PlanetMath: projective equivalence

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projective equivalence

(Definition)

Let be a ring with 1. Two -modules and are said to be projectively equivalent $A\sim B$ if there exist two projective -modules and such that

$\displaystyle A\oplus P\cong B\oplus Q.$

Remarks.

Projective equivalence is an equivalence relation.
Any projective module is projectively equivalent to the zero module.
(Schanuel's Lemma). Given two short exact sequences:
$\xymatrix{0\ar[r]&B_1\ar[r]&P\ar[r]&A_1\ar[r]&0}$

$\xymatrix{0\ar[r]&B_2\ar[r]&Q\ar[r]&A_2\ar[r]&0}$
with $A_1\sim A_2$ , then $B_1\sim B_2$ .
Schanuel's Lemma can be generalized. Given two projective resolutions:
$\xymatrix{\ldots\ar[r]^{p_3}&P_2\ar[r]^{p_2}&P_1\ar[r]^{p_1}&P_0\ar[r]^{p_0}&A_1\ar[r]&0}$

$\xymatrix{\ldots\ar[r]^{q_3}&Q_2\ar[r]^{q_2}&Q_1\ar[r]^{q_1}&Q_0\ar[r]^{q_0}&A_2\ar[r]&0}$
with $A_1\sim A_2$ , then $\operatorname{Ker}(p_n)\sim\operatorname{Ker}(q_n)$ for all $n\geq0$
The concept of projective equivalence between two modules can be generalized to any abelian categories having enough projectives.

"projective equivalence" is owned by CWoo.

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Also defines:

projectively equivalent, Schanuel's Lemma

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AMS MSC:	16E10 (Associative rings and algebras :: Homological methods :: Homological dimension)
	18G20 (Category theory; homological algebra :: Homological algebra :: Homological dimension)
	18G10 (Category theory; homological algebra :: Homological algebra :: Resolutions; derived functors)

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