projective equivalence

Let R be a ring with 1. Two R-modules A and B are said to be projectively equivalent AB if there exist two projective R-modules P and Q such that



  1. 1.

    Projective equivalence is an equivalence relationMathworldPlanetmath.

  2. 2.

    Any projective moduleMathworldPlanetmath is projectively equivalent to the zero moduleMathworldPlanetmath.

  3. 3.

    (Schanuel’s Lemma). Given two short exact sequencesMathworldPlanetmathPlanetmath:



    with A1A2, then B1B2.

  4. 4.

    Schanuel’s Lemma can be generalized. Given two projective resolutions:



    with A1A2, then Ker(pn)Ker(qn) for all n0

  5. 5.

    The concept of projective equivalence between two modules can be generalized to any abelian categoriesMathworldPlanetmathPlanetmathPlanetmath having enough projectives.

Title projective equivalence
Canonical name ProjectiveEquivalence
Date of creation 2013-03-22 14:50:13
Last modified on 2013-03-22 14:50:13
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 5
Author CWoo (3771)
Entry type Definition
Classification msc 16E10
Classification msc 18G20
Classification msc 18G10
Defines projectively equivalent
Defines Schanuel’s Lemma