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Orlov's functors in Macaulay2

Michael K. Brown, Souvik Dey, Geoffrey Fatin, Guanyu Li, Mahrud Sayrafi, Tim Tribone

Abstract

Given a commutative and graded Gorenstein ring $R$ with associated projective variety $X$, a theorem of Orlov gives fully faithful embeddings from the graded singularity category of $R$ to the derived category of $X$, or vice versa, depending on the degree of the canonical bundle of $X$. We describe algorithms for computing these embeddings that can be implemented in Macaulay2.

Orlov's functors in Macaulay2

Abstract

Given a commutative and graded Gorenstein ring with associated projective variety , a theorem of Orlov gives fully faithful embeddings from the graded singularity category of to the derived category of , or vice versa, depending on the degree of the canonical bundle of . We describe algorithms for computing these embeddings that can be implemented in Macaulay2.
Paper Structure (5 sections, 5 theorems, 12 equations)

This paper contains 5 sections, 5 theorems, 12 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Implementing Orlov's functors
  4. The functor $\Phi_t$
  5. The functor $\Psi_t$

Key Result

Theorem 1.1

If $a \ge 0$ (resp. $a \le 0$), then for each $t \in \mathbb{Z}$, there is a fully faithful functor $\Phi_t \colon \operatorname{D}_{\operatorname{gr}}^{\operatorname{sg}}(R) \hookrightarrow \operatorname{D}^{\operatorname{b}}(X)$ (resp. $\Psi_t \colon \operatorname{D}^{\operatorname{b}}(X) \hookrig

Theorems & Definitions (13)

  • Theorem 1.1: Orlov2009 Theorem 2.5
  • Theorem 2.1: Orlov2009 Theorem 2.5
  • Lemma 2.2
  • proof
  • Example 2.3
  • Example 2.4: BW24 Example 2.5
  • Example 2.5
  • Example 2.6
  • Proposition 3.1
  • proof
  • ...and 3 more