weak dimension of a module
Assume that is a ring. We will consider right -modules.
Definition 1. We will say that an -module is of weak dimension at most iff there exists a short exact sequence
Definition 2. We will say that an -module is of weak dimension iff but .
The weak dimension measures how far an -module is from being flat. Let as gather some known facts about the weak dimension:
Proposition 1. Assume that is a right -module. Then for some if and only if for any left -module we have
and there exists a left -module such that
where denotes the Tor functor.
Since every projective module is flat, then we can state simple observation:
Proposition 2. Assume that is a right -module. Then
where denotes the projective dimension of .
Generally these two dimension may differ.
|Title||weak dimension of a module|
|Date of creation||2013-03-22 19:18:40|
|Last modified on||2013-03-22 19:18:40|
|Last modified by||joking (16130)|