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On the Betti number of differential modules
Let R = k[x1 , …, xd] with k a field. A [special characters omitted]-graded differential R-module is a [special characters omitted]-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.