PlanetMath: projective dimension

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projective dimension

(Definition)

Let $\mathcal{A}$ be an abelian category and $M\in\operatorname{Ob}(\mathcal{A})$ such that a projective resolution of exists:

$\displaystyle \xymatrix{{\ldots}\ar[r]&P_n\ar[r]&{\ldots}\ar[r]&P_1\ar[r]&P_0\ar[r]&M\ar[r]&0}.$

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 $i\geq n$ ). If such a subset is non-empty, then the projective dimension of

is defined to be the smallest number

such that

$\displaystyle \xymatrix{0\ar[r]&P_d\ar[r]&{\ldots}\ar[r]&P_1\ar[r]&P_0\ar[r]&M\ar[r]&0}.$

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
$\displaystyle \xymatrix{0\ar[r]&P_d\ar[r]&{\ldots}\ar[r]&P_1\ar[r]&P_0\ar[r]&M\ar[r]&0}.$
Then $P_i\neq0$ for all $0\leq i\leq d$ .
$\operatorname{pd}(M)=0$ iff is a projective object.
In the (abelian) category of left (right) -modules, the projective dimension of a left (right) -module is denoted by $\operatorname{pd}_R(M)$ .

Likewise, given an abelian category and a object having at least one injective resolution. Then the injective dimension, denoted by $\operatorname{id}(N)$ is the minimum number such that

$\displaystyle \xymatrix{0\ar[r]&N\ar[r]&I_0\ar[r]&I_1\ar[r]&{\ldots}\ar[r]&I_d\ar[r]&0},$

if such an injective resolution exists. Otherwise, set $\operatorname{id}(N)=\infty$ . This is the dual notion of projective dimension.

"projective dimension" is owned by CWoo.

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Also defines:

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 4 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 2635 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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