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On the Betti number of differential modules
Let R = k[x1 , …, xd] with k a field. A Zd -graded differential R-module is a Zd -graded R-module D with a morphism δ : D → D such that δ2 = 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 δ/im δ of D is non-zero and finite dimensional over k then there is an inequality rank R D ≥ 2d. 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. ^
DeVries, Justin W, "On the Betti number of differential modules" (2011). ETD collection for University of Nebraska - Lincoln. AAI3450071.