exact sequences for modules with finite projective dimension

PropositionPlanetmathPlanetmath. Let R be a ring and M be a (left) R-module, such that proj dim(M)=n<. If


is an exact sequencePlanetmathPlanetmathPlanetmathPlanetmath of R-modules, such that each Pi is projective, then K is projective.

Proof. Since proj dim(M)=n<, then there exists exact sequence of R-modules


Note that sequencesPlanetmathPlanetmath


are projective resolutions of M. Let δ:PnPn-1 and β:PnPn-1 be maps take from these resolutions. Then generalized Schanuel’s lemma implies that kerδ and kerβ are projectively equivalent. But kerδK and kerβ=0. This means, that there are projective modulesMathworldPlanetmath P,Q such that


Therefore K is a direct summand of a free moduleMathworldPlanetmathPlanetmath (since Q is), which completesPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath the proof.

Title exact sequences for modules with finite projective dimension
Canonical name ExactSequencesForModulesWithFiniteProjectiveDimension
Date of creation 2013-03-22 19:04:55
Last modified on 2013-03-22 19:04:55
Owner joking (16130)
Last modified by joking (16130)
Numerical id 4
Author joking (16130)
Entry type Corollary
Classification msc 16E10
Classification msc 18G20
Classification msc 18G10