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A Semi-Orthogonal Decomposition Theorem for Weighted Blowups

Oliver Li

TL;DR

The paper addresses deriving a semi-orthogonal decomposition for weighted blowups along Koszul-regular centers on algebraic stacks, extending Orlov's classical result. The authors model the blowup as the Proj of an extended Rees algebra and leverage Koszul-regularity to control pushforwards and cohomology, combining Beilinson-type decompositions with conservative descent to pass from affine to global settings. The main contribution is a $|\mathbf{d}|=\sum d_i$-component decomposition of $\mathrm{D}(\widetilde{X})$ that specializes to known results for root stacks and ordinary blowups, while providing a framework applicable to a broad class of weighted centers. This advances the understanding of derived categories of weighted blowups and has potential applications in resolution of singularities and invariant computations in the stack setting.

Abstract

We establish a semi-orthogonal decomposition for the weighted blowup of an algebraic stack along a Koszul-regular weighted centre, generalising the classic result of Orlov. Our approach is based on the work of Bergh-Schnürer.

A Semi-Orthogonal Decomposition Theorem for Weighted Blowups

TL;DR

The paper addresses deriving a semi-orthogonal decomposition for weighted blowups along Koszul-regular centers on algebraic stacks, extending Orlov's classical result. The authors model the blowup as the Proj of an extended Rees algebra and leverage Koszul-regularity to control pushforwards and cohomology, combining Beilinson-type decompositions with conservative descent to pass from affine to global settings. The main contribution is a -component decomposition of that specializes to known results for root stacks and ordinary blowups, while providing a framework applicable to a broad class of weighted centers. This advances the understanding of derived categories of weighted blowups and has potential applications in resolution of singularities and invariant computations in the stack setting.

Abstract

We establish a semi-orthogonal decomposition for the weighted blowup of an algebraic stack along a Koszul-regular weighted centre, generalising the classic result of Orlov. Our approach is based on the work of Bergh-Schnürer.
Paper Structure (9 sections, 16 theorems, 44 equations)

This paper contains 9 sections, 16 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Related Results
  3. Acknowledgements
  4. Preliminaries
  5. Stacky $\mathop{\mathrm{Proj}}\nolimits$ and Weighted Blowups
  6. Triangulated Categories and Conservative Descent
  7. A Result on Pushforward
  8. Affine Case
  9. Reduction to the Affine Case

Key Result

Theorem 1.1

Fix $\mathbf{d} =(d_0,...,d_n)$ a tuple of positive integers. Let $X$ be a quasi-compact algebraic stack and let $\mathcal{A} = \bigoplus_{d\geq 0}\mathcal{I}_d t^d$ be a Koszul-regular Rees algebra (Definition definition-regularity) on $X$, that smooth-locally is of the form $\sum_{i = 0}^n (f_i, d Then there is a semi-orthogonal decomposition with $|\mathbf{d}| := \sum d_i$ summands: Moreover,

Theorems & Definitions (47)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2: QuekRydh
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.5
  • Definition 2.6
  • Remark 2.7
  • Definition 2.8
  • Example 2.9
  • ...and 37 more