On restricted powers of complete intersections

Abstract

A restricted dth power of an ideal I is obtained by restricting the exponent vectors allowed to appear on the “natural” generating set of I^d, for some integer d. In this paper, we study homological properties of restricted powers of complete intersections. We construct a generalization of the L-complex construction of Buchsbaum and Eisenbud. We use this resolution to compute an explicit basis for the Koszul homology which allows us to deduce that the quotient defined by any restricted dth power of a complete intersection is Golod, and construct an algebra structure on this complex.

KEYWORDS:

• DG-algebras
• free resolutions
• Golod rings
• Koszul homology
• restricted powers

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 13D02
• 13D07
• 13C13

Acknowledgments

Thanks to the anonymous referee for a very close reading, yielding many corrections and suggestions for improvement.