Fields Institute Communications Volume 00, 0000

Invariants of Singular Plane Curves

Ruth I. Michler †

Department of Mathematics, University of North Texas, Denton, Texas 76203-1430

                                   michler@unt.edu

Abstract. This paper illustrates the use of cyclic homology in the study of curve singularities. More specifically, it reinterprets the dimension of the first cyclic homology group of irreducible singular plane algebroid curves in terms of the r-invariant introduced by O. Zariski in [34] and the Tjurina number. A generalization for affine plane irreducible curves with a single singularity is given as well.

1 Introduction

The use of cyclic homology in the study of singularities was previously explored in the analytic setting by J. Brasselet and A. Legrand in [4]. In this paper we give a geometric interpretation of cyclic homology for irreducible singular plane algebroid curves, see Theorem 4.7. This result is the interpretation of the r-invariant for singular plane curves as defined by O. Zariski [32] in his 1973 monograph: “Modules pour les Branches Planes”, as the dimension of the completed first cyclic cohomology group HC_{1}(A)ˆ minus the Tjurina number of A. The notion of the r-invariant is also meaningful for affine irreducible curve singularities, see Definition 4.5. In the main result of this note we determine the r-invariant for irreducible affine curves with a single singularity at the origin that is analytically irreducible (Theorem 4.9). Its proof rests on the facts that the length c of the conductor ideal C of an irreducible algebraic curve A and the dimension of the torsion module of the Kähler module of Ω_{A/K}^{1} of differentials are preserved by localization and completion at the singular point of A.

In a series of papers several authors have computed cyclic homology of algebraic varieties, see [1], [2], [6], [11], [12], [18], [21], [30], and the author’s articles [24], [26]. In [26] the author showed that in the case of a plane singular curve A, the vector space dimensions of the odd Hodge components of cyclic homology equals the sum of dim_{K}(H_{dR}^{1}(A)) and the Tjurina number, a classical invariant of hypersurface singularities. In fact, by [25] the Tjurina number equals the vector space dimension

\begin{matrix}c \\ ◯ \end{matrix}0000 American Mathematical Society

1991 Mathematics Subject Classification. Primary 14F10, 32C20; Secondary 19D55.

The author was supported in part by TARP Grant #003594-0030b-1997. † She was killed in a traffic accident in Boston on November 1, 2000.

2 Ruth I. Michler †

of the torsion module Torsion(Ω_{A/K}^{1}) of the Kähler module Ω_{A/K}^{1} of differentials (Proposition 2.3).

The author first became aware of a connection of cyclic cohomology with classical invariants, when reading the paper by Heinrich [16], which contained a computation of the length of the torsion module of differentials using Berger’s dimension formula [3] for the torsion module Torsion(Ω_{A/K}^{1}). Heinrich started from a parametrization of the curve as opposed to an equation. The elimination of the parameter is feasible but tedious. Proposition 2.3 allows the application of computer algebra systems. This is illustrated in Example 2.4, where an elementary and short proof for Heinrich’s counterexample [16] to Azevedo’s conjecture is given.

By combining the author’s computation of the dimension of the torsion module and the Berger formula, a computation of the r-invariant of affine irreducible curve singularities becomes possible (Theorem 4.9). The missing step was a computation of the length of the conductor ideal in the non-local analytic case. This was done by H. Hironaka in his paper [17]. Instead of using Puiseux pairs he uses the multiplicity sequence of successive blow-ups at the singular points of the plane curve. Hironaka also proved that his definition of the constant c coincides with the length of the conductor ideal in the local analytic case. All these results are stated in section 4. The quoted results on cyclic homology can be found in section 3.

The algorithmic nature of the results of this note is demonstrated by calculating the r-invariant and related invariants for explicit examples of singular plane irreducible curves, see Examples 4.12, 4.13 and Proposition 5.1.

Concerning notation and terminology the reader is referred to the books by Brieskorn and Knörrer [5], Eisenbud [10], Vasconcelos [29] and C. Weibel [31].

2 Torsion module

We will now only consider plane singular curves: Let R := K[X, Y] where K is an algebraically closed field of characteristic zero and let the reduced, singular, plane curve C be defined by the polynomial F ∈ R. We recall the following.

Throughout this paper A denotes the coordinate ring of C, i.e.

A := \frac{K[X, Y]}{(F)} and C := Spec(A).

In the following we denote the curve C by its coordinate ring A.

Definition 2.1 The singular locus of the plane curve A as defined above is defined by:

Sing(A) := Spec(\frac{K[X, Y]}{(F, \frac{∂F}{∂X} , \frac{∂F}{∂Y})}).

We say that A has a singularity at the point P corresponding to the maximal ideal M if M ⊃ (F, \frac{∂F}{∂X} , \frac{∂F}{∂Y}).

We use the following definition of an isolated singularity:

Definition 2.2 [19] A reduced hypersurface A has an isolated singularity at P ∈ Spec(A) if there exists an element s ∈ A − P such that Sing(A_{s}) = {P_{s}}.

Throughout this article Ω_{A/K}^{1} denotes the module of Kähler differentials of the reduced curve A defined by the polynomial F ∈ R = K[X, Y], see Weibel [31], p. 294. Its torsion module is defined by

Torsion(Ω_{A/K}^{1}) = {m ∈ Ω_{A/K}^{1} | md = 0 for some non-zero divisor d ∈ A}.

One important invariant of singular plane curves is their Tjurina number, which is the K vector-space dimension of K[X, Y]/(F, \frac{∂F}{∂X} , \frac{∂F}{∂Y}).

In [25] the author has shown that Greuel’s [13] result holds in the non-algebroid case as well and we have:

Proposition 2.3 The Tjurina number of A is

dim_{K}[ Torsion (Ω_{A/K}^{1})] = dim_{K} \frac{K[X, Y]}{(F, \frac{∂F}{∂X} , \frac{∂F}{∂Y})}.

As an immediate consequence of this, the dimension of the torsion module Torsion(Ω_{A/K}^{1}) can be computed using computer algebra systems. See also [8], Exercise 7, p. 176, where an efficient algorithm is given for calculating the Tjurina number.

As an application of Proposition 2.3 we give a short proof for Heinrich’s counterexample [16] to the conjecture of Azevedo, which asserts:

If A is an irreducible plane algebroid curve, and A_{c} is the canonical branch in the equisingularity class of A as defined by Zariski [33], part II, p. 995, then

dim_{K}[Torsion(Ω_{A_{c}/K}^{1})] ≥ dim_{K}[Torsion(Ω_{A/K}^{1})].

Example 2.4 (Heinrich [16]) Consider the algebroid curve A given by the following parametrization:

a) X = t^{6}, Y = t^{9} + 2t^{10} − 2t^{11} ∈ K[[t]].

It has characteristic (6, 9, 10) in the sense of Zariski [33], and the canonical branch A_{c} with this characteristic is:

b) X = t^{6}, Y = t^{9} + t^{10} ∈ K[[t]].

Instead of using Berger’s dimension formula [3] for computing the vector space dimension of the two torsion modules Torsion(Ω_{A/K}^{1}) and Torsion(Ω_{A_{c}/K}^{1}) we eliminate in both curves the parameter t, and we obtain two reduced polynomials F, F_{c} ∈ K[X, Y] each of which defines a plane curve B, B_{c}, respectively, with a singularity at the origin.

Case a) By insertion of X = t^{6} and Y = t^{9} + 2t^{10} − 2t^{11} it can be checked that

F = Y^{6} − 3Y^{4}X^{3} + 3Y^{2}X^{6} + 32Y^{3}X^{5} − 96Y X^{8} − 96Y^{2}X^{7}
−192Y X^{9} − X^{9}(1 − 416X + 64X^{2}).

Hence

\frac{∂F}{∂X} = −9Y^{4}X^{2} + 18Y^{2}X^{5} + 160Y^{3}X^{4} − 768Y X^{7} − 672Y^{2}X^{6}
−1728Y X^{8} − 9X^{8}(1 − 46X + 64X^{2}) − X^{9}(−416 + 128X),

\frac{∂F}{∂Y} = 6Y^{5} − 12Y^{3}X^{3} + 6Y X^{6} + 96Y^{2}X^{5} − 96X^{8} − 192Y X^{7} − 192X^{9}

In particular, F has a singularity at the origin. Let M = (X, Y) be the corresponding maximal ideal of R = K[X, Y], and let m = M/(F) be the corresponding ideal in the singular locus of B = R/(F).

Let I = (F, \frac{∂F}{∂X} , \frac{∂F}{∂Y}), and let B_{m} be the localization of B at m. Then by Proposition 2.3 we have

dim_{K}[Torsion(Ω_{B/K}^{1})] = dim_{K}(R_{M}/I_{M}).

Applying Lazard’s algorithm, exercise 7 of Cox, Little and O’Shea [8], p. 176, the Tjurina number dim_{K}(R_{M}/I_{M}) is computed as follows. Let T be another indeterminate over K.

In S = K[T, X, Y] let “>“ be the monomial order lex with T > X > Y defined in [8], p. 159. Let

F^{h} = T^{5}Y^{6} − 3T^{4}X^{3}Y^{4} + 3T^{3}X^{6}Y^{2} + 32T^{3}X^{5}Y^{3} − T^{2}X^{9}
−96T^{2}X^{8}Y − 96T^{2}X^{7}Y^{2} + 416T X^{10} − 192T X^{9}Y − 64X^{11},

(\frac{∂F}{∂X})^{h} = −9T^{4}X^{2}Y^{4} + 18T^{3}X^{5}Y^{2} + 160T^{3}X^{4}Y^{3} − 9T^{2}X^{8} − 768T^{2}Y X^{7}
−672T^{2}X^{6}Y^{2} + 4160T X^{9} − 1278T X^{8}Y − 704X^{10},

(\frac{∂F}{∂Y} )^{h} = 6T^{5}Y^{5} − 12T^{4}X^{3}Y^{3} + 6T^{3}X^{6}Y + 96T X^{5}Y^{2} − 96T^{2}X^{8}
−192T^{2}X^{7}Y − 192T X^{9}

be the homogenizations of F, \frac{∂F}{∂X}, \frac{∂F}{∂Y} in S, respectively.

By Theorem 2 of Cox, Little and O’Shea [9], p. 367, the ideal

I^{h} = (F^{h}, (\frac{∂F}{∂X})^{h}, (\frac{∂F}{∂Y} )^{h})

of S has a reduced Gröbner basis G consisting of homogeneous polynomials. Using Mora’s normal form algorithm [8], p. 160 implemented in MACAULAY2 it follows that

G = {T^{2}X^{12}, T X^{14}, X^{18}, T^{2}X^{10}Y, T X^{13}Y, T^{3}X^{7}Y^{2}, T^{2}X^{9}Y^{2},
T^{3}X^{5}Y^{3}, T X^{12}Y^{3}, T^{4}X^{2}Y^{4}, T^{2}X^{8}Y^{4}, T^{5}Y^{5}, T X^{11}Y^{5}}

Setting now T = 1 we obtain by [8], p. 176 the Gröbner basis

H = {X^{12}, X^{10}Y, X^{7}Y^{2}, X^{5}Y^{3}, X^{2}Y^{4}, Y^{5}}

of I_{M} in R_{M}. Hence

dim_{K}[Torsion(Ω_{B/K}^{1})] = 36.

Case b) By insertion of X = t^{6} and Y = t^{9} + t^{10} it can be checked that

F_{c} = Y^{6} − 3Y^{4}X^{3} − 2Y^{3}X^{5} + 3Y^{2}X^{6} − 6Y X^{8} − X^{9} + X^{10}.

Using MAPLE and the lexicographical order y > x it follows that a Gröbner basis of the ideal I_{c} = (F_{c}, \frac{∂F_{c}}{∂X} , \frac{∂F_{c}}{∂Y}) of R = K[X, Y] is given by: Y^{5} − 2Y^{3}X^{3} − Y^{2}X^{5} + Y X^{6} − X^{8}, Y^{4}X^{2} + \frac{10}{9}Y^{3}X^{4} − 2Y^{2}X^{5} + \frac{16}{3}Y X^{7} − \frac{10}{9}X^{9} + X^{8}, Y^{3}X^{5} + 3Y X^{8} − X^{10}, Y^{2}X^{7} − \frac{30}{77}Y X^{9} − \frac{1}{77}X^{10}, Y X^{10} + \frac{8}{9}X^{11}, X^{12} + \frac{64}{27}X^{11}.

Let x and y be the images of X and Y in R_{M}, respectively. Then the two last equations imply that x^{11} = 0 and yx^{10} = 0 in R_{M}/(I_{c})_{M}. Since (B_{c})_{m} ≅ R_{M}/(I_{c})_{M} the other equations of the above Gröbner basis yield that

dim_{K}[Torsion(Ω_{(B_{c})/K}^{1})] = 35.

It follows from cases a) and b) that Heinrich’s example [16] is a counterexample to Azevedo’s conjecture.

3 Cyclic Homology

Throughout this section K denotes a field of characteristic 0. Let N ≥ 2 be an integer, and let F ∈ R = K[X_{1}, X_{2}, …, X_{N}] be a reduced polynomial in N indeterminates X_{1}, X_{2}, …, X_{N} over K defining a reduced affine hypersurface A = R/(F) in the affine space 𝔸^{n}. Let Ω_{A/K} be the Kähler module of differentials. Let Ω_{A/K}^{i} be the i-th exterior power of Ω_{A/K} for i = 1, 2, . . ..

The basic results on cyclic homology are explained in C. Weibel’s book [31], where the cyclic homology groups HC_{n}(A) for any K-algebra A are defined on p. 334. By a classical theorem of J. L. Loday each HC_{n}(A) has a direct decomposition

HC_{n}(A) = HC_{n}^{(1)}(A) ⊕ HC_{n}^{(2)}(A) ⊕ ⋯ ⊕ HC_{n}^{n}(A)

into its Hodge components HC_{n}^{(i)}(A), 1 ≤ i ≤ n, see [31], p. 352.

In [26] the author computed the cyclic homology of reduced affine hypersurfaces. She found:

Theorem 3.1 Let A be a reduced, affine hypersurface over K with only isolated singularities defined by a polynomial F ∈ K[X_{1}, X_{2}, …, X_{N}], and let n > N, then the Hodge-components of cyclic homology are given by:

HC_{n}^{(i)}(A) ≃ \{ \begin{matrix}Torsion (Ω_{A/K}^{N−1}) ⊕ H_{dR}^{N−1}(A) & if 2i − n = N − 1 \\ H_{dR}^{2i−n}(A) & otherwise \end{matrix}

Summing up the Hodge-components, we get:

HC_{n}(A) ≃ H_{dR}^{N}(A) ⊕ H_{dR}^{N−2}(A) ⊕ ⋯ if n ≡ N mod 2,

HC_{n}(A) ≃ Torsion (Ω_{A/K}^{N−1}) ⊕ H_{dR}^{N−1}(A) ⊕ H_{dR}^{N−3}(A) ⊕ ⋯ if n ≡ N − 1 mod 2.

Hence in [26] we obtained a similar formula as obtained in the smooth case by J.L. Loday and D. Quillen in [23] with an extra non-zero term Torsion (Ω_{A/K}^{N−1}) appearing for n ≥ N.

In our study of plane singular curves we need Theorem 3.1 only in the special case, where N = 2. Thus it suffices to define H_{dR}^{1}(A).

Definition 3.2 The naive de Rham cohomology is defined as

H_{dR}^{1}(A) = \frac{Ker(d : Ω_{A/K}^{1} ↦ Ω_{A/K}^{2})}{dA}

Cyclic homology was invented by A. Connes [7] in 1993 in his study of foliation spaces. J.L. Loday and D. Quillen [23] have computed cyclic homology of regular K-algebras A. Furthermore, for general K-algebras A Proposition 2.1.14 of Loday [22] states that

HC_{1}(A) = \frac{Ω_{A/K}^{1}}{dA} .

4 Integral closure and the r-invariant

The normalization Ã of the irreducible plane curve A is the spectrum of the integral closure of A in its quotient field Q.

Definition 4.1 Recall that the conductor ideal is defined as

C = (Ã : A) := {q ∈ A | Ãq ≤ A}

The integer c = dim_{K}(Ã/C) is called the length of the (global) conductor of the plane irreducible curve A.

The following subsidiary result is well known, see Kunz [20], p. 168 and p. 170.

Lemma 4.2 Let C be the conductor ideal of a plane singular curve A with integral closure Ã. Then the A-module Ã/C is supported on only a finite number of maximal ideals m of the singular locus Sing(A). Furthermore,

a) c = ∑_{m∈Sing(A)} c_{m}, where c_{m} = dim_{K}[(Ã/C)_{m}] for each m ∈ Sing(A).

b) c = 2 dim_{K}(Ã/A).

Proof a) By Corollary 6.4.1 of Vasconcelos [29], p. 164 the A-module Ã/C is only supported at the finitely many maximal ideals containing the singular locus Sing(A) of A. Note that the integral closure of the coordinate ring of A is also its desingularization. Hence

(Ã_{m} : C_{m})

has length 0 unless m lies in the singular locus. Since the singular locus of a plane curve consists of only finitely many points, we also have that the conductor ideal is only supported at finitely many points.

b) holds by a) and Gorenstein’s theorem, see Kunz [20], p. 170.

In his paper [17] H. Hironaka introduced a general method for the resolution of singularities of algebraic curves. In the special case of plane curves it yields the following formula for the computation of the local conductor lengths c_{m} of an irreducible singular curve A = K[X, Y]/(F).

Proposition 4.3 (Hironaka [17]) Let F ∈ R = K[X, Y] be an irreducible polynomial defining a plane singular curve A = R/(F). For every maximal ideal m in the singular locus Sing(A) of A let c_{m} be the conductor length of the local domain A_{m}. Then

c_{m} = ∑ e_{i}(e_{i} − 1)

where the multiplicities e_{i} are obtained from a finite sequence of blow-ups of A_{m}.

Lemma 4.4 Normalization commutes with completion.

Proof This lemma is a restatement of Theorem 33 of Zariski-Samuel [35], p. 320.

As a consequence, we have that the analytic and the algebraic definition of the length c of the conductor ideal C agree.

Definition 4.5 The r-invariant of the singular plane curve A is defined by:

r = c − dim_{K}[ Torsion (Ω_{A/K}^{1})],

where c is the dimension dim_{K}[Ã/C] of the residue class ring of the integral closure Ã of A by the conductor ideal C = (Ã : A) .

For singular plane algebroid curves this definition is due to O. Zariski [32], p. 35.

Remark 4.6 By Propositions 2.3 and 4.3 we can compute the r-invariant of a plane singular curve without knowing the Puiseux expansions and a parametrization of C as an algebroid curve. By using Hironaka’s result we circumvent the problem that an irreducible plane algebraic curve usually becomes reducible after localization and completion. It would then become necessary to work with each branch separately and then take care of intersections of branches. Note the fact that Puiseux expansions are only defined for branches.

Below and in the proof of Theorem 4.9 we use the “continuous” or completed module of Kähler differentials (Ω̂_{A/K}^{1}, d), as defined by Grothendieck in [14, 20.4.8.2]. Note that for (A, M) a complete local ring, A = K[[X, Y]]/(F), we have

Ω̂_{A/K}^{1} = (AdX ⊕ AdY)/AdF

(see Exercise 16.14d of D. Eisenbud’s [10] p. 413). We define completed de Rham cohomology, and completed cyclic homology analogously to Definition 3.2:

H_{dR}^{1}(A)ˆ = \frac{Ker(d : Ω̂_{A/K}^{1} ↦ Ω̂^{2})}{dA}, and HC_{1}(A)ˆ = \frac{Ω̂_{A/K}^{1}}{dA} .

Theorem 4.7 Let K be an algebraically closed field of characteristic zero, and let A = K[[X, Y]]/(F) be the coordinate ring of a plane algebroid curve with singularities, defined by an irreducible power series F ∈ K[[X, Y]]. Let (Ω̂_{A/K}^{1}, d) and (Ω̂_{Ã/K}^{1}, D) be the completed Kähler modules of continuous differentials with

canonical derivations d and D of A and its integral closure Ã, respectively. Then:

a) There is a unique A-module homomorphism σ : Ω̂_{A/K}^{1} → Ω̂_{Ã/K}^{1} such that Da = σda, for all a ∈ A, and the r-invariant of A is given by:

r = c − dim_{K}[Torsion(Ω̂_{A/K}^{1})] = dim_{K}[σ̄(HC_{1}(A)ˆ )].

where HC_{1}(A)ˆ is the completed first cyclic homology group, and σ̄ : Ω̂_{A/K}^{1}/dA → Ω̂_{Ã/K}^{1}/DA is the canonical residue class map induced by σ.

b) In particular,

dim_{K} HC_{1}(A)ˆ = r + dim_{K}[Torsion(Ω̂_{A/K}^{1})] = c, and
r = dim_{K} H_{dR}^{1}(A)ˆ .

Proof a) Let Ã be the integral closure of A in its quotient field Q. Let C = (Ã : A) be the conductor. Then

c = dim_{K}(Ã/C) = 2 dim_{K}(A/C) = 2 dim_{K}(Ã/A)

is finite, by Lemma 4.2.

By the Corollary of Zariski-Samuel [35], p. 307 it is well known that the integral closure Ã of A is a formal power series ring K[[t]] in one parameter t over K. Hence Ω̂_{Ã/K}^{1} = (Dt)Ã ≅ Ã, where the canonical derivation D : Ã → Ω̂_{Ã/K}^{1} is defined by

Dq = (∑_{i=1}^{∞} ia_{i}t^{i−1})Dt ∈ Ω̂_{Ã/K}^{1} for every q = ∑_{i=0}^{∞} a_{i}t^{i} ∈ Ã = K[[t]] with a_{i} ∈ K.

Furthermore, the derivation D : Ã → Ω̂_{Ã/K}^{1} is surjective, and it has kernel ker(D) = K ⊆ A ⊆ Ã. By naturality (see Eisenbud [10], p. 384) there exists a unique Amodule homomorphism σ : Ω̂_{A/K}^{1} → Ω̂_{Ã/K}^{1} such that Da = σda for all a ∈ A. Therefore σ induces an A-module homomorphism σ̄ : Ω̂_{A/K}^{1}/dA → Ω̂_{Ã/K}^{1}/DA such that

ADA/DA ≅ σ̄(Ω̂_{A/K}^{1}/dA) ≤ Ω̂_{Ã/K}^{1}/DA. (*)

Now Berger’s dimension formula stated in Korollar 2 of [3], p. 349 asserts that

dim_{K}[Torsion(Ω̂_{A/K}^{1})] = \frac{1}{2}c + dim_{K}[ÃDÃ/ADA],

where ADA is the image of σ, and ÃDÃ = Ω̂_{Ã/K}^{1}. By the third isomorphism theorem, we have

dim_{K}[ÃDÃ/ADA] = dim_{K}[ÃDÃ/DA] − dim_{K}[ADA/DA]. (**) As D : Ã → ÃDÃ = DtÃ is surjective, K-linear, with kernel Ker(D) = K ≤ A

it follows from Lemma 4.2 b) that

dim_{K}[ÃDÃ/DA] = dim_{K}[DÃ/DA] = dim_{K}[Ã/A] = \frac{1}{2}c.

Hence (*) and (**) imply that

dim_{K}[ÃDÃ/ADA] = \frac{1}{2}c − dim_{K}[σ̄(Ω̂_{A/K}^{1}/dA)], and

dim_{K}[Torsion(Ω̂_{A/K}^{1})] = c − dim_{K}[σ̄HC_{1}(A)ˆ ],

because Ω̂_{A/K}^{1}/dA = HC_{1}(A)ˆ by Proposition 2.1.14 of Loday [22], p. 58. Thus r = dim_{K}[σ̄HC_{1}(A)ˆ ].

b) Now Ã is a regular local ring. Thus Ω̂_{Ã/K}^{1} is a torsion-free A-module. Since σ : Ω̂_{A/K}^{1} → Ω̂_{Ã/K}^{1} is an A-module homomorphism it follows that Torsion(Ω̂_{A/K}^{1}) ≤ Ker(σ). If u = adx + bdy ∈ Ker(σ), a, b ∈ A, then

σ(u) = aσdx + bσdy = aDx + bDy = 0.

By the definition of Ω̂_{A/K}^{1} we have F_{x}dx+F_{y}dy = 0, where F_{x} = \frac{∂F}{∂X} and F_{y} = \frac{∂F}{∂Y}. Applying σ it follows that F_{x}Dx+F_{y}D_{y} = 0. Therefore \frac{a}{b} = \frac{F_{x}}{F_{y}} or \frac{b}{a} = \frac{F_{y}}{F_{x}} for b ≠ 0 or a ≠ 0, respectively, because A and Ã have the same quotient field, and Ω̂_{Ã/K}^{1} is a free Ã-module. In any case bF_{x} − aF_{y} = 0. Thus u = adx + bdy ∈ Torsion(Ω̂_{A/K}^{1}) by equation (∗) of [25], p. 83. Hence Ker(σ) = Torsion(Ω̂_{A/K}^{1}).

Suppose that u ∈ Torsion(Ω̂_{A/K}^{1}) ∩ dA. Then u = da for some a ∈ A and σ(u) = σda = Da = 0. Hence a ∈ Ker(D) = K by the definition of D, and u = da = 0. In particular,

[Torsion(Ω̂_{A/K}^{1}) + dA]/dA ≅ TorsionΩ̂_{A/K}^{1}.

From the definition of σ̄ follows that

σ̄(HC_{1}(A)ˆ ) ≅ [Ω̂_{A/K}^{1}/Torsion(Ω̂_{A/K}^{1})]/[(dA + (Torsion(Ω̂_{A/K}^{1}))/Torsion(Ω̂_{A/K}^{1})]
≅ [Ω̂_{A/K}^{1}/dA]/[(Torsion(Ω̂_{A/K}^{1}) + dA)/dA].

Hence r = dim_{K}[σ̄(HC_{1}(A)ˆ )] = dim_{K} HC_{1}(A)ˆ − dim_{K}[Torsion(Ω̂_{A/K}^{1})]. Since dim_{K} HC_{1}(A)ˆ = dim_{K}[Torsion(Ω̂_{A/K}^{1})]+dim_{K} H_{dR}^{1}(A)ˆ by Lemma 5 of [26], p. 296 we get r = dim_{K}[σ̄(HC_{1}(A)ˆ )] = dim_{K} H_{dR}^{1}(A)ˆ .

A geometric interpretation of this r-invariant is the K-vector space dimension of the space of non-exact regular differentials. Recall that a holomorphic differential ω ∈ Ω̂_{A/K}^{1} is exact if there exists an element u ∈ A such that du = ω.

Remark 4.8 Keep the notation of Theorem 4.7. Its assertion a) is by the isomorphism (*) equivalent to the following result of O. Zariski: The r-invariant r = c−dim_{K}[Torsion(Ω̂_{A/K}^{1})] is equal to the maximum number of linearly independent differentials in ADA ≤ Ω̂_{Ã/K}^{1} over K, no linear combination of which, with non-zero coefficients in K, is exact, see [34], p. 783.

From assertion b) of Theorem 4.7 and the main theorem of K. Saito [28] it follows immediately that the r-invariant of the irreducible plane algebroid curve A = K[[X, Y]]/(F) is zero if and only if for a suitable basis {x, y} of the maximal ideal m of A the equation of the curve A is of the form y^{n} = x^{m}, where (n, m) = 1. This statement is Theorem 4 of O. Zariski [34], p. 783. Hence assertion b) of Theorem 4.7 may be considered to be a generalisation of Zariski’s theorem.

The main result of this paper is the following identification of the r-invariant r with the dimension of the first continuous de Rham cohomology group of an analytically irreducible plane singular curve:

Theorem 4.9 Let F ∈ R = K[X, Y] be an irreducible polynomial defining a plane singular curve A = R/(F) with a single singularity at the origin m = (X, Y)/(F). Suppose that F remains irreducible when considered as an element of the power series ring K[[X, Y]]. Let Â_{m} be the completion of the local ring A_{m}. Then the r-invariant r of the plane irreducible curve A is given by:

r = dim_{K} H_{dR}^{1}(Â_{m})ˆ.

Proof Let Ã and Ã_{m} be the integral closures of A and A_{m}, respectively. Denote K[[X, Y]] by R̂. Let Â and Â_{m} be the completions of A and A_{m}, respectively. By Theorem 6 of Zariski-Samuel [35], p. 257 we have

Â_{m} ≅ R̂/(F).

In particular, by Theorem 4.7 b) the r-invariant r(Â_{m}) of the algebroid irreducible curve Â_{m} is

r(Â_{m}) = dim_{K} H_{dR}^{1}(Â_{m})ˆ .

Since m is the only singularity of A, we have

Torsion(Ω_{A/K}^{1}) ≅ Torsion(Ω_{A_{m}/K}^{1}) by [27], p. 640.

Now Theorem 2 of [25] asserts that

Torsion(Ω_{A/K}^{1}) ≅ (F : J)/J,

where J = (\frac{∂F}{∂X} , \frac{∂F}{∂Y}) is the Jacobian ideal of F.

Applying Proposition 2.3, since Torsion(Ω_{A_{m}/K}^{1}) = Torsion(Ω_{A/K}^{1}) ⊗_{A} A_{m} we get

dim_{K}[Torsion(Ω_{A_{m}/K}^{1})] = dim_{K}[Torsion(Ω_{A/K}^{1})] = dim_{K} \frac{R}{(F, J)},

and dim_{K}[Torsion(Ω_{A_{m}/K}^{1})] = dim_{K} \frac{R_{M}}{(F,J)_{M}}, where R_{M} denotes the localization of R = K[X, Y] at M = (X, Y).

A similar argument to the proof of Theorem 2 of [25], or using Torsion(Ω̂_{Â_{m}/K}^{1}) = Torsion(Ω_{A/K}^{1}) ⊗_{A} Â_{m} yields that

Torsion(Ω̂_{Â_{m}/K}^{1}) ≅ (F : JR̂_{M})/JR̂_{M}.

Since R̂_{M} ≅ R̂/F R̂ = K[[X, Y]]/F K[[X, Y]], it follows that

dim_{K}[Torsion(Ω̂_{Â_{m}/K}^{1})] = dim_{K}[(F : JR̂_{M})/JR̂_{M}] = dim_{K}[\frac{R̂}{F R̂ + JR̂}].

Now Proposition 2.11 of [8], p. 146 asserts that

dim_{K} \frac{K[X, Y]_{M}}{(F, J)K[X, Y]_{M}} = dim_{K} \frac{K[[X, Y]]}{(F K[[X, Y]] + JK[[X, Y]]}.

Therefore dim_{K}[Torsion(Ω_{A/K}^{1})] = dim_{K}[Torsion(Ω̂_{Â_{m}/K}^{1})].

Since m is the only singularity of F, Lemma 4.2 asserts that C = (Ã : A) = (Ã_{m} : A_{m}) is the conductor of the plane irreducible curve in its integral closure Ã, where Ã_{m} denotes the integral closure of the local domain A_{m}. From Lemma 4.4 it follows that CR̂ = (Ã_{m}R̂ : A_{m}R̂) is the conductor ideal of the completion Â_{m} of A_{m}. Hence CR̂ is the conductor ideal of the algebroid curve Â_{m} ≅ R̂/(F).

Since \frac{1}{2}c = dim_{K}(A/C) = dim_{K}(A_{m}/C_{m}) = dim_{K}(Â_{m}/CR̂), it follows that the r-invariant r of A equals the r-invariant r(Â_{m}) of Â_{m}. Hence

r = c − dim_{K}[Torsion(Ω_{A/K}^{1})] = r(Â_{m}) = dim_{K} H_{dR}^{1}(Â_{m})ˆ ,

which completes the proof.

Recall that a polynomial F ∈ K[X, Y] of degree d is called quasi-homogeneous with weights n, m if and only if it satisfies the generalized Euler equation

nX \frac{∂F}{∂X} + mY \frac{∂F}{∂Y} = dF

Corollary 4.10 Let F ∈ R = K[X, Y] be a quasi-homogeneous polynomial defining an irreducible plane curve A = R/(F) with a single singularity at the origin. Then its r-invariant r is zero.

Proof Each irreducible quasi-homogeneous polynomial F ∈ K[X, Y] remains irreducible by the criterion for irreducibility stated in Kunz [20], p. 158. Furthermore, the Poincaré lemma of Hartshorne [15], p. 53 and Proposition 7.1.1 of [15], p. 54 imply that the de Rham cohomology H_{dR}^{1}(Â_{m})ˆ = 0, where Â_{m} is the completion of the localization A_{m} of A at the origin m = (X, Y)/(F). Hence Theorem 4.9 asserts that

r = dim H_{dR}^{1}(Â_{m})ˆ = 0,

In [27] the author has proved that for singular plane curves A the torsion module Torsion(Ω_{A/K}^{1}) requires at most two generators and that this upper bound is sharp. There are many curves A with cyclic torsion module Torsion(Ω_{A/K}^{1}) which are not quasi-homogeneous, e.g. F = X^{7} + Y^{6} + X^{2}Y ∈ K[X, Y] as will be shown below.

Corollary 4.11 Let F ∈ R = K[X, Y] be an irreducible polynomial defining a plane singular curve A = R/(F) with a single singularity at the origin M = (X, Y). Suppose that F remains irreducible when considered as an element of the power series ring K[[X, Y]]. If the torsion module Torsion(Ω_{A/K}^{1}) of A is a cyclic A-module, then the r-invariant r = 0.

Proof Let m = (X, Y)/(F) = M/(F) be the maximal ideal of the singular locus Sing(A). Since Torsion(Ω_{A/K}^{1}) is a cyclic A-module, it follows from the proof of Theorem 3.1 of [27] that F ∈ J_{M}, where J = (\frac{∂F}{∂X} , \frac{∂F}{∂Y}) is the Jacobian ideal of F. Hence Saito’s main theorem [28], p. 123 asserts that F is locally analytically quasi-homogeneous in suitable coordinates of R̂_{M}. Thus dim_{K} H_{dR}^{1}(Â_{m})ˆ = 0 by the Poincaré lemma of [15], p. 53. Since F remains irreducible in K[[X, Y]], Theorem 4.9 asserts that r = 0.

Now follows an example of an irreducible polynomial F ∈ R = K[X, Y] defining a plane singular curve A = R/(F) with a single singularity at the origin such that its torsion module Torsion(Ω_{A/K}^{1}) is a cyclic A-module, but F does not remain irreducible in K[[X, Y]].

Example 4.12 Let F = X^{7} +Y^{6} +X^{2}Y ∈ R = K[X, Y]. It is easy to see that

F − \frac{1}{7}X \frac{∂F}{∂X} − \frac{1}{6}Y \frac{∂F}{∂Y} = \frac{23}{42}X^{2}Y.

Hence the origin m = (X, Y)/(F) is the only singular point of A = R/(F). Let M = (X, Y) and J = (\frac{∂F}{∂X} , \frac{∂F}{∂Y}). Then

\frac{1}{7} \frac{∂F}{∂X} − X^{4} \frac{∂F}{∂Y} = XY (\frac{2}{7} − 6Y^{5}X^{4}) ∈ J_{M}.

Hence XY ∈ J_{M}, and so F ∈ J_{M}. Let x, y be the images of X, Y in the local Artinian ring R_{M}/(F, J_{M}) = R_{M}/J_{M}. Then {1, y, y^{2}, y^{3}, y^{4}, x, x^{2}} is a K-vector space basis of R_{M}/J_{M}, and x^{2} generates the socle of R_{M}/J_{M}. Thus Torsion(Ω_{A/K}^{1}) is a cyclic A-module by [27], p. 644. Furthermore, dim_{K}[Torsion(Ω_{A/K}^{1})] = 7. Now Corollary 4.11 asserts that the r-invariant of A would be zero if F were irreducible in K[[X, Y]]. Then c = 7. However, c ≠ 7 because c is even by Lemma 4.2 b). Thus F does not remain irreducible in K[[X, Y]].

This section is concluded by the following example of a non quasi-homogeneous irreducible polynomial F ∈ K[X, Y] satisfying the hypothesis of Corollary 4.11.

Example 4.13 Let F = X^{6} + Y^{5} + X^{5}Y^{3} ∈ R = K[X, Y]. Then F remains irreducible in K[[X, Y]] by the irreducibility criterion of Kunz [20], p. 158, because the irreducible quasi-homogeneous polynomial X^{6} + Y^{5} is the quasi-homogeneous polynomial of least degree occurring in F. Clearly

F − \frac{1}{6}X \frac{∂F}{∂X} − \frac{1}{5}Y \frac{∂F}{∂Y} = −\frac{13}{30}X^{5}Y^{3}.

Hence the origin m = (X, Y)/(F) is the only singular point of A = R/(F). Let M = (X, Y) and J = (\frac{∂F}{∂X} , \frac{∂F}{∂Y}). Then

−\frac{1}{2}Y^{2} \frac{∂F}{∂X} + \frac{∂F}{∂Y} = 5Y^{4}(1 − \frac{1}{2}X^{4}Y) ∈ J_{M}.

Thus Y^{4}, X^{5}Y^{2} ∈ J_{M}, which implies F ∈ J_{M}. Let x, y be the images of X, Y in the local Artinian ring R_{M}/(F, J)_{M} = R_{M}/J_{M}. Then

{x^{4}, x^{4}y, x^{4}y^{2}, x^{5}} ∪ {x^{i}y^{j} | 0 ≤ i, j ≤ 3}

is a K-vector space basis of R_{M}/J_{M}. The socle of R_{M}/J_{M} is one-dimensional with basis vector x^{5}. Thus Torsion(Ω_{A/K}^{1}) is a cyclic A-module by [27], p. 644, and dim_{K}[Torsion(Ω_{A/K}^{1})] = 20.

Now Corollary 4.11 asserts that r = 0, and c = 20. Since A_{m} becomes regular after one blow up and the multiplicity e_{1} = 5, also Proposition 4.3 asserts that the conductor length c = 20.

5 The nodal cubic

In the final section we demonstrate the algorithmic nature of our results by computing the cyclic homology, the de Rham cohomology, the Tjurina number, and the r-invariant of the coordinate ring A of the nodal cubic F = Y^{2} − X^{2}(X + 1) ∈ R = K[X, Y], where K is a field of characteristic zero. In particular, it follows that cyclic homology does not commute with completion, in general.

This example was also studied in [12] (c.f. A.6.), where a different approach to finding HC_{n}(A) was suggested. We have:

Proposition 5.1 The coordinate ring A = R/(F) of the nodal cubic F has a single singularity at the origin. Furthermore:

a) The de Rham cohomology of A is given by:

H_{dR}^{0}(A) = K, H_{dR}^{1}(A) ≃ K.X^{2}dY + dA, H_{dR}^{2}(A) = 0.

b) Torsion (Ω_{A/K}^{1}) is a cyclic A-module with generator:

w = (\frac{3}{2}X + 1)Y dX − (X^{2} + X)dY, and dim_{K}[TorsionΩ_{A/K}] = 1

c) The cyclic homology has dimension dim_{K} HC_{1}(A) = 2.

d) The conductor length c = dim_{K}(Ã/C) = 2.

e) The r-invariant r = c − dim_{K}[Torsion(Ω_{A/K}^{1})] = 1.

f) If Â_{m} denotes the completion of A at the maximal ideal m = \frac{(X,Y)}{(F)} of A, then

dim_{K} HC_{1}(Â_{m})ˆ = 1.

Proof It is easy to see that the origin is the only singularity of A.

a) The coordinate ring of the nodal cubic is a domain. So we know that H_{dR}^{0}(A) = K. Moreover Ω_{A/K}^{2} = KdX ∧ dY, i.e. H_{dR}^{2}(A) = 0. Using the fact that

Y dY = \frac{(3X^{2} + 2X)}{2}dX

we have that

\frac{Ω_{A/K}^{1}}{dA} = \frac{(K.XdY + K.X^{2}dY + dA)}{dA}.

Hence:

H_{dR}^{1}(A) = Ker(d̃: Ω_{A/K}^{1}/dA → Ω_{A/K}^{2}) = (K.X^{2}dY + dA)/dA.

b) and c) Now Theorem 2 of [24], and Lemma 5 of [25] imply that

2 = dim_{K}[Ω_{A/K}^{1}] = dim_{K} HC_{1}(A) = dim_{K}[Torsion(Ω_{A/K}^{1})] + dim_{K} H_{dR}^{1}(A).

Hence a) implies dim_{K}[Torsion(Ω_{A/K}^{1})] = 1.

In particular, Torsion(Ω_{A/K}^{1}) is a cyclic A-module. In [26], p. 297 we showed that w is a generator.

d) In order to compute the conductor length c = dim_{K}(Ã/C) of A in its integral closure Ã we use the blow-up method of Proposition 4.3.

Let Y = XU. Then

Y^{2} − X^{2}(X + 1) = X^{2}U^{2} − X^{2}(X + 1) = X^{2}(U^{2} − X − 1),

which implies X = U^{2} − 1 and Y = U^{3} − U. Hence Ã = K[U], and dim_{K} Ã/A = 1. Thus c = 2 by Lemma 4.2.

e) follows immediately from c) and d).

f) In K[[X, Y]] each unit is a square. Therefore Y^{2} = X^{2}(X + 1) can be transformed into Y^{2} = T^{2} after a change of coordinates. Hence Â_{m} is the completion of a quasi-homogeneous affine curve. Thus H_{dR}^{1}(Â_{m})ˆ = 0 by Hartshorne’s Poincaré Lemma [15], p. 53. Now Theorem 3.1 asserts that dim_{K} HC_{1}(Â_{m})ˆ = 1, because completion preserves the dimension of the torsion module.

The Hodge components of the cyclic homology of the nodal cubic are given in the following result which immediately follows from Proposition 5.1 and Theorem 3.1.

Theorem 5.2 ([26]) Let A be the coordinate ring of the nodal cubic defined over an algebraically closed field K of characteristic zero by F = Y^{2} − X^{2}(X + 1) ∈ K[X, Y]. Then HC_{0}(A) = A and we have for l > 0 :

HC_{2l}(A) = H_{dR}^{0}(A) = K, and
HC_{2l+1}(A) ≃ Torsion (Ω_{A/K}^{1}) ⊕ H_{dR}^{1}(A) ≃ K ⊕ K.

DR. RUTH MICHLER, MARCH 8, 1967 - NOVEMBER 1, 2000

Dr. Ruth Michler died tragically on November 1, 2000, as the result of a traffic accident in Boston. She was run over by an articulating boom lift as she was waiting to cross a street. She was an Associate Professor at University of North Texas, and a Visiting Scholar at Northeastern University supported by a Professional Opportunities for Women in Research and Education (POWRE) grant from the National Science Foundation.

Among Ruth’s unpublished papers were a draft of this article, which she intended to submit to the Valuation Conference Proceedings, and also the calculations done using Macaulay II or MAPLE. Her father, Prof. Gerhard Michler, put these together; he wishes on Ruth’s behalf to thank Hal Schenck who checked Ruth’s calculations, and who also suggested the use of Lazard’s algorithm in Example 2.4.

Thanks are also due to Professor M. Lejeune-Jalabert for pointing out an error in an earlier version of this paper, to Professor C. Weibel for helpful comments, and to Professor A. Iarrobino for assistance.

Correspondence concerning the article, or requests for reprints may be sent to Prof. A. Iarrobino (e-mail iarrobin@neu.edu) or Prof. M. Levine (e-mail marc@neu.edu), at the Mathematics Department, 567 Lake Hall, Northeastern University, Boston, MA 02115.

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