
he
si ua ion
o
Example
8
.
Using
he
compu a ions
and
no a ions
o
ha
Example
8,
we
ob ain
he
ollowing
diag am
o
glueing
da a
whe e
QUANTUM
SECTIONS
AND
GAUGE
ALGEBRAS

71
1
U(g)
C{XZ
-1
,
ZX
-1
,YZ
-1
}
--
C{XZ
-1
,YZ
-1
}
--,
C[[ZY-1,YZ-1]][XY-1,7]
[XZ
-1
,
YZ
-1
]
=
XZ-1
C{YX
-1
,
ZX-1}

CpY-1]][XY-1,7]
[YX
-1
,
ZX
-1
]
=
(ZX-1)z
Only
he op
le
co ne
(glueing
he
en eloping
algeb a
o
he
excep-
ional
quan um
space)
esembles
he
commu a i e
case
.
The
o he
wo
co ne s
ha e
sh unk
in
dimension
.
I
may
be
help ul
in
unde s an_ding
hese
phenomena
o
look
a
he
pic u e
o
poin -modules
in
P oj(U(g))
pic u ed
in
he
usual
p
2
C[[ZY-1]][XY-1,YX-1,7]
he
y(x)-poin s
a e
z
=
0
wi hou
he
poin
a,
he
y(z)-poin s
a e
x
=
0
wi hou
he
poin
a,
he
y(y)-poin s
a e
a
.
z=0
Example
13
.

The
quan ized
Weyl
algeb a
A
l
(C,
q)
as a
gauge
algeb a
.
We
use
no a ion
and
calcula ions
as in
Example
9
.
Pu
R
=
Al
(C,
q)
.
Bo h
X
and
Y
a e
no malizing
in
he
quan um
plane
C
e
[X,Y],
so
we
may
de ine
a
"scheme"
s uc u e
on
P oj(R)
by
gi ing
712

L
.
LE
BRUYN,
F
.
VAN
OYSTAEYEN
he
ollowing
glueing
da a
.
C{XT
-1
,TX
-1
,YT
-1
}
--
{XT
-1
,YT
-1
}
--
.
C{XT-1,YT-1,TY-1}
XT
-1
,
YT
-1
-
qYT
-1
XT
-1
=
1
C{YX
-1
,TX
-1
}
C{XY
-1
,TY
-1
}
YX
-1
TX
-1
-qTX
-1
YX
-1
=(TX
-1
~
XY
-1
TY
-1
-
1-
'
TY
-1
XY
-1
=-Q(TY
-1
~
C{YX
-1
,
XY
-1
,TX
-1
}
and
again
one
can
isualize
' he
poin
modules
as
poin s
in
p
2
.
The
pic u e
co esponds
o
he
ac
ha
he associa ed
deg ee
3
di iso
is
T(T
Z
+
(q
-
1)XY)
whe e
C
is
he
conic
de ined
by
TZ
+
(q
-
1)XY
.
Y(T)
poin s a e
C
-
{a,
b}
Y(X)
poin s
a e
C
-
{b}
U
(T
=
0)
-
{b}
Y(Y)
poin s
a e
C
-
{a}
U
(T
=
0)
-
{b}
=0
The
scheme
s uc u e
o
A
l
(C,
q)
-
desc i
_
bed
abo e
is
c i ical
in de in-
ing
he
scheme
s uc u e
on
P oj
(W)
whe e
W
is
he
4-dimensional
quan-
um
space
o
he
Rees
ing
o
he
Wi en gauge
algeb as
.
Fo ,
i is
pos-
sible
o
change
he
pola iza ion
on
P oj(G(W))
as
in
[A ]
o
ob ain
P oj
(Al
(C,
q)
-) and
use
he
o egoing
in
o de
o
de ine
a scheme
s uc-
u e
on
P oj(G/W))
ha
is
hen
li ed
ia
quan um
sec ions
o
P oj(W)
as in
[LVW]
.
In
a
simila
way
one can
s udy
he
a -poin
modules
o
mul iplici y
n
(as in
[A ])
in
an
innocen
p ojec i e
quan um
space
by
glueing
oge he
he
n-dimensional
ep esen a ion
o
he
scheme
compo-
nen s
.
QUANTUM
SECTIONS
AND
GAUGE
ALGEBRAS

71
3
Re e en es
[A ]
M
.
ARTIN,
Geome y
o
Quan um
Planes,
P ep in
MIT
(1991)
.
[AS]
M
.
ARTIN
AND
W
.
SCHELTER,
G aded
Algeb as
o
Global
Di-
mension
Th ee,
Ad
.
Ma h
.
66
(1987),
171-216
.
[ATV1]
M
.
ARTIN,
J
.
TATE
AND
M
.
VAN
DEN
BERGH, Some
Al-
geb as
Associa ed
o
Au omo phisms
o
Ellip ic
Cu es,
in
"The
G o hendieck
Fes sch i ,
Vol
.
I,"
Bi khause
(1990),
pp
.
33-85
.
[ATV2]
M
.
ARTIN,
J
.
TATE
AND
M
.
VAN
DEN
BERGH,
Modules
o e
Regula
Algeb as
o
Dimension
Th ee,
MIT
.
[AVV]
M
.-J
.
ASENSIO,
M
.
VAN
DEN
BERGH
AND
F
.
VAN
OYS-
TAEYEN,
A
New
Algeb aic
App oach
o
Mic olocaliza ion
o
Fil e ed
Rings,
' ans
.
Ame
.
Ma h
.
Soc
.
[B]
J
.
E
.
BJóRK,
"Rings
o
Di e en ial
Ope a o s,"
Ma h
.
Lib a y
21,
No h
.
Holland,
Ams e dam,
1979
.
[LB]
L
.
LE BRUYN,
Gauge
Algeb as,
In p epa a ion
.
[LVO1]
LI
HUISHI
ANDF
.
VAN
OYSTAEYEN,
Za iskian
Fil a ions,
Comm
.
i
n
Algeb a
17(12)
(1989),
2945-2470
.
[LVO2]
LI
HUISHI
ANDF
.
VAN
OYSTAEYEN,
Global
Dimension and
Auslande
Regula i y
o
Rees
Rings,
Bull
.
Soc
.
Ma h
.
Belg
.
[LVW]
L
.
LE
BRUYN,
F
.
VAN
OYSTAEYEN
AND
L
.
WILLAERT,
Quan um
Sec ions
o
Schema ic
Algeb as,
UIA
p ep in ,
No embe
(1992)
.
[Man]
YU
1
MANIN,"Quan um
G oups
and
non
Commu a i e
Geom-
e y,"
Publ
.
Cen e
P ech
.
Ma h
.
Mon éal,
1988
.
[RVO1]
A
.
RADWAN
AND
F
.
VAN
OYSTAEYEN,
Cohe en
Shea es
o e
Mic os uc u e
Shea es,
UIA
p ep in
(1991),
P oceedings
o
he
Colma
mee ing
.
[RVO2]
A
.
RADWAN
AND F
.
VAN
OYSTAEYEN,
Mic o-s uc u e
Shea es
and
Quan um
Sec ions o e
Fo mal
Schemes,
UIA
p ep in
(1991),
o
appea
in
Bull
.
Soc
.
Ma h
.
Belg
.
(1993)
.
[SVO]
R
.
SALLAM
AND F
.
VAN
OYSTAEYEN,
A
Mic o-s uc u e
shea
and
Quan um
Sec ion
o e
a
P ojec i e
Scheme,
UIA
p ep in
(1990),
J
.
o
Algeb a
(1992)
.
[Sp ]
T
.
SPRINGER,
Algeb aic
Mic olocaliza ion,
in
"Sém
.
M
.
P
.
Malli-
a in,"
Lec u e
No es
in
Ma h
.,
Sp inge
Ve lag,
Be lin,
1984
.
[Schap]
P
.
SCHAPIRA,
"Mic odi e en ial
Sys ems
in he
Complex
Plane,"
G undleh en
de
Ma h
.
Wiss
.
269, Sp inge
Ve lag,
Be lin,
1985
.

714

L
.
LE
BRUYN,
F
.
VAN
OYSTAEYEN
[SS]
S
.
P
.
SMITH
AND
J
.
T
.
STAFFORD,
Regula i y
o
he
Fou
Di-
mensional
Sklyanin Algeb a, P ep in
Ann
.
A bo
(1990)
.
[Gin]
V
.
GINSBURG,
Cha ac e is ic
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2610
W¡I ijk
BELGIUM
Rebu
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14
d'Oc ub e
de
1991