Tame modules over arbitrary posets are proved to have finite encodings, fringe presentations, indicator resolutions, and primary decompositions, leading to proofs of two sheaf-theoretic conjectures.
Combinatorial resolutions of multigraded modules and multipersistent homology
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abstract
Let $ R=k[x_1...x_r]$ and $M$ a multigraded $R-$module. In this work we interpret $M$ as a multipersistent homology module and give a multigraded resolution of it. The construction involves cellular resolutions of monomial ideals and reflects the combinatorial structure of multipersistence homology modules. In the one critical case, a multifiltration is represented by a labelled cellular complex. A multipersistence homology module measures the defect of acyclicity of the associated multigraded cellular chain complex.
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Modules over posets: commutative and homological algebra
Tame modules over arbitrary posets are proved to have finite encodings, fringe presentations, indicator resolutions, and primary decompositions, leading to proofs of two sheaf-theoretic conjectures.