摘要
Let Ro be a Noetherian local ring and R a standard graded R0-algebra with maximal ideal ra and residue class field K = Rim. For a graded ideal I in R we show that for k 〉〉 0： （1） the Artin-Rees number of the syzygy modules of Ik as submodules of the free modules from a free resolution is constant, and thereby the Artin-Rees number can be presented as a proper replacement of regularity in the local situation; and （2） R is a polynomial ring over the regular Ro, the ring R/Ik is Golod, its Poincar4-Betti series is rational and the Betti numbers of the free resolution of K over R/I^k are polynomials in k of a specific degree. The first result is an extension of the work by Swanson on the regularity of I^k for k 〉〉 0 from the graded situation to the local situation. The polynomiality consequence of the second result is an analog of the work by Kodiyalam on the behaviour of Betti numbers of the minimal free resolution of R/Ik over R.
Let Ro be a Noetherian local ring and R a standard graded R0-algebra with maximal ideal ra and residue class field K = Rim. For a graded ideal I in R we show that for k 〉〉 0： （1） the Artin-Rees number of the syzygy modules of Ik as submodules of the free modules from a free resolution is constant, and thereby the Artin-Rees number can be presented as a proper replacement of regularity in the local situation; and （2） R is a polynomial ring over the regular Ro, the ring R/Ik is Golod, its Poincar4-Betti series is rational and the Betti numbers of the free resolution of K over R/I^k are polynomials in k of a specific degree. The first result is an extension of the work by Swanson on the regularity of I^k for k 〉〉 0 from the graded situation to the local situation. The polynomiality consequence of the second result is an analog of the work by Kodiyalam on the behaviour of Betti numbers of the minimal free resolution of R/Ik over R.
