minimal projective presentation

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

$\xymatrix{P_{1}\ar[r]^{p_{1}}&P_{0}\ar[r]^{p_{0}}&M\ar[r]&0}$

is called a minimal projective presentation of $M$ if both $p_{0}:P_{0}\to M$ and $p_{1}:P_{1}\to\mathrm{ker}p_{0}$ are projective covers.

Minimal projective presenetations are unique in the following sense: if

$\xymatrix{P_{1}\ar[r]^{p_{1}}&P_{0}\ar[r]^{p_{0}}&M\ar[r]&0\\ P^{\prime}_{1}\ar[r]^{p^{\prime}_{1}}&P^{\prime}_{0}\ar[r]^{p^{\prime}_{0}}&M% \ar[r]&0\\ }$

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

$\xymatrix{P_{1}\ar[r]^{p_{1}}\ar[d]^{a}&P_{0}\ar[r]^{p_{0}}\ar[d]^{b}&M\ar[r]% \ar[d]^{=}&0\\ P^{\prime}_{1}\ar[r]^{p^{\prime}_{1}}&P^{\prime}_{0}\ar[r]^{p^{\prime}_{0}}&M% \ar[r]&0\\ }$

were both $a, b$ are isomorphisms.

It can be shown, that if $R$ is a finite-dimensional algebra over a field $k$ , then every finitely generated $R$ -module $M$ admits minimal projective presentation (indeed, $R$ is semiperfect (http://planetmath.org/PerfectAndSemiperfectRings) 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