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We investigate when a commutative ring spectrum $R$ satisfies a homotopical version of local Gorenstein duality, extending the 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 $k = Q$ and $X$ a simply connected space which is Gorenstein in the sense of Dwyer, Greenlees, and Iyengar.
Cubical cochains are equipped with an associative product, dual to the Serre diagonal, lifting the graded ring structure in cohomology. In this work we introduce through explicit combinatorial methods an extension of this product to a full $E_infty$-
We use Segal-Mitchisons cohomology of topological groups to define a convenient model for topological gerbes. We introduce multiplicative gerbes over topological groups in this setup and we define its representations. For a specific choice of represe
Let $X$ be a topological space with Noetherian mod $p$ cohomology and let $C^*(X;mathbb{F}_p)$ be the commutative ring spectrum of $mathbb{F}_p$-valued cochains on $X$. The goal of this paper is to exhibit conditions under which the category of modul
We generalize Quillens $F$-isomorphism theorem, Quillens stratification theorem, the stable transfer, and the finite generation of cohomology rings from finite groups to homotopical groups. As a consequence, we show that the category of module spectr
We give a new proof of the equivalence between two of the main models for $(infty,n)$-categories, namely the $n$-fold Segal spaces of Barwick and the $Theta_{n}$-spaces of Rezk, by proving that these are algebras for the same monad on the $infty$-cat