Lech inequalities for deformations of singularities defined by power products of degree 2. (English) Zbl 1052.14008

Richter, Tim

Beitr. Algebra Geom. 43, No. 1, 33-37 (2002).

Summary: Using a result from B. Herzog [Kodaira-Spencer maps in local algebra. Lect. Notes Math. 1597 (1994; Zbl 0809.13011)] we prove the following. Let \((B_0,{\mathfrak n}_0)\) be an artinian local algebra of embedding dimension \(v\) over some field \(L\) with tangent cone \(\text{gr}(B_0)\cong L[X_1,\ldots ,X_v]/I_0\). Suppose the ideal \(I_0\) is generated by power products of degree 2. Then for every residually rational flat local homomorphism \((A,{\mathfrak m})\to (B,{\mathfrak n})\) 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 \( H_A^{v+1}\leq H_B^{v+1}\).

MSC:

14B12	Local deformation theory, Artin approximation, etc.
13H15	Multiplicity theory and related topics

Keywords:

Lech problem; \(L\)-algebras; local rings

Citations:

Zbl 0809.13011