Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 43 (2002), No. 1, 33-37.

Lech Inequalities for Deformations of Singularities Defined by Power Products of Degree 2

Tim Richter

Mathematisches Institut, Universität Leipzig Augustusplatz 10/11, 04109 Leipzig, Germany e-mail: richter@mathematik.uni-leipzig.de

Abstract. Using a result from Herzog [2] we prove the following. Let (B_{0}, 𝔫_{0}) be an artinian local algebra of embedding dimension v over some field L with tangent cone gr(B_{0}) ≅ L[X_{1}, …, X_{v}]/I_{0}. Suppose the ideal I_{0} is generated by power products of degree 2. Then for every residually rational flat local homomorphism (A, 𝔪) → (B, 𝔫) of local L-algebras that has a special fiber isomorphic to B_{0} the (v + 1)th sum transforms of the local Hilbert series of A and B satisfy the Lech inequality H_{A}^{v+1} ≤ H_{B}^{v+1}.

1. Notation

Throughout we fix a field L, an integer v ≥ 2, indeterminates \underline{X} = X_{1}, …, X_{v} and write R := L[[\underline{X}]] for the ring of formal power series and R_{0} := L[\underline{X}] for the polynomial ring.

Note that R_{0} and all the R_{0}-modules that will occur are canonically graded and furthermore admit a canonical ℤ^{v}-(multi)grading that refines the grading. If M is such an R_{0}-module , n ∈ ℤ and μ ∈ ℤ^{v}, we let M(n) and M(μ) denote the homogeneous parts of degree n and multidegree μ (e.g., R_{0}(μ) = L · X^{μ}). We will write M(< n) := ⨁_{m<n} M(m), M(≥ μ) := ⨁_{ν≥μ} M(ν) and similarly.

We use the term “local L-algebra” for a noetherian local L-algebra (A, 𝔪) such that L → A/𝔪 is an isomorphism. A deformation of a local L-algebra B_{0} is a flat local homomorphism of local L-algebras with special fiber isomorphic to B_{0}. In particular, any deformation will be residually rational, i.e., it induces a trivial extension of the residue fields.

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 2002 Heldermann Verlag

34 T. Richter: Lech Inequalities for Deformations …

If (A, 𝔪) is a local L-algebra, gr(A) denotes the tangent cone of A, which is the graded ring associated with the natural filtration of A by the powers of the maximal ideal. H_{A}^{i} is the ith sum transform of the local Hilbert series of A, i.e.,

H_{A}^{i} = (1 − T)^{−i} ∑_{j=0}^{∞} dim_{L}(𝔪^{j}/𝔪^{j+1})T^{j} = (1 − T)^{−i} ∑_{j=0}^{∞} dim_{L}(gr(A)(j))T^{j}.

We understand inequalities between formal power series in the “total” sense, i.e., ∑_{i=0}^{∞} a_{i}T^{i} ≤ ∑_{i=0}^{∞} b_{i}T^{i} ⟺ a_{i} ≤ b_{i} ∀i.

2. The Lech problem

In [3] C. Lech asks whether the multiplicities of any two local rings (A, 𝔪) and (B, 𝔫), connected by a flat local homomorphism A → B, satisfy the inequality e_{0}(A) ≤ e_{0}(B). A generalization is the question whether the analogous inequality

H_{A}^{d+i} ≤ H_{B}^{i}, (1)

always holds for some i, where d denotes the dimension of the special fiber B_{0} := B/𝔪B.

(1) has been shown to hold true for i = 1 if B_{0} has dimension zero and corresponds to a smooth point of the Hilbert scheme (see [1]) or, also for i = 1, if A → B is tangentially flat, i.e., induces a flat homomorphism of tangent cones (see [1], [2]). Little is known in between these two somewhat extreme situations. However, in ([2] Cor. 8.3 ) Herzog proved the following estimation:

Theorem. (Herzog) Let A → B be a deformation of a local L-algebra B_{0}. Then it holds

H_{A}^{1} · H_{B_{0}}^{0} ≤ H_{B}^{1} · ∏_{l=2}^{∞} (\frac{1 − T^{l}}{1 − T} )^{dim_{L}(T_{gr(B_{0})}^{1}(−l))} , (2)

where T_{gr(B_{0})}^{1}(−l) denotes the homogeneous part of degree −l of Schlessinger’s T^{1} of the tangent cone of B_{0}.

Remark. If T_{gr(B_{0})}^{1}(< −1) = 0 one immediately obtains H_{A}^{1} ≤ H_{B}^{1}; this is the tangentially flat situation. If the product on the right of (2) is not trivial, there are situations where it is small enough to allow for the conclusion of a Lech-type inequality from (2) (compare [2], 9.3). If gr(B_{0}) ≅ R_{0}/I_{0} for I_{0} generated by power products, gr(B_{0}) and hence T_{gr(B_{0})}^{1} are ℤ^{v}-graded and the determination of the dimensions of T_{gr(B_{0})}^{1}(−l) becomes a combinatorial problem. This problem does not appear to have an elegant solution. Therefore, we restrict ourselves to estimating dim_{L} T_{gr(B_{0})}^{1}(−l) in terms of the power products that generate I_{0} (see the Lemma below).

3. Estimates for dim_{L} T_{R_{0}/I_{0}}^{1}(−l)

Let S be a set of power products of degree d in the indeterminates \underline{X}.

Definition. For every k ∈ ℕ, k > 0 we define an equivalence relation ≡^{k} (“k-connectivity”) on S: we call s, t ∈ S k-connected (and write s ≡^{k} t) iff there exists a sequence s = s_{1}, …, s_{m} = t of elements of S such that

deg(gcd(s_{j}, s_{j+1})) ≥ d − k  for  j = 1, …, m − 1.

S splits into equivalence classes S = S_{k,1} ∪ … ∪ S_{k,n(S,k)} (called k-components in the following). We define

d_{k,j}^{S} := deg(gcd(S_{k,j}))  for  j = 1, …, n(S, k).

Whenever n < m, let the binomial coefficient \binom{n}{m} be zero.

Lemma. Let I_{0} be the ideal generated by S in R_{0}. Then with the above notation for all k ∈ ℕ it holds

dim_{L}(T_{R_{0}/I_{0}}^{1}(−(k + 1))) ≤ ∑_{j=1}^{n(S,k)} \binom{v + d_{j,k}^{S} − k − 2}{v − 1}.

Proof. By the exact sequence of graded modules and homomorphisms

Der_{L}(R_{0}, R_{0}/I_{0}) → Hom_{R_{0}}(I_{0}, R_{0}/I_{0}) → T_{R_{0}/I_{0}}^{1} → 0

and Der_{L}(R_{0}, R_{0}/I_{0})(l) = 0 if l < −1 we conclude

Hom_{R_{0}}(I_{0}, R_{0}/I_{0})(< −1) ≅ T_{R_{0}/I_{0}}^{1}(< −1).

For j = 1, …, n(S, k) let I_{j,k} ⊆ I_{0} denote the ideal generated by the elements of S_{j,k} in R_{0}. By applying Hom_{R_{0}}(·, R/I_{0}) to the surjection

⨁_{j=1}^{n(S,k)} I_{j,k} → I_{0} : (α_{1}, …, α_{n(S,k)}) ↦ ∑_{j=1}^{n(S,k)} α_{j}

we obtain an injection Hom_{R_{0}}(I_{0}, R_{0}/I_{0}) → ⨁_{j=1}^{n(S,k)} Hom_{R_{0}}(I_{j,k}, R_{0}/I_{0}). Thus it remains to prove that

dim_{L}(Hom_{R_{0}}(I_{j,k}, R_{0}/I_{0})(−(k + 1))) ≤ \binom{v + d_{j,k}^{S} − k − 2}{v − 1} (3)

for all k > 0, j = 1, …, n(S, k). Considering ℤ^{v}-gradings, there are vector space isomorphisms

Hom_{R_{0}}(I_{j,k}, R_{0}/I_{0})(−(k + 1)) ≅ ⨁_{\begin{matrix}μ∈ℤ^{v} \\ |μ|=−(k+1) \end{matrix}} Hom_{R_{0}}(I_{j,k}, R_{0}/I_{0})(μ)

for all j, k. Since there are exactly \binom{v+d_{j,k}^{S}−k−2}{v−1} multi-indices μ satisfying |μ| = −(k+1), \underline{X}^{μ}· gcd(S_{j,k}) ∈ R_{0}(≥ (0, …, 0)), equality will hold in (3) if for each μ ∈ ℤ^{v} satisfying |μ| = −(k + 1) it holds

(4) dim_{L}(Hom_{R_{0}}(I_{j,k}, R_{0}/I_{0})(μ)) = \begin{cases}1 & if \underline{X}^{μ} · gcd(S_{j,k}) ∈ R_{0}(≥ (0, …, 0)) \\ 0 & if \underline{X}^{μ} · gcd(S_{j,k}) ∉ R_{0}(≥ (0, …, 0)). \end{cases}

To prove (4), note that the relation module of I_{j,k} is generated by the pairwise relations

R_{0}^{t} ∋ r^{q,p} = (r_{1}^{q,p}, …, r_{t}^{q,p}), 1 ≤ q < p ≤ t

(5) r_{l}^{q,p} = \begin{cases}s_{p}/ gcd(s_{q}, s_{p}) & if l = q, \\ −s_{q}/ gcd(s_{q}, s_{p}) & if l = p, \\ 0 & otherwise \end{cases}

of the elements s_{1}, …, s_{t} of S_{j,k}. Thus we can identify g ∈ Hom_{R_{0}}(I_{j,k}, R_{0}/I_{0}) with (g_{1}, …, g_{t}) ∈ R_{0}^{t} (g_{p} arbitrary liftings of g(s_{p}) to R_{0}) for which

(6)  ∑_{l=1}^{t} g_{l}r_{l}^{p,q} ∈ I_{0} for all the r^{p,q}.

We fix j, k and μ ∈ ℤ^{v} such that |μ| = −(k + 1) and let g ∈ Hom_{R_{0}}(I_{j,k}, R_{0}/I_{0})(μ). Then g corresponds to

(a_{1} · s_{1} · \underline{X}^{μ}, …, a_{t} · s_{t} · \underline{X}^{μ})

with a_{p} ∈ L and a_{p} = 0 if s_{p} · \underline{X}^{μ} ∉ R_{0}(≥ (0, …, 0)). Furthermore, we have

a_{1} = … = a_{t}. (7)

Indeed, by k-connectivity of S_{j,k} there is u ∈ {1, …, t} with deg(gcd(s_{1}, s_{u})) ≥ d − k. By substituting q = 1, p = u into (6) we obtain

(a_{1} − a_{u}) · \frac{s_{1} · s_{u}}{gcd(s_{1}, s_{u})} · \underline{X}^{μ} ∈ I_{0}.

This monomial has degree less than d. Since I_{0} is generated by monomials of degree d, we obtain a_{1} = a_{u}. We use k-connectivity and iterate to obtain (7). Equation (4) follows, since \underline{X}^{μ} · gcd(S_{j,k}) ∈ R_{0}(≥ (0, …, 0)) iff for every element s_{p} of S_{j,k} it holds s_{p} · \underline{X}^{μ} ∈ R_{0}(≥ (0, …, 0)).

4. The estimates imply Lech-inequalities in the case I_{0} generated in degree 2

Theorem. Let B_{0} be an artinian local L-algebra of embedding dimension v such that gr(B_{0}) ≅ R_{0}/I_{0}, where I_{0} is generated by power products of degree 2. Then for every deformation A → B of B_{0} Lech’s inequality holds with i = v + 1,

H_{A}^{v+1} ≤ H_{B}^{v+1}.

Proof. Let S be the set of generators of I_{0}. Since B_{0} is artinian, {X_{1}^{2}, …, X_{v}^{2}} ⊆ S. One verifies that S is 2-connected. Therefore d_{1,k} = 0 for all k ≥ 2 and thus by the lemma dim_{L}(T_{R_{0}/I_{0}}^{1}(≤ −2)) = 0. Herzog’s theorem now reads (let ω(2) := dim_{L}(T_{gr(B_{0})}^{1}(−2))):

H_{A}^{1} · H_{B_{0}}^{0} ≤ H_{B}^{1} · (1 + T)^{ω(2)}

It remains to show that i. H_{B_{0}}^{0} ≥ (1 + T)^{ω(2)} ii. ω(2) ≤ v, since i. implies

H_{A}^{1} · (1 + T)^{ω(2)} ≤ H_{B}^{1} · (1 + T)^{ω(2)}

and multiplying with (1 − T^{2})^{−ω(2)} · (1 − T)^{−(v−ω(2))} (which has positive coefficients by ii.) yields H_{A}^{v+1} ≤ H_{B}^{v+1}.

Any 1-component of S either consists of a square of an indeterminate or it contains two or more squares of indeterminates, in which case its gcd is equal to 1. We may assume that the 1-components that have a gcd of degree 2 are {X_{1}^{2}}, …, {X_{w}^{2}}. The lemma gives

ω(2) ≤ w · \binom{v + 2 − 3}{v − 1} + (n(S, 1) − w) · \binom{v + 0 − 3}{v − 1} = w ≤ v

and proves ii. The residue classes of all the squarefree monomials in X_{1}, …, X_{w} are linearly independent elements of gr(B_{0}). Hence

H_{B_{0}}^{0}(n) ≥ \binom{w}{n} for 1 ≤ n ≤ w

and by the binomial theorem H_{B_{0}} ≥ (1 + T)^{w} which implies i.

References

[1] Herzog, Bernd: Local singularities such that all deformations are tangentially flat. Trans. Amer. Math. Soc 324(2) (1991), 555–601. Zbl 0728.13004

[2] Herzog, Bernd: Kodaira-Spencer maps in local algebra. Lecture Notes in Mathematics 1597. Springer-Verlag, Berlin Heidelberg 1994. Zbl 0809.13011

[3] Lech, Christer: Note on multiplicities of ideals. Ark. Mat. 4 (1960), 63–86. Zbl 0192.13901

[4] Schlessinger, Michael: Functors of artin rings. Trans. Amer. Math. Soc. 130 (1968), 205–222. Zbl 0167.49503

Received March 11, 2000