A non-commutative homogeneous coordinate ring for the degree six del Pezzo surface

0.18 MB·English

by  S. Paul Smith

A NON-COMMUTATIVE HOMOGENEOUS COORDINATE RING
FOR THE THIRD DEL PEZZO SURFACE
9
0 S.PAULSMITH
0
2
Abstract. Let R be the free C-algebra on x and y modulo the relations
n x5=yxy andy2=xyxendowedwiththeZ-gradingdegx=1anddegy=2.
u Let B3 denote the blow up of CP2 at three non-colliear points. The main
J result in this paper is that the category of quasi-coherent OB3-modules is
6 equivalent tothequotient ofthecategory ofZ-gradedR-modulesmodulothe
1 full subcategory of modules M such that for each m ∈ M, (x,y)nm = 0 for
n≫0. Thisreduces almostallrepresentation-theoretic questions aboutRto

] algebraic geometric questions about the del Pezzo surface B3. For example,
A thegenericsimpleR-modulehasdimensionsix. Furthermore,themainresult
R combined with results of Artin, Tate, and Van den Bergh, imply that R is a
noetheriandomainofglobaldimensionthree
.
h
t
a
m
[ 1. Introduction

2 We will work over the field of complex numbers.
v The surface obtained by blowing up P2 at three non-colinear points is, up to
1 isomorphism,independent of the points. It is called the third del Pezzo surface and
8 we will denote it by B .
4 3
Let R be the free C-algebra on x and y modulo the relations
2
6. (1-1) x5 =yxy and y2 =xyx.
0
Give R a Z-grading by declaring that
9
0 degx=1 and degy =2.
:
v
Inthispaperweprovethereisasurprisinglycloserelationshipbetweenthenon-
i
X commutative algebra R and the third del Pezzo surface. This relationship can be
r exploited to obtain a deep understanding of the representation theory of R.
a
Theorem 1.1. Let R be the non-commutative algebra C[x,y] with relations (1-1).
Let GrR be the category of Z-graded left R-modules. Then there is an equivalence
of categories
GrR
QcohB ≡
3 T
wheretheleft-handsideisthecategoryofquasi-coherentOB3-modulesandtheright-
hand side is the quotient category modulo the full subcategory T consisting of those
modules M such that for each m∈M, (x,y)nm=0 for n≫0.
Theorem 1.1 is a consequence of the following result.
1991 Mathematics Subject Classification. 14A22,16S38,16W50.
Key words and phrases. Homogeneous coordinatering,noncommutative, delPezzosurface.
Theauthor wassupportedbytheNationalScienceFoundation, AwardNo. 0602347.
1
2 S.PAULSMITH
Theorem 1.2. Let R be the non-commutative algebra C[x,y] with relations (1-1).
There is an automorphism σ of B having order six, and a line bundle L on B
3 3
such that R is isomorphic to the twisted homogeneous coordinate ring
B(B ,L,σ):= H0(B ,L )
3 3 n
n≥0
M
where
L :=L⊗(σ∗)L⊗···⊗(σ∗)n−1L.
n
Results of Artin, Tate, andVan den Berghnow imply that R is a 3-dimensional
Artin-Schelter regular algebra and therefore has the following properties.
Corollary 1.3. Let R be the algebra C[x,y] with relations (1-1). Then
(1) R is a left and right noetherian domain;
(2) R has global homological dimension 3;
(3) R is Auslander-Gorenstein and Cohen-Macaulay in the non-commutative
sense;
(4) the Hilbert series of R is the same as that of the weighted polynomial ring
on three variables of weights 1, 2, and 3;
(5) R is a finitely generated module over its center [8, Cor. 2.3];
(6) R(6) :=⊕∞n=0R6n is isomorphic to ∞n=0H0(B3O(−nKB3);
(7) SpecR(6) is the anti-canonical cone over B , i.e., the cone obtained by
3
L
collapsing the zero section of the total space of the anti-canonical bundle
over B .
3
This close connection between R and B means that almost all aspects of the
3
representation theory of R can be expressed in terms of the geometry of B . We
3
plan to address this question in another paper.
The justification for calling R a non-commutative homogeneous coordinate ring
for B is the similarity between the equivalence of categories in Theorem 1.1 and
3
following theorem of Serre:
if X ⊂Pn is the scheme-theoretic zero locus of a graded ideal I in
the polynomial ring S = C[x ,...,x ] with its standard grading,
0 n
and A=S/I, then there is an equivalence of categories
GrA
(1-2) QcohX ≡
T
where the right-handside is the quotient category of GrA, the cat-
egory of graded A-modules, by the full subcategory T of modules
supported at the origin.
Acknowledgments. Theauthorisgratefultothefollowingpeople: AmerIqbal
for directing the author to some papers in high energy physics where the algebra
R appears in a hidden form; Paul Hacking and Sa´ndor Kov´acs for passing on
some standard results about del Pezzo surfaces and algebraicgeometry; and Darin
Stephenson for telling the author that the algebra R is an iterated Ore extension,
andfor providingsome backgroundandguidance regardinghis two paperscited in
the bibliography.
A NON-COMMUTATIVE HOMOGENEOUS COORDINATE RING FOR THE THIRD DEL PEZZO SURFACE3
2. The non-commutative algebra R=C[x,y] with x5 =yxy and y2 =xyx
The following result is a straightforward calculation. The main point of it is to
show that R has the same Hilbert series as the weighted polynomial ring on three
variables of weights 1, 2, and 3.
Proposition 2.1 (Stephenson). The ring R:=C[x,y] with defining relations
x5 =yxy and y2 =xyx
is an iterated Ore extension of the polynomial ring C[w]. Explicitly, if ζ is a fixed
primitive 6th root of unity, then:
(1) R = C[w][z;σ][x;τ,δ] where σ ∈ AutC[z], τ ∈ AutC[w,z], and δ is the
τ-derivation defined by
σ(w)=ζw,
τ(w)=−ζ2w, τ(z)=ζz,
δ(w)=z, δ(z)=−w2;
(2) A set of defining relations of R=C[z,w,x] is given by
zw=ζwz,
xw =−ζ2wx+z,
xz =ζzx−w2;
(3) R has basis {wizjxk |i,j,k ≥0};
(4) R is a noetherian domain;
(5) The Hilbert series of R is (1−t)−1(1−t2)−1(1−t3)−1.
Proof. Define the elements
w := y−x2
z := xw+ζ2wx
= xy+ζ2yx−ζx3
of R. Since y belongs to the ring generated by x and w, C[x,y] = C[x,w] =
C[x,w,z]. It is easy to check that
(2-1) zw=ζwz, xw =z−ζ2wx, xz =ζzx−w2.
Let R′ be the free algebra Chw,x,zi modulo the relations in (2-1). We want to
show R′ is isomorphic to R. We already know there is a homomorphism R′ → R
andwe willnowexhibitahomomorphismR→R′ byshowingthereareelements x
andY inR′thatsatisfythedefiningrelationsforR. DefinetheelementY :=w+x2
in R′. A straightforwardcomputation in R′ gives
xwx−x2w =w2+wx2
so
Y2 =w2+x2w+wx2+x4 =xwx+x4 =xYx.
4 S.PAULSMITH
In the next calculation we make frequent use of the fact that 1−ξ+ξ2 =0. Deep
breath...
YxY =(w+x2)xw+wx3+x5
=(w+x2)(z−ζ2wx)+ wx3+x5
=x2z−ζ2x2wx+ wz−(cid:2)ζ2w2x+w(cid:3)x3+x5
=x(ζzx−w2)−ζ2(cid:2)x(z−ζ2wx)x+ wz−ζ2(cid:3)w2x+wx3+x5
=(ζ−ζ2)xzx−xw2−ζxwx2+ wz(cid:2)−ζ2w2x+wx3+x5 (cid:3)
=(ζ−ζ2)(ζzx−w2)x−(z−ζ2w(cid:2)x)w−ζ(z−ζ2wx)x2 (cid:3)
+ wz−ζ2w2x+wx3+x5
=(ζ2−ζ3)(cid:2)zx2−(ζ−ζ2)w2x−zw+(cid:3) ζ2wxw−ζzx2−wx3
+ wz−ζ2w2x+wx3+x5
=(ζ2−ζ3−(cid:2) ζ)zx2+ζ2wxw+ (1−(cid:3)ζ)wz−ζw2x+x5
=ζ2wxw+ (1−ζ)wz−ζw2x+(cid:2) x5 (cid:3)
=ζ2w(z−ζ(cid:2)2wx)+ −ζ2wz−ζw2x(cid:3)+x5
=x5. (cid:2) (cid:3)
Since YxY = x5, R is isomorphic to R′. Hence R is an iterated Ore extension as
claimed. The other parts of the proposition follow easily. (cid:3)
It is animmediate consequence of the relationsthat x6 =y3. Hence x6 is in the
center of R.
3. The del Pezzo surface B
3
LetB bethesurfaceobtainedbyblowingupthecomplexprojectiveplaneP2 at
3
threenon-collinearpoints. We willwriteE , E , andE for the exceptionalcurves
1 2 3
associated to the blowup.
The −1-curves on B lie in the following configuration
3
(3-1)
E2 E3?
(cid:127) ?
(cid:127) ?
(cid:127) ?
(cid:127) ?
(cid:127) ?
(cid:127) ?
(cid:127) ?
L1 (cid:127)(cid:127) ?? X=0
(cid:127) ?
(cid:127) ?
(cid:127) ?
(cid:127) ?
(cid:127) ?
(cid:127) ?
Z=0?(cid:127)?(cid:127)?(cid:127)?(cid:127)?(cid:127)?(cid:127)?(cid:127)?(cid:127)(cid:127) ?(cid:127)?(cid:127)?(cid:127)?(cid:127)?(cid:127)?(cid:127)?(cid:127)?Y=0
? (cid:127)
s=0 ?? (cid:127)(cid:127) u=0
? (cid:127)
? (cid:127)
? (cid:127)
? (cid:127)
? (cid:127)
? (cid:127)
E1 ?? (cid:127)(cid:127) t=0
? (cid:127)
? (cid:127)
? (cid:127)
? (cid:127)
? (cid:127)
(cid:127)
L3 L2
A NON-COMMUTATIVE HOMOGENEOUS COORDINATE RING FOR THE THIRD DEL PEZZO SURFACE5
where L , L , and L are the strict transforms of the lines in P2 spanned by the
1 2 3
points that are blown up. (The labeling of the equations for the −1-curves will be
justified shortly.)
The union of the −1-curvesis an anti-canonicaldivisor and is, of course, ample.
3.1. The Picard group of B . The Picard group of B is free abelian of rank
3 3
four. We will identify it with Z4 by using the ordered basis
H, −E , −E , −E
2 1 3
where the E s are the exceptional curves over the points blown up and H is the
i
strict transform of a line in P2 in general position, i.e., missing the points being
blown up. Thus
H =(1,0,0,0), E =(0,0,−1,0), E =(0,−1,0,0), E =(0,0,0,−1).
1 2 3
The canonical divisor K is −3H+E +E +E so the anti-canonical divisor is
1 2 3
−K =(3,1,1,1).
3.2. Cox’s homogeneous coordinate ring. By definition, Cox’s homogeneous
coordinate ring [5] for a complete smooth toric variety is
S := H0(X,L).
[L]M∈PicX
For the remainder of this paper S will denote Cox’s homogeneous coordinate ring
for B .
3
Let X,Y,Z,s,t,u be coordinate functions on C6. One can present B as a toric
3
variety by defining it as the orbit space
C6−W
B :=
3 (C×)4
where the irrelevant locus, W, is the union of nine codimension two subspaces,
namely
X =t=0 X =Y =0 s=t=0
(3-2) Y =s=0 Y =Z =0 u=t=0
Z =u=0 Z =X =0 s=u=0
and (C×)4 acts with weights
X 1 1 0 1
Y 1 0 1 1
Z 1 1 1 0
.
s 0 −1 0 0
t 0 0 −1 0
u 0 0 0 −1
Therefore S is the Z4-graded polynomial ring
S =C[X,Y,Z,s,t,u]
with the degrees of the generators given by their weights under the (C×)4 action.
It follows from Cox’s results [5, Sect. 3] that
Gr(S,Z4)
QcohB ≡
3 T
6 S.PAULSMITH
whereGr(S,Z4)isthecategoryofZ4-gradedS-modulesandTisthefullsubcategory
consisting of all direct limits of modules supported on W.
The labelling of the −1-curves the diagram (3-1) is explained by the existence
of the morphisms
(X,Y,Z,s,t,u)
B
~~}}}}}}}} 3AAAAAAAA (cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127) ???????(cid:31)(cid:31)
P2 P2 (Xsu,Ytu,Zst) (YZt,XZs,XYu)
that collapse the −1-curves.
3.3. An order six automorphism σ of B . The cyclic permutation of the six
3
−1-curves on B extends to a global automorphism of B of order six. We now
3 3
make this explicit.
The category of graded rings consists of pairs (A,Γ) consisting of an abelian
group Γ and a Γ-graded ring A. A morphism (f,θ):(A,Γ)→(B,Υ) consists of a
ring homomorphism f : A → B and a group homomorphism θ : Γ → Υ such that
f(A )⊂B for all i∈Γ.
i θ(i)
Let τ :S →S be the automorphism induced by the cyclic permutation
<
(3-3) X //u //Y //t //Z // s
τ
and let θ :Z4 →Z4 be left multiplication by the matrix
2 −1 −1 −1
1 −1 −1 0
θ = .
1 0 −1 −1
1 −1 0 −1
 
Then (τ,θ) : (S,Z4) → (S,Z4)is an automorphism in the category of graded
rings. Because the irrelevant locus (3-2) is stable under the action of τ, τ induces
an automorphism σ of B . It follows from the definition of τ that σ cyclically
3
permutes the six −1-curves
Since (τ,θ)6 =id(S,Z4) the order of σ divides six. But the action of σ on the set
of −1-curves has order six, so σ has order six as an automorphism of B .
3
3.4. Fix a primitive cube root of unity ω. The left action of θ on Z4 =PicB3 has
eigenvectors
1 3 0 0
1 1 1 1
v1 =1, v2 =1, v3 =ω, v4 =ω2,
1 1 ω2 ω
       
       
with corresponding eigenvalues −1, +1, ω2, ω.
4. A twisted hcr for B
3
4.1. A sequence of line bundles on B . We will blur the distinction between a
3
divisor D and the class of the line bundle O(D) in PicB .
3
We define a sequence of divisors: D is zero; D is the line L ; for n≥1
0 1 1
D :=(1+θ+···+θn−1)(D ).
n 1
A NON-COMMUTATIVE HOMOGENEOUS COORDINATE RING FOR THE THIRD DEL PEZZO SURFACE7
We will write L :=O(D ). Therefore
n n
L =L ⊗σ∗L ⊗...⊗(σ∗)n−1L .
n 1 1 1
For example,
O(D )=L =O(1,1,0,1) =O(L )
1 1 1
O(D )=L =O(1,1,0,0) =O(L −E )
2 2 1 3
O(D )=L =O(2,1,1,1) =O(L +L −E )
3 3 1 2 3
O(D )=L =O(2,1,0,1) =O(L +L −E −E )
4 4 1 2 1 3
O(D )=L =O(3,2,1,1) =O(L +L +L −E −E )
5 5 1 2 3 1 3
O(D )=L =O(3,1,1,1) =O(L +L +L −E −E −E )
6 6 1 2 3 1 2 3
=O(3H −E −E −E )
1 2 3
=O(−K).
Lemma 4.1. Suppose that m≥0 and 0≤r ≤5. Then
D =D −mK.
6m+r r
Proof. Since θ6 =1,
6m+r−1 m−1
θi =(1+θ+···+θ5) θ6j +θ6m(1+θ+···+θr−1)
i=0 j=0
X X
=(1+θ+···+θr−1)+m(1+θ+···+θ5)
where the sum 1+θ+···+θr−1 is empty and therefore equal to zero when r =0.
Therefore D =D +mD =D −mK, as claimed. (cid:3)
6m+r r 6 r
4.2. Vanishing results. For a divisor D on a smooth surface X, we write
hi(D):=dimHi(X,O (D)).
X
We need to know that h1(D)=h2(D)=0 for various divisors D on B .
3
IfD−Kisample,thentheKodairaVanishingTheoremimpliesthath0(K−D)=
h1(K−D)=0 and Serre duality then gives h2(D)=h1(D)=0.
The notational conventions in section 3.1 identify PicB with Z4 via
3
aH −cE −bE −dE ≡(a,b,c,d)
1 2 3
whereH isthestricttransformofalineinP2. TheintersectionformonB isgiven
3
by
H2 =1, E .E =−δ , H.E =0,
i j ij i
so the induced intersection form on Z4 is
(a,b,c,d)·(a′,b′,c′,d′)=aa′−bb′−cc′−dd′.
Lemma 4.2. Let D =(a,b,c,d)∈PicB ≡Z4. Suppose that
3
(4-1) (a+3)2 >(b+1)2+(c+1)2+(d+1)2
and
(4-2) b, c, d>−1, and a+1>b+c, b+d, c+d.
Then D−K is ample, whence h1(D)=h2(D)=0.
8 S.PAULSMITH
Proof. The effective cone is generated by L , L , L , E , E , and E so, by the
1 2 3 1 2 3
Nakai-Moishezoncriterion,D−K isampleifandonlyif(D−K)2 >0andD.L >0
i
andD.E >0 for all i. Now D−K =(a+3,b+1,c+1,d+1), so (D−K)2 >0 if
i
and only if (4-1) holds and (D−K).D′ > 0 for all effective D′ if and only if (4-2)
holds.
Hence the hypothesis that (4-1) and (4-2) hold implies that D −K is ample.
The Kodaira Vanishing Theorem now implies that h0(K −D) = h1(K −D) = 0.
Serre duality now implies that h2(D)=h1(D)=0. (cid:3)
Lemma 4.3. For all n≥0, h1(D )=h2(D )=0.
n n
Proof. The value of D for 0 ≤ n ≤ 6 is given explicitly in section 4.1. We also
n
note that D = D +D = (4,2,1,2). It is routine to check that conditions (4-1)
7 1 6
and (4-2) hold for D =D when n=0,2,3,4,5,6,7. Hence h1(D )=h2(D )=0
n n n
when n=0,2,3,4,5,6,7.
We now consider D which is the −1-curve X = 0. (Since (D −K).D = 0,
1 1 1
D −K is not ample so we can’t use Kodaira Vanishing as we did for the other
1
small values of n.) It follows from the exact sequence 0 → OB3 → OB3(D1) →
OD1(D1) → 0 that Hp(B3,OB3(D1)) ∼= Hp(B3,OD1(D1)) for p = 1,2. However,
O (D ) is the normal sheaf on D and as D can be contracted to a smooth
D1 1 1 1
point on the second del Pezzo surface, OD1(D1) ∼= OD1(−1). But D1 ∼= P1 so
Hp(B3,OD1(D1)) ∼= Hp(P1,O(−1)) which is zero for p = 1,2. It follows that
h1(D )=h2(D )=0.
1 1
Thush1(D )=h2(D )=0when0≤n≤7. WehavealsoshownthatD −K is
n n n
amplewhen2≤n≤7. We nowarguebyinduction. Supposen≥8andD −K
n−6
is ample. Now D −K =D −K−K. Since a sum of ample divisors is ample,
n n−6
D −K is ample. It follows that h1(D )=h2(D )=0. (cid:3)
n n n
4.3. The twisted homogeneous coordinate ring B(B ,L,σ). We assume the
3
reader is somewhat familiar with the notion of twisted homogeneous coordinate
rings. Standard references for that material are [3], [1], [2], and [4].
The notion of a σ-ample line bundle [3] plays a key role in the study of twisted
homogeneous coordinate rings. Because L is the anti-canonicalbundle and there-
6
fore ample, L is σ-ample. This allows us to use the results of Artin and Van den
1
Bergh in [3] to conclude that the twisted homogeneous coordinate ring
∞ ∞
B(B ,L,σ)= B = H0(B ,L )
3 n 3 n
n=0 n=0
M M
is such that
GrB
QcohB ≡
3 T
where T is the full subcategory of GrB consisting of those modules M such that
for each m ∈ M, B m = 0 for n ≫ 0. It then follows that B has a host of good
n
properties—see [3] for details.
We will now compute the dimensions h0(D ) of the homogeneous B of B. We
n n
will show that B has the same Hilbert series as the non-commutative ring R, i.e.,
the same Hilbert series as the weighted polynomial ring with weights 1, 2, and 3.
As usual we write χ(D)=h0(D)−h1(D)+h2(D). The Riemann-Roch formula is
χ(O(D))=χ(O)+ 1D·(D−K)
2
A NON-COMMUTATIVE HOMOGENEOUS COORDINATE RING FOR THE THIRD DEL PEZZO SURFACE9
where K denotes the canonical divisor. We have χ(OB3)=1 and KB23 =6.
Lemma 4.4. Suppose 0≤r≤5. Then
(m+1)(3m+r) if r 6=0,
h0(D )=
6m+r (3m2+3m+1 if r =0
and
∞
1
h0(D )tn = .
n (1−t)(1−t2)(1−t3)
n=0
X
Proof. Computations for 1≤r ≤5 give D2 =r−2 and D ·K =−r. Hence
r r
χ(D )=1+ 1(D −mK)·(D −(m+1)K)
6m+r 2 r r
=1+ 1 D2−(2m+1)D .K+6m(m+1)2
2 r r
=(3m+r)(m+1)
(cid:0) (cid:1)
for m≥0 and 1≤r ≤5. When r =0, D =0 so
r
χ(D )=3m2+3m+1.
6m
By Lemma 4.3, χ(D ) = h0(D ) for all n ≥ 0 so it follows from the formula for
n n
χ(D ) that
n
(4-3) h0(D )−h0(D )=n+6
n+6 n
for all n≥0.
To complete the proof of the lemma, it suffices to show that h0(D ) is the
n
coefficient of tn in the Taylor series expansion
∞
1
f(t):= = a tn.
(1−t)(1−t2)(1−t3) n
n=0
X
Because
∞
(1−t6)f(t)=(1−t+t2)(1−t)−2 =1+ ntn,
n=1
X
it follows that
∞
(1−t6)f(t)=a +a t+···+a t5+ (a −a )tn+6
0 1 5 n+6 n
n=0
X
∞
=1+t+2t+···+5t5+ ntn
n=6
X
∞
=1+t+2t+···+5t5+ (n+6)tn+6.
n=0
X
In particular, if 0≤r≤5, a =h0(D ). We now complete the proof by induction.
r r
Suppose we have proved that a = h0(D ) for i ≤ n + 5. By comparing the
i i
expressions in the Taylor series we see that
a =a +(n+6)=h0(D )+n+6=h0(D )
n+6 n n n+6
where the last equality is given by (4-3). (cid:3)
10 S.PAULSMITH
4.3.1. Remark. It wasn’t necessary to compute χ(D ) in the previous proof. The
n
proofonlyusedthefactthatχ(D )−χ(D )=n+6whichcanbeproveddirectly
n+6 n
as follows:
χ(D )−χ(D )= 1D ·(D −K)− 1D ·(D −K)
n+6 n 2 n+6 n+6 2 n n
= 1(D −D )·(D +D −K)
2 n+6 n n+6 n
=−K·(D −(m+1)K)
r
=6(m+1)−K·D
r
=n+6.
4.4. The isomorphism R → B(B ,L,σ). The ring B has the following basis
3
elements in the following degrees:
O(1,1,0,1) X
O(1,1,0,0) Xu Zt
O(2,1,1,1) XYu YZt XZs
O(2,1,0,1) XYtu YZt2 XZst X2su
O(3,2,1,1) XYZtu YZ2t2 XZ2st X2Zsu X2Yu2
O(3,1,1,1) XYZstu YZ2st2 XZ2s2t X2Zs2u X2Ysu2 XY2tu2 Y2Zt2u
Although B is a graded subspace of Cox’s homogeneous coordinate ring S,
the multiplication in B is not that in S.
The multiplication in B is Zhang’s twisted multiplication [11] with respect to the
automorphism τ defined in (3-3): the product in B of a∈B and b∈B is
m n
(4-4) a∗ b:=aτm(b).
B
To make it clear whether a product is being calculated in B or S we will write x
for X considered as an element of B and y for Zt considered as an element of B.
Therefore, for example,
x5 =Xτ(X)τ2(X)τ3(X)τ4(X)τ5(X)
=XuYtZ
=(Zt)Y(uX)
=Ztτ2(X)τ3(Zt)
=yxy
and
y2 =Ztτ2(Zt)=Zt(sX)=X(sZ)t=Xτ(zt)τ3(X)=xyx.
The following proposition is an immediate consequence of these two calculations.
Proposition 4.5. Let R be the free algebra Chx,yi modulo the relations x5 =yxy
and y2 =xyx. Then there is a C-algebra homomorphism
R=C[x,y]→B(B ,L,σ), x7→X, y 7→Zt.
3
Lemma 4.6. The homomorphism in Proposition 4.5 is an isomorphism in degrees
≤6.1
1We will eventually prove that the homomorphism is an isomorphism in all degrees but the
lowdegreecasesneedtobehandledseparately.