The sequential Cohen-Macaulayness of idealizations

Tóm tắt

Let  be a Noetherian local ring and  a finitely generated -module. The idealization , introduced by M. Nagata, has become a useful construction in commutative algebra. Recent work has characterized the approximate Cohen–Macaulayness of such idealizations via the length function associated with a good system of parameters. Motivated by these developments, we investigate via the sequential Cohen–Macaulayness of the idealization . We provide a characterization in terms of the length function with respect to a good system of parameters of the form , where . Furthermore, we provide equivalent conditions for  to be sequentially Cohen–Macaulay, expressed in terms of the length functions of both  and , and their respective dimension filtrations.