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

Using a result from Herzog [2] we prove the following. Let (B0, n0) be an artinian local algebra of embedding dimension v over some field L with tangent cone gr(B0) ∼= L[X1, . . . , Xv]/I0. Suppose the ideal I0 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 B0 the (v + 1)th sum transforms of the local Hilbert series of A and B satisfy the Lech inequality H A ≤ H v+1 B . 1. Notation Throughout we fix a field L, an integer v ≥ 2, indeterminates X = X1, . . . , Xv and write R := L[[X]] for the ring of formal power series and R0 := L[X] for the polynomial ring. Note that R0 and all the R0-modules that will occur are canonically graded and furthermore admit a canonical Z-(multi)grading that refines the grading. If M is such an R0-module , n ∈ Z and μ ∈ Z, we let M(n) and M(μ) denote the homogeneous parts of degree n and multidegree μ (e.g., R0(μ) = L ·X). We will write M(< n) := ⊕ m<nM(m), M(≥ μ) := ⊕ ν≥μM(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 B0 is a flat local homomorphism of local L-algebras with special fiber isomorphic to B0. In particular, any deformation will be residually rational, i.e., it induces a trivial extension of the residue fields. 0138-4821/93 $ 2.50 c © 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 i A is the ith sum transform of the local Hilbert series of A, i.e., H i A = (1− T ) −i ∞ ∑ j=0 dimL(m /m)T j = (1− T ) ∞ ∑ j=0 dimL(gr(A)(j))T . We understand inequalities between formal power series in the “total” sense, i.e., ∑∞ i=0 aiT i ≤ ∑∞ i=0 biT i ⇐⇒ ai ≤ bi ∀i.