LOCAL SINGULARITIES, FILTRATIONS AND TANGENTIAL FLATNESS

Jörg Jahnel^{∗}

Universität Göttingen  Université de Paris-Sud
Mathematisches Institut and Laboratoire de Mathématique
Bunsenstraße 3-5  Bâtiment 425
37073 Göttingen  91405 Orsay Cedex
Germany  France

Abstract

We consider local rings (B_{0}, n_{0}) all of whose deformations are tangentially flat when endowed with suitably constructed filtrations. Various examples of such singularities are constructed and applications to Lech-Hironaka type inequalities between the Hilbert functions of base and total space are discussed.

Introduction

Let (B_{0}, n_{0}) be a local ring. By a deformation of it we will mean a flat local homomorphism f : (A, m) ⟶ (B, n) of local rings whose special fiber B/mB is isomorphic to (B_{0}, n_{0}). As A/m ↪ B/mB this concept makes sense only in the case (B_{0}, n_{0}) contains a field.

In 1959 C. Lech [13] stated the problem whether the multiplicities of local rings (A, m) and (B, n) being base, respectively total space, of a deformation satisfy the inequality

e_{0}(A) ≤ e_{0}(B).  (1)

A generalization of this is the analogous inequality

H_{A}^{d+i} ≤ H_{B}^{i}  (2)

between sum transforms of the Hilbert functions, where d denotes the dimension of B_{0}. Recall that these are defined inductively by H_{A}^{j}(l) := ∑_{k=0}^{l} H_{A}^{j−1}(k) where H_{A}^{0} is the usual Hilbert function given by

H_{A}^{0}(l) := dim_{A/m} m^{l}/m^{l+1}.

By an inequality H ≤ H′ between two functions H, H′ : ℕ ⟶ ℕ we always mean the inequality in its total sense, i.e. H(l) ≤ H′(l) for all l. In 1970 H. Hironaka [9] asked whether (2) is always true for i = 1 as that would simplify his proof of the existence of a resolution of singularities in characteristic zero [8].

In a remarkable paper Larfeldt and Lech [12] showed that the problem of Hironaka is equivalent to the following statement: For every local ring A and every coheight one prime P in A the inequality H_{A_{P}}^{1} ≤ H_{A}^{0} is true. This one and its immediate corollaries are usually referred as Bennett’s inequality. Note that it generalizes Serre’s result that the localization of a regular local ring by a prime ideal is again regular. It is proven in the case A is excellent [2, 16].

∗This work was supported in part by a research scholarship of the Deutsche Forschungsgemeinschaft.

Unfortunately, Lech-Hironaka type inequalities are established in very few cases, only. The most interesting result in that direction is due to Lech himself. It says that H_{A}^{1} ≤ H_{B}^{1} in the case that the special fiber B_{0} is a zero dimensional complete intersection [14]. B. Herzog generalized this to the situation that B_{0} corresponds to a regular point [B_{0}] of the Hilbert scheme [5]. This includes all complete intersections and all singularities with embedding dimension less than 3.

A completely different approach is given in the paper [6]. B. Herzog asks the question for which local rings B_{0} all deformations are tangentially flat, by which one means that the induced homomorphism G(A) ⟶ G(B) between the associated graded rings is flat, where A and B are equipped with the canonical filtrations by powers of their maximal ideals. He proves a criterion expressing this property in terms of the normal module of B_{0}. Thus he is able to give a large class of local rings for all whose deformations Hironaka’s inequality is true even for i = 0.

The aim of this paper is to give a generalization of the second approach to more general filtrations. Recall that a homomorphism of filtered local rings f : (A, m, F_{A}) ⟶ (B, n, F_{B}) is a local homomorphism of local rings satisfying f(F_{A}^{i}) ⊆ F_{B}^{i}. Such a homomorphism is called tangentially flat, if the associated map G_{F_{A}}(A) ⟶ G_{F_{B}}(B) is flat.

We emphasize explicitly that we mainly consider the more general filtrations as a tool to obtain results on the usual Hilbert functions. For that we need a comparison between the Hilbert functions of a local ring induced by two different filtrations. One possibility to do that is given in Lemma 3.2.

The paper will be organized as follows. In section 1 we introduce the concept of a monomially filtered local ring and prove the basic features of them being necessary for what follows. Section 2 contains the main theorem of the paper. It expresses the property of a monomially filtered local ring to allow tangentially flat deformations only in terms of normal modules. In section 3 we give the application to Lech-Hironaka type inequalities. We include several examples which were not covered by the results known before. Section 4 is a little more technical. It deals with the problem that the normal modules we are forced to consider as a consequence of the main theorem can be highly complicated to compute. The experience says that it is possible practically only for singularities defined by monomials. By some modification of the deformation to the normal cone we show that in some sense it is actually possible to reduce the question to that case.

Throughout the paper we follow the standard assumptions and notations of commutative algebra as in [15] (unless stated otherwise). We will make extensive use of the volume [7], which is fundamental concerning tangential flatness for general filtrations. All local rings will be assumed Noetherian.

1. Fundamental Concepts

1.1 Definition. By a monomially filtered local ring we will mean a triple (B_{0}, y_{0}, E), where i) B_{0} is a local ring ii) y_{0} = (y_{10}, …, y_{r0}) is an r-tuple of elements generating n_{0}, the maximal ideal of B_{0}. iii) E = (E_{d})_{d∈ℕ} is a separated filtrating family, i.e. a sequence of subsets E_{d} ⊆ ℕ^{r}

satisfying the following conditions.

a) ℕ^{r} \ E_{1} is a finite set.

b) (a_{1}, …, a_{r}) ∈ E_{d} implies (b_{1}, …, b_{r}) ∈ E_{d} if b_{i} ≥ a_{i} for every i.

c) E_{i} ⊇ E_{i+1} for all i ∈ ℕ.

d) E_{i} + E_{j} ⊆ E_{i+j} for arbitrary i, j ∈ ℕ.

e) (0, …, 0) ∈ E_{0} \ E_{1}.

f) ⋂_{d} E_{d} = ∅ (separatedness).

1.2 Let (B_{0}, y_{0}, E) be a monomially filtered local ring. Associated to the data given, one has a filtration F_{B_{0}} on B_{0} defined by

F_{B_{0}}^{i} := < y_{10}^{a_{1}} · . . . · y_{r0}^{a_{r}} | (a_{1}, …, a_{r}) ∈ E_{i} > .

F_{B_{0}} is cofinite, i.e. the modules B/F_{B_{0}}^{i} are of finite length. A filtration generated by a separated filtrating family induces the canonical (n_{0}-adic) topology.

We will call a filtrating family finitely generated, if there is some d_{0} such that

E_{d} = ⋃_{i_{1}+ ... +i_{r}=d,i_{j}≤d_{0}} E_{i_{1}} + … + E_{i_{r}}.

Finitely generated filtrating families give rise to finitely generated filtrations.

1.3 Definition. Let (B_{0}, y_{0}, E) be a monomially filtered local ring. By a deformation of (B_{0}, y_{0}, E) we will mean a homomorphism f : (A, m, F_{A}) ⟶ (B, n, F_{B}) of filtered local rings such that

i) B/mB ≅ B_{0}.

ii) The filtration F_{B} is the sum [7] of a filtration defined by E and some lift y of the r-tuple y_{0} to B with the direct image f_{∗}(F_{A}), i.e.

F_{B}^{i} = ∑_{j+k=i,(a_{1}, ... ,a_{r})∈E_{k}} F_{A}^{j}y_{1}^{a_{1}} … y_{r}^{a_{r}}

iii) The homomorphism f ⊗ A/F_{A}^{1} : A/F_{A}^{1} ⟶ B/F_{A}^{1}B of filtered local rings is tangentially flat.

1.4 Remarks. a) We are interested in monomially filtered local rings having only tangentially flat deformations.

b) Without losing any of the results of this paper, condition i) could be weakened. It would be sufficient to require that only the completions of the local rings in question should be isomorphic. Then condition ii) has to be modified in the obvious manner.

c) For questions concerning tangential flatness by [7, 3.16] one is easily reduced to the case the base A is Artin. But in that case F_{A} induces the m-adic (discrete) topology on A and F_{B} defines the n-adic topology on B by construction. For that reason, from now on we are going to assume the following

1.5 Convention. All filtrations on local rings induce the canonical topology, i.e. the same topology as the filtration by powers of the maximal ideal. This is equivalent to say we restrict ourself to cofinite Artin-Rees filtrations with 1 ∉ F^{1} [7, 1.12, 1.14].

Note that Artin-Rees filtrations are not necessarily finitely generated, so the associated graded rings may be non-Noetherian. Weight filtrations with weight vectors having irrational coordinates are simple counterexamples. Nevertheless every standard base of an ideal, i.e. every system of elements whose initial forms generate the initial ideal, is a system of generators.

1.6 The following technical lemma is a generalization of [7, 5.9]. Lemma. Let (A, m) and (B, n) be complete filtered local rings and f : A ⟶ B be a deformation of the complete monomially filtered local ring (B_{0}, y_{0}, E). Then there exists a commutative diagram

\begin{matrix}A ⟶^{f} B \\ ↘g ↗h \\ R \end{matrix}

of homomorphisms of filtered complete local rings, where g is tangentially flat, h is surjective and F_{B} = h_{∗}F_{R}. The diagram can be adjusted such that h_{0} :=h⊗_{A}A/m : R/mR =:R_{0} ⟶B_{0} is isomorphic to the natural surjection L[[Y_{1}, …, Y_{r}]] ⟶ B_{0}, where Y_{i} ↦ y_{i0} and L is a coefficient field for B_{0}. Proof. Using the theory of Cohen rings [4, Ch. 0, §19] one sees there is a commutative diagram of local homomorphisms

A  ⟶^{f}  B
↑  ↑
C_{A}[[X_{1}, …, X_{s}]] ⟶^{q} C_{B}[[X_{1}, …, X_{s}, Y_{1}, …, Y_{r}]],

where C_{A} and C_{B} are Cohen rings, X_{i} and Y_{j} are indeterminates, q induces the identity on X_{i}, the vertical arrows are surjective and Y_{j} is mapped to y_{j0} under the canonical homomorphism onto B_{0}. Note that for C_{A} and C_{B} there are two cases. If the residue characteristic is p, they are both discrete valuation rings whose maximal ideals are generated by the prime number p. Otherwise they are fields of characteristic 0. We identify A with an appropriate factor ring of C_{A}[[X_{1}, …, X_{s}]] and obtain the commutative diagram

A  ⟶^{f}  B
∥  ↑ h
C_{A}[[X_{1}, …, X_{s}]]/I ⟶^{g} (C_{B}[[X_{1}, …, X_{s}]]/(q(I))) [[Y_{1}, …, Y_{r}]] =: R.

Clearly the filtrating family E defines a prefiltration F′_{R} on R. We put F_{R} = F′_{R} + g_{∗}F_{A}. One checks easily that this construction fulfills the assertions of the lemma.

The only point that requires a little bit work is to show g is tangentially flat. For that consider the natural factorization

A = C_{A}[[X_{1}, …, X_{s}]]/I ⟶^{i} C_{B}[[X_{1}, …, X_{s}]]/(q(I)) =: R̃ ⟶^{j} R̃[[Y_{1}, …, Y_{r}]] = R

and equip R̃ with FR̃ = i_{∗}F_{A}. Then F_{R}^{d} = ∑_{k=1}^{d} F k ; R̃ < Y_{1}^{a_{1}} · . . . · Y_{r}^{a_{r}} >_{(a_{1},...,a_{r})∈E_{d−k}}, hence

G(R)(d) = ⨁_{k=1}^{d} G(R̃)(k) < Y_{1}^{a_{1}} · . . . · Y_{r}^{a_{r}} >_{(a_{1},...,a_{r})∈E_{d−k}\E_{d−k+1}} .

In particular we get the identity H_{R}^{0} = H0 ; R̃H_{L[[Y_{1},...,Y_{r}]]}^{0} such that j is tangentially flat [7, 6.13]. On the other hand C_{B} is flat over C_{A} in any of the two cases above, thus R̃ is flat over A. Therefore F k ; R̃ = F_{A}^{k} ⊗_{A} R̃ and G(R̃) = G(A) ⊗_{A} R̃. Consequently, i is tangentially flat, too. □

2. Tangential flatness of all deformations and the normal module

2.1 Let R be a filtered local ring and I ⊆ R be an ideal. Then the normal module N_{I} = Hom_{R}(I, R/I) carries a natural filtration F_{N_{I}} = (F_{N_{I}}^{d})_{d∈Z} given by

F_{N_{I}}^{d} := {f ∈ N_{I} | ∀k ∈ N : f(I ∩ F_{R}^{k}) ⊆ F_{R}^{k+d} + I/I}.

Note also, if R is a graded ring and I ∈ R is a homogeneous ideal, then N_{I} = ⨁_{g∈Z} N_{I}(d) admits a natural grading.

2.2 Definition. Let (B_{0}, y_{0}, E) = (B_{0}, n_{0}) be a complete monomially filtered local ring containing a field. Choose a coefficient field B_{0}/n_{0} =: L ↪ B_{0}, let p : L[[Y_{1}, …, Y_{r}]] =: R_{0} ⟶ B_{0} be the canonical surjection and denote its kernel by I_{0}. We will call

N_{B_{0}} := N_{I_{0}}

the normal module of B_{0} and

N_{G(B_{0})} := N_{in(I_{0})}

the normal module of the associated graded ring to B_{0}. A standard argument gives that N_{B_{0}} and N_{G(B_{0})} are determined uniquely up to (non canonical) isomorphism respecting filtrations respectively gradings.

2.3 Proposition. Let f : (A, m, F_{A}) ⟶ (R, M, F_{R}) be a tangentially flat homomorphism of filtered local rings, t ∈ A an element in the socle, and I an ideal in R such that B := R/I is flat over A. Suppose \overline{B} := B/tB is tangentially flat over \overline{A} := A/tA and consider the following conditions.

i) N_{in(I_{0})}(< −ord_{F_{A}}(t)) = 0. Here R_{0} := R/mR and I_{0} := IR_{0}.

ii) For every ideal I′ in R such that I + tR = I′ + tR flatness of B′ := R/I′ over A implies tangential flatness. Then i) implies ii). Proof. We have to show that B′ = R/I′ is tangentially flat over A. For this by [7, 4.6] it suffices to show that there is a standard base of I_{0} which can be lifted to I′ preserving the orders. So let r_{0} = (r_{0λ})_{λ∈Λ} ∈ ∏_{λ∈Λ} R_{0} be a standard base of I_{0}. We put \overline{R} := R/tR and \overline{I} := I\overline{R} = I′\overline{R}. As \overline{B} = \overline{R}/\overline{I} is tangentially flat over \overline{A}, r_{0} can be lifted to a standard base \overline{r} ∈ ∏_{λ∈Λ} \overline{R} satisfying ord(\overline{r}) = ord(r_{0}). In particular the coordinates of \overline{r} generate \overline{I}. Further B′ is flat over A, so there is a lift r′ ∈ ∏_{λ∈Λ} R of \overline{r} whose coordinates generate I′. Write

r′ = r + ts_{0}, r ∈ ∏_{λ∈Λ} R, s_{0} ∈ ∏_{λ∈Λ} R_{0},

where ts_{0} is well-defined since t annihilates mR.

Let e := ord_{F_{A}}(t). If the lift r′ and its decomposition r′ = r + ts_{0} are such that

ord(s_{0}) ≥ ord(r_{0}) − (e),  (3)

then ord(r′) = ord(r_{0}), i.e. I′ satisfies condition (iii) of [7, 4.6] and B′ is tangentially flat over A.

So assume (3) is wrong and choose m to be the minimal number greater than e such that there exist λ ∈ Λ for which

ord(s_{0λ}) = ord(r_{0λ}) − m.

Denote the set of such indices by Λ_{m}. We are going to prove that in this situation the decomposition r′ = r + ts_{0} can be replaced by another one, say r′′ = r + ts′_{0}, which satisfies

ord(s′_{0λ}) ≥ ord(r′′_{0λ}) − m + 1

for λ ∈ Λ_{m} and s′_{0λ} = s′′_{0λ} otherwise.

Note that this will suffice to conclude the proof. Indeed after a finite number of steps the decomposition r′ = r + ts_{0} will be replaced by one which satisfies (3) for all indices λ ∈ Λ_{m} while the other coordinates remain unchanged. We continue by doing this process for the next m and obtain a limit for m → ∞ as every coordinate will eventually become stationary.

We shall use the following notations.

d_{λ} := ord(r_{0λ})
S_{0λ} := \{ \begin{matrix}s_{0λ} + F_{R_{0}}^{d_{λ}−m+1} if λ ∈ Λ_{m} \\ 0  otherwise \end{matrix}

We aim at showing that the system S_{0} ∈ ∏_{λ∈Λ} G(R_{0}) defines an element

< H_{0}, in(r_{0}) > ↦ < H_{0}, S_{0} > mod in(I_{0})  (4)

of the normal module

N_{in(I_{0})} = Hom_{G(R_{0})}(in(I_{0}), G(R_{0})/in(I_{0})).

Note r_{0} is a standard base of I_{0}, so the elements of in(I_{0}) can be written < H_{0}, in(r_{0}) > with H_{0} ∈ ⨁_{λ∈Λ} G(R_{0}).

The element (4), if existing, is homogeneous of degree −m as deg(S_{0λ})−deg(in(r_{0λ}))=−m for all λ where S_{0λ} ≠ 0. By assumption i) the element (4) has to be zero, i.e. all the coordinates of S_{0} must be in in(I_{0}). Since r_{0} is a standard base of I_{0} there exists a linear map A : ∏_{λ∈Λ} R_{0} ⟶ ⨁_{λ∈Λ_{m}} R_{0} such that the system s′_{0} := s_{0} − Ar_{0} satisfies

ord(s′_{0λ}) ≥ ord(r_{0λ}) − m + 1

for λ ∈ Λ_{m}. From the decomposition r′ = r + ts_{0} one obtains r′ = r + tAr_{0} + ts′_{0}. hence

r′′ := (E − tA)r′ = r + ts′_{0}.

Obviously r′′ mod t ∏_{λ∈Λ} R = r′ mod t ∏_{λ∈Λ} R and r′′ ∈ ∏_{λ∈Λ} I by construction. So r′′ is another lift of \overline{r} to a system of generators of I′. We may replace the decomposition r′ = r+ts_{0} by r′′ = r+ts′_{0}. The proof is reduced to the assertion that (4) gives a well-defined element in N_{in(I_{0})}.

So we have to show < H_{0}, S_{0} > ∈ in(I_{0}) for every system H_{0} ∈ ⨁_{λ∈Λ} G(R_{0}) satisfying < H_{0}, in(r_{0}) > = 0. We may assume the relation < H_{0}, in(r_{0}) > = 0 is homogeneous, i.e.

deg(H_{0λ}) + deg(in(r_{0λ})) = d,  (5)

whenever H_{0λ} ≠ 0. [7, 3.8] gives G(R_{0}) = G(\overline{R}/\overline{m}\overline{R}) = G(\overline{R})/in(\overline{m}\overline{R}) = G(\overline{R})/in(\overline{m})G(\overline{R}) as \overline{R} is tangentially flat over \overline{A}. So we may consider < H_{0}, in(r_{0}) > = 0 as a relation of in(\overline{r}) modulo in(\overline{m})G(\overline{R}). But G(\overline{B}) = G(\overline{R})/in(\overline{I}) = G(\overline{R})/in(\overline{r})G(\overline{R}) is flat over G(\overline{A}), so a standard argument shows the relation above can be lifted to one of in(\overline{r}). There is a system H ∈ ⨁_{λ∈Λ} G(R) of homogeneous elements satisfying

< H, in(r) > ≡ 0 mod in(tR) and H_{0} = (H mod in(m) ⨁_{λ∈Λ} G(R)).

Choose h ∈ ⨁_{λ∈Λ} R such that in(h) = H. Then < h, r > ∈ F_{R}^{d+1} +tR and, by homogeneity (5),

ord(h_{λ}) + ord(r_{0λ}) ≥ d

for all λ. As \overline{r} is a standard base of \overline{I} there is a system h′ ∈ ⨁_{λ∈Λ} R such that

< h, r > ≡ < h′, r > mod tR  (6)

and

ord(h′_{λ}) + ord(r_{0λ}) ≥ d + 1.  (7)

Let x_{0} ∈ R_{0} be such that < h−h′, r > = tx_{0}. Then ord_{F_{R}}(tx_{0}) = ord_{F_{R}}(< h−h′, r >) ≥ d and, since R is tangentially flat over A, [7, 5.10] implies

ord_{F_{R_{0}}} (x_{0}) ≥ d − e > d − m.

In the congruence (6) we may replace r by r′, i.e. < h − h′, r′ > ≡ 0 mod tR. As B′ = R/r′R is flat over A this relation modulo tR comes from a relation of r′ in R. There is a system y_{0} ∈ ⨁_{λ∈Λ} R_{0} such that

< h − h′ + ty_{0}, r′ > = 0.

That means, when indicating residue classes modulo mR by subscripts ”0”,

0 = < h − h′, r′ > + t < y_{0}, r_{0} >
= < h − h′, r > + < h − h′, ts_{0} > + t < y_{0}, r_{0} >
= t(x_{0}+ < h_{0} − h′_{0}, s_{0} > + < y_{0}, r_{0} >).

As R is flat over A this means nothing but x_{0}+ < h_{0} − h′_{0}, s_{0} > + < y_{0}, r_{0} > = 0 and

< h_{0}, s_{0} > − < h′_{0}, s_{0} > +x_{0} ∈ r_{0}R_{0} = I_{0}.

Taking initial forms on both sides will give the desired relation < H_{0}, S_{0} > ∈ in(I_{0}). To show this we have to estimate the orders of the terms < h_{0}, s_{0} >, < h′_{0}, s_{0} > and x_{0}. Recall for λ ∈ Λ_{m}

ord(h_{0λ}) + ord(s_{0λ}) ≥ ord(h_{0λ}) + ord(r_{0λ}) − m
≥ ord(h_{λ}) + ord(r_{λ}) − m
≥ d − m  (by (5)),
ord(h′_{0λ}) + ord(s_{0λ}) ≥ ord(h′_{0λ}) + ord(r_{0λ}) − m
≥ ord(h′_{λ}) + ord(r_{λ}) − m
≥ d + 1 − m  (by (7)).

As S_{0λ} = 0 for λ ∉ Λ_{m} we found

ord_{F_{R_{0}}} < h_{0}, s_{0} > ≥ d − m, ord_{F_{R_{0}}} < h′_{0}, s_{0} > ≥ d + 1 − m, ord_{F_{R_{0}}} x_{0} ≥ d + 1 − m

and, therefore

< h_{0}, s_{0} > +F_{R_{0}}^{d+1−m} ∈ in (I_{0}).

But this is just the relation < H_{0}, S_{0} > ∈ in (I_{0}) as for λ ∈ Λ_{m}

H_{0λ} = h_{0λ} + F_{R_{0}}^{d−d_{λ}+1} and
S_{0λ} = s_{0λ} + F_{R_{0}}^{d_{λ}−m+1}.
□

2.4 Proposition. Let f : (A, m, F_{A}) ⟶ (R, M, F_{R}) be a tangentially flat homomorphism of filtered local rings, t ∈ A an element in the socle, and I an ideal in R such that B := R/I is flat over A. We consider the following conditions.

i) For every ideal I′ in R such that I + tR = I′ + tR flatness of B′ := R/I′ over A implies tangential flatness.

ii) G_{F_{N_{I_{0}}}} (N_{I_{0}})(< −ord_{F_{A}}(t)) = 0. Here R_{0} := R/mR and I_{0} := IR_{0}. Then i) implies ii).

Proof. For simplicity denote ord_{F_{A}}(t) by e. We have to show F_{N_{I_{0}}}^{−e} = N_{I_{0}}, i.e. that every element g of the normal module N_{I_{0}} = Hom_{R_{0}}(I_{0}, R_{0}/I_{0}) satisfies the relation

g(I_{0} ∩ F_{R_{0}}^{k}) ⊆ F_{R_{0}}^{k−e} + I_{0}/I_{0}  (8)

for each k ∈ ℕ.

For this let r_{0} = (r_{0λ})_{λ∈Λ} be a standard base of I_{0} and (s_{0λ})_{λ∈Λ} ∈ ∏_{λ∈Λ} R be a system representing g.

g(< h_{0}, r_{0} >) = < h_{0}, s_{0} >

By assumption B = R/I is tangentially flat over A. So by [7, 4.6] there exists a lift r of r_{0} to a standard base of I such that ord(r) = ord(r_{0}). We put r′ := r + ts_{0} and let I′ := r′R be the ideal generated by the entries of r′. Then B′ = R/I′ is flat over A by the description of first-order deformation in terms of the normal module [1] and even tangentially flat by assumption. The ring B′ = R/I′ is constructed such that there are the identities t (s_{0λ} mod I_{0}) = −r_{λ} mod I′. But then [7, 5.10] implies that

e + ord_{F_{R_{0}/I_{0}}} (s_{0λ} mod I_{0}) = ord_{F_{R/I′}} (r_{λ} mod I′)
≥ ord_{F_{R}} (r_{λ})
= ord_{F_{R_{0}}} (r_{λ0})

giving the desired relation (8) as r_{0} is a standard base for I_{0}. □

2.5 Theorem. Let (B_{0}, y_{0}, E) be a monomially filtered local ring containing a field. Consider the following statements.

i) N_{G(B_{0}^{∧})}(< −1) = 0

ii) (B_{0}, y_{0}, E) has only tangentially flat deformations.

iii) G(N_{B_{0}^{∧}})(< −1) = 0 Then ii) implies iii). If E is finitely generated, then i) implies ii). Proof. i) ⟹ ii) Let f : (A, m, F_{A}) ⟶ (B, n, F_{B}) be a deformation of (B_{0}, y_{0}, E). We have to show f is tangentially flat. For this we may assume that A, B and B_{0} are complete. By [7, 3.16] it is sufficient to show that f ⊗_{A} A/J : A/J ⟶ B/JB is tangentially flat for every cofinite ideal J. We will proceed using induction on the length l := l(A/J). If l = 1 A/J is a field and f ⊗ A/J is obviously tangentially flat. So let l > 1.

First case: F_{A/J}^{1} = 0. Then F_{A}^{1} ⊆ J and f ⊗_{A} A/J is a surjective base change of f ⊗_{A} A/F_{A}^{1} being tangentially flat by assumption.

Second case: F_{A/J}^{1} ≠ 0. Then there is an element 0 ≠ t ∈ F_{A/J}^{1} in the socle of A/J. By induction hypothesis f ⊗_{A} (A/J)/t(A/J) : (A/J)/t(A/J) ⟶ (B/JB)/t(B/JB) is tangentially flat. By Lemma 1.6 there is a commutative diagram of homomorphisms of filtered local rings

A/J ⟶^{f⊗_{A}A/J} B/JB

↘g ↗h

R,

where g is tangentially flat, h is surjective, h_{0} : R_{0} ⟶ B_{0} is isomorphic to the natural surjection L[[Y_{1}, …, Y_{r}]] ⟶ B_{0} and F_{B/JB} = h_{∗}F_{R}. Write I := ker(h) and identify R_{0} with L[[Y_{1}, …, Y_{r}]]. So we are in the situation of Proposition 2.3 having

N_{in(I_{0})}(< −ord_{F_{A/J}} (t)) = N_{G(B_{0})}(< −ord_{F_{A/J}} (t))
⊆ N_{G(B_{0})}(< −1)
= 0.

f ⊗_{A} A/J : A/J ⟶ B/JB is tangentially flat in the second case, too.

ii) ⟹ iii) Assume as above B_{0} to be complete. Suppose condition iii) is wrong. We start with the trivial first-order deformation

L[[ε]]/(ε)^{2} =: A ⟶^{f} A ⊗_{L} B_{0} =: B,

where L ↪ B_{0} is a coefficient field. Equip A with any filtration and B with a filtration as described in Definition 1.3 ii). We have a natural commutative diagram

A ⟶^{f} B

↘g ↗h

R := A ⊗_{L} L[[Y_{1}, …, Y_{r}]]

and equip R with the sum of the filtration defined by E and Y_{1}, …, Y_{r} with g_{∗}(F_{A}). Then g is tangentially flat as a residually rational base change [7, 3.17] of L ⟶ L[[Y_{1}, …, Y_{r}]] being tangentially flat for trivial reason. Further h is surjective, h_{∗}F_{R} = F_{B} and the induced homomorphism h_{0} : R_{0} ⟶ B_{0} is isomorphic to the natural map L[[Y_{1}, …, Y_{r}]] ⟶ B_{0}. Proposition 2.4 proves there must be a factor ring of R giving rise to a non-tangentially flat deformation of B_{0}. □

2.6 Remark. Theorem 2.5 is the main result of this paper. Is is a natural generalization of [6, Thm. (2.5)]. The following examples are singularities allowing non-tangentially flat deformations when equipped with the canonical filtration. Nonetheless they admit this rather strong property with respect to some other filtrations which are not too artificial. In the next section we are going to give applications of this phenomenon to Lech-Hironaka type inequalities.

2.7 Example. Let B_{0} = R_{0}/I_{0}, where R_{0} = L[[X, Y, Z]] and I_{0} = (X^{2}, XY, Z^{2}), and let E be the filtrating family for the (X, Y, Z^{2})-adic filtration. Then (B_{0}, (\overline{X}, \overline{Y} , \overline{Z}), E) admits only tangentially flat deformations. Proof. We have to show N_{G(B_{0})}(< −1) = 0. Obviously G(R_{0}) = L[X, Y, Z] as an L-vector space, where the multiplication is given by

X^{a_{1}}Y^{b_{1}}Z^{c_{1}} · X^{a_{2}}Y^{b_{2}}Z^{c_{2}} = \{ \begin{matrix}X^{a_{1}+a_{2}}Y^{b_{1}+b_{2}}Z^{c_{1}+c_{2}} & if (9) is true \\ 0 & otherwise \end{matrix}

with

ord(X^{a_{1}}Y^{b_{1}}Z^{c_{1}}) + ord(X^{a_{2}}Y^{b_{2}}Z^{c_{2}}) = ord(X^{a_{1}+a_{2}}Y^{b_{1}+b_{2}}Z^{c_{1}+c_{2}}).  (9)

The last condition is easily reduced to [\frac{c_{1}}{2} ] + [\frac{c_{2}}{2} ] < 1. So we see (X^{2}, XY, Z^{2}) =: F is standard base for I_{0}. Therefore a homogeneous element of N_{G(B_{0})} of degree d is given by a triple (G_{1}, G_{2}, G_{3}) of elements of R_{0} of degrees d + 2, d + 2, d + 1, respectively, satisfying

< R, F > = 0 ⟹ < R, G > ∈ in(I_{0}).

The proof is then concentrated in the following table.

generators  F  X^{2}  XY  Z^{2}
a syzygy  R  Y  −X
N_{G(B_{0})}(< −1) G A_{1} + A_{2}Z B_{1} + B_{2}Z 0
0  0 Tab. 1

The polynomials in the first row of the table are the generators F_{i} of in(I_{0}). Below that we have written down a syzygy of these generators needed for the proof. The third row contains a presentation of all polynomials of degree 0, respectively −1 in G(R_{0}). Here A_{i} and B_{i} are coefficients from the ground field L. The syzygy requires A_{1}Y + A_{2}Y Z − B_{1}X − B_{2}XZ = 0 such that all G_{i} need to be zero. □

2.8 Example. Let B_{0} = R_{0}/I_{0}, where R_{0} = L[[X, Y, Z]] and

I_{0} = (X^{2}Y Z + XY^{2}Z, X^{2}Y Z + XY Z^{2}, X^{2}Y Z − Y^{2}Z^{2}),

and let E be the filtrating family for the (X, Y, Z)^{2}-adic filtration. Then (B_{0}, (\overline{X}, \overline{Y} , \overline{Z}), E) admits only tangentially flat deformations. Proof. For N_{G(B_{0})}(< −1) we get the following table.

generators F  X^{2}Y Z + XY^{2}Z  X^{2}Y Z + XY Z^{2}  X^{2}Y Z − Y^{2}Z^{2}
degrees  2  2  2
some syzygies R^{1}  XZ  −Y Z  −XZ
R_{2}  Z^{2}  −XZ  XZ
N_{G(B_{0})}(< −1)G A_{1}+B_{1}X+C_{1}Y +D_{1}Z A_{2}+B_{2}X+C_{2}Y +D_{2}Z A_{3}+B_{3}X+C_{3}Y +D_{3}Z Tab. 2

Here R_{1} gives A_{2} = C_{2} = D_{2} = 0 and R_{2} implies A_{1} = C_{1} = D_{1} = 0. But then we get A_{3} = D_{3} = 0 from R_{1} and C_{3} = 0 from R_{2}. Now R_{1} requires B_{2} = 0 while R_{2} implies B_{1} = 0. Finally B_{3} = 0 by R_{1} or R_{2}. □

2.9 We can mix the situation of a complete intersection being simple from the point of view of deformation theory with situations being trivial from the point of view of tangential flatness. Proposition. a) Let X_{1}, …, X_{t} be sets of indeterminates consisting of at least two elements each and Y_{1}, …, Y_{r} be indeterminates. Put B_{0} = R_{0}/I_{0}, where R_{0} = L[[X_{1}, …, X_{t}, Y_{1}, …, Y_{r}]] and I_{0} = (X_{1}^{e_{1}}, …, X_{t}^{e_{t}}, Y_{1}^{d_{1}}, …, Y_{r}^{d_{r}}). Let E be the filtrating family for the (X_{1}, …, X_{t}, Y_{1}^{d_{1}}, …, Y_{r}^{d_{r}})-adic filtration. Then (B_{0}, (\underline{X}, \underline{Y}), E) admits only tangentially flat deformations.

b) Instead of all monomials in X_{i} of a given degree one could have used t systems of power products as described in [6, Prop. (3.1)] and put

I_{0} = (s_{1}^{1}, …, s_{1}^{N_{1}}, …, s_{t}^{1}, …, s_{t}^{N_{t}}, Y_{1}^{d_{1}}, …, Y_{r}^{d_{r}}).

Proof. The system of generators given for I_{0} is a standard base. The monomials in \underline{Y} do not cause any difficulty as they are of degree 1. But for the monomials in \underline{X} one can argue as in [6, Prop. (3.1)]. □

2.10 Theorem 2.5 gives a necessary and a sufficient condition for a monomially filtered local ring to have tangentially flat deformations only. Unfortunately, they are not equivalent in general as the following example shows. Example. Let B_{0} = R_{0}/I_{0}, where R_{0} = L[[X, Y, Z]] and I_{0} = (X^{2}, XY, XZ, Y^{2}, Y Z, Z^{3}), and let E be the filtrating family for the (X, Y, Z^{2})-adic filtration. Then N_{G(B_{0})}(< −1) ≠ 0 but G(N_{B_{0}})(< −1) = 0. Proof. Let us show N_{G(B_{0})}(< −1) ≠ 0 first. G(B_{0}) is as in Example 2.7. So in order to get a standard base of I_{0} we have to add certain generators.

number  i  1 2 3 4 5 6 7 8 9
standard base F^{i} X^{2} XY XZ XZ^{2} Y^{2} Y Z Y Z^{2} Z^{3} Z^{4}
degree  2 2 1 2 2 1 2 1 2
syzygies  R_{ij}
an element from N^{G(B_{0})}(−2) G^{i} 0 0 0 0 0 0 0 0 Z
Tab. 3

We have to show R_{j9} · Z = < R_{j}, G > ∈ in(I_{0}) for every syzygy. But for a + b + c ≥ 2 the product X^{a}Y^{b}Z^{c} · Z is zero or a monomial whose sum of exponents is at least 3. In both cases it is in in(I_{0}). Further X · Z ∈ in(I_{0}), Y · Z ∈ in(I_{0}) and Z · Z = 0 ∈ in(I_{0}). So the only critical case is when R_{j9} contains a constant term. But this is impossible as Z · F_{8} = Z · Z^{3} = 0.

Consider G(N_{B_{0}})(< −1) now. We have to start with a system F of generators for I_{0}. For simplicity we choose the standard base above. An element in N_{B_{0}} is given by a 9-tuple G ∈ R_{0}^{9} satisfying < R, G > ∈ I_{0} for every syzygy of F. We obtain the following table.

number  i  1  2  3  4  5 6  7  8 9

system of generators F^{i} X^{2} XY XZ XZ^{2} Y^{2} Y Z Y Z^{2} Z^{3} Z^{4}

order  2  2  1  2  2 1  2  1 2
some syzygies  R^{1i} Z  −X
R_{2i}  Z  −X
R_{3i}  Z −Y
R_{4i} Z −X
R_{5i}  Z −Y
R_{6i}  Z −1
R_{7i} −Z^{2} X
N_{B_{0}}  G_{i} Tab. 4

We want to show ord(G_{i}) ≥ ord(F_{i}) − 1. This is trivial for i = 3, 6, 8. For the other i we need G_{i} cannot contain terms of the type C or CZ. For i = 1, 2, 4, 5 and 7 this clear from the syzygies R_{1}, R_{2}, R_{4}, R_{3}, respectively R_{5}. Consider i = 9. R_{6} would allow only that G_{9} contains CZ while C is in G_{8}. But this is forbidden by syzygy R_{7}. As F was chosen to be a standard base we have shown N_{B_{0}} = F_{N_{B_{0}}}^{−1} as desired. □

3. Lech-Hironaka type inequalities

3.1 Singularities with only tangentially flat deformations with respect to the canonical filtration give rise to a large class of homomorphisms for which the Lech-Hironaka problem has trivially a positive answer. We will show here that the local rings admitting only tangentially flat deformations for some other filtration give inequalities of exactly the same kind. The only difference is that one possibly gets some coefficients into the play.

3.2 Lemma. Let (R, M) be a local ring, f_{1}, …, f_{r} ∈ R[X] be polynomials without constant term and x ∈ M. Then for any d ∈ ℕ

l (R/(f_{1}(x^{d}), …, f_{r}(x^{d}))) ≤ d · l (R/(f_{1}(x), …, f_{r}(x)))

as soon as the right hand side is finite. Proof. It is sufficient to show the desired inequality for all factor rings R/M^{i}. So assume without restriction R to be Artin. Equip R with the (x)-adic filtration. Then S := G_{(x)}(R) is a factor ring of R_{0}[X] by some homogeneous ideal I, where R_{0} := R/(x) is an Artin local ring. Further we have

l (R/(f_{1}(x), …, f_{r}(x))) = l (G(R/(f_{1}(x), …, f_{r}(x))))
= l (G(R)/in(f_{1}(x), …, f_{r}(x))))
= l (R_{0}[X]/(I, H))

and the analogous result

l (R/(f_{1}(x^{d}), …, f_{r}(x^{d}))) = l (R_{0}[X]/(I, H^{(d)}))

for the other ideal in question.

We claim H^{(d)} is generated by {F (X^{d})|F ∈ H}. Clearly, if F ∈ H, i.e. if F is the homogeneous initial form of some element from (f_{1}(x), …, f_{r}(x)), then F (X^{d}) ∈ H^{(d)}. For the other direction let G ∈ H^{(d)}, i.e. the initial form of some element g ∈ (f_{1}(x^{d}), …, f_{r}(x^{d})). Then

g =  h_{1}^{0}(x^{d})f_{1}(x^{d}) + … + h_{r}^{0}(x^{d})f_{r}(x^{d})
+ x [h_{1}^{1}(x^{d})f_{1}(x^{d}) + … + h_{r}^{1}(x^{d})f_{r}(x^{d})]
⋮
+ x^{d−1}[h_{1}^{d−1}(x^{d})f_{1}(x^{d}) + … + h_{r}^{d−1}(x^{d})f_{r}(x^{d})],

whose initial form G is obviously in the ideal generated by the set given above.

We note, as everything is Artin, we could have dealt with power series in X instead of polynomials. As I is a homogeneous ideal we have a local homomorphism of local rings

R_{0}[[X]]/(I, H) ⟶^{X↦X^{d}} R_{0}[[X]]/(I, H^{(d)})

whose special fiber is of length d. Consequently, l(R_{0}[[X]]/(I, H^{(d)})) ≤ d · l(R_{0}[[X]]/(I, H)). □

3.3 Definition. Let (B_{0}, n_{0}) be a local ring of dimension d and I be an n_{0}-primary ideal in it. By the Hilbert-Serre theorem the Hilbert series of B_{0} with respect to I can be written in the form

H_{B_{0},I}^{1}(T) = \frac{P (T)}{(1 − T)^{d+1}} + ∑_{i=1}^{d} \frac{p_{i}(T)}{(1 − T)^{i}} ,

where P and p_{i} are polynomials with non-negative coefficients. Define the vage multiplicity ν_{I}(B_{0}) of B_{0} with respect to I as the maximal value of P (0) occurring in decompositions of the type above. Note, if B_{0} is Artin, then ν_{I}(B_{0}) = l(B_{0}/I). In any case ν_{I}(B_{0}) ≥ 1 as H_{B_{0},I}^{1} ≥ H_{B_{0},n_{0}}^{1} and \frac{1}{(1−T)^{d+1}} is the minimal Hilbert series possible in dimension d.

3.4 Theorem. Let B_{0} = R_{0}/I_{0}, where R_{0} = L[[X_{1}, …, X_{r}]], be a d-dimensional local ring and let E be a filtrating family such that (B_{0}, (\overline{X_{1}}, …, \overline{X_{r}}), E) admits only tangentially flat deformations. Assume (d_{1}, 0, …, 0), …, (0, …, 0, d_{r}) ∈ E_{1} (i.e. (X_{1}^{d_{1}}, …, X_{r}^{d_{r}}) ⊆ F_{R_{0}}^{1}). Further let I ∈ B_{0} be some ideal such that F_{B_{0}}^{i} ⊆ I^{i} for all i. Then for any deformation f : (A, m) ⟶ (B, n) of B_{0}

e(B) ≥ \frac{e_{I}(B_{0})}{d_{1} · . . . · d_{r}} e(A) and  (10)
H_{B}^{1}(i) ≥ \frac{ν_{I}(B_{0})}{d_{1} · . . . · d_{r}} H_{A}^{d+1}(i)  (11)

for all i. If B_{0} is Artin one has even

H_{B}^{1}(i) ≥ \frac{l(B_{0}/F_{B_{0}}^{1})}{d_{1} · . . . · d_{r}} H_{A}^{1}(i).  (12)

Proof. Equip A with the m-adic filtration and B with one of the filtrations described in Definition 1.3 ii). Clearly we obtain a deformation of the monomially filtered local ring (B_{0}, (\overline{X_{1}}, …, \overline{X_{r}}), E), which is tangentially flat by assumption. [7, 6.13] implies the equality H_{B,F_{B}}^{1} = H_{A}^{0}H_{B_{0},F_{B_{0}}}^{1} between Hilbert series.

The maximal ideal n in B is generated by some z_{1}, …, z_{s} ∈ f(m) and the lift (y_{10}, …, y_{r0}) chosen. We obtain F_{B}^{i} ⊇ (z_{1}, …, z_{s}, y_{10}^{d_{1}}, …, y_{r0}^{d_{r}})^{i} and

H_{B,F_{B}}^{1}(i) ≤ l(B/(z_{1}, …, z_{s}, y_{10}^{d_{1}}, …, y_{r0}^{d_{r}})^{i+1})
≤ d_{1} · . . . · d_{r} · l(B/(z_{1}, …, z_{s}, y_{10}, …, y_{r0})^{i+1})
= d_{1} · . . . · d_{r} · H_{B}^{1}(i),

where the middle inequality is nothing but an r-fold application of Lemma 3.2. Consequently,

H_{B}^{1} ≥ \frac{H_{B_{0},F_{B_{0}}}^{1}}{d_{1} · . . . · d_{r}} H_{A}^{0}.

Furthermore H_{B_{0},F_{B_{0}}}^{1} (T) ≥ H_{B_{0},I}^{1}(T) ≥ \frac{P (T)}{(1−T)^{d+1}} . So the estimate P (T) ≥ ν_{I}(B_{0}) implies assertion (11). On the other hand assertion (10) comes the fact that e_{I}(B_{0}) = P (1) and there are analogous formulas for e(A) and e(B). For (12) in the case B_{0} is Artin we simply use H_{B_{0},F_{B_{0}}}^{1} (T) ≥ l(B_{0}/F_{B_{0}}^{1})\frac{1}{1−T}. □

3.5 Remarks. a) In Example 2.7 one has H_{B}^{1} ≥ H_{A}^{2}. Indeed

l((L[[X, Y, Z]]/(X^{2}, XY, Z^{2}))/(X, Y, Z^{2})) = 2 and
l((L[[X, Y, Z]]/(X^{2}, XY, Z^{2}))/(X, Y, Z^{2})^{i+1}) = 2i + 4.

for i ≥ 1. Consequently H_{B_{0},F_{B_{0}}}^{1} (T) = \frac{2}{(1−T)^{2}} + \frac{4T}{1−T} .

b) For Example 2.8 we get H_{B}^{1} ≥ H_{A}^{3}. In fact

l(n_{0}^{i}/n_{0}^{i+1}) = \{ \begin{matrix}2i + 2 & if i ≤ 2 \\ 2i + 5 & otherwise \end{matrix}

as the monomials X^{a}Y^{b} and X^{a}Z^{b} are not affected by the relations given and among the (X^{a}Y^{b}Z^{c})Y Z there are the relations X^{2}(Y Z) = −XY (Y Z), X^{2}(Y Z) = −XZ(Y Z) and X^{2}(Y Z) = Y Z(Y Z) bringing every monomial into the form X^{i}(Y Z), Y^{i}(Y Z) or Z^{i}(Y Z). One easily computes H_{B_{0},F_{B_{0}}}^{1} (T) = \frac{8}{(1−T)^{3}} + \frac{6+3T +3T^{2}}{(1−T)^{2}} .

c) For deformations of the singularities from 2.9 we get H_{B}^{1} ≥ ν(L[[\underline{X}]]/(\underline{s})) · H_{A}^{d+1} and e(B) ≥ e(L[[\underline{X}]]/(\underline{s})) · e(A).

3.6 Example. Let B_{0} = R_{0}/I_{0}, where R_{0} = L[[X,Y,Z]] and I_{0} =(X^{2},XY, Y^{2},XZ^{2}, Y Z^{2},Z^{3}), and let E be the filtrating family for the (X, Y, Z^{2})-adic filtration. Then (B_{0}, (\overline{X}, \overline{Y} , \overline{Z}), E) admits only tangentially flat deformations. In particular, for every deformation f : A ⟶ B of B_{0} one has e(B) ≥ \frac{7}{2}e(A) and H_{B}^{1} ≥ H_{A}^{1}. Proof. We easily see l(B_{0}) = 7 and l(B_{0}/F_{B_{0}}^{1}) = 2. Tangential flatness of all deformations is shown in the table below.

standard base  X^{2}  XY  Y^{2}  XZ^{2}  Y Z^{2} Z^{3} Z^{4}
degrees  2  2  2  2  2  1  2
some syzygies  Y  −X
Z^{2}  −X
Y  −X
Z^{2}  −X
Z^{2}  −X
N_{G(B_{0})}(< −1) A_{1}+A_{2}Z B_{1}+B_{2}Z C_{1}+C_{2}Z D_{1}+D_{2}Z E_{1}+E_{2}Z 0 F_{1}+F_{2}Z
0  0  0  0  0  0 Tab. 5
□

3.7 Example. Let B_{0} = R_{0}/I_{0}, where R_{0} = L[[X, Y, Z]] and I_{0} = (X^{2}, XY, Y^{2}, XZ^{2}, Z^{3}), and let E be the filtrating family for the (X, Y, Z^{2})-adic filtration. Then (B_{0}, (\overline{X}, \overline{Y} , \overline{Z}), E) admits only tangentially flat deformations. In particular, for every deformation of B_{0} one has e(B) ≥ 4e(A) and H_{B}^{1} ≥ H_{A}^{1}. Proof. We have l(B_{0}) = 8 and l(B_{0}/F_{B_{0}}^{1}) = 2 and get the following table.

standard base  X^{2}  XY  Y^{2}  XZ^{2}  Z^{3}  Z^{4}
degrees  2  2  2  2  1  2
some syzygies  Y  −X
Z^{2}  −X
Y  −X
Z^{2}  −X
N_{G(B_{0})}(< −1) A_{1} + A_{2}Z B_{1} + B_{2}Z C_{1} + C_{2}Z D_{1} + D_{2}Z 0 E_{1} + E_{2}Z
0  0  0  0  0 Tab. 6
□

3.8 Example. Let B_{0} = R_{0}/I_{0}, where R_{0} = L[[X, Y, Z]] and I_{0} = (X^{3}, XZ, Y^{3}, Y^{2}Z, Z^{2}), and let E be the filtrating family for the (X^{2}, XY, Y^{2}, Z)-adic filtration. Then (B_{0},(\overline{X}, \overline{Y}, \overline{Z}),E) admits only tangentially flat deformations. For every deformation of B_{0} the inequalities e(B) ≥ \frac{11}{4}e(A) and H_{B}^{1} ≥ \frac{3}{4}H_{A}^{1} are true. Proof. To get a standard base we have to add further generators. The proof is concentrated in the table below.

standard base X^{3} XZ Y^{3} X^{4} X^{3}Y X^{2}Z XY Z XY^{3} Y^{4} Y^{2}Z Z^{2}

number  (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11)
degrees  1 1 1 2  2  2  2  2  2 2  2

some syzygies R^{1} XY −X^{2}

R_{2} XY −X^{2}
R_{3}  Z −Y^{2}
R_{4} −Z X^{2}
R_{5}  XY −X^{2}
R′_{5}  Y^{2} −XY
R_{6}  −Y^{2} XY
R′_{6}  −XY X^{2}
R_{7}  Z −XY
R_{8}  Z −Y^{2}
N_{G(B_{0})}(< −1)  0 0 0 0  0  0  0  0  0 0  0 Tab. 7

In principal for generators of degree 2 linear combinations of type A + BX + CY would be possible. R_{1} and R_{2} give 0 at number (6), R_{2} and R_{3} 0 at (7). Further, one considers R_{2} and R_{8} for number (10), R_{3} and R_{4} for (11), R_{5}, R′_{5} and R_{7} for numbers (4) and (5) and R_{6}, R′_{6} and R_{8} for (8) and (9). □

3.9 Example. Let B_{0} = R_{0}/I_{0}, where R_{0} =L[[X, Y, Z]] and I_{0} =(X^{3},XY^{2},XZ,Y^{3},Y^{2}Z,Z^{2}), and let E be the filtrating family for the (X^{2}, XY, Y^{2}, Z)-adic filtration. Then (B_{0},(\overline{X}, \overline{Y}, \overline{Z}),E) admits only tangentially flat deformations. For every deformation of B_{0} one has the inequalities e(B) ≥ \frac{9}{4}e(A) and H_{B}^{1} ≥ \frac{3}{4}H_{A}^{1}.

Proof. We obtain the table below.

standard base X^{3} XZ XY^{2} Y^{3} X^{2}Z XY Z Y^{2}Z X^{4} X^{3}Y X^{2}Y^{2}XY^{3} Y^{4} Z^{2}

number  (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13)
degrees  1 1 1 1 2 2 2 2 2 2 2 2 2

some syzygies Y^{2} −XY

XY −X^{2}
Y^{2} −XY
XY −X^{2}
Y^{2} −XY
XY −X^{2}
Y^{2} −XY
XY −X^{2}
Y^{2} −XY
XY −X^{2}
Y^{2} −XY
XY −X^{2}
R_{1}  X^{2}  −Z
R_{2}  X^{2}  −Z
R_{3}  Z  −X^{2}
R_{4}  Z  −XY
N_{G(B_{0})}(< −1)  0 0 0 0 0 0 0 0 0 0 0 0 0 Tab. 8

Trivially we have zeros at numbers (1) to (4). At the other generators there could be linear combinations like A + BX + CY. R_{1} and R_{2} yield that the coefficient of Y disappears at (5), (6), (8) and (9). The pairs of syzygies ((Y^{2}, −XY), (XY, −X^{2})) allow for the normal module only linear combinations of (X, Y) and (Y, 0). Consequently, we have 0 everywhere, maybe except at number (13). But there the zero comes from R_{3} and R_{4}. □

3.10 Example. Let B_{0} =R_{0}/I_{0}, where R_{0} =L[[X,Y,Z]] and I_{0} =(X^{3},X^{2}Y,Y^{3},XY Z,Y^{2}Z,Z^{2}), and let E be the filtrating family for the (X^{2}, XY, Y^{2}, Z)-adic filtration. Then (B_{0},(\overline{X}, \overline{Y}, \overline{Z}),E) admits only tangentially flat deformations. For every deformation f : A ⟶ B of B_{0} one has e(B) ≥ \frac{11}{4}e(A) and H_{B}^{1} ≥ \frac{3}{4}H_{A}^{1}. Proof. Here we obtain the following table.

standard base X^{3} X^{2}Y Y^{3} XY Z Y^{2}Z X^{4} X^{3}Y X^{2}Y^{2} XY^{3} Y^{4} Z^{2}

number  (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11)
degrees  1 1 1 1  2  2  2  2  2  2 2

some syzygies Y^{2} −XY

XY −X^{2}
Y^{2} −XY
XY −X^{2}
Y^{2} −XY
XY −X^{2}
Y^{2} −XY
XY −X^{2}
R_{1}  Z  −Y^{2}
R_{2}  Z  −XY
R_{3}  Y^{2}  −Z
R_{4}  XY  −Z
N_{G(B_{0})}(< −1)  0 0 0 0  0  0  0  0  0  0 0 Tab. 9

Here the pairs ((Y^{2}, −XY), (XY, −X^{2})) of syzygies allow only linear combinations of (X, Y) and (0, X). But R_{3} and R_{4} yield that the coefficients of X at (9) and (10) disappear. We get everything is zero except maybe the entries at numbers (4), (5) and (11). Considering the syzygies R_{1} and R_{2} completes the proof. □

3.11 Proposition (ramified coverings). Let e := (e_{1}, …, e_{r}) be some r-tuple of positive integers and consider the homomorphism R_{0} := L[[X_{1}, …, X_{r}]] ⟶ R_{0}^{e} := L[[X_{1}, …, X_{r}]] of local rings given by X_{i} ↦ X_{i}^{e_{i}}. For every R_{0}-module M denote the tensor product M ⊗_{R_{0}} R_{0}^{e} by M^{(e)}.

For some factor ring B_{0} = R_{0}/I_{0} and a filtrating family E assume G(R_{0}) to be Noetherian and N_{G_{F_{B_{0}}} (B_{0})}(< −1) = 0. Then N_{G_{F_{B_{0}} (e)}(B_{0}^{(e)})}(< −1) = 0. In particular B_{0}^{(e)} admits only tangentially flat deformations. Proof. R_{0}^{(e)} is flat over R_{0}. Therefore G_{F^{(e)}}(R_{0}^{(e)}) = G_{F} (R_{0}) ⊗_{R_{0}} R_{0}^{(e)} and R_{0}^{(e)} is even tangentially flat. Consequently,

G_{F^{(e)}}(B_{0}^{(e)}) = G_{F^{(e)}}(R_{0}^{(e)})/in_{F} (I_{0}) · G_{F^{(e)}}(R_{0}^{(e)}) = G_{F} (B_{0}) ⊗_{R_{0}} R_{0}^{(e)}

and, in particular, in_{F^{(e)}}(I_{0}^{(e)}) = in_{F} (I_{0}) ⊗_{R_{0}} R_{0}^{(e)}. For the normal modules one gets

N_{G(B_{0}^{(e)})}(B_{0}^{(e)}) = Hom_{G(R_{0})⊗_{R_{0}} R_{0}^{(e)}}(in_{F} (I_{0}) ⊗_{R_{0}} R_{0}^{(e)}, G_{F} (B_{0}) ⊗_{R_{0}} R_{0}^{(e)})
= N_{G(B_{0})}(B_{0}) ⊗_{R_{0}} R_{0}^{(e)}

as in_{F} (I_{0}) is finitely generated. The isomorphism constructed respects the gradings. □

3.12 Corollary. Let B_{0} = R_{0}/I_{0}, where R_{0} = L[[X_{1}, …, X_{r}]], be a d-dimensional local ring and let E be a finitely generated filtrating family such that N_{G(B_{0})}(< −1) = 0. Assume (d_{1}, 0, …, 0), …, (0, …, 0, d_{r}) ∈ E_{1}. Further let I ⊆ B_{0} be some ideal such that F_{B_{0}}^{i} ⊆ I^{i} for all i. Then for any deformation f : (A, m) ⟶ (B, n) of one of the B_{0}^{(e)}

e(B) ≥ \frac{e_{I}(B_{0})}{d_{1} · . . . · d_{r}} e(A) and
H_{B}^{1}(i) ≥ \frac{ν_{I}(B_{0})}{d_{1} · . . . · d_{r}} H_{A}^{d+1}(i)

for all i. If B_{0} is Artin one has even

H_{B}^{1}(i) ≥ \frac{l(B_{0}/F_{B_{0}}^{1})}{d_{1} · . . . · d_{r}} H_{A}^{1}(i).

4. Initial monomials

4.1 In the previous sections we mainly considered examples B_{0} = L[[X_{1}, …, X_{r}]]/I_{0}, where I_{0} was generated by monomials. That is because in other cases the computation of the normal module will usually become hopelessly complicated. Nevertheless based on the deformation to the normal cone there is a method to reduce the question whether the normal module of the associated graded ring to some B_{0} = R_{0}/I_{0} has non-zero elements in degrees less than −1 to the analogous question for some other ring B′_{0} = R_{0}/I′_{0} where I′_{0} is generated by monomials.

4.2 Definition. Let G be an ordered abelian group and H ⊆ G be a semigroup. An H-filtration on a ring R is a system F = (F^{g})_{g∈H} of ideals satisfying i) F^{g_{1}} ⊇ F^{g_{2}} if g_{1} ⪯ g_{2}, ii) F^{g_{1}} · F^{g_{2}} ⊆ F^{g_{1}+g_{2}}.

The associated H-graded ring to R is given by

G_{F} (R) := ⨁_{g∈H} F^{g}/( ∑_{g≺g′} F^{g′}).

We note that the concept of an initial ideal carries over directly to the situation of an H-filtered ring. Any element in R has a well-defined order in H ∪ {∞}.

On R let F′ be a filtration and F an H-filtration. F is said to refine F′ if for every d ∈ ℕ there is a k ∈ H such that F^{′d} = F^{k}.

By a filtration on an R-module M we always mean a G-filtration, i.e. a system (F_{M}^{g})_{g∈G} of submodules satisfying

i) F_{M}^{g_{1}} ⊇ F_{M}^{g_{2}} if g_{1} ⪯ g_{2},

ii) F^{g} · F_{M}^{g′} ⊆ F_{M}^{g+g′}. Then G_{F_{M}} (M) := ⨁_{g∈G} F_{M}^{g}/(∑_{g≺g′} F_{M}^{g′}) is endowed with a natural structure of a graded module over G(R).

4.3 Example. Let R_{0} := L[[X_{1}, …, X_{r}]]. Then put G := ℤ^{r} and H := ℕ^{r} and equip G with an ordering inducing the order type ω ≅ (ℕ, ≤) on H. We get an H-filtration given by

F^{g} := ((y_{01}^{g_{1}} · . . . · y_{0r}^{g_{r}}) | g ⪯ (g_{1}, …, g_{r})).

If l(F^{g}/ ∑_{g≺g′} F^{g′}) = 1 for each g ∈ H the direct image p_{∗}F under the natural map to a monomially filtered local ring B_{0} will be called a full monomial filtration. Note initial ideals with respect to full monomial filtrations are generated by monomials.

Let F′ be the weight filtration defined by (q_{1}, …, q_{r}) on (B_{0}, y_{0}). Then there exists a full monomial filtration F refining F′. Indeed put

(g_{1}, …, g_{r}) ≺ (g′_{1}, …, g′_{r}) := { g_{1}q_{1} + … + g_{r}q_{r} < g′_{1}q_{1} + … + g′_{r}q_{r} or g_{1}q_{1} + … + g_{r}q_{r} = g′_{1}q_{1} + … + g′_{r}q_{r} and (g_{1}, …, g_{r}) precedes (g′_{1}, …, g′_{r}) lexicographically.

Analogously for the (y_{01}^{a_{1}}, …, y_{0r}^{a_{r}})-adic filtration one can choose the ordering

(g_{1}, …, g_{r}) ≺ (g′_{1}, …, g′_{r}) := \{ ⌊\frac{g_{1}}{a_{1}} ⌋ + … + ⌊\frac{g_{r}}{a_{r}} ⌋ < ⌊\frac{g′_{1}}{a_{1}} ⌋ + … + ⌊\frac{g′_{r}}{a_{r}} ⌋ or ⌊\frac{g_{1}}{a_{1}} ⌋ + … + ⌊\frac{g_{r}}{a_{r}} ⌋ = ⌊\frac{g′_{1}}{a_{1}} ⌋ + … + ⌊\frac{g′_{r}}{a_{r}} ⌋ and (g_{1}, …, g_{r}) precedes (g′_{1}, …, g′_{r}) lexicographically.

4.4 Proposition. Let R_{0} = (L[[X_{1}, …, X_{r}]] and (R_{0}/I_{0}, (X_{1}, …, X_{r}), E) be a monomially filtered local ring, where G(R_{0}) is supposed Noetherian. Assume F is a full monomial filtration refining F_{R_{0}}.

Abusing notation we consider G_{F} (R_{0}) ≅ L[X_{1}, …, X_{r}] as a subring of R_{0}. Let I′_{0} := in_{F} (I_{0})R_{0} be the ideal generated by all initial forms of the elements i_{0} ∈ I_{0}. The following implication is true:

N_{in_{F_{R_{0}}}} (I′_{0})(< −1) = 0 ⟹ N_{in_{F_{R_{0}}}} (I_{0})(< −1) = 0.

Proof. Apply the Lemma below to R := G_{F} (R_{0}). □

4.5 Lemma. Let G be an ordered abelian group and H ⊆ G a semigroup, which generates G and is isomorphic to ℕ as an ordered set. Consider an H-filtered Noetherian ring R and an ideal I ⊆ R. Assume G_{F_{R}}(R) is Noetherian, too. Then the normal module N_{I} carries a filtration given by F_{N_{I}}^{g} := {f ∈ N_{I} | ∀k ∈ H : f(I ∩ F_{R}^{k}) ⊆ ∑_{k+g⪯l}(F_{R}^{l} + I/I)}.

There is a graded injective homomorphism of degree zero

G(N_{I}) ⟶ N_{in_{F_{R}} (I)}.  (13)

Moreover, if R = ⨁_{k∈ℕ} R(k) is an ℕ-graded ring, I = ⨁_{k∈ℕ} I(k) a homogeneous ideal and the filtration F_{R} is compatible with the graded structure

F_{R}^{d} = ⨁_{k∈ℕ} F_{R}^{d}(k), F_{R}^{d}(k) := F_{R}^{d} ∩ R(k),

then the ℕ-graded structure on R induces ℤ-graded structures on G_{F_{R}}(R), in_{F_{R}}(I), G(N_{I}), and N_{in_{F_{R}} (I)}, respectively, such that (13) is also graded of degree zero with respect to these additional graded structures. In particular, if

N_{in_{F_{R}} (I)}(< −1) = 0

and F_{R} as well as the filtration induced by F_{R} on R/I are separated, then N_{I}(< −1) = 0. Proof. This is a generalized version of [6, Lemma (4.7)]. Define a G-filtration on R by F_{R}^{g} := ∑_{g′∈H,g⪯g′} F_{R}^{g′}. Then the construction of (13) and the proof of injectivity carry over immediately when using the successor of some element d in G is replaced by the consideration of a sum indexed by all elements strictly succeeding d. N_{in_{F_{R}} (I)}(< −1) = 0 immediately implies G(N_{I}(< −1)) = 0. In order to deduce N_{I}(< −1) = 0 we need every non-zero element g ∈ N_{I} admits a non-zero initial form.

As R and G_{F_{R}}(R) are Noetherian, there exists a finite standard base (r_{1}, …, r_{s}) generating I (and not containing 0). The orders of its elements will be denoted d_{1}, …, d_{s} ∈ H. Since H is a system of generators for G there exists one a ∈ G such that a − d_{i} ∈ H for every i. Thus the filtration on N_{I} lives on −a + H only, which is isomorphic to ℕ as an ordered set. As R/I is separated, 0 ≠ g ∈ N_{I} admits a well-defined order in −a + H and a non-zero initial form. □

4.6 Remark. In the special case of a weight filtration with rational weights (q_{1}, …, q_{r}), which is in fact the most interesting one for applications, there is another proof for Proposition 4.4, which is more geometric in nature and therefore may be more enlightening. Consider the following weighted version of the deformation to the normal cone (cf. [3, Ch. 5]).

Let I_{0} = (f_{1}, …, f_{s}). We choose an integer M such that all Mq_{i} become natural numbers. Define I_{(λ)} := (f_{1}^{(λ)}, …, f_{s}^{(λ)}), where f_{i}^{(λ)}(X_{1}, …, X_{r}) := f_{i}(λ^{Mq_{1}}X_{1}, …, λ^{Mq_{r}}X_{r}). We obtain an algebraic family

X = Spec L[λ, X_{1}, …, X_{r}]/I ⟶ A^{1} = Spec L[λ],

whose fibers are isomorphic to Spec R_{0}/I′_{0} if λ = 0 and to Spec R_{0}/I_{0} otherwise. Going over to the associated graded objects defines a family of weighted affine cones over A^{1}. Then the semicontinuity argument from [6, Proposition (4.5)] carries over immediately.

4.7 Example. Let B_{0} = R_{0}/I_{0}, where R_{0} = L[[X, Y, Z]] and I_{0} = (X^{2}, XY, Y^{2}, Z^{3}+XZ), and let E be the filtrating family for the (X, Y, Z^{3})-adic filtration. Then (B_{0}, (\overline{X}, \overline{Y} , \overline{Z}), E) admits only tangentially flat deformations. In particular, for every deformation f : A ⟶ B of B_{0} one has e(B) ≥ 3e(A) and H_{B}^{1} ≥ H_{A}^{1}. Proof. We deal with the weight filtration defined by (1, 1, \frac{1}{3}). So consider the refining full monomial filtration described in Example 4.3. By Proposition 2.9 it is sufficient to show I′_{0} = (X^{2}, XY, Y^{2}, Z^{3}), i.e. that (X^{2}, XY, Y^{2}, Z^{3} + XZ) is a standard base for I_{0}. Consider the table below.

generators of I_{0}  s X^{2} XY Y^{2} Z^{3}(+XZ)

generators of the syzygy module r^{1} Y −X

(except trivial syzygies)  r^{2}  Y −X
Tab. 10

We get < r^{1}, s > = < r^{2}, s > = 0 and see s is a standard base for I_{0} [6, Lemma (4.3)]. □

4.8 Example. Let B_{0} = R_{0}/I_{0}, where R_{0} = L[[X, Y, Z]] and

I_{0} = (X^{2} + Y Z^{2}, XY, Y^{2} + Y Z^{2}, XZ^{2}, Z^{3} + Y Z^{2}),

and let E be the filtrating family for the (X, Y, Z^{2})-adic filtration. Then (B_{0}, (\overline{X}, \overline{Y} , \overline{Z}), E) admits only tangentially flat deformations. In particular, for every deformation f : A ⟶ B of B_{0} one has e(B) ≥ 4e(A) and H_{B}^{1} ≥ H_{A}^{1}. Proof. Here we deal with the weight filtration defined by (1, 1, \frac{1}{3}). So consider the refining full monomial filtration from Example 4.3. We aim at showing that the system of generators given for I_{0} is a standard base. Note the monomials written first in the sums above always precede the other summand (Z comes after X and Y). We get the following table.

generators of I_{0}  s X^{2}(+Y Z^{2}) XY Y^{2}(+Y Z^{2}) XZ^{2} Z^{3}(+Y Z^{2}) < r^{j}, s >
generators of the  r^{1}  Y  −X  Y^{2}Z^{2}

syzygy module

(except trivial syzygies) r^{2}  Z^{2}  −X  Y Z^{4}
r^{3}  Y  −X  −XY Z^{2}
r^{4}  Z^{2}  −Y  0
r^{5}  Z  −X  −XY Z^{2} Tab. 11

The error terms occurring can be expressed by linear combinations in s such that ord_{F} (h_{i}^{j}) + ord_{F} (s_{i}) succeeds (in (ℤ^{3}, ≺)) the syzygy order of r^{j}.

generators of I_{0} s X^{2}(+Y Z^{2}) XY Y^{2}(+Y Z^{2}) XZ^{2} Z^{3}(+Y Z^{2}) < r^{j}, s >
coefficients  h^{1}  Z^{2} · \frac{1}{1−Z}  −Y Z · \frac{1}{1−Z} Y^{2}Z^{2}
h^{2}  −Z^{3} · \frac{1}{1−Z}  Y Z · \frac{1}{1−Z}  Y Z^{4}
h^{3}  −Y  −XY Z^{2}
h^{4}  0
h^{5}  −Y  −XY Z^{2} Tab. 12

By Example 3.7 the proof is complete. □

4.9 Example. Let B_{0} = R_{0}/I_{0}, where R_{0} = L[[X, Y, Z]] and

I_{0} = (X^{3} + X^{2}Y + Y Z, XY^{2} + Y Z, XZ + Y Z, Y^{3} + Y Z, Y^{2}Z, Z^{2}),

and let E be the filtrating family for the (X^{2}, XY, Y^{2}, Z)-adic filtration. Then (B_{0},(\overline{X},\overline{Y}, \overline{Z}),E) admits only tangentially flat deformations. In particular, for every deformation f : A ⟶ B of B_{0} one has e(B) ≥ \frac{9}{4}e(A) and H_{B}^{1} ≥ \frac{3}{4}H_{A}^{1}. Proof. We have to consider the weight filtration defined by (\frac{1}{2}, \frac{1}{2}, 1) and the corresponding full monomial filtration F. In order to show we have already a standard base for I_{0} one obtains the table below.

generators of I_{0} s  X^{3}  XY^{2} XZ  Y^{3} Y^{2}Z Z^{2}  < r^{j}, s >
(+X^{2}Y+Y Z) (+Y Z) (+Y Z) (+Y Z)
generators of the r^{1}  Y^{2}  −X^{2}  X^{2}Y^{3}−X^{2}Y Z

syzygy module +Y^{3}Z (except trivial r^{2} −Z X^{2} −Y Z^{2} syzygies)

r^{3}  Z  −Y^{2}  −Y^{3}Z+Y Z^{2}
r^{4}  Y  −X  −XY Z+Y^{2}Z
r^{5}  Z  −X  Y Z^{2}
r^{6}  Y^{2}  −X  Y^{3}Z
r^{7}  Z  −X  Y Z^{2}
r^{8}  Z −Y  Y Z^{2}
r^{9}  Z −Y^{2}  0 Tab. 13

The error terms are expressed by linear combinations of s as follows.

generators of I_{0} s  X^{3}  XY^{2} XZ  Y^{3} Y^{2}Z Z^{2}  < r^{j}, s >
(+X^{2}Y+Y Z) (+Y Z) (+Y Z) (+Y Z)
coefficients h^{1} XY −XY Y X^{2}Y^{3}−X^{2}Y Z
+Y^{3}Z
h^{2} −Y −Y Z^{2}
h^{3}  −Y Y −Y^{3}Z+Y Z^{2}
h^{4}  −Y  2  −XY Z+Y^{2}Z
h^{5}  Y  Y Z^{2}
h^{6}  Y  Y^{3}Z
h^{7}  Y  Y Z^{2}
h^{8}  Y  Y Z^{2}
h^{9}  0 Tab. 14

We are reduced to the situation of Example 3.9. □

4.10 Proposition. Let (B_{0}, y_{0}) be a local ring with a system of generators for its maximal ideal n_{0}. There exists some integer d with the following property: Let d′ ≥ d and E be the filtrating family for the n_{0}^{d′}-adic filtration. Then (B_{0}, y_{0}, E) admits only tangentially flat deformations. Proof. Write down the canonical factorization B_{0} = L[[X_{1}, …, X_{r}]]/I_{0}. There is one full monomial filtration being a refinement for all filtrations in question. We obtain an ideal I′_{0} generated by monomials and apply the Lemma below. □

4.11 Lemma. Let B_{0} = R_{0}/I_{0}, where R_{0} = L[[X_{1}, …, X_{r}]] and I_{0} is generated by monomials of degrees at most d, and let E be the filtrating family for the (X_{1}, …, X_{r})^{d}- adic filtration. Then N_{G(B_{0})}(< −1) = 0. In particular, (B_{0}, (\overline{X_{1}}, …, \overline{X_{r}}), E) admits only tangentially flat deformations. Proof. In order to get a standard base to the given set of generators for I_{0} we have to add all their multiples of degrees up to d. So we find a system of generators for in(I_{0}) ⊆ G(R_{0}) consisting of homogeneous elements in degrees 0 and 1 only. Consequently, the normal module N_{G(B_{0})} has non-zero elements in degrees ≥ −1 only. □

4.12 Question. For some monomially filtered local ring B_{0} consider the set of all points (q_{1}, …, q_{r}) ∈ ℝ_{+}^{d} such that B_{0} admits only tangentially flat deformations with respect to the filtrating family for the weight filtration defined by (q_{1}, …, q_{r}). This set is non-empty. What is its geometric nature?

4.13 Corollary. Let B_{0} be some d-dimensional local ring containing a field. Then there exists some ε > 0 such that every deformation F : A ⟶ B of B_{0} satisfies H_{B}^{1} ≥ εH_{A}^{d+1}.

Acknowledgment

In the last couple of years the author was intensively encouraged by B. Herzog (Stockholm and Leipzig) to write down these results. Without that this paper probably would never have appeared. He wishes to express his deep gratitude at this place.

References

[1] Artin, M.: Lectures on deformations of singularities, Tata Institute of Fundamental Research, Bombay 1976

[2] Bennett, B. M.: On the characteristic functions of a local ring, Ann. Math. 91(1970) 25-87

[3] Fulton, W.: Intersection theory, Springer, Ergebnisse, 3. Folge, Band 2, Berlin 1984

[4] Grothendieck, A., Dieudonné, J.: Étude locale des schémas et des morphismes de schémas, Publ. Math. IHES 20(1964), 24(1965), 28(1966), 32(1967)

[5] Herzog, B.: Lech-Hironaka inequalities for flat couples of local rings, manuscripta math. 68(1990)351-371

[6] Herzog, B.: Local singularities such that all deformations are tangentially flat, Trans. Amer. Math. Soc. 324(1991)555-601

[7] Herzog, B.: Kodaira-Spencer maps in local algebra, Lecture Notes Math. 1597, Springer 1994

[8] Hironaka, H.: Resolution of singularities of an algebraic variety over a field of characteristic zero, Annals of Math. 79(1964)109-326

[9] Hironaka, H.: Certain numerical characters of singularities, J. Math. Kyoto Univ. 10(1970)151-187

[10] Jahnel, J.: Tangentiale Flachheit und Lech-Hironaka-Ungleichungen, Doktorarbeit, 94 S., Jena 1991

[11] Jahnel, J.: Lech’s conjecture on deformations of singularities and second Harrison cohomology, J. London Math. Soc. 51(1995)27-40

[12] Larfeldt, T., Lech, C.: Analytic ramifications and flat couples of local rings, Acta Math. 146(1981)201-208

[13] Lech, C.: Note on multiplicities of ideals, Ark. för mat. 4(1958-62)63-86

[14] Lech, C.: Inequalities related to certain couples of local rings, Acta Math. 112(1964) 69-89

[15] Matsumura, H.: Commutative ring theory, Cambridge Univ. Press 1986

[16] Singh, B.: Effect of a permissible blowing up on the local Hilbert functions, Invent. Math. 26(1974)201-212

References

[1] Artin, M.: Lectures on deformations of singularities, Tata Institute of Fundamental Research, Bombay 1976

[2] Bennett, B. M.: On the characteristic functions of a local ring, Ann. Math. 91(1970)25-87

[3] Fulton, W.: Intersection theory, Springer, Ergebnisse, 3. Folge, Band 2, Berlin 1984 [4] Grothendieck, A., Dieudonné, J.: Étude locale des schémas et des morphismes de

schémas, Publ. Math. IHES 20(1964), 24(1965), 28(1966), 32(1967)

[5] Herzog, B.: Lech-Hironaka inequalities for flat couples of local rings, manuscripta math. 68(1990)351-371

[6] Herzog, B.: Local singularities such that all deformations are tangentially flat, Trans. Amer. Math. Soc. 324(1991)555-601

[7] Herzog, B.: Kodaira-Spencer maps in local algebra, Lecture Notes Math. 1597, Springer 1994

[8] Hironaka, H.: Resolution of singularities of an algebraic variety over a field of characteristic zero, Annals of Math. 79(1964)109-326

[9] Hironaka, H.: Certain numerical characters of singularities, J. Math. Kyoto Univ. 10 (1970)151-187

[10] Jahnel, J.: Tangentiale Flachheit und Lech-Hironaka-Ungleichungen, Doktorarbeit, 94 S., Jena 1991

[11] Jahnel, J.: Lech’s conjecture on deformations of singularities and second Harrison cohomology, J. London Math. Soc. 51(1995)27-40

[12] Larfeldt, T., Lech, C.: Analytic ramifications and flat couples of local rings, Acta Math. 146(1981)201-208

[13] Lech, C.: Note on multiplicities of ideals, Ark. för mat. 4(1958-62)63-86

[14] Lech, C.: Inequalities related to certain couples of local rings, Acta Math. 112(1964)69-89

[15] Matsumura, H.: Commutative ring theory, Cambridge Univ. Press 1986

[16] Singh, B.: Effect of a permissible blowing up on the local Hilbert functions, Invent. Math. 26(1974)201-212