Hochschild Homology and Cohomology of Klein Surfaces
Within the framework of deformation quantization, a first step towards the study of star-products is the calculation of Hochschild cohomology. The aim of this article is precisely to determine the Hochschild homology and cohomology in two cases of algebraic varieties. On the one hand, we consider si...
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| description | Within the framework of deformation quantization, a first step towards the study of star-products is the calculation of Hochschild cohomology. The aim of this article is precisely to determine the Hochschild homology and cohomology in two cases of algebraic varieties. On the one hand, we consider singular curves of the plane; here we recover, in a different way, a result proved by Fronsdal and make it more precise. On the other hand, we are interested in Klein surfaces. The use of a complex suggested by Kontsevich and the help of Groebner bases allow us to solve the problem.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 4 (2008), 064, 26 pages
Hochschild Homology and Cohomology
of Klein Surfaces?
Frédéric BUTIN
Université de Lyon, Université Lyon 1, CNRS, UMR5208, Institut Camille Jordan,
43 blvd du 11 novembre 1918, F-69622 Villeurbanne-Cedex, France
E-mail: butin@math.univ-lyon1.fr
URL: http://math.univ-lyon1.fr/∼butin/
Received April 09, 2008, in final form September 04, 2008; Published online September 17, 2008
Original article is available at http://www.emis.de/journals/SIGMA/2008/064/
Abstract. Within the framework of deformation quantization, a first step towards the
study of star-products is the calculation of Hochschild cohomology. The aim of this article
is precisely to determine the Hochschild homology and cohomology in two cases of algebraic
varieties. On the one hand, we consider singular curves of the plane; here we recover, in
a different way, a result proved by Fronsdal and make it more precise. On the other hand,
we are interested in Klein surfaces. The use of a complex suggested by Kontsevich and the
help of Groebner bases allow us to solve the problem.
Key words: Hochschild cohomology; Hochschild homology; Klein surfaces; Groebner bases;
quantization; star-products
2000 Mathematics Subject Classification: 53D55; 13D03; 30F50; 13P10
1 Introduction
1.1 Deformation quantization
Given a mechanical system (M, F(M)), where M is a Poisson manifold and F(M) the algebra of
regular functions on M , it is important to be able to quantize it, in order to obtain more precise
results than through classical mechanics. An available method is deformation quantization,
which consists of constructing a star-product on the algebra of formal power series F(M)[[~]].
The first approach for this construction is the computation of Hochschild cohomology of F(M).
We consider such a mechanical system given by a Poisson manifold M , endowed with a Poisson
bracket {·, ·}. In classical mechanics, we study the (commutative) algebra F(M) of regular
functions (i.e., for example, C∞, holomorphic or polynomial) on M , that is to say the observables
of the classical system. But quantum mechanics, where the physical system is described by
a (non commutative) algebra of operators on a Hilbert space, gives more correct results than its
classical analogue. Hence the importance to get a quantum description of the classical system
(M, F(M)), such an operation is called a quantization.
One option is geometric quantization, which allows us to construct in an explicit way a Hilbert
space and an algebra of operators on this space (see the book [10] on the Virasoro group and
algebra for a nice introduction to geometric quantization). This very interesting method presents
the drawback of being seldom applicable.
That is why other methods, such as asymptotic quantization and deformation quantization,
have been introduced. The latter, described in 1978 by F. Bayen, M. Flato, C. Fronsdal,
A. Lichnerowicz and D. Sternheimer in [5], is a good alternative: instead of constructing an
?This paper is a contribution to the Special Issue on Deformation Quantization. The full collection is available
at http://www.emis.de/journals/SIGMA/Deformation Quantization.html
mailto:butin@math.univ-lyon1.fr
http://math.univ-lyon1.fr/~butin/
http://www.emis.de/journals/SIGMA/2008/064/
http://www.emis.de/journals/SIGMA/Deformation_Quantization.html
2 F. Butin
algebra of operators on a Hilbert space, we define a formal deformation of F(M). This is given
by the algebra of formal power series F(M)[[~]], endowed with some associative, but not always
commutative, star-product,
f ∗ g =
∞∑
j=0
mj(f, g)~j , (1)
where the maps mj are bilinear and where m0(f, g) = fg. Then quantization is given by the
map f 7→ f̂ , where the operator f̂ satisfies f̂(g) = f ∗ g.
In which cases does a Poisson manifold admit such a quantization? The answer was given by
Kontsevich in [11]: in fact he constructed a star-product on every Poisson manifold. Besides,
he proved that if M is a smooth manifold, then the equivalence classes of formal deformations
of the zero Poisson bracket are in bijection with equivalence classes of star-products. Moreover,
as a consequence of the Hochschild–Kostant–Rosenberg theorem, every Abelian star-product is
equivalent to a trivial one.
In the case where M is a singular algebraic variety, say
M = {z ∈ Cn / f(z) = 0},
with n = 2 or 3, where f belongs to C[z] – and this is the case which we shall study – we
shall consider the algebra of functions on M , i.e. the quotient algebra C[z] / 〈f〉. So the above
mentioned result is not always valid. However, the deformations of the algebra F(M), defined
by the formula (1), are always classified by the Hochschild cohomology of F(M), and we are
led to the study of the Hochschild cohomology of C[z] / 〈f〉.
1.2 Cohomologies and quotients of polynomial algebras
We shall now consider R := C[z1, . . . , zn] = C[z], the algebra of polynomials in n variables with
complex coefficients. We also fix m elements f1, . . . , fm of R, and we define the quotient algebra
A := R / 〈f1, . . . , fm〉 = C[z1, . . . , zn] / 〈f1, . . . , fm〉.
Recent articles were devoted to the study of particular cases, for Hochschild as well as for
Poisson homology and cohomology:
C. Roger and P. Vanhaecke, in [16], calculate the Poisson cohomology of the affine
plane C2, endowed with the Poisson bracket f1 ∂z1 ∧ ∂z2 , where f1 is a homogeneous
polynomial. They express it in terms of the number of irreducible components of the sin-
gular locus {z ∈ C2 / f1(z) = 0} (in this case, we have a symplectic structure outside the
singular locus), the algebra of regular functions on this curve being the quotient algebra
C[z1, z2] / 〈f〉.
M. Van den Bergh and A. Pichereau, in [18, 13] and [14], are interested in the case
where n = 3 and m = 1, and where f1 is a weighted homogeneous polynomial with
an isolated singularity at the origin. They compute the Poisson homology and cohomol-
ogy, which in particular may be expressed in terms of the Milnor number of the space
C[z1, z2, z3] / 〈∂z1f1, ∂z2f1, ∂z3f1〉 (the definition of this number is given in [3]).
Once more in the case where n = 3 and m = 1, in [2], J. Alev and T. Lambre compare the
Poisson homology in degree 0 of Klein surfaces with the Hochschild homology in degree 0
of A1(C)G, where A1(C) is the Weyl algebra and G the group associated to the Klein
surface. We shall give more details about those surfaces in Section 4.1.
In [1], J. Alev, M.A. Farinati, Th. Lambre and A.L. Solotar establish a fundamental result:
they compute all the Hochschild homology and cohomology spaces of An(C)G, where An(C)
Hochschild Homology and Cohomology of Klein Surfaces 3
is the Weyl algebra, for every finite subgroup G of Sp2nC. It is an interesting and classical
question to compare the Hochschild homology and cohomology of An(C)G with the Poisson
homology and cohomology of the ring of invariants C[x, y]G, which is a quotient algebra
of the form C[z] / 〈f1, . . . , fm〉.
C. Fronsdal studies in [8] Hochschild homology and cohomology in two particular cases:
the case where n = 1 and m = 1, and the case where n = 2 and m = 1. Besides, the
appendix of this article gives another way to calculate the Hochschild cohomology in the
more general case of complete intersections.
In this paper, we propose to calculate the Hochschild homology and cohomology in two
particularly important cases.
• The case of singular curves of the plane, with polynomials f1 which are weighted homo-
geneous polynomials with a singularity of modality zero: these polynomials correspond
to the normal forms of weighted homogeneous functions of two variables and of modality
zero, given in the classification of weighted homogeneous functions of [3] (this case already
held C. Fronsdal’s attention).
• The case of Klein surfaces XΓ which are the quotients C2 / Γ, where Γ is a finite subgroup
of SL2C (this case corresponds to n = 3 and m = 1). The latter have been the subject of
many works; their link with the finite subgroups of SL2C, with the Platonic polyhedra,
and with McKay correspondence explains this large interest. Moreover, the preprojective
algebras, to which [6] is devoted, constitute a family of deformations of the Klein surfaces,
parametrized by the group which is associated to them: this fact justifies once again the
calculation of their cohomology.
The main result of the article is given by two propositions:
Proposition 1. Given a singular curve of the plane, defined by a polynomial f ∈ C[z], of
type Ak, Dk or Ek. For j ∈ N, let HHj (resp. HHj) be the Hochschild cohomology (resp.
homology) space in degree j of A := C[z] / 〈f〉, and let ∇f be the gradient of f . Then HH0 '
HH0 ' A, HH1 ' A ⊕ Ck and HH1 ' A2 / (A∇f), and for all j ≥ 2, HHj ' HHj ' Ck.
Proposition 2. Let Γ be a finite subgroup of SL2C and f ∈ C[z] such that C[x, y]Γ ' C[z] / 〈f〉.
For j ∈ N, let HHj (resp. HHj) be the Hochschild cohomology (resp. homology) space in degree j
of A := C[z] / 〈f〉, and let ∇f be the gradient of f . Then HH0 ' HH0 ' A, HH1 ' (∇f ∧
A3) ⊕ Cµ and HH1 ' ∇f ∧A3, HH2 ' A⊕Cµ and HH2 ' A3 / (∇f ∧A3), and for all j ≥ 3,
HHj ' HHj ' Cµ, where µ is the Milnor number of XΓ.
For explicit computations, we shall make use of, and develop a method suggested by M. Kont-
sevich in the appendix of [8].
We will first study the case of singular curves of the plane in Section 3: we will use this
method to recover the result that C. Fronsdal proved by direct calculations. Then we will
refine it by determining the dimensions of the cohomology and homology spaces by means of
multivariate division and Groebner bases.
Next, in Section 4, we will consider the case of Klein surfaces XΓ. For j ∈ N, we denote
by HHj the Hochschild cohomology space in degree j of XΓ. We will first prove that HH0
identifies with the space of polynomial functions on the singular surface XΓ. We will then prove
that HH1 and HH2 are infinite-dimensional. We will also determine, for j greater or equal
to 3, the dimension of HHj , by showing that it is equal to the Milnor number of the surface XΓ.
Finally, we will compute the Hochschild homology spaces.
In Section 1.3 we begin by recalling important classical results about deformations.
4 F. Butin
1.3 Hochschild homology and cohomology and deformations of algebras
Consider an associative C-algebra, denoted by A. The Hochschild cohomological complex of A is
C0(A) d(0)
// C1(A) d(1)
// C2(A) d(2)
// C3(A) d(3)
// C4(A) d(4)
// . . . ,
where the space Cp(A) of p-cochains is defined by Cp(A) = 0 for p ∈ −N∗, C0(A) = A, and for
p ∈ N∗, Cp(A) is the space of C-linear maps from A⊗p to A. The differential d =
⊕∞
i=0 d(p) is
given by
∀ f ∈ Cp(A), d(p)f(a0, . . . , ap) = a0f(a1, . . . , ap)
−
p−1∑
i=0
(−1)if(a0, . . . , aiai+1, . . . , ap) + (−1)pf(a0, . . . , ap−1)ap.
We may write it in terms of the Gerstenhaber bracket1 [·, ·]G and of the product µ of A, as
follows
d(p)f = (−1)p+1[µ, f ]G.
Then we define the Hochschild cohomology of A as the cohomology of the Hochschild coho-
mological complex associated to A, i.e. HH0(A) := Ker d(0) and for p ∈ N∗, HHp(A) :=
Ker d(p)/ Im d(p−1).
We denote by C[[~]] (resp. A[[~]]) the algebra of formal power series in the parameter ~, with
coefficients in C (resp. A). A deformation of the map µ is a map m from A[[~]]×A[[~]] to A[[~]]
which is C[[~]]-bilinear and such that
∀ (s, t) ∈ A[[~]]2, m(s, t) = st mod ~A[[~]],
∀ (s, t, u) ∈ A[[~]]3, m(s, m(t, u)) = m(m(s, t), u).
This means that there exists a sequence of bilinear maps mj from A×A to A of which the first
term m0 is the product of A and such that
∀ (a, b) ∈ A2, m(a, b) =
∞∑
j=0
mj(a, b)~j ,
∀ n ∈ N,
∑
i+j=n
mi(a,mj(b, c)) =
∑
i+j=n
mi(mj(a, b), c),
that is to say
∑
i+j=n
[mi,mj ]G = 0.
We say that (A[[~]],m) is a deformation of the algebra (A,µ). We say that the deformation is
of order p if the previous formulae are satisfied (only) for n ≤ p.
The Hochschild cohomology plays an important role in the study of deformations of the
algebra A, by helping us to classify them. In fact, if π ∈ C2(A), we may construct a first order
deformation m of A such that m1 = π if and only if π ∈ Ker d(2). Moreover, two first order
1Recall that for F ∈ Cp(A) and H ∈ Cq(A), the Gerstenhaber product is the element F • H ∈ Cp+q−1(A)
defined by F •H(a1, . . . , ap+q−1) =
∑p−1
i=0 (−1)i(q+1)F (a1, . . . , ai, H(ai+1, . . . , ai+q), ai+q+1, . . . , ap+q−1), and the
Gerstenhaber bracket is [F, H]G := F •H − (−1)(p−1)(q−1)H • F . See for example [9], and [4, page 38].
Hochschild Homology and Cohomology of Klein Surfaces 5
deformations are equivalent2 if and only if their difference is an element of Im d(1). So the set
of equivalence classes of first order deformations is in bijection with HH2(A).
If m =
∑p
j=0 mj~j , mj ∈ C2(A) is a deformation of order p, then we may extend m to
a deformation of order p + 1 if and only if there exists mp+1 such that
∀ (a, b, c) ∈ A3,
p∑
i=1
(mi(a,mp+1−i(b, c))−mi(mp+1−i(a, b), c)) = −d(2)mp+1(a, b, c),
i.e.
p∑
i=1
[mi, mp+1−i]G = 2 d(2)mp+1.
According to the graded Jacobi identity for [·, ·]G, the last sum belongs to Ker d(3). So HH3(A)
contains the obstructions to extend a deformation of order p to a deformation of order p + 1.
The Hochschild homological complex of A is
. . . d5 // C4(A)
d4 // C3(A)
d3 // C2(A)
d2 // C1(A)
d1 // C0(A),
where the space of p-chains is given by Cp(A) = 0 for p ∈ −N∗, C0(A) = A, and for p ∈ N∗,
Cp(A) = A⊗A⊗p. The differential d =
⊕∞
i=0 dp is given by
dp (a0 ⊗ a1 ⊗ · · · ⊗ ap) = a0a1 ⊗ a2 ⊗ · · · ⊗ ap
+
p−1∑
i=1
(−1)ia0 ⊗ a1 ⊗ · · · ⊗ aiai+1 ⊗ · · · ⊗ ap + (−1)papa0 ⊗ a1 ⊗ · · · ⊗ ap−1.
We define the Hochschild homology of A as the homology of the Hochschild homological complex
associated to A, i.e. HH0(A) := A / Im d1 and for p ∈ N∗, HHp(A) := Ker dp / Im dp+1.
2 Presentation of the Koszul complex
We recall in this section some results about the Koszul complex used below (see the appendix
of [8]).
2.1 Kontsevich theorem and notations
As in Section 1.2, we consider R = C[z] and (f1, . . . , fm) ∈ Rm, and we denote by A the quotient
R / 〈f1, . . . , fm〉. We assume that we have a complete intersection, i.e. the dimension of the set
of solutions of the system {f1(z) = · · · = fm(z) = 0} is n−m.
We consider the differential graded algebra
T̃ = A[η1, . . . , ηn; b1, . . . , bm] =
C[z1, . . . , zn]
〈f1, . . . , fm〉
[η1, . . . , ηn; b1, . . . , bm],
2Two deformations m =
∑p
j=0 mj ~j , mj ∈ C2(A) and m′ =
∑p
j=0 m′
j ~j , m′
j ∈ C2(A) are called equivalent
if there exists a sequence of linear maps ϕj from A to A of which the first term ϕ0 is the identity of A and such
that
∀ a ∈ A, ϕ(a) =
∞∑
j=0
ϕj(a)~j ,
∀ n ∈ N,
∑
i+j=n
ϕi(mj(a, b)) =
∑
i+j+k=n
m′
i(ϕj(a), ϕk(b)).
6 F. Butin
where ηi := ∂
∂zi
is an odd variable (i.e. the ηi’s anticommute), and bj an even variable (i.e. the
bj ’s commute).
T̃ is endowed with the differential
d
T̃
=
n∑
j=1
m∑
i=1
∂fi
∂zj
bi
∂
∂ηj
,
and the Hodge grading, defined by deg(zi) = 0, deg(ηi) = 1, deg(bj) = 2.
We may now state the main theorem which will allow us to calculate the Hochschild coho-
mology:
Theorem 1 (Kontsevich). Under the previous assumptions, the Hochschild cohomology of A
is isomorphic to the cohomology of the complex (T̃ , d
T̃
) associated with the differential graded
algebra T̃ .
Remark 1. Theorem 1 may be seen as a generalization of the Hochschild–Kostant–Rosenberg
theorem to the case of non-smooth spaces.
There is no element of negative degree. So the complex is as follows
T̃ (0)
0̃ // T̃ (1)
d
(1)
T̃ // T̃ (2)
d
(2)
T̃ // T̃ (3)
d
(3)
T̃ // T̃ (4)
d
(4)
T̃ // . . . .
For each degree p, we choose a basis Bp of T̃ (p). For example for p = 0, . . . , 3, we may take
T̃ (0) = A,
T̃ (1) = Aη1 ⊕ · · · ⊕Aηn,
T̃ (2) = Ab1 ⊕ · · · ⊕Abm ⊕
⊕
i<j
A ηiηj ,
T̃ (3) =
⊕
i=1,...,m
j=1,...,n
A biηj ⊕
⊕
i<j<k
A ηiηjηk.
Below we shall make use of the explicit matrices MatBp,Bp+1(d
(p)
T̃
).
Set H0 := A, H1 := Ker d
(1)
T̃
and for j ≥ 2, Hp := Ker d
(p)
T̃
/ Im d
(p−1)
T̃
. According to Theo-
rem 1, we have, for p ∈ N, HHp(A) ' Hp.
There is an analogous of Theorem 1 for the Hochschild homology. We consider the complex
Ω̃ = A[ξ1, . . . , ξn; a1, . . . , am],
where ξi is an odd variable and aj an even variable. Ω̃ is endowed with the differential
d
Ω̃
=
n∑
i=1
m∑
j=1
∂fj
∂zi
ξi
∂
∂aj
,
and the Hodge grading, defined by deg(zi) = 0, deg(ξi) = −1, deg(aj) = −2.
Theorem 2 (Kontsevich). Under the previous assumptions, the Hochschild homology of A is
isomorphic to the cohomology of the complex (Ω̃, d
Ω̃
)
. . .
d
(−5)
Ω̃ // Ω̃(−4)
d
(−4)
Ω̃ // Ω̃(−3)
d
(−3)
Ω̃ // Ω̃(−2)
d
(−2)
Ω̃ // Ω̃(−1)
d
(−1)
Ω̃ // Ω̃(0) .
Hochschild Homology and Cohomology of Klein Surfaces 7
For each degree p, we will choose a basis Vp of Ω̃(p) and we will make use of the explicit
matrices MatVp,Vp+1(d
(p)
Ω̃
). Set L0 := A / Im d
(−1)
Ω̃
, and for p ≥ 1, L−p := Ker d
(−p)
Ω̃
/ Im d
(−p−1)
Ω̃
.
According to Theorem 2, we have, for p ∈ N, HHp(A) ' L−p.
For each ideal J of C[z], we denote by JA the image of J by the canonical projection
C[z] → A = C[z]/〈f1, . . . , fm〉.
Similarly if (g1, . . . , gr) ∈ Ar we denote by 〈g1, . . . , gr〉A the ideal of A generated by (g1, . . . , gr).
Besides, if g ∈ C[z], and if J is an ideal of C[z], we set
AnnJ(g) := {h ∈ C[z] / hg = 0 mod J}.
In particular, g does not divide 0 in C[z] / J if and only if AnnJ(g) = J . Finally, we denote
by ∇g the gradient of a polynomial g ∈ C[z].
From now on, we consider the case m = 1 and set f := f1. Moreover, we use the notation ∂j
for the partial derivative with respect to zj .
2.2 Particular case where n = 1 and m = 1
In the case where n = 1 and m = 1, according to what we have seen, we have for p ∈ N,
T̃ (2p) = Abp
1, T̃ (2p + 1) = Abp
1η1, Ω̃(−2p) = Aap
1, Ω̃(−2p− 1) = Aap
1ξ1.
We deduce
H0 = L0 = A, H1 = {gη1 / g ∈ Aand g∂1f = 0},
and for p ∈ N∗,
H2p =
Abp
1
{g(∂1f)bp
1 / g ∈ A}
, and H2p+1 = {gbp
1η1 / g ∈ A and g∂1f = 0}.
Similarly, for p ∈ N∗,
L−2p = {gap
1 / g ∈ A and g∂1f = 0},
and for p ∈ N,
L−2p−1 =
Aap
1ξ1
{g(∂1f)ap
1ξ1 / g ∈ A}
.
Now if f = zk
1 , then
H0 = L0 = A = C[z1] / 〈zk
1 〉 ' Ck,
H1 = {gη1 / g ∈ A and kgzk−1
1 = 0} ' Ck−1, L−1 =
Aξ1
{g(kzk−1
1 )ξ1 / g ∈ A}
' Ck−1,
and for p ∈ N∗,
H2p ' L−2p−1 ' Abp
1
{g(kzk−1
1 )bp
1 / g ∈ A}
' Ck−1,
and for p ∈ N∗,
H2p+1 ' L−2p ' {gbp
1η1 / g ∈ A and kgzk−1
1 = 0} ' Ck−1.
See [12] for a similar calculation.
8 F. Butin
3 Case n = 2, m = 1. Singular curves of the plane
3.1 Singular curves of the plane
In this section, we recall a result about the weighted homogeneous functions, given in [3,
page 181].
Theorem 3 (Classif ication of weighted homogeneous functions, [3]). The weighted
homogeneous functions of two variables and of modality zero reduce, up to equivalence, to the
following list of normal forms
Type Ak Dk E6 E7 E8
Normal form zk+1
1 + z2
2 z2
1z2 + zk−1
2 z3
1 + z4
2 z3
1 + z1z
3
2 z3
1 + z5
2
The singularities of types Ak, Dk, E6, E7, E8 are called simple singularities. In the two
following sections, we will study the Hochschild cohomology of C[z] / 〈f〉, where f is one of the
normal forms of the preceding table.
3.2 Description of the cohomology spaces
With the help of Theorem 1 we calculate the Hochschild cohomology of A := C[z1, z2] / 〈f〉,
where f ∈ C[z1, z2]. We begin by making cochains and differentials explicit, by using the
notations of Section 2.1.
The various spaces of the complex are given by
T̃ (0) = A, T̃ (5) = Ab2
1η1 ⊕Ab2
1η2,
T̃ (1) = Aη1 ⊕Aη2, T̃ (6) = Ab3
1 ⊕Ab2
1η1η2,
T̃ (2) = Ab1 ⊕Aη1η2, T̃ (7) = Ab3
1η1 ⊕Ab3
1η2,
T̃ (3) = Ab1η1 ⊕Ab1η2, T̃ (8) = Ab4
1 ⊕Ab3
1η1η2,
T̃ (4) = Ab2
1 ⊕Ab1η1η2, T̃ (9) = Ab4
1η1 ⊕Ab4
1η2,
i.e., for an arbitrary p ∈ N∗,
T̃ (2p) = Abp
1 ⊕Abp−1
1 η1η2,
and for an arbitrary p ∈ N,
T̃ (2p + 1) = Abp
1η1 ⊕Abp
1η2.
As in [8], we denote by ∂
∂ηk
the partial derivative with respect to the variable ηk, for k ∈ {1, 2}.
So, for {k, l} = {1, 2}, we have
∂
∂ηk
(ηk ∧ ηl) = 1 ∧ ηl = −ηl ∧ 1,
hence
d
(2)
T̃
(ηkηl) = − ∂f
∂zk
b1ηl +
∂f
∂zl
b1ηk.
The matrices of d
T̃
are therefore given by
MatB2p,B2p+1
(
d
(2p)
T̃
)
=
(
0 ∂2f
0 −∂1f
)
,
Hochschild Homology and Cohomology of Klein Surfaces 9
MatB2p+1,B2p+2
(
d
(2p+1)
T̃
)
=
(
∂1f ∂2f
0 0
)
.
We deduce a simpler expression for the cohomology spaces
H0 = A,
H1 = {g1η1 + g2η2 / (g1, g2) ∈ A2 and g1∂1f + g2∂2f = 0}
'
{
g =
(
g1
g2
)
∈ A2 /g · ∇f = 0
}
.
For p ∈ N∗,
H2p =
{g1b
p
1 + g2b
p−1
1 η1η2 / (g1, g2) ∈ A2 and g2∂1f = g2∂2f = 0}
{(g1∂1f + g2∂2f)bp
1 / (g1, g2) ∈ A2}
'
{
g =
(
g1
g2
)
∈ A2 / g2∂1f = g2∂2f = 0
}
{(
g · ∇f
0
)
/g ∈ A2
}
' A
〈∂1f, ∂2f〉A
⊕ {g ∈ A / g ∂1f = g∂2f = 0},
H2p+1 =
{g1b
p
1η1 + g2b
p
1η2 / (g1, g2) ∈ A2 and g1∂1f + g2∂2f = 0}
{g2(∂2fbp
1η1 − ∂1fbp
1η2) / g2 ∈ A}
'
{
g =
(
g1
g2
)
∈ A2 /g · ∇f = 0
}
{
g2
(
∂2f
−∂1f
)
/ g2 ∈ A
} .
Remark 2. We recover a result of [8] (here, we use the notations of [8]). According to Theo-
rem 3.8 of [8], we have Hoch2p = Hoch2p,p ⊕ Hoch2p,p+1 and Hoch2p+1 = Hoch2p+1,p+1, so
Hoch2p,k = 0 if k /∈ {p, p + 1}, and Hoch2p+1,k = 0 if k 6= p + 1. By using Section 4.1 of [8], we
deduce H2p,k = 0 if k /∈ {p, p + 1}, and H2p+1,k = 0 if k 6= p + 1. Hence H2p = H2p,p ⊕H2p,p+1
and H2p+1 = H2p+1,p+1. So Theorem 4.9 of [8] gives the cohomology spaces which we have just
obtained.
It remains to determine these spaces more explicitly. This will be done in the two following
sections.
3.3 Explicit calculations in the particular case where f has separate variables
In this section, we consider the polynomial f = a1z
k
1 +a2z
l
2, with k ≥ 2, l ≥ 2, and (a1, a2)∈(C∗)2.
The partial derivatives of f are ∂1f = ka1z
k−1
1 and ∂2f = la2z
l−1
2 .
We already have
H0 = C[z1, z2] / 〈a1z
k
1 + a2z
l
2〉.
Besides, as f is weighted homogeneous, Euler’s formula gives 1
kz1∂1f + 1
l z2∂2f = f . So we
have the inclusion 〈f〉 ⊂ 〈∂1f, ∂2f〉, hence
A
〈∂1f, ∂2f〉A
' C[z1, z2]
〈∂1f, ∂2f〉
' Vect
(
zi
1z
j
2 / i ∈ [[0, k − 2]], j ∈ [[0, l − 2]]
)
.
10 F. Butin
But ∂1f and f are relatively prime, just as ∂2f and f are, hence if g ∈ A satisfies g∂1f = 0
mod 〈f〉, then g ∈ 〈f〉, i.e. g is zero in A. So,
H2p ' Vect
(
zi
1z
j
2 / i ∈ [[0, k − 2]], j ∈ [[0, l − 2]]
)
' C(k−1)(l−1).
We now determine the set{
g =
(
g1
g2
)
∈ A2 /g · ∇f = 0
}
.
First we have
〈f, ∂1f〉 = 〈a1z
k
1 + a2z
l
2, z
k−1
1 〉 = 〈zl
2, z
k−1
1 〉.
So the only monomials which are not in this ideal are the elements zi
1z
j
2 with i ∈ [[0, k − 2]] and
j ∈ [[0, l − 1]]. Every polynomial P ∈ C[z] may be written in the form
P = αf + β∂1f +
∑
i=0,...,k−2
j=0,...,l−1
aijz
i
1z
j
2,
with α, β ∈ C[z] and aij ∈ C. Therefore, the polynomials P ∈ C[z] such that P∂2f ∈ 〈f, ∂1f〉
are the elements
P = αf + β∂1f +
∑
i=0,...,k−2
j=1,...,l−1
aijz
i
1z
j
2.
So we have calculated Ann〈f,∂1f〉(∂2f). Let g =
(
g1
g2
)
∈ A2 satisfy the equation
g · ∇f = 0 mod 〈f〉. (2)
Then we have
g2∂2f = 0 mod 〈f, ∂1f〉,
i.e. g2 ∈ Ann〈f,∂1f〉(∂2f), i.e. again
g2 = αf + β∂1f +
∑
i=0,...,k−2
j=1,...,l−1
aijz
i
1z
j
2, with (α, β) ∈ C[z]2.
It follows that
g1∂1f + αf∂2f + β∂1f∂2f +
∑
i=0,...,k−2
j=1,...,l−1
aijz
i
1z
j
2∂2f ∈ 〈f〉.
From the equality z2∂2f = lf − l
kz1∂1f , one deduces
∂1f
g1 + β∂2f −
l
k
∑
i=0,...,k−2
j=1,...,l−1
aijz
i+1
1 zj−1
2
∈ 〈f〉,
Hochschild Homology and Cohomology of Klein Surfaces 11
i.e.
g1 = −β∂2f +
l
k
∑
i=0,...,k−2
j=1,...,l−1
aijz
i+1
1 zj−1
2 + δf, with δ ∈ C[z].
Then we verify that the elements g1 and g2 obtained in this way are indeed solutions of equa-
tion (2).
Finally, we have{
g ∈ A2 /g · ∇f = 0
}
=
−β
(
∂2f
−∂1f
)
+
∑
i=0,...,k−2
j=1,...,l−1
aijz
i
1z
j−1
2
(
l
kz1
z2
) /
β ∈ A and aij ∈ C
.
We immediately deduce the cohomology spaces of odd degree:
∀ p ≥ 1, H2p+1 ' C(k−1)(l−1),
H1 ' C(k−1)(l−1) ⊕ C[z1, z2]/〈a1z
k
1 + a2z
l
2〉,
where the direct sum results from the following argument: if we have
−β
(
∂2f
−∂1f
)
=
∑
i=0,...,k−2
j=1,...,l−1
aijz
i
1z
j−1
2
(
l
kz1
z2
)
mod 〈f〉,
then
v := −βla2z
l−1
2 −
∑
i=0,...,k−2
j=1,...,l−1
l
k
aijz
i+1
1 zj−1
2 ∈ 〈f〉.
And by a Euclidian division in (C[z2]) [z1], we may write β = fq + r, where the z1-degree of r
is smaller or equal to k − 1. So the z1-degree of v is also smaller or equal to k − 1, thus v ∈ 〈f〉
implies β = 0 and aij = 0.
Remark 3. We obtain in particular the cohomology for the cases where f = zk+1
1 +z2
2 (k ∈ N∗),
f = z3
1 + z4
2 and f = z3
1 + z5
2 . These cases correspond respectively to the weighted homogeneous
functions of types Ak, E6 and E8 given in Theorem 3.
The table below summarizes the results we have just obtained for the three particular cases
H0 H1 Hp, p ≥ 2
Ak C[z] / 〈zk+1
1 + z2
2〉 C[z] / 〈zk+1
1 + z2
2〉 ⊕ Ck Ck
E6 C[z] / 〈z3
1 + z4
2〉 C[z] / 〈z3
1 + z4
2〉 ⊕ C6 C6
E8 C[z] / 〈z3
1 + z5
2〉 C[z] / 〈z3
1 + z5
2〉 ⊕ C8 C8
The cases where f = z2
1z2 + zk−1
2 and f = z3
1 + z1z
3
2 , i.e. respectively Dk and E7, will be
studied in the next section.
12 F. Butin
3.4 Explicit calculations for Dk and E7
To study these particular cases, we use the following result about Groebner bases (Theorem 4).
First, recall the definition of a Groebner basis. For g ∈ C[z], we denote by lt(g) its leading
term (for the lexicographic order). Given a non-trivial ideal J of C[z], a Groebner basis of J is
a finite subset GJ of J \ {0} such that for all f ∈ J \ {0}, there exists g ∈ GJ such that lt(g)
divides lt(f). See [15] for more details.
Definition 1. Let J be a non-trivial ideal of C[z] and let GJ := [g1, . . . , gr] be a Groebner basis
of J . We call set of the GJ -standard terms, the set of all monomials of C[z] that are not divisible
by any of lt(g1), . . . , lt(gr).
Theorem 4 (Macaulay). The set of the GJ -standard terms forms a basis of the quotient vector
space C[z] / J .
3.4.1 Case of f = z2
1z2 + zk−1
2 , i.e. Dk
Here we have
f = z2
1z2 + zk−1
2 , ∂1f = 2z1z2 and ∂2f = z2
1 + (k − 1)zk−2
2 .
A Groebner basis of the ideal 〈f, ∂2f〉 is
B := [b1, b2] = [z2
1 + (k − 1)zk−2
2 , zk−1
2 ].
So the set of the standard terms is
{zi
1z
j
2 / i ∈ {0, 1} and j ∈ [[0, k − 2]]}.
We may now solve the equation p∂1f = 0 in C[z] / 〈f, ∂2f〉. In fact, by writing
p :=
∑
i=0,1
j=0,...,k−2
aijz
i
1z
j
2,
the equation becomes
q :=
∑
i=0,1
j=0,...,k−2
aijz
i+1
1 zj+1
2 ∈ 〈f, ∂2f〉.
We look for the normal form of the element q modulo the ideal 〈f, ∂2f〉.
The multivariate division of q by B is q = q1b1 + q2b2 + r with r =
∑k−3
j=0 a0,jz1z
j+1
2 .
Thus the solution in C[z] / 〈f, ∂2f〉 is
p = a0,k−2z
k−2
2 +
k−2∑
j=0
a1,jz1z
j
2.
But the equation
g · ∇f = 0 mod 〈f〉
yields
g1∂1f = 0 mod 〈f, ∂2f〉,
Hochschild Homology and Cohomology of Klein Surfaces 13
i.e.
g1 = αf + β∂2f + azk−2
2 +
k−2∑
j=0
bjz1z
j
2, with (α, β) ∈ C[z]2 and a, bj ∈ C.
Hence
g2∂2f + β∂1f ∂2f + azk−2
2 ∂1f +
k−2∑
j=0
bjz1z
j
2∂1f ∈ 〈f〉.
And with the equalities,
zk−1
2 =
1
2− k
(f − z2∂2f) = − 1
2− k
z2∂2f mod 〈f〉,
and
k − 2
2
z1∂1f + z2∂2f = (k − 1)f (Euler),
we obtain
∂2f
g2 + β∂1f −
2a
2− k
z1z2 +
k−2∑
j=0
bj
2
2− k
zj+1
2
∈ 〈f〉.
i.e.,
g2 = −β∂1f +
2a
2− k
z1z2 −
k−2∑
j=0
bj
2
2− k
zj+1
2 + δf, with δ ∈ C[z].
So {
g ∈ A2 /g · ∇f = 0
}
=
β
(
∂2f
−∂1f
)
+ a
(
zk−2
2
2
2−kz1z2
)
+
k−2∑
j=0
bjz
j
2
(
z1
− 2
2−kz2
) /
β ∈ A, a, bj ∈ C
.
On the other hand, a Groebner basis of 〈∂1f, ∂2f〉 is [z2
1 + (k − 1)zk−2
2 , z1z2, z
k−1
2 ], thus
C[z] / 〈∂1f, ∂2f〉 ' Vect
(
z1, 1, z2, . . . , z
k−2
2
)
.
Let us summarize (by using, for the direct sum, the same argument as in Section 3.2):
H0 = C[z] / 〈z2
1z2 + zk−1
2 〉,
H1 ' C[z] / 〈z2
1z2 + zk−1
2 〉 ⊕ Ck,
H2p ' Ck,
H2p+1 ' Ck.
3.4.2 Case of f = z3
1 + z1z3
2, i.e. E7
Here we have ∂1f = 3z2
1 + z3
2 and ∂2f = 3z1z
2
2 . A Groebner basis of the ideal 〈f, ∂1f〉 is
[3z2
1 + z3
2 , z1z
3
2 , z
6
2 ], and a Groebner basis of 〈∂1f, ∂2f〉 is [3z2
1 + z3
2 , z1z
2
2 , z
5
2 ]. By an analogous
proof, we obtain
H0 = C[z] / 〈z3
1 + z1z
3
2〉,
H1 ' C[z] / 〈z3
1 + z1z
3
2〉 ⊕ C7,
H2p ' C7,
H2p+1 ' C7.
14 F. Butin
3.5 Homology
The study is the same as the one of the Hochschild cohomology: to get the Hochschild homology
is equivalent to compute the cohomology of the complex (Ω̃, d
Ω̃
) described in Section 2.1. We
have Ω̃(0) = A, Ω̃(−2p) = Aap
1 ⊕ Aap−1
1 ξ1ξ2 for p ∈ N∗, and Ω̃(−2p − 1) = Aap
1ξ1 ⊕ Aap
1ξ2 for
p ∈ N. This defines the bases Vp. The differential is d
Ω̃
= (ξ1∂1f + ξ2∂2f) ∂
∂a1
.
So we obtain, for p ∈ N∗, the matrices
MatV−2p,V−2p+1
(
d
(−2p)
Ω̃
)
=
(
p∂1f 0
p∂2f 0
)
,
and
MatV−2p−1,V−2p
(
d
(−2p−1)
Ω̃
)
=
(
0 0
−p∂2f p∂1f
)
.
The cohomology spaces read as
L0 = A, L−1 =
A2
{g∇f / g ∈ A}
.
For p ∈ N∗,
L−2p '
{(
g1
g2
)
∈ A2
/
pg1∂1f = pg1∂2f = 0
}
{(
0
−(p + 1)g1∂2f + (p + 1)g2∂1f
) / (
g1
g2
)
∈ A2
}
' {g ∈ A / g∂1f = g∂2f = 0} ⊕ A
〈∇f〉A
.
For p ∈ N∗,
L−2p−1 '
{(
g1
g2
)
∈ A2
/
− pg1∂2f + pg2∂1f = 0
}
{(
(p + 1)g1∂1f
(p + 1)g1∂2f
) / (
g1
g2
)
∈ A2
} ' {g ∈ A2 / det(∇f, g) = 0}
{g∇f / g ∈ A}
.
From now on, we assume that f has separate variables, or f is of type Dk or E7. Then we
have {g ∈ A / g∂1f = g∂2f = 0} = {0}, and according to Euler’s formula, for p ∈ N, L−2p '
A
〈∇f〉A ' C[z]
〈∇f〉 . For the computation of {g ∈ A2 / det(∇f,g) = 0} and A2
{g∇f / g∈A} , we proceed
with Groebner bases as in Section 3.3. For example, we do it for f = z2
1z2 + zk−1
2 (i.e. type Dk).
Let g ∈ A2 be such that det(∇f,g) = 0. Then g2∂1f = 0 mod 〈f, ∂2f〉, i.e., according to
Section 3.4.1,
g2 = αf + β∂2f + azk−2
2 +
k−2∑
j=0
bjz1z
j
2,
with (α, β) ∈ C[z]2 and a, bj ∈ C. Hence
−g1∂2f + αf∂1f + β∂2f∂1f + azk−2
2 ∂1f +
k−2∑
j=0
bjz1z
j
2∂1f ∈ 〈f〉.
Hochschild Homology and Cohomology of Klein Surfaces 15
With the equalities,
zk−1
2 =
1
2− k
(f − z2∂2f) = − 1
2− k
z2∂2f mod 〈f〉,
and
k − 2
2
z1∂1f + z2∂2f = (k − 1)f (Euler),
we obtain
∂2f
−g1 + β∂1f −
2a
2− k
z1z2 +
k−2∑
j=0
bj
2
2− k
zj+1
2
∈ 〈f〉.
i.e.,
g1 = β∂1f −
2a
2− k
z1z2 +
k−2∑
j=0
bj
2
2− k
zj+1
2 + δf, with δ ∈ C[z].
So {
g ∈ A2 / det(∇f, g) = 0
}
=
β∇f + a
(
− 2
2−kz1z2
zk−2
2
)
+
k−2∑
j=0
bjz
j
2
(
2
2−kz2
z1
) /
β ∈ A, a, bj ∈ C
.
We have {g∇f / g ∈ A} ⊂ {g ∈ A2 / det(∇f,g) = 0}, thus
dim
(
A2 / {g∇f / g ∈ A}
)
≥ dim
(
A2 / {g ∈ A2 / det(∇f,g) = 0}
)
.
Since A2 / {g ∈ A2 / det(∇f,g) = 0} ' {det(∇f,g) /g ∈ A2}, and since the map
g ∈ A 7→ det
(
∇f,
(
g
0
))
∈ {det(∇f,g) /g ∈ A2}
is injective, we deduce that A2 / {g∇f / g ∈ A} is infinite-dimensional.
We collect in the following table the results for the Hochschild homology in the various cases
Type HH0 = A HH1 HHp, p ≥ 2
Ak C[z] / 〈zk+1
1 + z2
2〉 A2 / A∇f Ck
Dk C[z] / 〈z2
1z2 + zk−1
2 〉 A2 / A∇f Ck
E6 C[z] / 〈z3
1 + z4
2〉 A2 / A∇f C6
E7 C[z] / 〈z3
1 + z1z
3
2〉 A2 / A∇f C7
E8 C[z] / 〈z3
1 + z5
2〉 A2 / A∇f C8
4 Case n = 3, m = 1. Klein surfaces
4.1 Klein surfaces
Given a finite group G acting on Cn, we associate to it, according to Erlangen program of
Klein, the quotient space Cn/G, i.e. the space whose points are the orbits under the action
16 F. Butin
of G; it is an algebraic variety, and the polynomial functions on this variety are the polynomial
functions on Cn which are G-invariant. In the case of SL2C, invariant theory allows us to
associate a polynomial to any finite subgroup, as explained in Proposition 4. Thus, to every
finite subgroup of SL2C is associated the zero set of this polynomial; it is an algebraic variety,
called a Klein surface.
In this section we recall some results about these surfaces. See the references [17] and [7] for
more details.
Proposition 3. Every finite subgroup of SL2C is conjugate to one of the following groups:
• Ak (cyclic), k ≥ 1, |Ak| = k;
• Dk (dihedral), k ≥ 1, |Dk| = 4k;
• E6 (tetrahedral), |E6| = 24;
• E7 (octahedral), |E7| = 48;
• E8 (icosahedral), |E8| = 120.
Proposition 4. Let G be one of the groups of the preceding list. The ring of invariants is the
following
C[x, y]G = C[e1, e2, e3] = C[e1, e2]⊕ e3C[e1, e2] ' C[z1, z2, z3] / 〈f〉,
where the invariants ej are homogeneous polynomials, with e1 and e2 algebraically independent,
and where f is a weighted homogeneous polynomial with an isolated singularity at the origin.
These polynomials are given in the following table.
We call Klein surface the algebraic hyper-surface defined by {z ∈ C3 / f(z) = 0}.
G e1, e2, e3 f C[z1, z2, z3] / 〈∂1f, ∂2f, ∂3f〉
Ak
e1 = xk
e2 = yk
e3 = xy
−k(z1z2 − zk
3 ) Vect(1, z3, . . . , zk−2
3 )
dim = k − 1
Dk
e1 = x2k+1y +(−1)k+1xy2k+1
e2 = x2k + (−1)ky2k
e3 = x2y2
λk((−1)kz2
1+(−1)k+1z2
2z3+4zk+1
3 )
with λk = 2k(−1)k+1
Vect(1, z2, z3, . . . , zk−1
3 )
dim = k + 1
E6
e1 = 33y8x4 − y12 + 33y4x8 − x12
e2 = 14y4x4 + x8 + y8
e3 = x5y − xy5
4(z2
1 − z3
2 + 108z4
3) Vect(1, z2, z3, z2z3, z2z2
3 , z2
3)
dim = 6
E7
e1 = −34x5y13−yx17+34y5x13+xy17
e2 = −3y10x2 + 6y6x6 − 3y2x10
e3 = 14y4x4 + x8 + y8
8(3z2
1 − 12z3
2 + z2z3
3) Vect(1, z2, z2
2 , z3, z2z3, z2
2z3, z2
3)
dim = 7
E8
e1 = x30+522x25y5−10 005x20y10
−10 005x10y20−522x5y25+y30
e2 = x20 − 228x15y5 + 494x10y10
+ 228x5y15 + y20
e3 = x11y + 11x6y6 − xy11
10(−z2
1 + z3
2 + 1 728z5
3)
Vect(zi
2zj
3) i=0,1,
j=0,...,3
dim = 8
Before carrying on with our study, we make a digression in order to draw a parallel bet-
ween the Poisson and the Hochschild cohomologies of Klein surfaces, by recalling the result of
A. Pichereau.
Theorem 5 (Pichereau). Consider the Poisson bracket defined on C[z1, z1, z3] by
{·, ·}f = ∂3f∂1 ∧ ∂2 + ∂1f∂2 ∧ ∂3 + ∂2f∂3 ∧ ∂1 = i(df)(∂1 ∧ ∂2 ∧ ∂3),
Hochschild Homology and Cohomology of Klein Surfaces 17
where i is the contraction of a multiderivation by a differential form. Denote by HP ∗
f (resp.
HP f
∗ ) the Poisson cohomology (resp. homology) for this bracket. Under the previous assump-
tions, the Poisson cohomology HP ∗
f and the Poisson homology HP f
∗ of (C[z1, z1, z3] / 〈f〉, {·, ·}f )
are given by
HP 0
f = C, HP 1
f ' HP 2
f = {0},
HP f
0 ' HP f
2 ' C[z1, z2, z3] / 〈∂1f, ∂2f, ∂3f〉,
dim(HP f
1 ) = dim(HP f
0 )− 1,
HP f
j = HP j
f = {0} if j ≥ 3.
The algebra C[x, y] is a Poisson algebra for the standard symplectic bracket {·, ·}std. As G is
a subgroup of the symplectic group Sp2C (since Sp2C = SL2C), the invariant algebra C[x, y]G is
a Poisson subalgebra of C[x, y]. The following proposition allows us to deduce, from Theorem 5,
the Poisson cohomology and homology of C[x, y]G for the standard symplectic bracket.
Proposition 5. With the choice made in the preceding table for the polynomial f , the isomor-
phism of associative algebras
π : (C[x, y]G, {·, ·}std) → (C[z1, z1, z3]/〈f〉, {·, ·}f ), ej 7→ zj
is a Poisson isomorphism.
In the sequel, we will calculate the Hochschild cohomology of C[z1, z1, z3]/〈f〉, and we will
immediately deduce the Hochschild cohomology of C[x, y]G, with the help of the isomorphism π.
Note that the fact that π preserves the Poisson structures has no incidence on the computation
of the Hochschild cohomology. Therefore, so as to simplify the calculations, we may replace the
polynomial f by a simpler one, given in the following table
G Ak Dk E6 E7 E8
f z2
1 + z2
2 + zk
3 z2
1 + z2
2z3 + zk
3 z2
1 + z3
2 + z4
3 z2
1 + z3
2 + z2z
3
3 z2
1 + z3
2 + z5
3
Indeed, the linear maps defined by
C[z] → C[z],
(z1, z2, z3) 7→ (α1z1, α2z2, α3z3),
(z1, z2, z3) 7→ (α1(z1 + z2), α2(z1 + z2), α3z3)
are isomorphisms of associative algebras.
4.2 Description of the cohomology spaces
We consider now the case A := C[z1, z2, z3], / 〈f〉 and we want to calculate the Hochschild
cohomology of A. We use the notations of Section 2.1, but we change the ordering of the basis:
we shall take (η1η2, η2η3, η3η1) instead of (η1η2, η1η3, η2η3). The different spaces of the complex
are now given by
T̃ (0) = A,
T̃ (1) = Aη1 ⊕Aη2 ⊕Aη3,
T̃ (2) = Ab1 ⊕Aη1η2 ⊕Aη2η3 ⊕Aη3η1,
T̃ (3) = Ab1η1 ⊕Ab1η2 ⊕Ab1η3 ⊕Aη1η2η3,
18 F. Butin
T̃ (4) = Ab2
1 ⊕Ab1η1η2 ⊕Ab1η2η3 ⊕Ab1η3η1,
T̃ (5) = Ab2
1η1 ⊕Ab2
1η2 ⊕Ab2
1η3 ⊕Ab1η1η2η3,
i.e., for an arbitrary p ∈ N∗,
T̃ (2p) = Abp
1 ⊕Abp−1
1 η1η2 ⊕Abp−1
1 η2η3 ⊕Abp−1
1 η3η1,
and
T̃ (2p + 1) = Abp
1η1 ⊕Abp
1η2 ⊕Abp
1η3 ⊕Abp−1
1 η1η2η3.
We have
∂
∂η1
(η1 ∧ η2 ∧ η3) = 1 ∧ η2 ∧ η3 = η2 ∧ η3 ∧ 1,
thus
d
(3)
T̃
(η1η2η3) =
∂f
∂z1
b1η2η3 +
∂f
∂z2
b1η3η1 +
∂f
∂z3
b1η1η2.
The matrices of d
T̃
are therefore given by
MatB1,B2(d
(1)
T̃
) =
∂1f ∂2f ∂3f
0 0 0
0 0 0
0 0 0
,
∀ p ∈ N∗, MatB2p,B2p+1(d
(2p)
T̃
) =
0 ∂2f 0 −∂3f
0 −∂1f ∂3f 0
0 0 −∂2f ∂1f
0 0 0 0
,
∀ p ∈ N∗, MatB2p+1,B2p+2(d
(2p+1)
T̃
) =
∂1f ∂2f ∂3f 0
0 0 0 ∂3f
0 0 0 ∂1f
0 0 0 ∂2f
.
We deduce
H0 = A,
H1 = {g1η1 + g2η2 + g3η3 / (g1, g2, g3) ∈ A3 and g1∂1f + g2∂2f + g3∂3f = 0}
'
g =
g1
g2
g3
∈ A3 /g · ∇f = 0
,
H2 =
g0b1 + g3η1η2 + g1η2η3 + g2η3η1
/ (g0, g1, g2, g3) ∈ A4 and
g3∂2f − g2∂3f = g1∂3f − g3∂1f
= g2∂1f − g1∂2f = 0
{(g1∂1f + g2∂2f + g3∂3f)b1 , / (g1, g2, g3) ∈ A3}
'
g =
g0
g1
g2
g3
∈ A4
/
∇f ∧
g1
g2
g3
= 0
/{(
g · ∇f
03,1
)
/g ∈ A3
}
Hochschild Homology and Cohomology of Klein Surfaces 19
' A
〈∂1f, ∂2f, ∂3f〉A
⊕ {g ∈ A3 /∇f ∧ g = 0}.
For p ≥ 2,
H2p =
g0b
p
1 + g3b
p−1
1 η1η2 + g1b
p−1
1 η2η3 + g2b
p−1
1 η3η1
/ (g0, g1, g2, g3) ∈ A4 and
g3∂2f − g2∂3f
= g1∂3f − g3∂1f
= g2∂1f − g1∂2f = 0
{
(g1∂1f + g2∂2f + g3∂3f)bp
1
+ g0(∂3fbp−1
1 η1η2 + ∂1fbp−1
1 η2η3 + ∂2fbp−1
1 η3η1)
/
(g0, g1, g2, g3) ∈ A3
}
'
g =
g0
g1
g2
g3
∈ A4
/
∇f ∧
g1
g2
g3
= 0
/
g · ∇f
g0 ∂1f
g0 ∂2f
g0 ∂3f
/g ∈ A3 and g0 ∈ A
' A
〈∂1f, ∂2f, ∂3f〉A
⊕ {g ∈ A3 /∇f ∧ g = 0}
{g∇f / g ∈ A}
.
For p ∈ N∗,
H2p+1 =
g1b
p
1η1 + g2b
p
1η2 + g3b
p
1η3 + g0b
p−1
1 η1η2η3
/ (g0, g1, g2g3) ∈ A4 and
g1∂1f + g2∂2f + g3∂3f = 0,
g0∂3f = g0∂1f = g0∂2f = 0
{
(g3∂2f − g2∂3f)bp
1η1 + (g1∂3f − g3∂1f)bp
1η2
+(g2∂1f − g1∂2f)bp
1η3
/
(g1, g2, g3) ∈ A3
}
'
g1
g2
g3
g0
∈ A4
/
∇f ·
g1
g2
g3
= 0
g0 ∂3f = g0 ∂1f = g0 ∂2f = 0
/{(
∇f ∧ g
0
) /
g ∈ A3
}
'
{
g ∈ A3 /∇f · g = 0
}
{∇f ∧ g /g ∈ A3}
⊕ {g ∈ A / g∂3f = g∂1f = g∂2f = 0}.
The following section will allow us to make those various spaces more explicit.
4.3 Explicit calculations in the particular case where f has separate variables
In this section, we consider the polynomial f = a1z
i
1 + a2z
j
2 + a3z
k
3 , with 2 ≤ i ≤ j ≤ k and
aj ∈ C∗. Its partial derivatives are ∂1f = ia1z
i−1
1 , ∂2f = ja2z
j−1
2 and ∂3f = ka3z
k−1
3 .
We already have
H0 = C[z1, z2, z3] / 〈a1z
i
1 + a2z
j
2 + a3z
k
3 〉.
Moreover, as f is weighted homogeneous, Euler’s formula gives
1
i
z1∂1f +
1
j
z2∂2f +
1
k
z3∂3f = f.
So we have the inclusion 〈f〉 ⊂ 〈∂1f, ∂2f, ∂3f〉, thus
A
〈∂1f, ∂2f, ∂3f〉A
' C[z1, z2, z3]
〈∂1f, ∂2f, ∂3f〉
' Vect (zp
1z
q
2z
r
3 / p ∈ [[0, i− 2]], q ∈ [[0, j − 2]], r ∈ [[0, k − 2]]) .
20 F. Butin
Finally, as ∂1f and f are relatively prime, if g ∈ A verifies g∂1f = 0 mod 〈f〉, then g ∈ 〈f〉,
i.e. g is zero in A.
Now we determine the setg =
g1
g2
g3
∈ A3 /g · ∇f = 0
.
First we have
〈f, ∂1f, ∂2f〉 = 〈a1z
i
1 + a2z
j
2 + a3z
k
3 , zi−1
1 , zj−1
2 〉 = 〈zi−1
1 , zj−1
2 , zk
3 〉.
Thus the only monomials which are not in this ideal are the elements zp
1z
q
2z
r
3 with p ∈ [[0, i− 2]],
q ∈ [[0, j − 2]], and r ∈ [[0, k − 1]].
So every polynomial P ∈ C[z] may be written in the form
P = αf + β∂1f + γ∂2f +
∑
p=0,...,i−2
q=0,...,j−2
r=0,...,k−1
apqrz
p
1z
q
2z
r
3.
The polynomials P ∈ C[z] such that P∂3f ∈ 〈f, ∂1f, ∂2f〉 are therefore the following ones
P = αf + β∂1f + γ∂2f +
∑
p=0,...,i−2
q=0,...,j−2
r=1,...,k−1
apqrz
p
1z
q
2z
r
3.
So we have calculated Ann〈f,∂1f,∂2f〉(∂3f). The equation
g · ∇f = 0 mod 〈f〉
leads to g3 ∈ Ann〈f,∂1f,∂2f〉(∂3f), i.e.
g3 = αf + β∂1f + γ∂2f +
∑
p=0,...,i−2
q=0,...,j−2
r=1...k−1
apqrz
p
1z
q
2z
r
3,
with (α, β, γ) ∈ C[z]3. Hence
g2∂2f + γ∂2f∂3f +
∑
p=0,...,i−2
q=0,...,j−2
r=1,...,k−1
apqrz
p
1z
q
2z
r
3∂3f ∈ 〈f, ∂1f〉.
Thus, according to Euler’s formula,
∂2f
g2 + γ∂3f −
k
j
∑
p=0,...,i−2
q=0...j−2
r=1,...,k−1
apqrz
p
1z
q+1
2 zr−1
3
∈ 〈f, ∂1f〉.
Since Ann〈f,∂1f〉(∂2f) = 〈f, ∂1f〉, this equation is equivalent to
g2 = −γ∂3f +
k
j
∑
p=0,...,i−2
q=0,...,j−2
r=1,...,k−1
apqrz
p
1z
q+1
2 zr−1
3 + δf + ε∂1f,
Hochschild Homology and Cohomology of Klein Surfaces 21
with δ, ε ∈ C[z]. It follows that
g1∂1f + β∂1f∂3f + ε∂1f∂2f
+
∑
p=0,...,i−2
q=0,...,j−2
r=1,...,k−1
apqrz
p
1z
q
2z
r
3∂3f +
k
j
∑
p=0,...,i−2
q=0,...,j−2
r=1,...,k−1
apqrz
p
1z
q+1
2 zr−1
3 ∂2f ∈ 〈f〉.
And, according to Euler’s formula,
∂1f
g1 + β∂3f + ε∂2f −
k
i
∑
p=0,...,i−2
q=0,...,j−2
r=1,...,k−1
apqrz
p+1
1 zq
2z
r−1
3
∈ 〈f〉,
i.e.
g1 = −β∂3f − ε∂2f +
k
i
∑
p=0,...,i−2
q=0,...,j−2
r=1,...,k−1
apqrz
p+1
1 zq
2z
r−1
3 + ηf,
with η ∈ C[z]. Finally{
g ∈ A3 /g · ∇f = 0
}
=
∇f ∧
−γ
β
−ε
+
∑
p=0,...,i−2
q=0,...,j−2
r=1,...,k−1
apqrz
p
1z
q
2z
r−1
3
k
i z1
k
j z2
z3
/
(β, γ, ε) ∈ A3 and apqr ∈ C
.
We deduce immediately the cohomology spaces of odd degrees
∀ p ≥ 1, H2p+1 ' C(i−1)(j−1)(k−1),
H1 ' ∇f ∧ (C[z] / 〈f〉)3 ⊕ C(i−1)(j−1)(k−1).
It remains to determine the setg =
g1
g2
g3
∈ A3 /∇f ∧ g = 0
.
Let g ∈ A3 be such that ∇f ∧ g = 0. This means that, modulo 〈f〉, g verifies the system
∂2fg3 − ∂3fg2 = 0, ∂3fg1 − ∂1fg3 = 0, ∂1fg2 − ∂2fg1 = 0.
The first equation gives, modulo 〈f, ∂2f〉, ∂3fg2 = 0. Now Ann〈f,∂2f〉(∂3f) = 〈f, ∂2f〉, therefore
g2 = αf + β∂2f . Hence
∂2f(g3 − β∂3f) = 0 mod 〈f〉,
i.e. g3 = γf + β∂3f . Finally, we obtain
∂3f(g1 − β∂1f) = 0 mod 〈f〉,
i.e. g1 = δf + β∂1f . So, {g ∈ A3 /∇f ∧ g = 0} = {β∇f / β ∈ A}.
22 F. Butin
We deduce the cohomology spaces of even degrees (for the direct sum, we use the same
argument as in Section 3.2)
∀ p ≥ 2, H2p ' A / 〈∂1f, ∂2f, ∂3f〉 ' C[z] / 〈zi−1
1 , zj−1
2 , zk−1
3 〉
' Vect (zp
1z
q
2z
r
3 / p ∈ [[0, i− 2]], q ∈ [[0, j − 2]], r ∈ [[0, k − 2]])
' C(i−1)(j−1)(k−1),
H2 ' {β∇f / β ∈ A} ⊕ C(i−1)(j−1)(k−1)
' C[z] / 〈a1z
i
1 + a2z
j
2 + a3z
k
3 〉 ⊕ C(i−1)(j−1)(k−1).
Remark 4. We also have
∇f ∧ (C[z] / 〈f〉)3 ' (C[z] / 〈f〉)3 / {g /∇f ∧ g = 0} = (C[z] / 〈f〉)3 / (C[z] / 〈f〉)∇f.
Moreover the map
(C[z] / 〈f〉)2 → ∇f ∧ (C[z] / 〈f〉)3 ,
(
g1
g2
)
7→ ∇f ∧
g1
g2
0
is injective, thus ∇f ∧ (C[z] / 〈f〉)3 is infinite-dimensional.
Remark 5. In particular, we obtain the cohomology for the cases where f = z2
1 + z2
2 + zk
3 ,
f = z2
1 + z3
2 + z4
3 and f = z2
1 + z3
2 + z5
3 . These cases correspond respectively to the types Ak, E6
and E8 of the Klein surfaces.
The following table sums up the results of those three special cases:
H0 H1 H2 Hp, p ≥ 3
Ak C[z] / 〈z2
1 + z2
2 + zk
3 〉 ∇f ∧ (C[z] / 〈f〉)3 ⊕ Ck−1 C[z] / 〈z2
1 + z2
2 + zk
3 〉 ⊕ Ck−1 Ck−1
E6 C[z] / 〈z2
1 + z3
2 + z4
3〉 ∇f ∧ (C[z] / 〈f〉)3 ⊕ C6 C[z] / 〈z2
1 + z3
2 + z4
3〉 ⊕ C6 C6
E8 C[z] / 〈z2
1 + z3
2 + z5
3〉 ∇f ∧ (C[z] / 〈f〉)3 ⊕ C8 C[z] / 〈z2
1 + z3
2 + z5
3〉 ⊕ C8 C8
The cases where f = z2
1 + z2
2z3 + zk
3 and f = z2
1 + z3
2 + z2z
3
3 , i.e. respectively Dk and E7 are
studied in the following section.
4.4 Explicit calculations for Dk and E7
4.4.1 Case of f = z2
1 + z2
2z3 + zk
3 , i.e. Dk
In this section, we consider the polynomial f = z2
1 +z2
2z3 +zk
3 , with k ≥ 3. Its partial derivatives
are ∂1f = 2z1, ∂2f = 2z2z3 and ∂3f = z2
2 + kzk−1
3 .
We already have
H0 = C[z] / 〈z2
1 + z2
2z3 + zk
3 〉.
Besides, since f is weighted homogeneous, Euler’s formula gives
k
2
z1∂1f +
k − 1
2
z2∂2f + z3∂3f = kf. (3)
Thus, we have the inclusion 〈f〉 ⊂ 〈∂1f, ∂2f, ∂3f〉. Moreover, a Groebner basis of 〈∂1f, ∂2f, ∂3f〉
is [zk
3 , z2z3, z
2
2 + kzk−1
3 , z1], therefore
A
〈∂1f, ∂2f, ∂3f〉A
' C[z1, z2, z3]
〈∂1f, ∂2f, ∂3f〉
' Vect
(
z2, 1, z3, . . . , z
k−1
3
)
.
Hochschild Homology and Cohomology of Klein Surfaces 23
Finally, as ∂1f and f are relatively prime, if g ∈ A verifies g∂1f = 0 mod 〈f〉, then g ∈ 〈f〉,
i.e. g is zero in A, thus {g ∈ A / g∂3f = g∂1f = g ∂2f = 0} = 0.
Now we determine the setg =
g1
g2
g3
∈ A3 /g · ∇f = 0
.
A Groebner basis of 〈f, ∂1f, ∂3f〉 is [z1, z
k
3 , z2
2 + kzk−1
3 ], thus a basis of C[z] / 〈f, ∂1f, ∂3f〉 is
{zi
2z
j
3 / i ∈ {0, 1}, j ∈ [[0, k − 1]]}. We have already solved the equation p∂2f = 0 in this space;
the solutions of this equation in C[z] / 〈f, ∂1f, ∂3f〉 are of the form
p = a0,k−1z
k−1
3 +
k−1∑
j=0
a1,jz2z
j
3,
where a0,k−1, a1,j ∈ C.
Let g =
g1
g2
g3
∈ A3 satisfy the equation
g · ∇f = 0 mod 〈f〉.
Then we have
g2∂2f = 0 mod 〈f, ∂1f, ∂3f〉,
hence
g2 = αf + β∂1f + γ∂3f + azk−1
3 +
k−1∑
j=0
bjz2z
j
3,
with (α, β, γ) ∈ C[z]3. And
g3∂3f + γ∂3f∂2f + azk−1
3 ∂2f +
k−1∑
j=0
bjz2z
j
3∂2f ∈ 〈f, ∂1f〉. (4)
Now according to Euler’s formula (3) and the equality
zk
3z2 =
1
1− k
(
z2f − z2z3∂3f −
1
2
z2z1∂1f
)
= − 1
1− k
z2z3∂3f mod 〈f, ∂1f〉,
Equation (4) becomes
∂3f
g3 + γ∂2f −
2a
1− k
z2z3 −
k−1∑
j=0
bj
2
k − 1
zj+1
3
∈ 〈f, ∂1f〉.
As Ann〈f,∂1f〉(∂3f) = 〈f, ∂1f〉, this equation is equivalent to
g3 = −γ∂2f +
2a
1− k
z2z3 +
k−1∑
j=0
bj
2
k − 1
zj+1
3 + δf + ε∂1f,
24 F. Butin
with δ, ε ∈ C[z]. We find
g1 = −β∂2f − ε∂3f +
k−1∑
j=0
bj
k
k − 1
z1z
j
3 +
a
1− k
z2z1 + ηf,
with η ∈ C[z]. Finally, we have{
g ∈ A3 /g · ∇f = 0
}
=
∇f ∧
γ
ε
−β
+
k−1∑
j=0
bj
k
k−1z1z
j
3
z2z
j
3
− 2
1−kzj+1
3
+ a
1
1−kz2z1
zk−1
3
2
1−kz2z3
/
(β, γ, ε) ∈ A3
and a, bj ∈ C
,
as well as cohomology spaces of odd degrees (for the direct sum, we use the same argument as
in Section 3.2)
∀ p ≥ 1, H2p+1 ' Ck+1,
H1 ' ∇f ∧ (C[z] / 〈f〉)3 ⊕ Ck+1.
To show {g ∈ A3 /∇f ∧ g = 0} =
{
fg + β∇f /g ∈ A3, β ∈ A
}
, we proceed as in the case
of separate variables. We deduce the cohomology spaces of even degrees
∀ p ≥ 2, H2p ' A / 〈∂1f, ∂2f, ∂3f〉 ' Vect
(
z2, 1, z3, . . . , z
k−1
3
)
' Ck+1,
H2 ' {β∇f / β ∈ A} ⊕ Ck+1 ' C[z] / 〈z2
1 + z2
2z3 + zk
3 〉 ⊕ Ck+1.
4.4.2 Case of f = z2
1 + z3
2 + z2z3
3, i.e. E7
Here we have ∂1f = 2z1, ∂2f = 3z2
2 + z3
3 and ∂3f = 3z2z
2
3 . The proof is similar to that of the
previous cases. A Groebner basis of 〈∂1f, ∂2f, ∂3f〉 is [z5
3 , z2z
2
3 , 3z2
2+z3
3 , z1]. Similarly, a Groebner
basis of 〈f, ∂1f, ∂2f〉 is [z6
3 , z2z
3
3 , 3z2
2 + z3
3 , z1]. We obtain the following results
∀ p ≥ 1, H2p+1 ' C7,
H1 ' ∇f ∧ (C[z] / 〈f〉)3 ⊕ C7,
∀ p ≥ 2, H0 = C[z] / 〈z2
1 + z3
2 + z2z
3
3〉,
H2p ' A / 〈∂1f, ∂2f, ∂3f〉 ' Vect
(
z2, z
2
2 , 1, z3, z
2
3 , z
3
3 , z
4
3
)
' C7,
H2 ' {β∇f / β ∈ A} ⊕ C7 ' C[z] / 〈z2
1 + z3
2 + z2z
3
3〉 ⊕ C7.
Remark 6. In all the previously studied cases, there exists a triple (i, j, k) such that {i, j, k} =
{1, 2, 3}, and such that the map
C[z] / 〈∂1f, ∂2f, ∂3f〉 → {g ∈ C[z] / 〈f, ∂jf, ∂kf〉 / g∂if = 0},
P mod 〈∂1f, ∂2f, ∂3f〉 7→ ziP mod 〈f, ∂jf, ∂kf〉
is an isomorphism of vector spaces.
4.5 Homology
The study is the same as the one of the Hochschild cohomology, and we proceed as in Section 3.5.
Here, we have
Ω̃(0) = A, Ω̃(−1) = Aξ1 ⊕Aξ2 ⊕Aξ3,
Hochschild Homology and Cohomology of Klein Surfaces 25
∀ p ∈ N∗, Ω̃(−2p) = Aap
1 ⊕Aap−1
1 ξ1ξ2 ⊕Aap−1
1 ξ2ξ3 ⊕Aap−1
1 ξ3ξ1,
∀ p ∈ N∗, Ω̃(−2p− 1) = Aap
1ξ1 ⊕Aap
1ξ2 ⊕Aap
1ξ3 ⊕Aap−1
1 ξ1ξ2ξ3.
This defines the bases Vp. The differential is d
Ω̃
= (ξ1∂1f + ξ2∂2f + ξ3∂3f) ∂
∂a1
.
By setting Df := (∂3f ∂1f ∂2f), we deduce the matrices
MatV−2,V−1
(
d
(−2)
Ω̃
)
=
(
∇f 03,3
)
,
∀ p ≥ 2, MatV−2p,V−2p+1
(
d
(−2p)
Ω̃
)
=
(
∇f 03,3
0 (p− 1)Df
)
,
∀ p ≥ 1, MatV−2p−1,V−2p
(
d
(−2p−1)
Ω̃
)
=
0 0 0 0
−p∂2f p∂1f 0 0
0 −p∂3f p∂2f 0
p∂3f 0 −p∂1f 0
.
The cohomology spaces read as
L0 = A, L−1 =
A3
{g∇f / g ∈ A}
,
L−2 = {g ∈ A / g∂1f = g∂2f = g∂3f = 0} ⊕ A3
{∇f ∧ g /g ∈ A3}
.
For p ≥ 2,
L−2p ' {g ∈ A / g∂1f = g∂2f = g∂3f = 0} ⊕ {g ∈ A3 /g · ∇f = 0}
{∇f ∧ g /g ∈ A3}
.
For p ∈ N∗,
L−2p−1 ' {g ∈ A3 /∇f ∧ g = 0}
{g∇f / g ∈ A}
⊕ A
〈∇f〉A
.
From now on, we assume that either f has separate variables, or f is of type Dk or E7. Then
we have {g ∈ A / g∂1f = g∂2f = g∂3f = 0} = {0}, and according to Euler’s formula,
A
〈∇f〉A
' C[z]
〈∇f〉
.
Most of the spaces have already been computed in Sections 4.3 and 4.4. In particular, we have
A3 / A∇f ' ∇f ∧A3. Moreover, {∇f ∧ g /g ∈ A3} ⊂ {g ∈ A3 /g · ∇f = 0}, thus
dim
(
A3 / {∇f ∧ g /g ∈ A3}
)
≥ dim
(
A3 / {g ∈ A3 /g · ∇f = 0}
)
.
And A3 / {g ∈ A3 /g · ∇f = 0} ' {g · ∇f /g ∈ A3}. Since the map
g ∈ A 7→
g
0
0
· ∇f ∈ {g · ∇f /g ∈ A3}
is injective, A3 / {∇f ∧ g /g ∈ A3} is infinite-dimensional.
In the following table we collect the results for the Hochschild homology in the various cases
Type HH0 = A HH1 HH2 HHp, p ≥ 3
Ak C[z] / 〈z2
1 + z2
2 + zk
3 〉 ∇f ∧A3 A3 / (∇f ∧A3) Ck−1
Dk C[z] / 〈z2
1 + z2
2z3 + zk
3 〉 ∇f ∧A3 A3 / (∇f ∧A3) Ck+1
E6 C[z] / 〈z2
1 + z3
2 + z4
3〉 ∇f ∧A3 A3 / (∇f ∧A3) C6
E7 C[z] / 〈z2
1 + z3
2 + z2z
3
3〉 ∇f ∧A3 A3 / (∇f ∧A3) C7
E8 C[z] / 〈z2
1 + z3
2 + z5
3〉 ∇f ∧A3 A3 / (∇f ∧A3) C8
26 F. Butin
Acknowledgements
I would like to thank my thesis advisors Gadi Perets and Claude Roger for their efficient and
likeable help, for their great availability, and for the time that they devoted to me all along
this study. I also thank Daniel Sternheimer who paid attention to my work and the referees for
their relevant remarks and their judicious advice. And I am grateful to Serge Parmentier for
the rereading of my English text.
References
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92 (1998), 605–635.
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http://arxiv.org/abs/math-ph/0507021
http://arxiv.org/abs/q-alg/9709040
http://arxiv.org/abs/math.QA/0511201
1 Introduction
1.1 Deformation quantization
1.2 Cohomologies and quotients of polynomial algebras
1.3 Hochschild homology and cohomology and deformations of algebras
2 Presentation of the Koszul complex
2.1 Kontsevich theorem and notations
2.2 Particular case where n=1 and m=1
3 Case n=2, m=1. Singular curves of the plane
3.1 Singular curves of the plane
3.2 Description of the cohomology spaces
3.3 Explicit calculations in the particular case where f has separate variables
3.4 Explicit calculations for D_k and E_7
3.4.1 Case of f=z_1^2z_2+z_2^{k-1}, i.e. D_k
3.4.2 Case of f=z_1^3+z_1z_2^3, i.e. E_7
3.5 Homology
4 Case n=3, m=1. Klein surfaces
4.1 Klein surfaces
4.2 Description of the cohomology spaces
4.3 Explicit calculations in the particular case where f has separate variables
4.4 Explicit calculations for D_k and E_7
4.4.1 Case of f=z_1^2+z_2^2z_3+z_3^k, i.e. D_k
4.4.2 Case of f=z_1^2+z_2^3+z_2z_3^3, i.e. E_7
4.5 Homology
References
|
| id | nasplib_isofts_kiev_ua-123456789-149019 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T17:43:30Z |
| publishDate | 2008 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Butin, F. 2019-02-19T13:06:06Z 2019-02-19T13:06:06Z 2008 Hochschild Homology and Cohomology of Klein Surfaces / F. Butin // Symmetry, Integrability and Geometry: Methods and Applications. — 2008. — Т. 4. — Бібліогр.: 18 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 53D55; 13D03; 30F50; 13P10 https://nasplib.isofts.kiev.ua/handle/123456789/149019 Within the framework of deformation quantization, a first step towards the study of star-products is the calculation of Hochschild cohomology. The aim of this article is precisely to determine the Hochschild homology and cohomology in two cases of algebraic varieties. On the one hand, we consider singular curves of the plane; here we recover, in a different way, a result proved by Fronsdal and make it more precise. On the other hand, we are interested in Klein surfaces. The use of a complex suggested by Kontsevich and the help of Groebner bases allow us to solve the problem. This paper is a contribution to the Special Issue on Deformation Quantization. I would like to thank my thesis advisors Gadi Perets and Claude Roger for their ef ficient and likeable help, for their great availability, and for the time that they devoted to me all along this study. I also thank Daniel Sternheimer who paid attention to my work and the referees for their relevant remarks and their judicious advice. And I am grateful to Serge Parmentier for the rereading of my English text. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Hochschild Homology and Cohomology of Klein Surfaces Article published earlier |
| spellingShingle | Hochschild Homology and Cohomology of Klein Surfaces Butin, F. |
| title | Hochschild Homology and Cohomology of Klein Surfaces |
| title_full | Hochschild Homology and Cohomology of Klein Surfaces |
| title_fullStr | Hochschild Homology and Cohomology of Klein Surfaces |
| title_full_unstemmed | Hochschild Homology and Cohomology of Klein Surfaces |
| title_short | Hochschild Homology and Cohomology of Klein Surfaces |
| title_sort | hochschild homology and cohomology of klein surfaces |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/149019 |
| work_keys_str_mv | AT butinf hochschildhomologyandcohomologyofkleinsurfaces |