Commutative algebra

Yoshifumi Tsuchimoto

DEFINITION 11.1   Let $A$ be a commutative ring. Let $M$ be an $A$ -module. An element $x\in A$ is said to be $M$ -regular if

$$0 \to M \overset{x\times}{\to} M$$

is exact.

PROPOSITION 11.2   Let $A$ be a ring, $M,N$ two $A$ -modules, and $x\in A$ . Suppose that $x$ is both $A$ -regular and $M$ -regular, and that $xN=0$ . Set $B=A/x A$ and $\bar M=M/xM$ . Then:

1. $\operatorname{Tor}^A_n(M,B)=0$ for all $n>0$ .

2. $\operatorname{Ext}_A^n(M,N)\cong \operatorname{Ext}_B^n (\bar M, N)$ for all $n\geq 0$ .
3. $\operatorname{Tor}^A_n(M,N)\cong \operatorname{Tor}^B_n (\bar M, N)$ for all $n\geq 0$ .