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Local Gorenstein duality in chromatic group cohomology

Luca Pol, Jordan Williamson

TL;DR

The paper develops a chromatic, homotopy-theoretic version of local Gorenstein duality for cochains $C^*(BG;R)$ with coefficients in complex orientable ring spectra $R$. It combines algebraic groundwork on graded Gorenstein rings and dimension theory with topological tools (Venkov’s theorem, Borel spectra, and unitary generation) to prove that LGD holds for a broad class of $R$ (including Lubin–Tate theories, $K$-theory, and tmf variants) and for finite groups $G$, with explicit shift formulas. A key methodological feature is the ascent along finite, relatively Gorenstein maps and a descent principle that transfers LGD along descendable maps, enabling LGD for KO, ku, and related spectra. The results yield concrete shift computations for many examples and establish a unified framework connecting algebraic Gorenstein phenomena with spectral dualities in cochain theories, with implications for chromatic phenomena and modular representation-theoretic dualities in topology.

Abstract

We consider local Gorenstein duality for cochain spectra $C^*(BG;R)$ on the classifying spaces of compact Lie groups $G$ over complex orientable ring spectra $R$. We show that it holds systematically for a large array of examples of ring spectra $R$, including Lubin-Tate theories, topological $K$-theory, and various forms of topological modular forms. We also prove a descent result for local Gorenstein duality which allows us to access further examples.

Local Gorenstein duality in chromatic group cohomology

TL;DR

The paper develops a chromatic, homotopy-theoretic version of local Gorenstein duality for cochains with coefficients in complex orientable ring spectra . It combines algebraic groundwork on graded Gorenstein rings and dimension theory with topological tools (Venkov’s theorem, Borel spectra, and unitary generation) to prove that LGD holds for a broad class of (including Lubin–Tate theories, -theory, and tmf variants) and for finite groups , with explicit shift formulas. A key methodological feature is the ascent along finite, relatively Gorenstein maps and a descent principle that transfers LGD along descendable maps, enabling LGD for KO, ku, and related spectra. The results yield concrete shift computations for many examples and establish a unified framework connecting algebraic Gorenstein phenomena with spectral dualities in cochain theories, with implications for chromatic phenomena and modular representation-theoretic dualities in topology.

Abstract

We consider local Gorenstein duality for cochain spectra on the classifying spaces of compact Lie groups over complex orientable ring spectra . We show that it holds systematically for a large array of examples of ring spectra , including Lubin-Tate theories, topological -theory, and various forms of topological modular forms. We also prove a descent result for local Gorenstein duality which allows us to access further examples.

Paper Structure

This paper contains 19 sections, 47 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Local Gorenstein duality in algebra
  3. Equicodimensionality and the dimension formula
  4. Graded Gorenstein rings
  5. Arithmetic of shifts
  6. Local Gorenstein duality for ring spectra
  7. Local cohomology and localization
  8. Local Gorenstein duality and consequences
  9. Proving local Gorenstein duality
  10. Venkov's theorem
  11. Borel equivariant spectra
  12. Borel spectra and cochains
  13. Orientability
  14. Generating modules over Borel spectra
  15. Local Gorenstein duality for cochains
  16. ...and 4 more sections

Key Result

Theorem 1

Let $G$ be a finite group and $R$ be a commutative ring spectrum with $\pi_*R$ even, noetherian, and of finite global dimension. Then the cochain spectrum $C^*(BG;R)$ has local Gorenstein duality and the shift function on maximal ideals $\mathfrak{m}$ of $R^*(BG)$ takes the form where $\nu$ denotes the Gorenstein shift function of the graded ring $\pi_*R$.

Theorems & Definitions (119)

  • Theorem
  • Theorem
  • Definition 2.1
  • Example 2.2
  • Lemma 2.3: BrunsHerzog
  • Remark 2.4
  • Proposition 2.5
  • proof
  • Lemma 2.6
  • proof
  • ...and 109 more