Exterior Depth and Exterior Generic Annihilator Numbers

Abstract

We study the exterior depth of an E-module and its exterior generic annihilator numbers. For the exterior depth of a squarefree E-module, we show how it relates to the symmetric depth of the corresponding S-module and classify those simplicial complexes having a particular exterior depth in terms of their exterior algebraic shifting. We define exterior annihilator numbers analogously to the annihilator numbers over the polynomial ring introduced by Trung [19] and Conca et al. [9]. In addition to a combinatorial interpretation of the annihilator numbers, we show how they are related to the symmetric Betti numbers and the Cartan–Betti numbers, respectively. We finally conclude with an example which shows that neither the symmetric nor the exterior generic annihilator numbers are minimal among the annihilator numbers with respect to a sequence.

Key Words:

• Annihilator numbers
• Betti numbers
• Depth
• Exterior algebra

2000 Mathematics Subject Classification:

• Primary 05E45
• Secondary 13A99, 13D02, 05E40, 16E99

ACKNOWLEDGMENT

The main work of this article was made while the authors were participating at the summer school of PRAGMATIC 2008 in Catania, Sicily. The authors thank the organizers of PRAGMATIC 2008 for support and hospitality. They are grateful to Jürgen Herzog and Volkmar Welker for their suggestion to study this topic and for their helpful discussions. Further thanks go to Eran Nevo for pointing out the idea of Theorem 2.9 to us and to the referee for very careful reading and for the proof of Theorem 4.6.

Notes

Communicated by R. Villarreal.