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arXiv · 2309.11780

Geometric Extensions

Chris Hone·Geordie Williamson
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Abstract

We prove that the derived direct image of the constant sheaf with field coefficients under any proper map with smooth source contains a canonical summand. This summand, which we call the geometric extension, only depends on the generic fibre. For resolutions we get a canonical extension of the constant sheaf. When our coefficients are of characteristic zero, this summand is the intersection cohomology sheaf. When our coefficients are finite we obtain a new object, which provides interesting topological invariants of singularities and topological obstructions to the existence of morphisms. The geometric extension is a generalization of a parity sheaf. Our proof is formal, and also works with coefficients in modules over suitably finite ring spectra.

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Chris Hone, Geordie Williamson. 2023-09-21. Geometric Extensions. https://arxiv.org/abs/2309.11780

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