exact sequences for modules with finite projective dimension
Proposition. Let R be a ring and M be a (left) R-module, such that proj dim(M)=n<∞. If
0→K→Pn→⋯→P0→M→0 |
is an exact sequence 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
0→P′n→⋯→P′0→M→0, |
Note that sequences
Pn→⋯→P0→M→0; |
P′n→⋯→P0→M→0, |
are projective resolutions of M. Let δ:Pn→Pn-1 and β:P′n→P′n-1 be maps take from these resolutions. Then generalized Schanuel’s lemma implies that kerδ and kerβ are projectively equivalent. But kerδ≃K and kerβ=0. This means, that there are projective modules P,Q such that
K⊕P≃Q. |
Therefore K is a direct summand of a free module (since Q is), which completes
the proof. □
Title | exact sequences for modules with finite projective dimension |
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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 |