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Almost equivalences between Tamarkin category and Novikov sheaves

Tatsuki Kuwagaki

Abstract

We revisit the relationship between the Tamarkin's extra variable $t$ and Novikov rings. We prove that the equivariant version of Tamarkin category is almost equivalent (in the sense of almost mathematics) to the category of derived complete modules over the Novikov ring.

Almost equivalences between Tamarkin category and Novikov sheaves

Abstract

We revisit the relationship between the Tamarkin's extra variable and Novikov rings. We prove that the equivariant version of Tamarkin category is almost equivalent (in the sense of almost mathematics) to the category of derived complete modules over the Novikov ring.
Paper Structure (33 sections, 24 theorems, 95 equations)

This paper contains 33 sections, 24 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Why is the theorem important?
  4. Symplectic geometry
  5. Algebraic asymptotic analysis
  6. Novikov rings
  7. Definition
  8. Modules over the Novikov ring
  9. Real valuation
  10. Derived complete modules
  11. Derived completeness
  12. Almost modules
  13. Almost isomorphism in $\mathrm{Mod}(R)$
  14. Almost isomorphisms between $\mathcal{C}$-modules
  15. Almost equivalence
  16. ...and 18 more sections

Key Result

Theorem 1.1

We have an almost embedding The almost images are sheaves valued in derived complete modules over $\Lambda_0$.

Theorems & Definitions (75)

  • Theorem 1.1
  • Remark 1.2
  • Definition 2.1
  • Example 2.2: The universal Novikov ring
  • Example 2.3
  • Example 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • Example 2.7
  • ...and 65 more