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

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:

Abstract: \font\eufm=eufm10 Using a result from Herzog [H] we prove the following. Let $(B_0,\hbox{\eufm n}_0)$ be an artinian local algebra of embedding dimension $v$ over some field $L$ with tangent cone $ 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 $(A,\hbox{\eufm m})\to (B,\hbox{\eufm n})$ 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}\le H_B^{v+1}$.

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

Keywords: Lech problem; Lech inequalities; deformation of singularities; L-algebras; local rings

Classification (MSC2000): 14B12; 13H15

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