THE RESIDUALS OF LEX PLUS POWERS IDEALS AND THE EISENBUD-GREEN HARRIS CONJECTURE

摘要

The n-type vectors introduced by Geramita, Harima, and Shin are in 1-1 correspondence with the Hilbert functions of Artinian lex ideals. Letting A = {a_1,… ,a_n} define the degrees of a regular sequence, we construct lpp_≤(A)-vectors which are in 1-1 correspondence with the Hilbert functions of certain lex plus powers ideals (depending on A). This construction enables us to show that the residual of a lex plus powers ideal in an appropriate regular sequence is again a lex plus powers ideal. We then use this result to show that the Eisenbud-Green-Harris conjecture is equivalent to showing that lex plus powers ideals have the largest last graded Betti numbers (it is well known that the Eisenbud-Green-Harris conjecture is equivalent to showing that lex plus powers ideals have the largest first graded Betti numbers).