Portal de Periódicos da CAPES

$\rho$-Borel principal ideals

1997; Duke University Press; Volume: 41; Issue: 1 Linguagem: Inglês

10.1215/ijm/1255985847

1945-6581

Annetta Aramova, Jürgen Herzog,

Rings, Modules, and Algebras

Pardue calls a monomial ideal satisfying this combinatorial condition p-Borel re- gardless of the characteristic of K.It is pretty obvious that p-Borel ideals have a much richer structure than the corresponding stable ideals, and of course are considerably more difficult to treat.At present not too much is known about their structure.For example one does not know the regularity of these ideals, let alone their resolution.Among the p-Borel ideals the principal ones are the most simple.Let u be a monomial; then (u) denotes the smallest p-Borel ideal which contains u.The ideal (u) is called p-Borel principal with Borel generator u.In his thesis Pardue conjectures a formula for the regularity of a p-Borel principal ideal, and proves his conjecture in the case that at most two variables (in successive order) divide u.As one of our main results in this paper we show in Section 3 that Pardue's formula is indeed a lower bound for the regularity of a p-Borel principal ideal.We prove this by exhibiting certain Koszul cycles which we discover in Section of this paper where we succeed in computing the Koszul homology of a p-Borel principal Cohen-Macaulay ideal.It is noted by Pardue 12] that a p-Borel principal ideal (u) is Cohen-Macaulay if and only if the Borel generator is of the form u x.In Section 2 we give the explicit minimal free resolution of p-Borel principal Cohen-Macaulay ideals.Pardue's and our results can only be the begin in the study of p-Borel ideals.From our point of view the most challenging tasks to be accomplished in this theory are the following: (i) prove Pardue's conjecture concerning the regularity of p-Borel principal ideals, (ii) compute the Koszul homology of these ideals, or even better their resolution, and (iii) give bounds for the regularity of general p-Borel ideals.1.The Koszul homology of Cohen-Macaulay p-Borel principal ideals