ON COHEN–MACAULAY MODULES ON SURFACE SINGULARITIES

YURIY A. DROZD, GERT-MARTIN GREUEL, AND IRINA KASHUBA

Introduction

During the study of Cohen–Macaulay modules on curve singularities (cf. [15, 9, 12, 7]) it was proved that such these singularities split into three classes:

• Cohen–Macaulay finite, having only finitely many indecomposable Cohen–Macaulay modules;

• Cohen–Macaulay tame, i.e., such that, for each fixed r, the indecomposable Cohen–Macaulay modules of rank r form a finite set of 1-parameter families;

• Cohen–Macaulay wild, that can be characterized in two ways:

– “geometrically” as those having n-parameter families of nonisomorphic indecomposable Cohen–Macaulay modules for arbitrary large n,

– “algebraically” as such that for every finitely generated algebra Λ there is an exact functor from the category of finite dimensional Λ-modules to the category of Cohen–Macaulay modules over this singularity which maps non-isomorphic modules to non-isomorphic ones and indecomposable to indecomposable.

The latter “versality” property shows that the study of modules in the wild case is extremely complicated needs indeed essentially new and highly non-trivial approach.

Moreover, it turned out that this “trichotomy” is closely related to the position of a curve singularity in the well-known Arnold’s list of singularities having good deformation properties (cf. [1]). Namely:

• a singularity is Cohen–Macaulay finite if and only if it dominates one of the simple plane curves singularities A_{n}, D_{n}, E_{6}, E_{7}, E_{8} ;

• a singularity is Cohen–Macaulay tame if and only if it dominates

one of the “serial” unimodal singularities T_{pq}. (Remind that a singularity (X, x) dominates (Y, y) if there is a birational surjection (X, x) → (Y, y) .)

In [10, 4] it was proved that a normal surface singularity (in characteristic 0) is Cohen–Macaulay finite if and only if it is a quotient singularity. In [16] it was shown that a simple elliptic singularity is always Cohen–Macaulay tame. In both cases a complete description of Cohen–Macaulay modules over such singularities was obtained. Moreover, in [16] Kahn elaborated a general method relating Cohen– Macaulay modules over normal surface singularities to vector bundles over some projective curves (usually singular and even non-reduced).

The first author was partially supported by CRDF Award UM2-2094.

In this paper we use the Kahn correspondence and the results of [8] to prove the following main theorems:

• every cusp singularity (cf. [14, 17]) is Cohen–Macaulay tame;

• a minimally elliptic singularity (cf. [20]) that is neither simple

elliptic nor cusp one is Cohen–Macaulay wild. As a corollary we get that any log-canonical surface singularity (cf. [18]) is Cohen–Macaulay tame (or finite, as quotient singularities are also log-canonical). There is an evidence that any other normal surface singularity is Cohen–Macaulay wild. We also give a description of Cohen–Macaulay modules over cusp singularities and use it to provide a description of Cohen–Macaulay modules over curve singularities T_{pq} (there was no such description in [7], the proof of their tameness was indirect).

1. Generalities

In what follows we use the following

Definitions and Notations 1.1. • A surface singularity means a spectrum X = Spec A, where A is a local henselian noetherian ring of Krull dimension 2. We denote by 𝔪 = 𝔪_{A} the maximal ideal of A and by k = A/𝔪 the residue field. We always suppose that the field k is algebraically closed and that Artin’s approximation theorem [2] is valid for A (e.g., A is complete, or is an henselization of an affine k-algebra, or is the local ring of a point of a complex analytic variety).

• Such a singularity is called normal if the ring A is normal, i.e., integral and integrally closed in its field of fractions Q. In what follows we only consider normal surface singularities. Remind that any normal surface singularity is isolated, i.e., 𝔪 is the unique singular point of it.

• A resolution of a normal surface singularity is a birational proper map π : X̃ → X, where X̃ is smooth, which induces an isomorphism X̃ \ E →^{∼} X \ { 𝔪 }, where E = π^{−1}(𝔪) . E is called the exceptional curve on X̃ (it is indeed a projective curve over k ). Denote by E_{i} ( i = 1, …, s) the irreducible components of E. Put also X̆ = X \ { 𝔪 }.

• A resolution π as above is called minimal if it cannot be decomposed as

X̃ ⟶^{π_{1}} X′ ⟶^{π_{2}} X ,

where X′ is also smooth. It is known (cf., e.g., [21]) that π is minimal if and only if neither of the components E_{i} is a smooth rational curve with the self-intersection index E_{i} · E_{i} = −1 .

• A cycle on X̃ is a divisor of the shape Z = ∑_{i=1}^{s} k_{i}E_{i}. Such a cycle is called positive if all k_{i} > 0 . We write Z ≤ Z′, where Z′ = ∑_{i=1}^{s} k′_{i}E_{i} if k_{i} ≤ k′_{i} for all i.

• The fundamental cycle, denoted by Z_{0} , is the minimal positive cycle (with respect to the above defined order) such that Z_{0} ·E_{i} ≤ 0 for all i.

• A positive cycle Z is called a weak reduction cycle if the sheaf O_{Z}(−Z) is generically generated by its global section (i.e., generated outside a finite set) and H^{1}(E, O_{Z}(−Z)) = 0 . Here, as usually, we write O_{Z} = OX̃/OX̃(−Z) and consider Z as a closed subscheme of X̃.

• A reduction cycle is a weak reduction cycle Z such that the sheaf ω_{Z}^{∨} is generically generated by global sections, where ω_{Z} = ωX̃ ⊗_{OX̃} O_{Z}(−Z) is the dualizing sheaf for Z and M^{∨}, where M is a coherent sheaf on a scheme S, denotes Hom_{O_{S}}(M, O_{S}) .

• One calls a surface singularity rational if H^{1}(X̃, OX̃) = 0 and minimally elliptic if it is Gorenstein (i.e., inj.dim_{A}A = 2 ) and H^{1}(X̃, OX̃) ≃ k .

• A simple elliptic singularity is such one that E is a smooth curve of genus 1.

• We call E a cyclic configuration in the following cases:

(i) s = 1 , E is rational and has a unique singular point which is a simple node;

(ii) s = 2 , E_{1} ≃ E_{2} ≃ ℙ^{1} and they intersect transversally in

(exactly) 2 points; (iii) s > 2 , E_{i} ≃ ℙ^{1} for all i, E_{i} · E_{i+1} = 1 , E_{s} · E_{1} = 1 , and

E_{i} · E_{j} = 0 otherwise. For a cyclic configuration we put E_{s+k} = E_{k} for all integers k.

• A cusp singularity is such one that E is a cyclic configuration. Note that both simple elliptic and cusp singularities are minimally elliptic. Moreover, for all of them Z_{0} = E.

• Denote by Coh S the category of coherent sheaves on a scheme S and by VB(S) the category of vector bundles on S, i.e., locally free coherent sheaves.

• For a surface singularity X = Spec A denote by CM(X) , or by CM(A) , the category of (maximal) Cohen–Macaulay modules over the ring A. If it is normal, Cohen–Macaulay modules coincide with reflexive modules, i.e., such A-modules M that M^{∨∨} ≃ M , where M^{∨} = Hom_{A}(M, A) . We always identify an A-module M with its “sheafification” M̃, which is a quasicoherent sheaf on X.

Remind the main result of Kahn [16, Theorem 1.4] concerning the relations between the vector bundles on a reduction cycle and the Cohen– Macaulay modules over a normal surface singularity.

Theorem 1.2. Let π : X̃ → X be a resolution of a normal surface singularity with the exceptional curve E and a reduction cycle Z. Denote by R_{Z} : CM(X) → VB(Z) the functor which is the composition of the functors

CM(X) → VB(X̃) : M ↦ (π^{∗}M)^{∨∨}

and

VB(X̃) → VB(Z) : F ↦ F ⊗_{OX̃} O_{Z} .

This functor maps non-isomorphic objects to non-isomorphic ones, and a vector bundle V ∈ VB(Z) is isomorphic to R_{Z}M for some M if and only if it is generically generated by global sections and there is an extension of V to a vector bundle V_{2} on 2Z such that the exact sequence

0 ⟶ V(−Z) ⟶ V_{2} ⟶ V ⟶ 0 ,

induces a monomorphism H^{0}(E, V(Z)) → H^{1}(E, V) .

We denote by VB^{K}(Z) the full subcategory of VB(Z) consisting of the vector bundles satisfying the latter Kahn’s conditions. Note that obviously rk R_{Z}M = rk M for any Cohen–Macaulay module M .

For minimally elliptic singularities one can give a simpler description of the latter category (cf. [16, Theorem 2.1]).

Theorem 1.3. Let π : X̃ → X be a minimal resolution of a minimal elliptic singularity. Then the fundamental cycle Z = Z_{0} is a reduction cycle and the category VB^{K}(Z) consists of the vector bundles of the form nO_{Z} ⊕ G, where G satisfies the following conditions:

1. G is generically generated by global sections,

2. H^{1}(E, G) = 0 ,

3. dim_{k} H^{0}(E, G(Z)) ≤ n. In particular, indecomposable objects of VB^{K}(Z) are the following:

• the trivial line bundle O_{Z} ,

• nO_{Z} ⊕ G, where G is an indecomposable vector bundle satisfying the above conditions (1),(2) and n = dim_{k} H^{0}(E, G(Z)) .

Remark. In the paper [16] Theorems 1.2 and 1.3 are proved in the “geometric” case, when A is a k-algebra and char k = 0 , but one easily verifies that all proofs can be easily extended to the general situation. Those results of Grauert–Riemenschneider [11] as well as of Laufer [20] used in [16] are also valid in this “abstract” context.

2. Cohen–Macaulay modules on cusp singularities

In this section E denotes a cyclic configuration, ν : Ẽ → E the normalization of E and E_{i} ( i = 1, …, s) the irreducible components of Ẽ. We put E_{s+k} = E_{k} for all integers k, O = O_{E}, Õ = OẼ, O_{i} = O_{E_{i}}, and always identify Õ and O_{i} with their images under ν_{∗}. If s > 2 , denote by p_{i} ( i = 1, …, s) the unique intersection point of E_{i} and E_{i+1} . Denote also by S = { p_{1}, p_{2}, …, p_{s} } the set of singular point of E and choose their indices in such a way that ν^{−1}(p_{i}) ⊂ E_{i} ∪E_{i+1} . Note that the latter preimage always consists of 2 points, which we denote by p′_{i} , p′′_{i+1} so that p′_{i} ∈ E_{i} and p′′_{i+1} ∈ E_{i+1} . Let J be the conductor of Õ in O, i.e., the biggest Õ-ideal contained in O. Then O_{i}/J O_{i} = k(p′_{i}) ⊕ k(p′′_{i}) .

Remind the description of vector bundles on a cyclic configuration given in [8]. Call an s-sequence any sequence of integers d = (d_{1}, …, d_{rs}) of length multiple of s. Call this sequence aperiodic if it cannot be obtained by repetition of a shorter s-sequence. Call an s-shift of d any sequence d^{k} = (d_{ks+1}, …, d_{rs}, d_{1}, …, d_{ks}) .

Theorem 2.1. Indecomposable vector bundles on a cyclic configuration are in one-to-one correspondence with the triples B = (d, m, λ) , where m is a positive integer, λ ∈ k \ 0 and d is a non-periodic s-sequence defined up to an s-shift.

We denote the vector bundle corresponding to the triple B by G(B) . The precise construction of this sheaf is the following:

1. Let d = (d_{1}, …, d_{rs}) . Put G_{i} = mO_{i}(d_{i}) and G̃(B) = ⨁_{i=1}^{rs} G_{i}. Note that G_{i}/J G_{i} ≃ mk(p′_{i}) ⊕ mk(p′′_{i}) . Put also

F (B) = G̃(B)/J G̃(B) ≃ ⨁_{i=1}^{rs}(mk(p′_{i}) ⊕ mk(p′′_{i})) .

As F (B) is a sky-scraper sheaf, we identify it with the vector space of its global sections.

2. Choose a k-bases of this factor { e′_{ik}, e′′_{ik} } ( k = 1, …, m) in such a way that { e′_{ik} } form a basis of mk(p′_{i}) and { e′′_{ik} } form a basis of mk(p′′_{i}) .

3. Define the elements e_{ik} ∈ F (B) ( i = 1, …, rs, k = 1, …, m) in the following way:

e_{ik} = \begin{cases}e′_{ik} + e′′_{i+1,k} & if i ≠ rs , \\ e′_{rs,k} + λe′′_{1k} + e′′_{1,k−1} & if i = rs, k ≠ 1 , \\ e′_{rs,1} + λe_{11} & if i = rs, k = 1 . \end{cases}

Denote by G(B) the subspace of F (B) with the basis

{ e_{ik} | i = 1, …, rs, k = 1, …, m } .

4. Now G(B) is defined as the preimage of G(B) in G̃(B) . Note that rk G(B) = mr.

To use Kahn’s Theorem we have to calculate the cohomologies of the sheaves G(B) . Let H(B) = G̃(B)/G(B) = F (B)/G(B) . The exact sequence 0 → G(B) → G̃(B) → H(B) → 0 gives the exact sequence of cohomologies

0 → H^{0}(G(B)) → H^{0}(G̃(B)) ⟶^{h(B)} H(B) → H^{1}(G(B)) → H^{1}(G̃(B)) → 0

(we identify H(B) with the space of its global sections and omit “E” in the notation of cohomologies). One knows that dim_{k} H^{0}(G̃(B)) = m ∑_{i=1}^{rs}(d_{i} + 1)^{+} and dim_{k} H^{1}(G̃(B)) = m ∑_{i=1}^{rs}(d_{i} + 1)^{−}, where

k^{+} = \begin{cases}k & if k > 0 , \\ 0 & otherwise \end{cases} and k^{−} = \begin{cases}−k & if k < 0 , \\ 0 & otherwise , \end{cases} ,

while dim_{k} T = mrs. So we only have to calculate rk h(B) . Denote by f the natural homomorphism H^{0}(G̃(B)) → F (B) . Then rk h(B) = dim_{k}(Im f + G(B)) − dim_{k} G(B) .

Identifying E_{i} with ℙ^{1} we can suppose that p′_{i} = (1 : 0) and p′′_{i} = (0 : 1) . If d_{i} > 0 , a basis { g_{0}, …, g_{d_{i}} } in H^{0}(O_{i}(d_{i})) can be chosen in such a way that g_{0}(p′_{i}) = 1, g_{1}(p′′_{i}) = 1 , while all other values g_{j}(p′_{i}) and g_{j}(p′′_{i}) equal 0 , so the basic elements e′_{ik} and e′′_{ik} can be chosen as the images of g_{0} and g_{1} respectively. Hence, Im f is generated by the set { e′_{ik}, e′′_{ik} | d_{i} > 0 } ∪ { e′_{ik} + e′′_{ik} | d_{i} = 0 }.

Call a subsequence p = (d_{k+1}, …, d_{k+l}) , where 0 ≤ k < rs and 1 ≤ l ≤ rs, a positive part of d if d_{k+j} ≥ 0 for j = 1, …, l and either l = rs or both d_{k} < 0 and d_{k+l+1} < 0 . For such a positive part put θ(p) = l if either l = rs or p = (0, …, 0) and θ(p) = l + 1 otherwise. Denote by P the set of all positive parts of d and put θ(d) = ∑_{p∈P} θ(p) . Put also δ(d, λ) = 1 if d = (0, …, 0) and λ = 1 and δ(d, λ) = 0 otherwise. The preceding consideration and a rather simple straightforward calculation show that, for B = (d, m, λ) ,

rk h(B) = mθ(d) − δ(d, λ) ,

whence the following:

Theorem 2.2. For the vector bundle G = G(d, m, λ) ,

dim_{k} H^{0}(E, G) = m (∑_{i=1}^{r} s(d_{i} + 1)^{+} − θ(d)) + δ(d, λ)

dim_{k} H^{1}(E, G) = m (∑_{i=1}^{r} s(d_{i} + 1)^{−} + rs − θ(d)) + δ(d, λ) .

We write d > 0 if d_{i} ≥ 0 for all i = 1, …, rs and at least one of them is strictly positive. Denote by 0 the zero sequence (0, …, 0) of length s (the unique non-periodic zero s-sequence).

Corollary 2.3. A vector bundle G = G(d, m, λ) satisfies Kahn’s conditions (1),(2) of Theorem 1.3 if and only if either d > 0 or d = 0 and λ ≠ 1 .

Suppose now that the cycle configuration E is actually the exceptional curve of a cusp singularity X = Spec A. If s > 1 , E_{i} are the components of E, so, the self-intersection indices E_{i} · E_{i} are well defined. Note that E_{i} · E_{i} ≤ −2 and at least one of them is smaller than −2 . Let b_{i} = −E · E_{i} = −E_{i} · E_{i} − 2 . If s = 1 , put b_{1} = −E · E. Let b^{r} = (b_{1}, …, b_{s}, …, b_{1}, …, b_{s}) ( r times). We write b for b^{1} Put also n(G) = dim_{k} H^{0}(E, G(E)) . Remind that in this case one can choose E itself as a reduction cycle.

Corollary 2.4. 1. Suppose that a vector bundle G = G(d, m, λ) satisfies the Kahn conditions (1),(2). Then

n(G) = m (∑_{i=1}^{rs}(d_{i} − b_{i} + 1)^{+} − θ(d − b^{r})) + δ(d − b^{r}, λ)

and n(G)O ⊕G ≃ R_{E}M(d, m, λ) , where M(d, m, λ) is a Cohen– Macaulay A-module of rank rs + n(G) .

2. All Cohen–Macaulay modules M(d, m, λ) are indecomposable and every indecomposable Cohen–Macaulay A-module is isomorphic either to A or to one of the modules M(d, m, λ) .

Note that here either d > 0 or d = 0 and λ ≠ 1 .

3. Lifting families

Vector bundles on cyclic configurations form natural 1-parametric families. We are going to show that these families can be lifted to 1parametric families of Cohen–Macaulay modules on cusp singularities. To do it, we first prove analogues of Kahn’s results for families of modules and vector bundles. For the sake of simplicity, we suppose from now on that A is a fK-algebra and all other algebras and schemes are k-algebras and k-schemes. Moreover, we only consider k-schemes S such that k(x) = k for each closed point x ∈ S (for instance, schemes of finite type over k ). We write ⊗, dim, ×, etc., instead of ⊗_{k}, dim_{k}, ×_{Spec k} , etc.

Definition 3.1. Let S, T be k-schemes, p be the projection of S×T onto T and g_{t}, where t is a closed point of T , be the embedding S ≃ S × t → S × T .

• A family of sheaves over S based on T is, by definition, a coherent sheaf over S × T flat over T .

• Such a family is called a family of vector bundles (respectively, of (maximal) Cohen–Macaulay sheaves if it is a locally free sheaf over S × T (respectively, a sheaf of (maximal) Cohen–Macaulay modules).

• If M is a family of sheaves over S based on T and t is a closed point of T , denote by M(t) the sheaf g_{t}^{∗}M.

• We say that a family F is generically generated over S if there is an open subset U ⊆ S such that S \ U is a finite set of closed points and the restriction onto U × T of the natural homomorphism p^{∗} p_{∗} F → F is an epimorphism.

Proposition 3.2. Suppose that T is a Cohen–Macaulay k-scheme. A coherent sheaf M on S × T flat over T is a family of (maximal) Cohen–Macaulay modules if and only if M(t) is a sheaf of (maximal) Cohen–Macaulay modules over S for each closed point t ∈ T .

Proof. As all these conditions are local, we can suppose that S = Spec A, T = Spec B, where A, B are local noetherian Cohen–Macaulay k-algebras. Denote by m and n respectively there maximal ideals and choose a B-sequence (b_{1}, b_{2}, …, b_{m}) in n , where m = Kr. dim B. Then p = m ⊗ B ⊕ A ⊗ n is the maximal ideal of the local ring C = A⊗B and C/p = k . If M is a Cohen–Macaulay C-module flat over B, (b_{1}, b_{2}, …, b_{m}) is an M-sequence and M = M/ (b_{1}, b_{2}, …, b_{m}) M is a Cohen–Macaulay A-module that has a finite filtration with all factors isomorphic to M(n) = M/nM. Now Ext_{A}^{i}(k, M) = 0 easily implies Ext_{A}^{i}(k, M(n)) = 0 for i < Kr. dim A, hence, M(n) is a Cohen–Macaulay A-module.

On the contrary, suppose that M(n) is a Cohen–Macaulay Amodule and M is flat over B. Then M is also a Cohen–Macaulay A-module. Moreover, if b ∈ B is a non-zero-divisor, then it is also a non-zero-divisor on M and M/bM is flat over B/bB. Therefore, (b_{1}, b_{2}, …, b_{m}) is an M-sequence. Put A′ = C/ (b_{1}, b_{2}, …, b_{m}) C. It is an A-algebra finitely generated as A-module and M is a finitely generated A′-module. As A-module, it has a finite filtration with all factors isomorphic to M(n) , so it is a Cohen–Macaulay module. Then it is also a Cohen–Macaulay A′-module [22, Proposition IV.11], hence, M is a Cohen–Macaulay C-module.

Proposition 3.3. Suppose that T is a smooth k-scheme. A coherent sheaf M on S × T flat over T is a family of vector bundles if and only if M(t) is a vector bundle over S for each closed point t ∈ T .

We return to Definitions and Notations 1.1 and prove some results analogous to those of Kahn about the relations between Cohen–Macaulay modules and vector bundles. In this case we call families of sheaves on X families of A-modules. A family of vector bundles over X̃ based on a scheme T is called full if it is isomorphic to ((π × 1)^{∗}M)^{∨∨}, where M is a family of Cohen–Macaulay A-modules based on T .

Proposition 3.4. (Cf. [16, Proposition 1.2].) Suppose that a family F of vector bundles over X̃ based on a smooth scheme T satisfies the following conditions:

1. F is generically generated over X.

2. The restriction map p_{∗} F → p_{∗} ι^{∗}F is an epimorphism (or, the

same, the map R^{1}p_{∗}^{E} F → R^{1}p_{∗} F is a monomorphism). Then F is full.

Here ι denotes the embedding X̆ × T → X̃ × T , p_{∗}^{E}F is the “direct image with support E ”, i.e., the kernel of the map p_{∗} F → p_{∗} ι^{∗}F and R^{1}p_{∗}^{E} is the first derived functor of p_{∗}^{E}.

Proof. Obviously, this claim is local, so one can suppose that T = Spec B. Thus, p_{∗} F can be identified with the B-module H^{0}(X̃ × T, F) . Put M = (π × 1)_{∗}F , F′ = (π × 1)^{∗}M/(torsion) and consider F′ as a subsheaf of F in a natural way. Then M is flat over T , Cohen–Macaulay as so is F , and coherent as π is proper. Moreover, as F is generically generated over X̃, codim Supp F/F′ ≥ 2 . But X̃×T is normal, hence, any sheaf of the form N^{∨} is completely defined by its stalks at the points of height 1. In particular, ((π × 1)^{∗}M)^{∨} ≃ (F′)^{∨} ≃ F^{∨}, whence F ≃ F^{∨∨} ≃ ((π × 1)^{∗}M)^{∨∨}.

Proposition 3.5. (Cf. [16, Proposition 1.6].) Suppose that Z is a weak reduction cycle on X̃, F is a family of vector bundles on X̃ based on a smooth scheme T and V = F ⊗_{OX̃} O_{Z} . If V is generically generated by global sections and the map p_{∗} V(Z) → R^{1}p_{∗} V induced by the exact sequence

0 ⟶ V(−Z) ⟶ V_{2} ⟶ V ⟶ 0 ,

where V_{2} = F ⊗_{OX̃} O_{2Z} , is a monomorphism, then F is full.

Proof. Again one can suppose that T is affine; then one simply has to follow literally the proof of Kahn.

Proposition 3.6. (Cf. [16, Proposition 1.9].) Suppose that Z is a weak reduction cycle on X̃ and a family V of vector bundles over Z based on a smooth affine scheme T of finite type satisfies the following conditions:

1. V is generically generated over Z.

2. There exists an extension of V to a family of vector bundles V_{2}

on 2Z such that the induced map p_{∗} V(Z) → R^{1}p_{∗} V is injective. Then there is a full family of vector bundles F over X̃ such that V ≃ F ⊗_{O_{tX}} O_{Z} . If, moreover, Z is a reduction cycle, for any other family of vector bundles F′ over X̃ such that V ≃ F′ ⊗_{O_{tX}} O_{Z} there is an open subset U ⊆ T such that F|_{U} ≃ F′|_{U} .

Proof. As H^{2}(E × T, N) = 0 for each coherent sheaf over Z × T , any family of vector bundles over 2Z can be lifted to a family of vector bundles over nZ for every n, hence, to a family of vector bundles over X̃. (Remind that we suppose that the Artin’s Approximation holds.) Proposition 3.5 implies that F is full.

If Z is a reduction cycle, [16, Proposition 1.9] implies that F(g) ≃ F′(g) , where g is the generic point of T , whence that last assertion follows.

For minimally elliptic singularities one gets a simpler version.

Proposition 3.7. (Cf. [16, Theorem 2.1].) Let π : X̃ → X be the minimal resolution of a minimally elliptic singularity and Z be the fundamental cycle on X̃. Suppose that a family of vector bundles G on Z based on a smooth affine scheme T of finite type satisfies the following conditions:

1. G is generically generated over Z.

2. R^{1}p_{∗}(E, G) = 0 .

3. p_{∗}(E, G(Z)) = P is locally free over T .

Then there is a full family of vector bundles F over X̃ such that F ⊗_{OX̃} O_{Z} ≃ G ⊕ O_{Z} ⊗ P. Moreover, if F′ is another family with this property, there is an open subset U ⊆ T such that F|_{U} ≃ F′|_{U} .

Proof. Again we can use the arguments of the corresponding proof from [16, Page 151] to show that the family V = G ⊕ O_{Z} ⊗ P satisfies the conditions of Proposition 3.6.

For cusp singularities we can now precise Corollary 2.4 in the following way. (We use the notations of Section 2.)

Corollary 3.8. Let X = Spec A be a cusp singularity. For each nonperiodic s-sequence d ≥ 0 there is a family of Cohen–Macaulay Amodules M(d) based on k \ { 0, 1 } if d = 0 or d = b and on k \ 0 otherwise such that M(d, m, λ) ≃ M ⊗_{O_{T}} L(m, λ) for every m ∈ ℕ and every λ ∈ T , where L(m, λ) denotes the unique indecomposable sheaf on T with Supp L(m, λ) = { λ } and dim L(m, λ)_{λ} = m.

Proof. Consider the family of vector bundles over Ẽ: G̃ = ⨁_{i=1}^{r} sO_{i} ⊗ O_{T} . Define G(d) as the preimage in G̃ of the sub-bundle in G̃/J G̃ generated by the elements e′_{i} + e′′_{i+1} for i ≠ rs and the elements e′_{rs} + te_{1} , where t denotes the coordinate on T and e′_{i}, e′′_{i} are defined as on page 5. It is easy to see that G(d) ⊗_{O_{T}} L(m, λ) ≃ G(d, m, λ) . Corollaries 2.3 and 2.4 shows that such a family satisfies the conditions of Proposition 3.7. Hence, it gives a full family of vector bundles F over X̃ such that (F ⊗_{O_{T}} L(m, λ)) ⊗_{OX̃} O_{E} ≃ mn(d)O_{E} ⊕ G(d, mλ) , where n(d) = ∑_{i=1}^{rs}(d_{i} − b_{i} + 1) − θ(d − b^{r}) . Therefore, we obtain a family M(d) of Cohen–Macaulay A-modules just as we need.

Note that among M(d) ⊗_{O_{T}} L(m, d) one obtains all indecomposable Cohen–Macaulay A-modules except M(0, m, 1) and M(b, m, 1) .

4. Cohen–Macaulay tame and wild singularities

We provide now some formal definitions of “tame” and “wild” singularities (with respect to the classification of Cohen–Macaulay modules), analogous to the usual ones from the representation theory of algebras. Again we consider the case of k-algebras.

Definition 4.1. Let X = Spec A be a surface singularity. It is said to be Cohen–Macaulay tame if it is not Cohen–Macaulay finite and for every r there is a finite set 𝔐_{r} = { M_{r}^{α} } of families of Cohen– Macaulay A-modules such that:

• each M_{r}^{α} is based on a smooth curve C_{r}^{α};

• each indecomposable Cohen–Macaulay A-module of rank r is isomorphic to M_{r}^{α}(t) for some index α and some closed point t ∈ C_{r}^{α}.

Note that in the second condition one can replace “each indecomposable module” by “each indecomposable module except a finite set of them.”

Example 4.2. 1. Corollary 3.8 shows immediately that every cusp

singularity is Cohen–Macaulay tame.

2. Just in the same way, the results of [16] show that every simple elliptic singularity is also Cohen–Macaulay tame (the corresponding families are based either on Pic^{0} E or on Pic^{0} E \ { 1 }).

Remark 4.3. Denote by Pic^{0} E the subgroup of the Picard group of a (singular) curve E consisting of the classes of all line bundles L such that the degree of the restriction of L onto every irreducible component of E equals 0. One can easily see that Pic^{0} E ≃ k\{ 0 } for every cusp singularity. So both examples above have a lot of common features. Nevertheless, there is one that provides an essential difference. Indeed, Kahn’s description shows that there is a positive integer N such that for each fixed r the set 𝔐_{r} has at most Nr elements. On the contrary, for cusp singularities card 𝔐_{r} grows exponentially when r → ∞. They say that simple elliptic singularities are Cohen– Macaulay tame of bounded growth, while cusp singularities are Cohen– Macaulay tame of unbounded growth.

Definition 4.4. Let X = Spec A be a surface singularity. It is said to be Cohen–Macaulay wild if for every finitely generated (not necessarily commutative) k-algebra Λ there is an exact functor F : mod Λ → CM(A) such that:

• if L is indecomposable, so is F L;

• if L ≄ L′, also F L ≄ F L′.

Remark 4.5. It is well known (cf. [6]) that to prove that X is Cohen– Macaulay wild one only has to find such a functor F for one of the following algebras:

(i) k⟨ x, y ⟩, the free non-commutative algebra with 2 generators; (ii) k[ x, y] , the polynomial algebra with 2 generators; (iii) k[[ x, y]] , the power series algebra with 2 generators. In case (ii) such a functor is always given by a family of Cohen– Macaulay modules based on 𝔸^{2} . This family is then called strict. In view of (iii), it is even enough to construct a strict family of Cohen– Macaulay modules based on any open subset U ⊆ 𝔸^{2} .

Obviously, if X is Cohen–Macaulay wild it has families M of Cohen–Macaulay modules based on arbitrary algebraic variety T such that all modules M(t) with t ∈ T are indecomposable and pairwise non-isomorphic.

Proposition 4.6. Suppose that a minimally elliptic singularity is neither simple elliptic nor cusp. Then it is Cohen–Macaulay wild.

Proof. Remind [8] that E is vector bundle wild if E is neither smooth rational or elliptic curve, nor cyclic or linear configuration, where a linear configuration is a curve E such that all its components E_{i} ( i = 1, …, s) are smooth rational, E_{i} ∩ E_{j} = 0 if j ≠ i ± 1 , while E_{i} and E_{i+1} intersect transversally in exactly one point for i = 1, …, s − 1 . The latter case is impossible as X is not rational. So, if X is neither simple elliptic nor cusp, the fundamental cycle Z is vector bundle wild, i.e., there is a strict family of vector bundles G over Z based on 𝔸^{2} . By Serre’s Theorems [13, Theorems II.5.17 and III.5.2], there is an integer n such that G(n) is generated by global sections and R^{1}p_{∗} G(n) = 0 . As G(n) is obviously also strict, one can suppose that G itself has these properties. Moreover, p_{∗} G is coherent, hence, there is an open affine subset U ⊆ 𝔸^{2} such that p_{∗} G|U is locally free. Replace 𝔸^{2} by U (cf. Remark 4.5). By Proposition 3.7, there is a full family F of vector bundles over X̃ based on U such that F ⊗_{OX̃} O_{Z} ≃ G ⊕ O_{Z} ⊗ p_{∗} G. Let F = (π^{∗}M)^{∨∨}, where M is a family of Cohen– Macaulay modules. As G is strict, Kahn’s theorem (Theorem 1.3) imply that M is also strict, hence, X is Cohen–Macaulay wild.

Consider one procedure that allows to obtain new Cohen–Macaulay tame singularities.

Definition 4.7. Let X = Spec A, Y = Spec B be normal surface singularities, X → Y be a finite morphism given by an embedding B ⊆ A. Call it:

• a split extension if A ≃ B ⊕ A/B as B-module;

• a weakly separable extension if the epimorphism of A-bimodules A ⊠ A → A induced by the natural homomorphism A ⊗_{B} A → A splits.

Here and further on, for every B-module N we put A ⊠ N = (A ⊗_{B} N)^{∨∨} (as A-module).

Proposition 4.8. (We keep the notations of the latter definition.)

1. If X → Y is a split extension and X is Cohen–Macaulay tame, so is also Y .

2. If X → Y is a weakly separable extension and Y is Cohen– Macaulay tame, so is also X.

Proof. 1. Suppose that this extension is split. If N is any Cohen– Macaulay B-module, then it is a direct summand of A ⊗_{B} N , hence, of A⊠N as of B-module. Suppose that A is Cohen–Macaulay tame. For every family M_{r}^{α} from Definition 4.1 let t_{r}^{α} be the general point of the curve C_{r}^{α} and M_{r}^{α}(t_{r}^{α}) = ⨁_{β} M̃_{r}^{αβ}, where M̃_{r}^{αβ} are indecomposable B ⊗ k(t_{r}^{α})-modules. There is an open subset U_{r}^{α} ⊆ C_{r}^{α} and a decomposition M_{r}^{α}|_{U} ≃ ⨁_{β} N_{r}^{αβ} (as of family of B-modules) such that N_{r}^{αβ}(x) is indecomposable for all x ∈ U_{r}^{α}. Certainly, for every integer m there is only a finite set of N_{r}^{αβ} which are of rank m.

Let N be any indecomposable Cohen–Macaulay B-module. Then it is a direct summand of an indecomposable Cohen–Macaulay A-module M . If rk M = r, M ≃ M_{r}^{α}(x) for some α and some x ∈ C_{r}^{α}. If x ∈ U_{r}^{α}, N ≃ N_{r}^{αβ}(x) for some β, while there are only finitely many B-direct summands of M_{r}^{α}(x) with x ∉ U_{α}^{r}. Therefore, B is Cohen– Macaulay tame.

2. Suppose that this extension is weakly separable. If M is any Cohen–Macaulay A-module, one easily verifies that (A ⊠ M)^{∨} ≃ ((A ⊠ A) ⊗_{A} M)^{∨}. Hence, M^{∨} is isomorphic to a direct summand of (A ⊠ M)^{∨} and M is isomorphic to a direct summand of A ⊠ M . So, there is an indecomposable Cohen–Macaulay B-module N such that M is isomorphic to a direct summand of A ⊠ N . Now the same considerations as in the part 1 above show that if B is Cohen– Macaulay tame, so is A.

The most interesting example of split and weakly separable extensions are obtained from actions of finite groups.

Proposition 4.9. Let G be a finite group of automorphisms of the algebra A, B = A^{G} be its subalgebra of G-invariants.

1. If char k does not divide card G, the extension B ⊆ A is split.

2. If every prime ideal 𝔭 ⊂ A is unramified, the extension B ⊆ A

is weakly separable. Remind that an ideal 𝔭 is unramified if 𝔭A_{𝔭} = (𝔭 ∩ B)A_{𝔭}.

In the situation of this proposition we say that the singularity Y = Spec B is a quotient of the singularity X = Spec A by the group G.

Proof. 1. We get a splitting f : A → B putting f(a) = \frac{1}{n} ∑_{σ∈G} σ(a) , where n = card G.

2. As (A ⊗_{B} A)^{∨} ≃ Hom_{B}(A, A) , the second claim follows from the isomorphism Hom_{B}(A, A) ≃ A[G] (cf. [3]).

Call a surface singularity Y elliptic-quotient if it is a quotient of either a simple elliptic or a cusp singularity by a group G such that char k does not divide card G. If char k = 0 , elliptic-quotient singularities coincide with those log-canonical ones that are not quotient singularities (cf. [18]). Corollary 4.10. Every elliptic-quotient surface singularity is Cohen– Macaulay tame.

The preceding consideration gives rise to a general conjecture concerning Cohen–Macaulay tame and wild singularities. Conjecture 4.11. If a surface singularity is neither quotient nor ellipticquotient, it is Cohen–Macaulay wild.

5. Curve singularities T_{pq}

Important example of cusp singularities for char k = 0 are the “serial” unimodal singularities [1] T_{pqr}: A = k[[x, y, z]]/(x^{p} + y^{q} + z^{r} + αxyz) , where r ≤ p ≤ q and 1/p+1/q+1/r < 1 (in this case all values of α lead to isomorphic algebras). Note that if 1/p + 1/q + 1/r = 1 , the corresponding singularity is simple elliptic except some special values of α. If r = 2 , Cohen–Macaulay modules over this singularity are closely related to those over the curve singularity T_{pq}. Indeed, in this case the singularity T_{pq2} can be rewritten in the form: A = k[[x, y, z]]/(z^{2} + x^{p} + y^{q} + βx^{2}y^{2}) for β = −α^{2}/4 . Therefore, one can use the Knörrer correspondence (cf. [19, 23]) described in the following proposition to relate Cohen–Macaulay modules over A and over A/(z^{2}) = k[[x, y]]/(x^{p} + y^{q} + βx^{2}y^{2}) . Proposition 5.1. Let f ∈ k[[x_{1}, x_{2}, …, x_{n}]] be a non-invertible element, A = k[[x_{0}, x_{1}, …, x_{n}]]/(x_{0}^{2}+f) and A′ = k[[x_{1}, x_{2}, …, x_{n}]]/(f) . For every Cohen–Macaulay module M over A let rest M = M/zM and M^{σ} be the A-module that coincide with M as a group, while the action of the elements of A is given by the rule: a ·_{M^{σ}} v = a^{σ} ·_{M} v, where g(x_{0}, x_{1}, …, x_{n})^{σ} = g(−x_{0}, x_{1}, x_{2}, …, x_{n}) . Then:

• every indecomposable Cohen–Macaulay A′-module is a direct summand of rest M for an indecomposable Cohen–Macaulay A-module M and M ≃ A if and only if rest M ≃ A′;

• if M is an indecomposable Cohen–Macaulay A-module such that M ≄ M^{σ}, rest M is also indecomposable;

• if M ≄ A is an indecomposable Cohen–Macaulay A-module such that M ≃ M^{σ}, then rest M is a direct sum of two non-isomorphic indecomposable Cohen–Macaulay A′-modules, which we denote by rest_{1} M and rest_{2} M .

• if M, N are non-isomorphic indecomposable Cohen–Macaulay Amodules and M ≄ N^{σ}, then the indecomposable A′-modules obtained from M and from N as described above are also nonisomorphic.

Obviously, always rest M ≃ rest(M^{σ})′ and A^{σ} ≃ A.

So, to describe the Cohen–Macaulay modules over the curve singularity T_{pq} we have to know when M^{σ} ≃ M for a Cohen–Macaulay module M over the surface singularity T_{pqr}. The automorphism g → g^{σ} induces an automorphism of the minimal resolution X̃, hence, an automorphism of the exceptional curve E and of the category of vector bundles over E. We denote all these automorphisms by σ too. The following result is immediate.

Proposition 5.2. R_{E}M^{σ} ≃ (R_{E}M)^{σ} for every Cohen–Macaulay Amodule M .

From the description of the minimal resolutions of T_{pqr} given, for instance, in [17, 20], one can deduce the following shape of the exceptional curve E for the minimal resolution of T_{pq2} and for the action of the automorphism σ on it.

Proposition 5.3. (We use the notations from Section 2.)

1. If p = 3 (hence, q ≥ 7 ), then s = q − 6 . If q > 7 , E has 1 component E_{1} with self-intersection −3 and s − 1 components E_{2}, …, E_{s} with self-intersection −2 ; E_{1}^{σ} = E_{1} and E_{i}^{σ} = E_{s+2−i}. If q = 7 , E is irreducible with one node and E · E = −1 . In both cases σ(p′′_{1}) = p′_{1} .

2. If p = 4 (hence, q ≥ 5 ), then s = q−4 . If q > 5 , E has 1 component E_{1} with E_{1} · E_{1} = −4 and s − 1 components E_{2}, …, E_{s} with E_{i} · E_{i} = −2 . If q = 5 , E is irreducible with one node and E · E = −2 . The action of σ is the same as in the previous case.

3. If p ≥ 5 , then s = p + q − 8 . Put t = p − 3 . E has 2 components E_{1} and E_{t} with E_{i} · E_{i} = −3 and s − 2 components with E_{i} · E_{i} = −2 (no one if p = q = 5 ). E_{i}^{σ} = E_{t+1−i} if 1 ≤ s ≤ t and E_{i}^{σ} = E_{s+t+1−i} if t < i ≤ s; σ(p′′_{1}) = p′_{t}.

Given an s-sequence d = (d_{1}, d_{2}, …, d_{rs}) , denote by d^{σ} the sequence (d′_{1}, d′_{2}, …, d′_{rs}) obtained from d by the following procedure (we keep the notations of Proposition 5.3):

• if p = 3 or p = 4 , then d′_{1} = d_{1} and d′_{i} = d_{rs+2−i} for 1 < i ≤ rs;

• if p ≥ 5 , then d′_{i} = d_{t+1−i} for 1 ≤ i ≤ t and d′_{i} = d_{rs+t+1−i} for

t < i ≤ rs. Call an s-sequence d σ-symmetric if d^{σ} = d^{k}, an s-shift of d (cf. page 5). Then the description of Cohen–Macaulay modules from Section 2 implies the following results.

Corollary 5.4. Let M = M(d, m, λ) be an indecomposable Cohen– Macaulay module over the surface singularity T_{pq2} (cf. Corollary 2.4). Then M^{σ} ≃ M(d^{σ}, m, 1/λ) . In particular, M ≃ M^{σ} if and only if the sequence d is σ-symmetric and λ = ±1 .

Corollary 5.5. The indecomposable Cohen–Macaulay modules over the curve singularity A′ = k[[x, y]]/(x^{p} + y^{q} + βx^{2}y^{2}) of type T_{pq} ( q ≥ p, 1/p + 1/q < 1/2 ) are the following:

• N(d, m, λ) = rest M(d, m, λ) , where either the sequence d is not σ-symmetric or λ ≠ ±1 ;

• N_{i}(d, m, ±1) = rest_{i} M(d, m, ±1) ( i = 1, 2 ), where d is σsymmetric and either d ≠ 0 or λ = −1 ;

• regular module A′. The only isomorphisms between so defined Cohen–Macaulay modules are N(d, m, λ) ≃ N(d^{σ}, m, 1/λ) .

Note that the modules M(d, m, λ) form rational families based on k \ { 0 } if d is not σ-symmetric and on k \ { 0, 1, −1 } if d is σsymmetric.

One can consider higher-dimensional hypersurface singularities of type T_{pqr} given by the equation x^{p} + y^{q} + z^{r} + αxyz + ∑_{i=1}^{k} t_{i}^{2} = 0 for some k (maybe, 0). Due to the Knörrer correspondence, all of them are Cohen–Macaulay tame.

Conjecture 5.6. If an isolated hypersurface singularity is neither simple nor of type T_{pqr}, it is Cohen–Macaulay wild.

If Conjectures 4.11 and 5.6 are true, they imply finite/tame/wild trichotomy and an explicit description both of tame surface and hypersurface singularities and of Cohen–Macaulay modules in tame case.

6. Auslander–Reiten quivers

In this section we calculate Auslander–Reiten quivers for Cohen– Macaulay modules and related vector bundles. Remind that an Auslander–Reiten sequence in a category of modules or of vector bundles is a non-split exact sequence

0 ⟶ M′ ⟶^{g} N ⟶^{f} M ⟶ 0 ,

where M, M′ are indecomposable, such that any homomorphism N′ → M that is not a split epimorphism factors through f (equivalently, any homomorphism M′ → N′ that is not a split monomorphism factors through g). In [5] they proved that for any indecomposable vector bundle G on a projective curve E there is an Auslander–Reiten sequence

0 ⟶ ω_{E} ⊗_{O_{E}} G ⟶ E ⟶ G ⟶ 0 .

In particular, if E is a cyclic configuration, then, using the notation of Section 2, ω_{E} ≃ O_{E} ≃ G(0, 1, 1) and if G = G(d, m, λ) , there is an exact sequence

0 ⟶ G ⟶ E ⟶ G ⟶ 0 ,

where

E ⊗_{O_{E}} G(d, m, λ) ≃ \begin{cases}G(d, 2, λ) & if m = 1 , \\ G(d, m + 1, λ) ⊕ G(d, m − 1, λ) & if m > 1 , \end{cases}

that is easily recognized as Auslander–Reiten sequence (if char k = 0 it follows immediately from [5], and the combinatoric description of VB(X) in [8] does not depend on the characteristic). In particular, the Auslander–Reiten quiver of the category of vector bundles over a cyclic configuration is a disjoint union of simple tubes

T (d, λ) : G(d, 1, λ) ⇆ G(d, 2, λ) ⇆ G(d, 3, λ) ⇆ …

where d runs through non-periodic s-sequences and λ ∈ k \ { 0 }.

Passing from vector bundles to Cohen–Macaulay modules, we use the following result of Kahn:

Proposition 6.1. [16, Theorem 3.1] If X is a minimally elliptic singularity, Z is its fundamental cycle and 0 → M → N → M → 0 is an Auslander–Reiten sequence in CM(X) , then the exact sequence 0 → R_{Z}M → R_{Z}N → R_{Z}M → 0 is the direct sum of an Auslander–Reiten sequence 0 → G → F → G → 0 and a split sequence 0 → nO_{Z} → 2nO_{Z} → nO_{Z} → 0 . (Note that, as minimally elliptic singularity is Gorenstein, any Auslander–Reiten sequence sequence in CM(X) is of the form 0 → M → N → M → 0 , cf. [5, Theorem I.3.1].)

Corollary 6.2. Auslander–Reiten sequence of Cohen–Macaulay modules over a cusp singularity are the following:

0 → M(d, 1, λ) → M(d, 2, λ) → M(d, 1, λ) → 0 if d ≠ b or λ ≠ 1 ; 0 → M(b, 1, 1) → A ⊕ M(b, 2, 1) → M(b, 1, 1) → 0 ; 0 → M(d, m, λ) → M(d, m + 1, λ) ⊕ M(d, m − 1, λ) → M(d, m, λ) → 0 if m > 1 .

In particular, Auslander–Reiten quiver of the category CM(X) , where X is a cusp singularity, is a disjoint union of simple tubes

𝔗(d, λ) : M(d, 1, λ) ⇆ M(d, 2, λ) ⇆ M(d, 3, λ) ⇆ …

where d ≠ b or λ ≠ 1 , and one “special” tube

𝔗(b, 1) : A ⇆ M(b, 1, 1) ⇆ M(b, 2, 1) ⇆ M(b, 3, 1) ⇆ …

enlarged by the regular module A that is both projective and extinjective in this category. Remind that in M(d, m, λ) always either d > 0 or λ ≠ 1 .

At last, passing from the singularities T_{pq2} to T_{pq}, one only has to note that Knörrer correspondence maps an Auslander–Reiten sequence 0 → M → N → M → 0 either to an Auslander–Reiten sequence 0 → rest M → rest N → rest M → 0 (if M^{σ} ≄ M ) or to the direct sum of Auslander–Reiten sequences 0 → rest_{2} M → N′ → rest_{1} M and 0 → rest_{1} M → N′′ → rest_{2} M → 0 if M ≃ M^{σ} (cf. [19, Corollary 2.10]). In particular, considering the Auslander–Reiten sequence 0 → M(b, 1, 1) → A ⊕ M(b, 2, 1) → M(b, 1, 1) → 0 , we see that A′ must be in the middle term of the Auslander–Reiten sequence starting either from N_{1}(b, 1, 1) or from N_{2}(b, 1, 1) . We choose the second possibility (it only depends on the notation). We also choose the notations in such a way that N_{i}(d, m + 1, ±1) were a direct summand of the middle term of the Auslander–Reiten sequence starting from N_{i}(d, m, ±1) ( i = 1, 2 ). Then the Auslander–Reiten quiver is in this case a disjoint union of simple tubes

𝔗̃(d, λ) : N(d, 1, λ) ⇆ N(d, 2, λ) ⇆ N(d, 3, λ) ⇆ …

for non- σ-symmetric d or for λ ≠ 1 ; regular tubes 𝔗̃(d, ±1) of period 2 for σ-symmetric d ( d ≠ b or λ = −1 ):

[figure: N_{1}(d, 1, ±1); N_{1}(d, 2, ±1); N_{1}(d, 3, ±1); N_{2}(d, 1, ±1); N_{2}(d, 2, ±1); N_{2}(d, 3, ±1); ⋯; ⋯]

and one “special” tube 𝔗̃(b, 1) of period 2 enlarged by the regular module A′:

[figure: N_{1}(b, 1, 1); N_{1}(b, 2, 1); N_{1}(b, 3, 1); N_{2}(b, 1, 1); N_{2}(b, 2, 1); N_{2}(b, 3, 1); ⋯; ⋯; A′]

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Max-Plank-Institut für Mathematik and Kyiv Taras Shevchenko University

E-mail address: yuriy@drozd.org

Universität Kaiserslautern

E-mail address: greuel@mathematik.uni-kl.de
E-mail address: kashuba@ime.usp.br