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Preservation for generation along the structure morphism of coherent algebras over a scheme

Anirban Bhaduri, Souvik Dey, Pat Lank

Abstract

This work demonstrates classical generation is preserved by the derived pushforward along the structure morphism of a noncommutative coherent algebra to its underlying scheme. Additionally, we establish that the Krull dimension of a variety over a field is a lower bound for the Rouquier dimension of the bounded derived category associated with a noncommutative coherent algebra on it. This is an extension of a classical result of Rouquier to the noncommutative context.

Preservation for generation along the structure morphism of coherent algebras over a scheme

Abstract

This work demonstrates classical generation is preserved by the derived pushforward along the structure morphism of a noncommutative coherent algebra to its underlying scheme. Additionally, we establish that the Krull dimension of a variety over a field is a lower bound for the Rouquier dimension of the bounded derived category associated with a noncommutative coherent algebra on it. This is an extension of a classical result of Rouquier to the noncommutative context.
Paper Structure (4 sections, 10 theorems, 9 equations)

This paper contains 4 sections, 10 theorems, 9 equations.

Table of Contents

  1. Introduction
  2. Generation
  3. Noncommutative schemes
  4. Descent along structure morphism

Key Result

Theorem A

(see Theorem thm:descent_for_coherent_algebras) Let $X$ be a Noetherian $J\textrm{-}2$ scheme of finite Krull dimension. Suppose $\mathcal{A}$ is a coherent $\mathcal{O}_X$-algebra with full support and canonical map $\pi \colon \mathcal{O}_X \to \mathcal{A}$. If $G$ is classical generator for $D^b_

Theorems & Definitions (32)

  • Theorem A
  • Corollary B
  • Theorem C
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • Example 2.7
  • Definition 2.8
  • ...and 22 more