Parity sheaves and Smith theory

Abstract Let p be a prime number and let X complex algebraic variety with an action of ℤ / p ⁢ {\mathbb{Z}/p\mathbb{Z}} . We develop the theory parity complexes in certain 2-periodic localization equivariant constructible derived category D b ( X , stretchy="false">) {D^{b}_{\mathbb{Z}/p\mathbb{Z}}(X,\mathbb{Z}_{p})} Under assumptions, we use this to define functor from sheaves on fixed-point locus {X^{\mathbb{Z}/p\mathbb{Z}}} This may thought as categorification Smith theory. When is affine Grassmannian associated some reductive group, our gives geometric construction Frobenius-contraction recently defined by M. Gros Kaneda via Satake equivalence.