A Non-Existence Theorem for Almost Split Sequences

Abstract

Let k be a field, Q a quiver with countably many vertices and I an ideal of kQ such that kQ/I is a spectroid. In this note, we prove that there is no almost split sequence ending at an indecomposable not finitely presented representation of the bound quiver (Q, I). We then get that an indecomposable representation M of (Q, I) is the ending term of an almost split sequence if and only if it is finitely presented and not projective. The dual results are also true.

Key Words:

• Almost split sequences
• Finitely co-presented representations
• Finitely presented representations
• Locally finite dimensional representations
• Representations of quivers

2000 Mathematics Subject Classification:

• 16G20
• 16G70

ACKNOWLEDGMENTS

The author is thankful to S. Liu and V. Shramchenko for financial support while doing a postdoctorate at the Université de Sherbrooke.

Notes

Communicated by D. Zacharia.