sigma08-064. 353KB Jun 04 2011 12:10:18 AM

Symmetry, Integrability and Geometry: Methods and Applications

SIGMA 4 (2008), 064, 26 pages

Hochschild Homology and Cohomology
of Klein Surfaces⋆
Fr´ed´eric BUTIN
Universit´e de Lyon, Universit´e 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

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 =

∞
X

mj (f, g)~j ,

(1)

j=0

where the maps mj are bilinear and where m0 (f, g) = f g. Then quantization is given by the
map f 7→ fb, where the operator fb satisfies fb(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] / hf i. 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] / hf i.

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 / hf1 , . . . , fm i = C[z1 , . . . , zn ] / hf1 , . . . , fm i.
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 singular 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 ] / hf i.
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 cohomology, which in particular may be expressed in terms of the Milnor number of the space
C[z1 , z2 , z3 ] / h∂z1 f1 , ∂z2 f1 , ∂z3 f1 i (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 Sp2n C. 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] / hf1 , . . . , fm i.
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 homogeneous 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 SL2 C (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 SL2 C, 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 HH j (resp. HHj ) be the Hochschild cohomology (resp.
homology) space in degree j of A := C[z] / hf i, and let ∇f be the gradient of f . Then HH 0 ≃
HH0 ≃ A, HH 1 ≃ A ⊕ Ck and HH1 ≃ A2 / (A∇f ), and for all j ≥ 2, HH j ≃ HHj ≃ Ck .
Proposition 2. Let Γ be a finite subgroup of SL2 C and f ∈ C[z] such that C[x, y]Γ ≃ C[z] / hf i.
For j ∈ N, let HH j (resp. HHj ) be the Hochschild cohomology (resp. homology) space in degree j
of A := C[z] / hf i, and let ∇f be the gradient of f . Then HH 0 ≃ HH0 ≃ A, HH 1 ≃ (∇f ∧
A3 ) ⊕ Cµ and HH1 ≃ ∇f ∧ A3 , HH 2 ≃ A ⊕ Cµ and HH2 ≃ A3 / (∇f ∧ A3 ), and for all j ≥ 3,
HH j ≃ 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. Kontsevich 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 HH j the Hochschild cohomology space in degree j of XΓ . We will first prove that HH 0
identifies with the space of polynomial functions on the singular surface XΓ . We will then prove
that HH 1 and HH 2 are infinite-dimensional. We will also determine, for j greater or equal
to 3, the dimension of HH j , 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
C 0 (A)

d(0)

/ C 1 (A)

d(1)

/ C 2 (A)

d(2)

/ C 3 (A)

d(3)

/ C 4 (A)

d(4)

/ ...,

where the space C p (A) of p-cochains is defined by C p (A) = 0 for p ∈ −N∗ , C 0 (A) =LA, and for
(p) is
p ∈ N∗ , C p (A) is the space of C-linear maps from A⊗p to A. The differential d = ∞
i=0 d
given by
∀ f ∈ C p (A),

d(p) f (a0 , . . . , ap ) = a0 f (a1 , . . . , ap )
−

p−1
X

(−1)i f (a0 , . . . , ai ai+1 , . . . , ap ) + (−1)p f (a0 , . . . , ap−1 )ap .

i=0

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 cohomological complex associated to A, i.e. HH 0 (A) := Ker d(0) and for p ∈ N∗ , HH p (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[[~]],
3

∀ (s, t, u) ∈ A[[~]] ,

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 ,
∀ n ∈ N,

m(a, b) =
X

∞
X

mj (a, b)~j ,

j=0

mi (a, mj (b, c)) =

i+j=n

that is to say

X

X

mi (mj (a, b), c),

i+j=n

[mi , mj ]G = 0.

i+j=n

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 ∈ C 2 (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
1

Recall that for F ∈ C p (A) and P
H ∈ C q (A), the Gerstenhaber product is the element F • H ∈ C p+q−1 (A)
i(q+1)
defined by F • H(a1 , . . . , ap+q−1 ) = p−1
F (a1 , . . . , ai , H(ai+1 , . . . , ai+q ), ai+q+1 , . . . , ap+q−1 ), and the
i=0 (−1)
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 HH 2 (A).
Pp
j
2
If m =
j=0 mj ~ , mj ∈ C (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
p
X

3

∀ (a, b, c) ∈ A ,

(mi (a, mp+1−i (b, c)) − mi (mp+1−i (a, b), c)) = −d(2) mp+1 (a, b, c),

i=1

i.e.

p
X

[mi , mp+1−i ]G = 2 d(2) mp+1 .

i=1

According to the graded Jacobi identity for [·, ·]G , the last sum belongs to Ker d(3) . So HH 3 (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

d4

/ C4 (A)

/ C3 (A)

d3

/ C2 (A)

d2

/ C1 (A)

d1

/ C0 (A),

where the space of p-chains is given by L
Cp (A) = 0 for p ∈ −N∗ , C0 (A) = A, and for p ∈ N∗ ,
⊗p
Cp (A) = A ⊗ A . The differential d = ∞
i=0 dp is given by
dp (a0 ⊗ a1 ⊗ · · · ⊗ ap ) = a0 a1 ⊗ a2 ⊗ · · · ⊗ ap
+

p−1
X

(−1)i a0 ⊗ a1 ⊗ · · · ⊗ ai ai+1 ⊗ · · · ⊗ ap + (−1)p ap a0 ⊗ a1 ⊗ · · · ⊗ ap−1 .

i=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 / hf1 , . . . , fm i. 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
C[z1 , . . . , zn ]
Te = A[η1 , . . . , ηn ; b1 , . . . , bm ] =
[η1 , . . . , ηn ; b1 , . . . , bm ],
hf1 , . . . , fm i

P
P
Two deformations m = pj=0 mj ~j , mj ∈ C 2 (A) and m′ = pj=0 m′j ~j , m′j ∈ C 2 (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
2

∀ a ∈ A,
∀ n ∈ N,

ϕ(a) =
X
i+j=n

∞
X

ϕj (a)~j ,

j=0

ϕi (mj (a, b)) =

X
i+j+k=n

m′i (ϕj (a), ϕk (b)).

6

F. Butin

where ηi := ∂z∂ i is an odd variable (i.e. the ηi ’s anticommute), and bj an even variable (i.e. the
bj ’s commute).
Te is endowed with the differential
dTe =

n X
m
X
∂fi ∂
bi
,
∂zj ∂ηj
j=1 i=1

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 cohomology:
Theorem 1 (Kontsevich). Under the previous assumptions, the Hochschild cohomology of A
is isomorphic to the cohomology of the complex (Te, dTe ) associated with the differential graded
algebra Te.

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
e
0

Te(0)

/ Te(1)

(1)
T

de

(2)
T

de

/ Te(2)

/ Te(3)

(3)
T

de

/ Te(4)

(4)
T

de

/ ... .

For each degree p, we choose a basis Bp of Te(p). For example for p = 0, . . . , 3, we may take
Te(0) = A,
Te(1) = Aη1 ⊕ · · · ⊕ Aηn ,

Te(2) = Ab1 ⊕ · · · ⊕ Abm ⊕
Te(3) =

M

A bi ηj ⊕

i=1,...,m
j=1,...,n

M

A ηi ηj ,

i