Let R=K[x1,…,xn]be the polynomial ring in n variables over a field K and I be a matroidal ideal of R.We show that I is sequentially Cohen-Macaulay if and only if the Alexander dual Iy has linear quotients.As a conseq...Let R=K[x1,…,xn]be the polynomial ring in n variables over a field K and I be a matroidal ideal of R.We show that I is sequentially Cohen-Macaulay if and only if the Alexander dual Iy has linear quotients.As a consequence,I is sequentially Cohen-Macaulay if and only if I is shellable.展开更多
Let (A,m) be a commutative quasi-local ring with non-zero identity and M be an Artinian A-module with dim M = d. If I is an ideal of A with l(0 :m I)<∞, then we show that for a minimal reduction J of I,(0 : m JI)=...Let (A,m) be a commutative quasi-local ring with non-zero identity and M be an Artinian A-module with dim M = d. If I is an ideal of A with l(0 :m I)<∞, then we show that for a minimal reduction J of I,(0 : m JI)=(0 :m I^2) if and only if l(0:M I^n+1)=l(0:m J)^(n+d/d)-l(0 :M J)/(0 :M I))(n+d-1/d-1) for all n≥> 0. Moreover, we study the dual of Burch's inequality. In particular, the Burch's inequality becomes an equality if G(I,M) is co-Cohen-Macaulay.展开更多
Let a be an ideal of a commutative Noetherian ring R and M be a finitely generated R-module of dimension d. We characterize Cohen-Macaulay rings in term of a special homological dimension. Lastly, we prove that if R i...Let a be an ideal of a commutative Noetherian ring R and M be a finitely generated R-module of dimension d. We characterize Cohen-Macaulay rings in term of a special homological dimension. Lastly, we prove that if R is a complete local ring, then the Matlis dual of top local cohomology module Ha^d（M） is a Cohen-Macaulay R-module provided that the R-module M satisfies some conditions.展开更多
Let （R, m） be a Cohen-Macaulay local ring of dimension d, C a canonical R-module and M an almost Cohen-Macaulay R-module of dimension n and of depth t. We prove that dim Extd-n R（M,C） = n and if n ≤ 3 then Extd-n...Let （R, m） be a Cohen-Macaulay local ring of dimension d, C a canonical R-module and M an almost Cohen-Macaulay R-module of dimension n and of depth t. We prove that dim Extd-n R（M,C） = n and if n ≤ 3 then Extd-n（M,C） is an almost Cohen-Macaulay R-module. In particular, if n = d ≤ 3 then HomR（M, C） is an almost Cohen-Macaulay R-module. In addition, with some conditions, we show that Ext1R（M, C） is also almost Cohen-Macaulay. Finally, we study the vanishing Ext2R （Extd-n （M, C）, C） and Ext2R （Extd-n（M, C）, C）.展开更多
文摘Let R=K[x1,…,xn]be the polynomial ring in n variables over a field K and I be a matroidal ideal of R.We show that I is sequentially Cohen-Macaulay if and only if the Alexander dual Iy has linear quotients.As a consequence,I is sequentially Cohen-Macaulay if and only if I is shellable.
文摘Let (A,m) be a commutative quasi-local ring with non-zero identity and M be an Artinian A-module with dim M = d. If I is an ideal of A with l(0 :m I)<∞, then we show that for a minimal reduction J of I,(0 : m JI)=(0 :m I^2) if and only if l(0:M I^n+1)=l(0:m J)^(n+d/d)-l(0 :M J)/(0 :M I))(n+d-1/d-1) for all n≥> 0. Moreover, we study the dual of Burch's inequality. In particular, the Burch's inequality becomes an equality if G(I,M) is co-Cohen-Macaulay.
文摘Let a be an ideal of a commutative Noetherian ring R and M be a finitely generated R-module of dimension d. We characterize Cohen-Macaulay rings in term of a special homological dimension. Lastly, we prove that if R is a complete local ring, then the Matlis dual of top local cohomology module Ha^d（M） is a Cohen-Macaulay R-module provided that the R-module M satisfies some conditions.
文摘Let （R, m） be a Cohen-Macaulay local ring of dimension d, C a canonical R-module and M an almost Cohen-Macaulay R-module of dimension n and of depth t. We prove that dim Extd-n R（M,C） = n and if n ≤ 3 then Extd-n（M,C） is an almost Cohen-Macaulay R-module. In particular, if n = d ≤ 3 then HomR（M, C） is an almost Cohen-Macaulay R-module. In addition, with some conditions, we show that Ext1R（M, C） is also almost Cohen-Macaulay. Finally, we study the vanishing Ext2R （Extd-n （M, C）, C） and Ext2R （Extd-n（M, C）, C）.
