# weak global dimension

Let $R$ be a ring. The (right) weak global dimension of $R$ is defined as

$$\mathrm{w}.\mathrm{gl}.\mathrm{dim}R=\mathrm{sup}\{{\mathrm{wd}}_{R}M|M\text{is a right module}\}.$$ |

Unlike global dimension of $R$ the definition of the weak global dimension is left/right symmetric^{}. This follows from the fact that for every left module $M$ and right module $N$ there is an isomorphism^{}

$${\mathrm{Tor}}_{n}^{R}(M,N)\simeq {\mathrm{Tor}}_{n}^{R}(N,M).$$ |

Thus we simply say that $R$ has the weak global dimension. Note that this does not hold for Ext functors, so (generally) the definition of global dimension is not left/right symmetric.

The following proposition^{} is a simple consequence of the fact that every projective module^{} is flat:

Proposition. For any ring $R$ we have

$$\mathrm{w}.\mathrm{gl}.\mathrm{dim}R\u2a7d\mathrm{min}\{\mathrm{l}.\mathrm{gl}.\mathrm{dim}R,\mathrm{r}.\mathrm{gl}.\mathrm{dim}R\},$$ |

where $\mathrm{l}.\mathrm{gl}.\mathrm{dim}$ and $\mathrm{r}.\mathrm{gl}.\mathrm{dim}$ denote the left global dimension and right global dimension respectively.

Title | weak global dimension |
---|---|

Canonical name | WeakGlobalDimension |

Date of creation | 2013-03-22 19:18:42 |

Last modified on | 2013-03-22 19:18:42 |

Owner | joking (16130) |

Last modified by | joking (16130) |

Numerical id | 4 |

Author | joking (16130) |

Entry type | Definition |

Classification | msc 16E05 |