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Piecewise linear sheaves
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On a finite-dimensional real vector space, we give a microlocal characterization of (derived) piecewise linear sheaves (PL sheaves) and prove that the triangulated category of such sheaves is generated by sheaves associated with convex polyhedra. We then give a similar theorem for PL gamma-sheaves, that is, PL sheaves associated with the gamma-topology, for a closed convex polyhedral proper cone gamma. Our motivation is that convex polyhedra may be considered as building blocks for higher dimensional barcodes.
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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.
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