🤖 AI Summary
The absence of a computable microlocal theory for sheaves on finite posets and simplicial complexes hinders applications in discrete geometry and representation theory.
Method: Integrating discrete Morse theory with microlocal sheaf theory, the authors develop a computable framework within the derived category. Their approach combines Alexandrov topology-based sheaf theory, critical point analysis of discrete Morse functions, algorithms for minimal injective resolutions, and functorial computations in derived categories.
Contribution/Results: They introduce the notion of “discrete microsupport”, rigorously adapting microlocal theory to discrete topological structures. They establish a uniqueness theorem for minimal injective resolutions of sheaf complexes and derive asymptotically tight bounds for minimal resolutions of constant sheaves. Furthermore, they prove a microlocal version of the discrete Morse homology theorem and associated inequalities. These results provide novel microlocal tools and a computational paradigm for discrete geometry and representation theory.
📝 Abstract
We establish several results combining discrete Morse theory and microlocal sheaf theory in the setting of finite posets and simplicial complexes. Our primary tool is a computationally tractable description of the bounded derived category of sheaves on a poset with the Alexandrov topology. We prove that each bounded complex of sheaves on a finite poset admits a unique (up to isomorphism of complexes) minimal injective resolution, and we provide algorithms for computing minimal injective resolution of an injective complex, as well as several useful functors between derived categories of sheaves. For the constant sheaf on a simplicial complex, we give asymptotically tight bounds on the complexity of computing the minimal injective resolution using those algorithms. Our main result is a novel definition of the discrete microsupport of a bounded complex of sheaves on a finite poset. We detail several foundational properties of the discrete microsupport, as well as a microlocal generalization of the discrete homological Morse theorem and Morse inequalities.
