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Orlov's theorem over a quasiexcellent ring

Fei Peng

TL;DR

The paper generalizes Orlov's representability theorem to smooth proper tame algebraic stacks over a quasiexcellent base ring $R$ of finite Krull dimension, under the condition that $X$ has a resolution property and an ample line bundle. It adapts Kawamata and Canonaco–Stellari's approach by developing resolutions of the diagonal, generating vector bundles, and ample sequences to construct a Fourier–Mukai kernel $K$ with $F\cong\Phi_K$, with $K$ unique up to quasi-isomorphism. The method relies on strong generation and saturation results for derived categories of coherent sheaves on tame stacks, as well as a Postnikov-system construction to produce the kernel. The results hold broadly, including new cases over $\mathbb{Z}$, and yield a relative Bondal–Orlov-type reconstruction as a corollary under mild hypotheses. An appendix addresses the resolution property for generically tame stacks, broadening applicability to stacks with quasi-projective coarse moduli spaces.

Abstract

Following the approach of Kawamata and Canonaco-Stellari, we establish Orlov's representability theorem for smooth tame Deligne-Mumford stacks with projective coarse moduli spaces over a quasiexcellent ring of finite Krull dimension. This generalizes a previous result of Canonaco-Stellari for smooth projective varieties over a field.

Orlov's theorem over a quasiexcellent ring

TL;DR

The paper generalizes Orlov's representability theorem to smooth proper tame algebraic stacks over a quasiexcellent base ring of finite Krull dimension, under the condition that has a resolution property and an ample line bundle. It adapts Kawamata and Canonaco–Stellari's approach by developing resolutions of the diagonal, generating vector bundles, and ample sequences to construct a Fourier–Mukai kernel with , with unique up to quasi-isomorphism. The method relies on strong generation and saturation results for derived categories of coherent sheaves on tame stacks, as well as a Postnikov-system construction to produce the kernel. The results hold broadly, including new cases over , and yield a relative Bondal–Orlov-type reconstruction as a corollary under mild hypotheses. An appendix addresses the resolution property for generically tame stacks, broadening applicability to stacks with quasi-projective coarse moduli spaces.

Abstract

Following the approach of Kawamata and Canonaco-Stellari, we establish Orlov's representability theorem for smooth tame Deligne-Mumford stacks with projective coarse moduli spaces over a quasiexcellent ring of finite Krull dimension. This generalizes a previous result of Canonaco-Stellari for smooth projective varieties over a field.

Paper Structure

This paper contains 6 sections, 18 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Resolutions of the diagonal
  3. Generating sheaves and ample sequences
  4. Producing a Fourier--Mukai kernel
  5. Proof of Theorem \ref{['1.1']}
  6. tame stacks and the resolution property

Key Result

Theorem 1.1

Let $R$ be a quasiexcellent ring of finite Krull dimension. Let $\mathcal{X},\mathcal{Y}$ be smooth proper tame algebraic stacks over $R$. Let $\pi\colon\mathcal{X}\to X$ be the associated coarse moduli space. Let $F\colon D_{\operatorname{\operatorname{Coh}}}^{b}(\mathcal{X})\to D_{\operatorname{Co for all $\mathcal{A},\mathcal{B}\in\operatorname{Coh}(\mathcal{X})$ and $i<0$. If $\mathcal{X}$ has

Theorems & Definitions (36)

  • Theorem 1.1
  • Proposition 2.1
  • proof
  • Corollary 2.2
  • Definition 3.1: Huy06
  • Definition 3.2: OS03
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 26 more