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A note on explicit homological invariants of graded double Ore extensions

Andrés Rubiano

Abstract

We compute explicit homological invariants of a trimmed graded double Ore extension of the quantum plane. For a pilot family of type (14641), we determine the minimal graded free resolution and graded Betti numbers of the trivial right module and also compute linear resolutions for two natural cyclic right modules. This provides a concrete link between the PBW structure of the algebra and the homological behavior of its natural quotients.

A note on explicit homological invariants of graded double Ore extensions

Abstract

We compute explicit homological invariants of a trimmed graded double Ore extension of the quantum plane. For a pilot family of type (14641), we determine the minimal graded free resolution and graded Betti numbers of the trivial right module and also compute linear resolutions for two natural cyclic right modules. This provides a concrete link between the PBW structure of the algebra and the homological behavior of its natural quotients.

Paper Structure

This paper contains 13 sections, 9 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Double Ore extensions
  4. The graded setting
  5. Minimal graded free resolutions and Betti numbers
  6. The pilot family and its PBW structure
  7. The family Lg over the quantum plane
  8. PBW basis and normal monomials
  9. Explicit minimal resolutions
  10. The trivial right module
  11. Natural cyclic right modules
  12. Homological consequences
  13. Future work

Key Result

Proposition 2.2

Let $B=R_P[y_1,y_2;\sigma,\delta,\tau]$ be a right double extension of $R$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (21)

  • Definition 2.1: ZZ08; CLM11
  • Proposition 2.2: CLM11
  • Proposition 2.3: ZZ09
  • Definition 2.4: Peeva2011
  • Remark 2.5
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Theorem 4.1
  • ...and 11 more