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Atiyah duality for motivic spectra

Toni Annala, Marc Hoyois, Ryomei Iwasa

Abstract

We prove that Atiyah duality holds in the $\infty$-category of non-$\mathbb A^1$-invariant motivic spectra over arbitrary derived schemes: every smooth projective scheme is dualizable with dual given by the Thom spectrum of its negative tangent bundle. The Gysin maps recently constructed by L. Tang are a key ingredient in the proof. We then present several applications. First, we study $\mathbb A^1$-colocalization, which transforms any module over the $\mathbb A^1$-invariant sphere into an $\mathbb A^1$-invariant motivic spectrum without changing its values on smooth projective schemes. This can be applied to all known $p$-adic cohomology theories and gives a new elementary approach to "logarithmic" or "tame" cohomology theories; it recovers for instance the logarithmic crystalline cohomology of strict normal crossings compactifications over perfect fields and shows that the latter is independent of the choice of compactification. Second, we prove a motivic Landweber exact functor theorem, associating a motivic spectrum to any graded formal group law classified by a flat map to the moduli stack of formal groups. Using this theorem, we compute the ring of $\mathbb P^1$-stable cohomology operations on the algebraic K-theory of qcqs derived schemes, and we prove that rational motivic cohomology is an idempotent motivic spectrum.

Atiyah duality for motivic spectra

Abstract

We prove that Atiyah duality holds in the -category of non--invariant motivic spectra over arbitrary derived schemes: every smooth projective scheme is dualizable with dual given by the Thom spectrum of its negative tangent bundle. The Gysin maps recently constructed by L. Tang are a key ingredient in the proof. We then present several applications. First, we study -colocalization, which transforms any module over the -invariant sphere into an -invariant motivic spectrum without changing its values on smooth projective schemes. This can be applied to all known -adic cohomology theories and gives a new elementary approach to "logarithmic" or "tame" cohomology theories; it recovers for instance the logarithmic crystalline cohomology of strict normal crossings compactifications over perfect fields and shows that the latter is independent of the choice of compactification. Second, we prove a motivic Landweber exact functor theorem, associating a motivic spectrum to any graded formal group law classified by a flat map to the moduli stack of formal groups. Using this theorem, we compute the ring of -stable cohomology operations on the algebraic K-theory of qcqs derived schemes, and we prove that rational motivic cohomology is an idempotent motivic spectrum.
Paper Structure (11 sections, 58 theorems, 193 equations)

This paper contains 11 sections, 58 theorems, 193 equations.

Table of Contents

  1. Introduction
  2. Atiyah duality
  3. Background on Gysin maps
  4. The geometric evaluation map
  5. Duals of projective spaces
  6. Ambidexterity and Atiyah duality
  7. Applications
  8. $\mathbb A^1$-colocalization and logarithmic cohomology theories
  9. Algebraic cobordism via Grassmannians
  10. Motivic Landweber exactness
  11. Operations in algebraic K-theory and rational motivic cohomology

Key Result

Theorem 1.1

Let $f\colon X\to S$ be a smooth projective morphism between derived schemes.

Theorems & Definitions (159)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4: Computing the $\mathbb A^1$-colocalization
  • Theorem 1.5: Examples of $\mathbf 1_{\mathbb A^1}$-modules
  • Remark 1.6
  • Theorem 1.7: Comparison with rigid and logarithmic crystalline cohomology, Proposition \ref{['prop:crys']}
  • Theorem 1.8: Homological Conner–Floyd isomorphism, Corollary \ref{['cor:conner-floyd']}
  • Theorem 1.9: Motivic Landweber exact functor theorem, Theorem \ref{['thm:LEFT']}
  • Theorem 1.10
  • ...and 149 more