weak dimension of a module

Assume that $R$ is a ring. We will consider right $R$-modules.

Definition 1. We will say that an $R$-module $M$ is of weak dimension at most $n\in \mathbb{N}$ iff there exists a short exact sequence

such that each $F_i$ is a flat module. In this case we write ${\mathrm{wd}}_{R}M⩽n$ (also we say that $M$ is of finite weak dimension). If such short exact sequence does not exist, then the weak dimension is defined as infinity, ${\mathrm{wd}}_{R}M=\mathrm{\infty }$.

Definition 2. We will say that an $R$-module $M$ is of weak dimension $n\in \mathbb{N}$ iff ${\mathrm{wd}}_{R}M⩽n$ but ${\mathrm{wd}}_{R}M⩽̸n-1$.

The weak dimension measures how far an $R$-module is from being flat. Let as gather some known facts about the weak dimension:

Proposition 1. Assume that $M$ is a right $R$-module. Then ${\mathrm{wd}}_{R}M=n$ for some $n\in \mathbb{N}$ if and only if for any left $R$-module $N$ we have

$${\mathrm{Tor}}_{n+1}^R(M,N)=0$$

and there exists a left $R$-module $N^{\prime }$ such that

$${\mathrm{Tor}}_n^R(M,N^{\prime })\ne 0,$$

where $\mathrm{Tor}$ denotes the Tor functor.

Since every projective module is flat, then we can state simple observation:

Proposition 2. Assume that $M$ is a right $R$-module. Then

$${\mathrm{wd}}_{R}M⩽{\mathrm{pd}}_{R}M,$$

where ${\mathrm{pd}}_{R}M$ denotes the projective dimension of $M$.

Generally these two dimension may differ.