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Let R = k[x1, ..., xn] 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 rankR 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