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Countability of relative Fourier-Mukai partners

Riku Kurama

Abstract

Anel and Toën proved that a smooth projective complex variety has only countably many smooth projective Fourier-Mukai partners up to isomorphism. This is generalized in the Stacks Project to the case where the varieties are smooth proper over an arbitrary algebraically closed field. This note will upgrade the proof of the latter reference to show that a smooth proper scheme over a noetherian base has only countably many relative Fourier-Mukai partners up to isomorphism.

Countability of relative Fourier-Mukai partners

Abstract

Anel and Toën proved that a smooth projective complex variety has only countably many smooth projective Fourier-Mukai partners up to isomorphism. This is generalized in the Stacks Project to the case where the varieties are smooth proper over an arbitrary algebraically closed field. This note will upgrade the proof of the latter reference to show that a smooth proper scheme over a noetherian base has only countably many relative Fourier-Mukai partners up to isomorphism.

Paper Structure

This paper contains 3 sections, 6 theorems, 11 equations.

Table of Contents

  1. Introduction
  2. Approximation and Models
  3. Proof of the theorem

Key Result

Theorem 1.1

Let $X$ be a smooth proper $k$-scheme, where $k$ is an algebraically closed field field. Then, $X$ has at most countably many smooth proper Fourier-Mukai partners up to isomorphism.

Theorems & Definitions (10)

  • Theorem 1.1: Toen,stacks-project
  • Theorem 1.2: Kurama
  • Theorem 1.3
  • Lemma 3.1: see stacks-project
  • proof
  • Lemma 3.2: see stacks-project
  • proof
  • Proposition 3.3: see stacks-project
  • proof
  • proof : Proof of Theorem \ref{['mainTh']}