# 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

 $\xymatrix{0\ar[r]&F_{n}\ar[r]&F_{n-1}\ar[r]&\cdots\ar[r]&F_{1}\ar[r]&F_{0}\ar[% r]&M\ar[r]&0}$

such that each $F_{i}$ is a flat module. In this case we write $\mathrm{wd}_{R}M\leqslant n$ (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, $\mathrm{wd}_{R}M=\infty$.

Definition 2. We will say that an $R$-module $M$ is of weak dimension $n\in\mathbb{N}$ iff $\mathrm{wd}_{R}M\leqslant n$ but $\mathrm{wd}_{R}M\not\leqslant n-1$.

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

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

 $\mathrm{Tor}_{n}^{R}(M,N^{\prime})\neq 0,$

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

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

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

Generally these two dimension may differ.

Title weak dimension of a module WeakDimensionOfAModule 2013-03-22 19:18:40 2013-03-22 19:18:40 joking (16130) joking (16130) 4 joking (16130) Derivation msc 16E05