projective dimension
Let be an abelian category and such that a projective resolution of exists:
Among all the projective resolutions of , consider the subset consisting of those projective resolutions that contain only a finite number of non-zero projective objects (there exists a non-negative integer such that for all ). If such a subset is non-empty, then the projective dimension of is defined to be the smallest number such that
We denote this by . If this subset is empty, then we define .
Remarks.
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1.
In an abelian category having enough projectives, the projective dimension of an object always exists (whether it is finite or not).
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2.
If and
Then for all .
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3.
iff is a projective object.
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4.
In the (abelian) category of left (right) -modules, the projective dimension of a left (right) -module is denoted by .
Likewise, given an abelian category and a object having at least one injective resolution. Then the injective dimension, denoted by is the minimum number such that
if such an injective resolution exists. Otherwise, set . This is the dual notion of projective dimension.
Title | projective dimension |
---|---|
Canonical name | ProjectiveDimension |
Date of creation | 2013-03-22 14:50:56 |
Last modified on | 2013-03-22 14:50:56 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 7 |
Author | CWoo (3771) |
Entry type | Definition |
Classification | msc 16E10 |
Classification | msc 13D05 |
Classification | msc 18G20 |
Defines | injective dimension |