Local Gorenstein duality

Abstract: There are several equivalent characterizatons of local commutative Gorenstein rings, which, in stable homotopy theory, have inspired the notions of Gorenstein ring spectrum (Dwyer–Greenlees–Iyengar [4]) and Gorenstein duality [6]. We investigate when a commutative ring spectrum $R$ satisfies a homotopical version of local Gorenstein duality introduced by Barthel–Heard–Valenzuela [2], extending the notion previously studied by Greenlees, and which has structural implications on the homotopy groups of the ring spectrum.

In order to do this, we prove an ascent theorem for local Gorenstein duality along morphisms of $k$-algebras [1]. 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 [5] and $p$-local finite groups [3] as well as for $k = \mathbb Q$ and $X$ a simply connected space which is Gorenstein in the sense of Dwyer, Greenlees and Iyengar.

References:

[1] Tobias Barthel, Natàlia Castellana, Drew Heard, and Gabriel Valenzuela, J. Pure Appl. Algebra 225(2) (2021).

[2] Tobias Barthel, Drew Heard, and Gabriel Valenzuela, Local duality for structured ring spectra, J. Pure Appl. Algebra 222(2) (2018), 433–463.

[3] Carles Broto, Ran Levi, and Bob Oliver, The homotopy theory of fusion systems, J. Amer. Math. Soc. 16(4) (2003), 779–856.

[4] William Dwyer, John Greenlees, and Srikanth Iyengar, Duality in algebra and topology, Adv. Math. 200(2) (2006), 357–402.

[5] William Dwyer and Clarence Wilkerson, Homotopy fixed-point methods for Lie groups and finite loop spaces, Ann. of Math. (2) 139(2) (1994), 395–442.

[6] John Greenlees, Homotopy invariant commutative algebra over fields, In: Building Bridges between Algebra and Topology, Advanced Courses in Math. CRM Barcelona, pages 1031–69, Birkhäuser/Springer, Cham, 2018.

Mathematics

Audience: researchers in the topic

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Opening Workshop (IRP Higher Homotopy Structures 2021, CRM-Bellaterra)

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