PlanetMath: syzygy

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syzygy

(Definition)

Let $\mathcal{A}$ be an abelian category. Let $M\in\operatorname{Ob}(\mathcal{A})$ . If there is an exact sequence

$\displaystyle \xymatrix{0\ar[r]&S\ar[r]&P_n\ar[r]&{\cdots}\ar[r]&P_1\ar[r]&P_0\ar[r]&M\ar[r]&0},$

where each

is a projective object in $\mathcal{A}$ , then we call

th syzygy of

If is itself projective, then the projective dimension of , $\operatorname{pd}(M)$ , is less than or equal to .

Remark.

The word “syzygy” is used in astromony to describe three celestial objects (usually the Sun, Moon and Earth, or the Sun, Earth and Moon) being collinear.
Any two th syzygies of a given object are projectively equivalent.

"syzygy" is owned by CWoo.

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Cross-references: projectively equivalent, collinear, objects, projective dimension, projective object, exact sequence, abelian category
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This is version 4 of syzygy, born on 2004-11-24, modified 2006-06-19.
Object id is 6524, canonical name is Syzygy.
Accessed 2121 times total.

Classification:

AMS MSC:	13D02 (Commutative rings and algebras :: Homological methods :: Syzygies and resolutions)
	16E05 (Associative rings and algebras :: Homological methods :: Syzygies, resolutions, complexes)
	18G10 (Category theory; homological algebra :: Homological algebra :: Resolutions; derived functors)

Pending Errata and Addenda

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