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Some memos on Stable Symplectic Structured Space II: Symplectic motives

Eita Haibara

TL;DR

This work develops a framework for symplectic motives over $\mathcal{G}_{\mathbb{S}}^{alg}$-schemes by integrating scheme theory, relative symplectic K-theory, and motivic homotopy concepts. It defines symplectic objects in stable categories, constructs a category of symplectic motives using global Thom spectra, and establishes a symplectic Nisnevich site to support descent and gluing. A stabilized, weight-graded category $\mathrm{Mot}_{\mathrm{symp}}^{\mathrm{stab}}(\mathcal{X})$ is introduced along with a bigraded suspension framework to define symplectic motivic cohomology via maps to a symplectic motivic sphere. The paper also connects relative symplectic K-theory to mapping spectra in idempotent-complete settings and discusses the topological Dennis trace to THH, illustrating with CP$^1$-style examples. Overall, it lays groundwork for a new direction in symplectic motives and motivates further development in a subsequent full publication.

Abstract

These memos include the research on $\mathcal{G}_{\mathbb{S}}^{alg}$-scheme theory, the definition of symplectic motives over $\mathcal{G}_{\mathbb{S}}^{alg}$-schemes and symplectic motivic cohomology. This presents a new research direction.

Some memos on Stable Symplectic Structured Space II: Symplectic motives

TL;DR

This work develops a framework for symplectic motives over -schemes by integrating scheme theory, relative symplectic K-theory, and motivic homotopy concepts. It defines symplectic objects in stable categories, constructs a category of symplectic motives using global Thom spectra, and establishes a symplectic Nisnevich site to support descent and gluing. A stabilized, weight-graded category is introduced along with a bigraded suspension framework to define symplectic motivic cohomology via maps to a symplectic motivic sphere. The paper also connects relative symplectic K-theory to mapping spectra in idempotent-complete settings and discusses the topological Dennis trace to THH, illustrating with CP-style examples. Overall, it lays groundwork for a new direction in symplectic motives and motivates further development in a subsequent full publication.

Abstract

These memos include the research on -scheme theory, the definition of symplectic motives over -schemes and symplectic motivic cohomology. This presents a new research direction.
Paper Structure (10 sections, 5 theorems, 24 equations)

This paper contains 10 sections, 5 theorems, 24 equations.

Table of Contents

  1. Scheme theory
  2. Scheme theory
  3. Morphisms of schemes
  4. Symplectic K-Theory
  5. Relative Symplectic K-Theory
  6. Applying the Topological Dennis Trace
  7. Symplectic motives over $\mathop{\mathrm{\mathcal{G}_{\mathop{\mathrm{\mathbb{S}}}\nolimits} ^{alg}}}\nolimits$-schemes
  8. Category of motives
  9. Symplectic Nisnevich topology
  10. Symplectic motivic cohomology

Key Result

Proposition 1.2.2

Since $\beta_{(M,A)}$ is an effective epimorphism, $\mathop{\mathrm{\mathcal{O}_{\mathop{\mathrm{\mathcal{X}}}\nolimits}}}\nolimits(M,A)$ is the cofiber of $\mathcal{I}_{(M,A)} \to f^{*}\mathcal{O}_{\mathcal{Y}}(M,A)$. If $\mathcal{O}_{\mathcal{Y}}(M,A)$ and $\mathop{\mathrm{\mathcal{O}_{\mathop{\ma

Theorems & Definitions (29)

  • Definition 1.1.1
  • Definition 1.1.2
  • Example 1.1.3
  • Definition 1.2.1
  • Proposition 1.2.2
  • Definition 1.2.3
  • Proposition 1.2.4
  • Definition 2.1.1
  • Proposition 2.1.2
  • Proposition 2.1.3
  • ...and 19 more