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A remark on Continuous K-theory and Fourier-Sato transform

Bingyu Zhang

TL;DR

The paper addresses the computation of universal localizing invariants for categories of sheaves with microsupport constraints by harnessing the Fourier-Sato-Tamarkin transform to relate constrained microlocal categories to unconstrained ones. It provides a conic generalization of Efimov's no-support theorem and shows that the invariant of Sh_{V×X}(V; C) is governed by compactly supported cohomology Gamma_c(X; U_loc(C)), with a vanishing result for nonzero convex cones. An explicit application to the Novikov toric scheme identifies almost QCoh with a constrained sheaf category and gives a concrete formula for its invariant in terms of Gamma_c over the fan. The work also develops C-valued microlocal sheaf theory and demonstrates how these tools yield new computational access to noncommutative motives and related analytic-geometric situations, paving the way for further microlocal invariants with general coefficients.

Abstract

In this note, we prove a generalization of Efimov's computation for the universal localizing invariant of categories of sheaves with certain microsupport constraints. The proof is based on certain categorical equivalences given by the Fourier-Sato transform, which is different from the original proof. As an application, we compute the universal localizing invariant of the category of almost quasi-coherent sheaves on the Novikov toric scheme introduced by Vaintrob.

A remark on Continuous K-theory and Fourier-Sato transform

TL;DR

The paper addresses the computation of universal localizing invariants for categories of sheaves with microsupport constraints by harnessing the Fourier-Sato-Tamarkin transform to relate constrained microlocal categories to unconstrained ones. It provides a conic generalization of Efimov's no-support theorem and shows that the invariant of Sh_{V×X}(V; C) is governed by compactly supported cohomology Gamma_c(X; U_loc(C)), with a vanishing result for nonzero convex cones. An explicit application to the Novikov toric scheme identifies almost QCoh with a constrained sheaf category and gives a concrete formula for its invariant in terms of Gamma_c over the fan. The work also develops C-valued microlocal sheaf theory and demonstrates how these tools yield new computational access to noncommutative motives and related analytic-geometric situations, paving the way for further microlocal invariants with general coefficients.

Abstract

In this note, we prove a generalization of Efimov's computation for the universal localizing invariant of categories of sheaves with certain microsupport constraints. The proof is based on certain categorical equivalences given by the Fourier-Sato transform, which is different from the original proof. As an application, we compute the universal localizing invariant of the category of almost quasi-coherent sheaves on the Novikov toric scheme introduced by Vaintrob.

Paper Structure

This paper contains 5 sections, 18 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. Sheaves and microsupport
  3. Fourier-Sato-Tamarkin transform
  4. Localizing invariants
  5. Further questions

Key Result

Theorem 1.1

Let $X$ be a locally compact Hausdorff space and $\underline{\mathcal{C}}$ be a presheaf on $X$ with values in ${\operatorname{Cat^{dual}_{st}}}$. Then the category ${\operatorname{Sh}}(X;\underline{\mathcal{C}})$ is dualizable stable and we have the following natural isomorphism in ${\operatorname{

Theorems & Definitions (37)

  • Theorem 1.1: Efimov-K-theory
  • Theorem 1.2: Efimov-K-theory
  • Theorem 1.3: \ref{['Theorem: main thm']} (2-b) below
  • Remark 1
  • Remark 2
  • Theorem 1.4: \ref{['Theorem: main thm']} (1) below
  • Remark 3
  • Theorem 1.5
  • Corollary 1
  • Remark 4
  • ...and 27 more