Local Gorenstein duality for cochains on spaces

Tobias Barthel,Natàlia Castellana,Drew Heard,Gabriel Valenzuela

doi:10.1016/j.jpaa.2020.106495

Abstract

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.

Highlights

We investigate when a commutative ring spectrum R satisfies a homotopical version of local Gorenstein duality, extending the notion previously studied by Greenlees
Our examples will mostly come from ring spectra of the form R = C∗(X; k), the ring spectrum of k-valued cochains on a suitable space X
A consequence of the structural implications mentioned earlier is that H∗(BG; Fp) is Cohen–Macaulay if and only if H∗(BG; Fp) is Gorenstein, a result originally shown by Benson and Carlson [BC94]

Summary

Gorenstein ring spectra

We first review the notions of Gorenstein ring spectrum, relative Gorenstein morphism, and Gorenstein duality, as studied by Dwyer, Greenlees, and Iyengar [DGI06]. We say that a morphism R → k of ring spectra is Gorenstein of shift a if it is proxy-regular and there is an equivalence HomR(k, R) ≃ Σak of k-modules. Using the Gorenstein orientable condition there are equivalences of right E-modules HomR(k, Cellk(R)) ≃ HomR(k, R) ≃ Σa HomR(k, Homk(R, k)). A k-algebra R is said to satisfy Poincare duality of dimension a if there is an equivalence R → Σa Homk(R, k) of R-modules. By [DGI06, Proposition 8.12], R is Gorenstein of shift a and we get orientability by applying HomR(k, −) to the equivalence R ≃ Σa Homk(R, k). We say that R satisfies local Gorenstein duality with shift a if, for each p ∈ Spech(π∗R) of dimension d,4 there is an equivalence ΓpR ≃ Σa+dTR(Ip). Let R be a ring spectrum satisfying local Gorenstein duality of shift a. This is exactly the claim that HomS(R, S) is relatively Gorenstein of shift s − r

Gorenstein ascent

Examples

Local Gorenstein duality for p-local compact groups