Mathematics

CLC number: O153.3

Absolutely Clean N-Complexes

绝对Clean N-复形

College of Mathematics and Computer Science, Northwest Minzu University, Lanzhou 730030, Gansu, China

^† Corresponding author. E-mail: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 10 April 2024

Abstract

The notion of absolutely clean N-complexes is studied. We show that an N-complex $X$ is absolutely clean if and only if $X$ is N-exact and $Z_n^i(X)$ is an absolutely clean module for each $n\in \mathbf{\mathbb{Z}}$ and $i=\mathrm{1,2},\cdots ,N.$ In particular, we prove that a bounded above N-complex $X$ is absolutely clean if and only if $X_n$ is an absolutely clean module for each $n\in \mathbf{\mathbb{Z}}$. We also show that under certain hypotheses, an N-complex $X$ is Gorenstein AC-injective if and only if $Z_n^t(X)$ is a Gorenstein AC-injective module for each $n\in \mathbf{\mathbb{Z}}$ and $t=\mathrm{1,2},\cdots ,N$.

摘要

研究了绝对clean N-复形的概念，证明了N-复形$X$是绝对clean的当且仅当$X$是N-正合的并且对任意的$n\in \mathbf{\mathbb{Z}}$和$i=\mathrm{1,2},\cdots ,N$，$Z_n^i(X)$是绝对clean模。特别地，证明了上有界N-复形$X$是绝对clean的当且仅当$X$是N-正合的并且对任意的$n\in \mathbf{\mathbb{Z}}$，$X_n$是绝对clean 模。证明了一定条件下N-复形$X$是Gorenstein AC-内射的当且仅当对任意的$n\in \mathbf{\mathbb{Z}}$和$t=\mathrm{1,2},\cdots ,N$，$Z_n^t(X)$是Gorenstein AC-内射模。