minimal projective presentation

Let R be a ring and M a (right) module over R. A short exact sequenceMathworldPlanetmathPlanetmath of modules


is called a minimal projective presentation of M if both p0:P0M and p1:P1kerp0 are projective covers.

Minimal projective presenetations are unique in the following sense: if


are both minimal projective presentations of M, then this diagram can be completed to the following commutativePlanetmathPlanetmathPlanetmathPlanetmath one:


were both a,b are isomorphismsMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath.

It can be shown, that if R is a finite-dimensional algebra over a field k, then every finitely generatedMathworldPlanetmathPlanetmathPlanetmath R-module M admits minimal projective presentation (indeed, R is semiperfect ( in this case).

Title minimal projective presentation
Canonical name MinimalProjectivePresentation
Date of creation 2013-03-22 19:18:00
Last modified on 2013-03-22 19:18:00
Owner joking (16130)
Last modified by joking (16130)
Numerical id 4
Author joking (16130)
Entry type Definition
Classification msc 16D40