Testing flatness and computing rank of a module using syzygies

Oswaldo Lezama¹

¹ Departamento de Matemáticas Universidad Nacional de Colombia Bogotá, Colombia

Colloquium Mathematicum, Tome 117 (2009) no. 1, pp. 65-79.

Résumé

Using syzygies computed via Gröbner bases techniques, we present algorithms for testing some homological properties for submodules of the free module $A^m$, where $A=R[x_1,\dots,x_n]$ and $R$ is a Noetherian commutative ring. We will test if a given submodule $M$ of $A^m$ is flat. We will also check if $M$ is locally free of constant dimension. Moreover, we present an algorithm that computes the rank of a flat submodule $M$ of $A^m$ and also an algorithm that computes the projective dimension of an arbitrary submodule of $A^m$. All algorithms are illustrated with examples.

Détail

DOI : 10.4064/cm117-1-4

Keywords: using syzygies computed via bner bases techniques present algorithms testing homological properties submodules module where dots noetherian commutative ring test given submodule flat check locally constant dimension moreover present algorithm computes rank flat submodule algorithm computes projective dimension arbitrary submodule algorithms illustrated examples