Flatness and Shipley’s algebraicization theorem

Abstract

We provide an enhancement of Shipley's algebraicization theorem which behaves better in the context of commutative algebras. This involves defining flat model structures as in Shipley and Pavlov-Scholbach, and showing that the functors still provide Quillen equivalences in this refined context. The use of flat model structures allows one to identify the algebraic counterparts of change of groups functors, as demonstrated in forthcoming work of the author.

Tensor-triangular rigidity in chromatic homotopy theory

“…Since π * (E) = π * (R) which is rational, it follows that E is a commutative HQ-algebra (as HQ is the rational sphere spectrum). By Shipley's algebraicization theorem [28,Theorem 1.2] (see also [31,Theorem 7.2] and [18, §7.1.2]), there are commutative DGAs Θ(E) and Θ(R) such that H * (Θ(E)) = π * (R) = H * (Θ(R)), together with symmetric monoidal equivalences Mod(E) ≃ Mod(Θ(E)) and Mod(R) ≃ Mod(Θ(R)).…”

Adelic models of tensor-triangulated categories

“…We note that Shipley's algebraicization theorem still holds in the flat model structure by [43]. To simplify notation we call the (underlying) category on the right A t (T).…”