CANCELLATION IN RESOLUTIONS AND LEVEL ALGEBRAS

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 C.C.A) which describes how to construct all the type vectors (in codimension 3) with fixed socle degree and fixed value in that degree.

Key Words:

• h-vector
• Hilbert function
• Level algebra
• n-type vector
• Resolution

Mathematics Subject Classification:

• 13D02
• 13D40
• 13E10

ACKNOWLEDGMENTS

We wish to acknowledge the interesting and useful conversations we have had about level algebras with M. Boji, A. Bigatti, A. Conca, A. Iarrobino, and G. Valla as well as with out long-time collaborators T. Harima, J. C. Migliore, and Y. S. Shin.