# 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^{}

$$\text{xymatrix}0\text{ar}[r]\mathrm{\&}{F}_{n}\text{ar}[r]\mathrm{\&}{F}_{n-1}\text{ar}[r]\mathrm{\&}\mathrm{\cdots}\text{ar}[r]\mathrm{\&}{F}_{1}\text{ar}[r]\mathrm{\&}{F}_{0}\text{ar}[r]\mathrm{\&}M\text{ar}[r]\mathrm{\&}0$$ |

such that each ${F}_{i}$ is a flat module^{}. In this case we write ${\mathrm{wd}}_{R}M\u2a7dn$ (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 ${\mathrm{wd}}_{R}M\u2a7dn$ but ${\mathrm{wd}}_{R}M\u2a7d\u0338n-1$.

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

$${\mathrm{wd}}_{R}M\u2a7d{\mathrm{pd}}_{R}M,$$ |

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

Generally these two dimension may differ.

Title | weak dimension of a module |
---|---|

Canonical name | WeakDimensionOfAModule |

Date of creation | 2013-03-22 19:18:40 |

Last modified on | 2013-03-22 19:18:40 |

Owner | joking (16130) |

Last modified by | joking (16130) |

Numerical id | 4 |

Author | joking (16130) |

Entry type | Derivation^{} |

Classification | msc 16E05 |