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Lech's Conjecture on Deformations of Singularities and Second Harrison Cohomology
Author(s) -
Jahnel Jörg
Publication year - 1995
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/51.1.27
Subject(s) - lift (data mining) , cohomology , mathematics , conjecture , pure mathematics , gravitational singularity , dimension (graph theory) , algebraic number , singularity , algebra over a field , mathematical analysis , computer science , data mining
Let B 0 be a local singularity of dimension d . Then we consider the problem of Lech, whether for every deformation ( A, m ) → ( B, n ) of B 0 the inequality H A d + 1⩽ H B 1 between the Hilbert functions is true, and give a positive answer in the case that the formal versal deformation of B 0 is a base change of an algebraic family ( R, M ) → ( S, N ), where R is regular and dim S = dim R + d . So one would hope to lift versal deformations in that way. There are obstructions against this in certain second Harrison cohomology groups.