On the Betti Number of Differential Modules

Authors

Justin DeVries, University of Nebraska-LincolnFollow

First Advisor

Srikanth B. Iyengar

Date of this Version

2011

Comments

A DISSERTATION Presented to the Faculty of The Graduate College at the University of Nebraska In Partial Fulfillment of Requirements For the Degree of Doctor of Philosophy, Major: Mathematics, Under the Supervision of Professor Srikanth B. Iyengar. Lincoln, Nebraska: May, 2011
Copyright 2011 Justin W. DeVries

Abstract

Let R = k[x₁, ..., x_n] with k a field. A multi-graded differential R-module is a multi-graded R-module D with an endomorphism d such that d² = 0. This dissertation establishes a lower bound on the rank of such a differential module when the underlying R-module is free. We define the Betti number of a differential module and use it to show that when the homology ker d/im d of D is non-zero and finite dimensional over k then there is an inequality rank_R D ≥ 2ⁿ. This relates to a problem of Buchsbaum, Eisenbud and Horrocks in algebra and conjectures of Carlsson and Halperin in topology.

Motivated by some steps of this work, further results are proved relating the homotopical Loewy length, derived Loewy length and generalized Loewy length.

Adviser: Srikanth B. Iyengar