Exterior depth and exterior generic annihilator numbers

TL;DR

This paper investigates the properties of exterior depth and generic annihilator numbers of E-modules, establishing their relationships with symmetric depth, Betti numbers, and shifting, and providing combinatorial interpretations and examples.

Contribution

It introduces the concept of exterior annihilator numbers, relates exterior depth to symmetric depth, and explores their connections with Betti numbers and shifting.

Findings

Exterior depth relates to symmetric depth of associated modules.

Exterior annihilator numbers have combinatorial interpretations.

Neither symmetric nor exterior generic annihilator numbers are minimal among all 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 shifting. We define exterior annihilator numbers analogously to the annihilator numbers over the polynomial ring introduced by Trung and Conca, Herzog and Hibi. 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.