COMMUTATIVE ALGEBRA

YOSHIFUMI TSUCHIMOTO

11. regular element

Definition 11.1. Let A be a commutative ring. Let M be an Amodule. An element x ∈ A is said to be M-regular if

0 → M →^{x×} M

is exact.

Proposition 11.2. Let A be a ring, M, N two A-modules, and x ∈ A. Suppose that x is both A-regular and M-regular, and that xN = 0. Set B = A/xA and M̄ = M/xM. Then:

(1) Tor_{n}^{A}(M, B) = 0 for all n > 0.

(2) Ext_{A}^{n}(M, N) ≅ Ext_{B}^{n}(M̄, N) for all n ≥ 0.

(3) Tor_{n}^{A}(M, N) ≅ Tor_{n}^{B}(M̄, N) for all n ≥ 0.