We characterize pure lexsegment complexes which are Cohen-Macaulay in arbitrary codimension. More precisely, we prove that any lexsegment complex is Cohen- Macaulay if and only if it is pure and its 1-dimensional link...We characterize pure lexsegment complexes which are Cohen-Macaulay in arbitrary codimension. More precisely, we prove that any lexsegment complex is Cohen- Macaulay if and only if it is pure and its 1-dimensional links are connected, and that a lexsegment flag complex is Cohen-Macaulay if and only if it is pure and connected. We show that any non-Cohen-Macaulay lexsegment complex is a Buchsbaum complex if and only if it is a pure disconnected flag complex. For t≥2, a lexsegment complex is strictly Cohen-Macaulay in codimension t if and only if it is the join of a lexsegment pure discon- nected flag complex with a （t - 2）-dimensional simplex. When the Stanley-Reisner ideal of a pure lexsegment complex is not quadratic, the complex is Cohen-Macaulay if and only if it is Cohen-Macaulay in some codimension. Our results are based on a characterization of Cohen-Macaulay and Buchsbaum lexsegment complexes by Bonanzinga, Sorrenti and Terai.展开更多
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 syzy...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.展开更多
Let C be a semidualizing module for a commutative ring R. It is shown that the ：IC-injective dimension has the ability to detect the regularity of R as well as the Pc-projective dimension. It is proved that if D is d...Let C be a semidualizing module for a commutative ring R. It is shown that the ：IC-injective dimension has the ability to detect the regularity of R as well as the Pc-projective dimension. It is proved that if D is dualizing for a Noetherian ring R such that idR（D） = n 〈 ∞, then ：ID-idR（F） ≤ n for every flat R-module F. This extends the result due to Enochs and Jenda. Finally, over a Noetherian ring R, it is shown that if M is a pure submodule of an R-module N, then/TC-idR（M） ≤ IC-idR（N）. This generalizes the result of Enochs and Holm.展开更多
文摘We characterize pure lexsegment complexes which are Cohen-Macaulay in arbitrary codimension. More precisely, we prove that any lexsegment complex is Cohen- Macaulay if and only if it is pure and its 1-dimensional links are connected, and that a lexsegment flag complex is Cohen-Macaulay if and only if it is pure and connected. We show that any non-Cohen-Macaulay lexsegment complex is a Buchsbaum complex if and only if it is a pure disconnected flag complex. For t≥2, a lexsegment complex is strictly Cohen-Macaulay in codimension t if and only if it is the join of a lexsegment pure discon- nected flag complex with a （t - 2）-dimensional simplex. When the Stanley-Reisner ideal of a pure lexsegment complex is not quadratic, the complex is Cohen-Macaulay if and only if it is Cohen-Macaulay in some codimension. Our results are based on a characterization of Cohen-Macaulay and Buchsbaum lexsegment complexes by Bonanzinga, Sorrenti and Terai.
文摘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.
文摘Let C be a semidualizing module for a commutative ring R. It is shown that the ：IC-injective dimension has the ability to detect the regularity of R as well as the Pc-projective dimension. It is proved that if D is dualizing for a Noetherian ring R such that idR（D） = n 〈 ∞, then ：ID-idR（F） ≤ n for every flat R-module F. This extends the result due to Enochs and Jenda. Finally, over a Noetherian ring R, it is shown that if M is a pure submodule of an R-module N, then/TC-idR（M） ≤ IC-idR（N）. This generalizes the result of Enochs and Holm.
