Algebraic Shifting and Exterior and Symmetric Algebra Methods

Abstract

We define and study Cartan–Betti numbers of a graded ideal J in the exterior algebra over an infinite field which include the usual graded Betti numbers of J as a special case. Following ideas of Conca regarding Koszul–Betti numbers over the symmetric algebra, we show that Cartan–Betti numbers increase by passing to the generic initial ideal and the squarefree lexsegement ideal, respectively. Moreover, we characterize the cases where the inequalities become equalities. As combinatorial applications of the first part of this note and some further symmetric algebra methods we establish results about algebraic shifting of simplicial complexes and use them to compare different shifting operations. In particular, we show that each shifting operation does not decrease the number of facets, and that the exterior shifting is the best among the exterior shifting operations in the sense that it increases the number of facets the least.

Key Words:

• Algebraic shifting
• Arithmetic degree
• Betti numbers
• Cartan homology
• Generic initial ideal

2000 Mathematics Subject Classification:

• Primary 13F55
• Secondary 13D02

ACKNOWLEDGMENTS

The authors would like to thank the referee for the careful reading and the very helpful comments.

The third author gratefully acknowledges partial support by the FIRB Research Project “Teoria dell'intersezione e applicazioni computazionali.”

Notes

Communicated by R. H. Villarreal.