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projective dimension
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(Definition)
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Let $\mathcal{A}$ be an abelian category and $M\in\operatorname{Ob}(\mathcal{A})$ such that a projective resolution of $M$ exists:
Among all the projective resolutions of $M$ , 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 $n$ such that $P_i=0$ for all $i\geq n$ ). If such a subset is non-empty, then the projective
dimension of $M$ is defined to be the smallest number $d$ such that
We denote this by $\operatorname{pd}(M)=d$ . If this subset is empty, then we define $\operatorname{pd}(M)=\infty$ .
Remarks.
- In an abelian category having enough projectives, the projective dimension of an object always exists (whether it is finite or not).
- If $\operatorname{pd}(M)=d$ and
Then $P_i\neq0$ for all $0\leq i\leq d$ .
- $\operatorname{pd}(M)=0$ iff $M$ is a projective object.
- In the (abelian) category of left (right) $R$ -modules, the projective dimension of a left (right) $R$ -module $M$ is denoted by $\operatorname{pd}_R(M)$ .
Likewise, given an abelian category and a object $N$ having at least one injective resolution. Then the injective dimension, denoted by $\operatorname{id}(N)$ is the minimum number $d$ such that
if such an injective resolution exists. Otherwise, set $\operatorname{id}(N)=\infty$ . This is the dual notion of projective dimension.
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"projective dimension" is owned by CWoo.
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injective dimension |
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Cross-references: injective resolution, right, category, abelian, iff, object, enough projectives, integer, projective objects, number, finite, contain, subset, projective resolution, abelian category
There are 5 references to this entry.
This is version 4 of projective dimension, born on 2004-11-23, modified 2007-03-05.
Object id is 6519, canonical name is ProjectiveDimension.
Accessed 3747 times total.
Classification:
| AMS MSC: | 13D05 (Commutative rings and algebras :: Homological methods :: Homological dimension) | | | 16E10 (Associative rings and algebras :: Homological methods :: Homological dimension) | | | 18G20 (Category theory; homological algebra :: Homological algebra :: Homological dimension) |
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Pending Errata and Addenda
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![$\displaystyle \xymatrix{{\ldots}\ar[r]&P_n\ar[r]&{\ldots}\ar[r]&P_1\ar[r]&P_0\ar[r]&M\ar[r]&0}.$](http://images.planetmath.org:8080/cache/objects/6519/js/img1.png "$\displaystyle \xymatrix{{\ldots}\ar[r]&P_n\ar[r]&{\ldots}\ar[r]&P_1\ar[r]&P_0\ar[r]&M\ar[r]&0}.$"){kind=small}
![$\displaystyle \xymatrix{0\ar[r]&P_d\ar[r]&{\ldots}\ar[r]&P_1\ar[r]&P_0\ar[r]&M\ar[r]&0}.$](http://images.planetmath.org:8080/cache/objects/6519/js/img2.png "$\displaystyle \xymatrix{0\ar[r]&P_d\ar[r]&{\ldots}\ar[r]&P_1\ar[r]&P_0\ar[r]&M\ar[r]&0}.$"){kind=small}
![$\displaystyle \xymatrix{0\ar[r]&P_d\ar[r]&{\ldots}\ar[r]&P_1\ar[r]&P_0\ar[r]&M\ar[r]&0}.$](http://images.planetmath.org:8080/cache/objects/6519/js/img3.png "$\displaystyle \xymatrix{0\ar[r]&P_d\ar[r]&{\ldots}\ar[r]&P_1\ar[r]&P_0\ar[r]&M\ar[r]&0}.$"){kind=small}
![$\displaystyle \xymatrix{0\ar[r]&N\ar[r]&I_0\ar[r]&I_1\ar[r]&{\ldots}\ar[r]&I_d\ar[r]&0},$](http://images.planetmath.org:8080/cache/objects/6519/js/img4.png "$\displaystyle \xymatrix{0\ar[r]&N\ar[r]&I_0\ar[r]&I_1\ar[r]&{\ldots}\ar[r]&I_d\ar[r]&0},$"){kind=small}