弱Koszul模的若干注记

设M=i≥0Mi是弱Koszul模,则对任意的i≥0,证明了〈Mi〉\-i\]是Koszul模.特别地,若M=i≥0Mi是由0次生成的分次模,则M是Koszul模当且仅当M是弱Koszul模,当且仅当对任意的i≥0,〈Mi〉-i]是Koszul模.进一步证明了M=i≥0Mi是弱Koszul模当且仅当对任意的i≥0,有〈Ji M/Ji+1 M〉\-i\]是Koszul模,其中J是Koszul代数A的分次Jacobson根.

Notes on weakly Koszul modules

Some results on weakly Koszul modules are given.Let M=i≥0Mi be a weakly Koszul module.Then for all i≥0,we prove that 〈Mi〉\-i\] is a Koszul module.In particular,if M=i≥0 Mi is generated in degree 0 as a graded module,then M is the Koszui if and only if M is the weakly Koszul,if and only if for all i≥0,〈Mi〉\ -i\] is the Koszul.As a corollary,we have proved that M=i≥0 Mi is a weakly Koszul,if and only if for all i≥0,〈JiM/Ji+1M〉\-i\] is the Koszul,where J is the graded Jacobson radical of the Koszul algebra A.