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Linear strands of multigraded free resolutions

Michael K. Brown, Daniel Erman

Abstract

We develop a notion of linear strands for multigraded free resolutions, and we prove a multigraded generalization of Green's Linear Syzygy Theorem.

Linear strands of multigraded free resolutions

Abstract

We develop a notion of linear strands for multigraded free resolutions, and we prove a multigraded generalization of Green's Linear Syzygy Theorem.
Paper Structure (8 sections, 14 theorems, 33 equations)

This paper contains 8 sections, 14 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. Background on the multigraded BGG correspondence
  3. Computing the unit and counit of the multigraded BGG adjunction
  4. Existence and uniqueness of maximal strongly linear subcomplexes
  5. Proof of Theorem \ref{['maxlinear']}(1)
  6. Proof of Theorem \ref{['maxlinear']}(2)
  7. Linear strands of multigraded free resolutions
  8. A multigraded Linear Syzygy Theorem

Key Result

Theorem 1.3

Let ${\bf k}$ be a field, and suppose $S = {\bf k}[x_0, \dots, x_n]$ is positively graded by an abelian group $A$. Let $M$ be a finitely generated graded $S$-module that is generated in degree $\mathbf{0}$ and $F$ its minimal free resolution.

Theorems & Definitions (46)

  • Definition 1.2
  • Theorem 1.3: see Theorem \ref{['maxlinear']} below
  • Theorem 1.4: Multigraded Linear Syzygy Theorem
  • Corollary 1.5
  • Proposition 2.2: hhw Propositions 3 and 4
  • Lemma 3.1
  • proof
  • Remark 4.1
  • Theorem 4.3
  • Example 4.4
  • ...and 36 more