Local Gorenstein duality for cochains on spaces

Tobias Barthel, Natalia Castellana, Drew Heard, Gabriel Valenzuela

Published 2020-01-08Version 1

We investigate when a commutative ring spectrum $R$ satisfies a homotopical version of local Gorenstein duality, extending the absolute notion previously studied by Greenlees. In order to do this, we prove an ascent theorem for local Gorenstein duality along morphisms of $k$-algebras. Our main examples are of the form $R = C^*(X;k)$, the ring spectrum of cochains on a space $X$ for a field $k$. In particular, we establish local Gorenstein duality in characteristic $p$ for $p$-compact groups and $p$-local finite groups as well as for $k = \mathbb{Q}$ and $X$ a simply connected space which is Gorenstein in the sense of Dwyer, Greenlees, and Iyengar.