2023/09/21 by Hone, Chris, Williamson, Geordie
#Algebraic Geometry (math.AG) #FOS: Mathematics #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2309.11780
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.