# 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 MinimalProjectivePresentation 2013-03-22 19:18:00 2013-03-22 19:18:00 joking (16130) joking (16130) 4 joking (16130) Definition msc 16D40