Geometric Extensions

TL;DR

This paper introduces the geometric extension, a canonical summand in the derived direct image of constant sheaves under proper maps, generalizing intersection cohomology and providing new invariants for singularities.

Contribution

It proves the existence of the geometric extension as a canonical summand, extending known sheaf concepts to broader contexts with new topological invariants.

Findings

The geometric extension exists for proper maps with smooth sources.

In characteristic zero, it coincides with intersection cohomology.

For finite coefficients, it yields new topological invariants.

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.