A difficult open problem (even in codimension 3) asks: what can be the Hilbert function of a standard graded Artinian level algebra? In this paper we solve the related problem (in codimension 3): What are the Hilbert functions H for which the minimal resolution of the lex-segment ideal with Hilbert function H permit (at least theoretically) enough cancellation to give a possible resolution for a level algebra? This gives a necessary (but not sufficient, as we show by example) response to the original open problem. The answer is given in terms of type vectors (which are equivalent to, but different from, Hilbert functions of Artinian algebras). We also give an algorithm (implemented in CoCoA) which describes how to construct all the type vectors (in codimension 3) with fixed socle degree and fixed value in that degree.
Cancellation in resolutions and level algebras
LORENZINI, Anna
2005
Abstract
A difficult open problem (even in codimension 3) asks: what can be the Hilbert function of a standard graded Artinian level algebra? In this paper we solve the related problem (in codimension 3): What are the Hilbert functions H for which the minimal resolution of the lex-segment ideal with Hilbert function H permit (at least theoretically) enough cancellation to give a possible resolution for a level algebra? This gives a necessary (but not sufficient, as we show by example) response to the original open problem. The answer is given in terms of type vectors (which are equivalent to, but different from, Hilbert functions of Artinian algebras). We also give an algorithm (implemented in CoCoA) which describes how to construct all the type vectors (in codimension 3) with fixed socle degree and fixed value in that degree.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
