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Relative Monoidal Bondal-Orlov

Artan Sheshmani, Angel Toledo

Abstract

In this article we study a relative monoidal version of the Bondal-Orlov reconstruction theorem. We establish an uniqueness result for tensor triangulated category structures $(\boxtimes,\mathbb{1})$ on the derived category $D^{b}(X)$ of a variety $X$ which is smooth projective and faithfully flat over a quasi-compact quasi-separated base scheme $S$ in the case where the fibers $X_{s}$ over any point $s\in S$ all have ample (anti-)canonical bundles. To do so we construct a stack $Γ$ of dg-bifunctors which parametrize the local homotopical behaviour of $\boxtimes$, and we study some of its properties around the derived categories of the fibers $X_{s}$.

Relative Monoidal Bondal-Orlov

Abstract

In this article we study a relative monoidal version of the Bondal-Orlov reconstruction theorem. We establish an uniqueness result for tensor triangulated category structures on the derived category of a variety which is smooth projective and faithfully flat over a quasi-compact quasi-separated base scheme in the case where the fibers over any point all have ample (anti-)canonical bundles. To do so we construct a stack of dg-bifunctors which parametrize the local homotopical behaviour of , and we study some of its properties around the derived categories of the fibers .

Paper Structure

This paper contains 9 sections, 39 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on tensor triangulated categories
  3. From X to fibers
  4. Enhancing tensor structures
  5. The theory of dg-enhancements
  6. Lifting tensor products
  7. Descent of dg-categories
  8. Stacks of tensor product structures
  9. Examples

Key Result

Theorem 1.1

Let $S$ be a noetherian Artin stack with affine diagonal and let $X,Y\to S$ be flat, proper and relative algebraic spaces. Assume also that for all $s\in S$ the fibers $X_{s}, Y_{s}$ are projective, connected and Gorenstein and that $X_{s}$ has either ample or anti-ample canonical bundle. Then $X\co

Theorems & Definitions (101)

  • Theorem 1.1: calabrese2018relative, Theorem 6.2
  • Theorem 1.2
  • Corollary 1.3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6
  • Theorem 2.7
  • ...and 91 more