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Dimension theory of noncommutative curves

Anirban Bhaduri, Isaac Goldberg, Antonios-Alexandros Robotis

Abstract

We compute several types of dimension for the bounded derived categories of coherent sheaves of orbifold curves. This completes the calculation of these dimensions for derived categories of noncommutative curves in the sense of Reiten-van den Bergh. Along the way we construct stability conditions for orbifold curves. We also obtain a characterisation of orbifold curves with hereditary tilting bundle in terms of diagonal dimension.

Dimension theory of noncommutative curves

Abstract

We compute several types of dimension for the bounded derived categories of coherent sheaves of orbifold curves. This completes the calculation of these dimensions for derived categories of noncommutative curves in the sense of Reiten-van den Bergh. Along the way we construct stability conditions for orbifold curves. We also obtain a characterisation of orbifold curves with hereditary tilting bundle in terms of diagonal dimension.

Paper Structure

This paper contains 12 sections, 37 theorems, 35 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Orbifold curves
  3. Definition and first properties
  4. Homological properties
  5. Stability on orbifold curves
  6. Global Generation
  7. Chen-Ruan cohomology of orbifold curves and orbifold Chern character
  8. Rouquier dimension of orbifold curves
  9. Diagonal dimension of orbifold curves
  10. Serre dimension and global dimension of orbifold curves
  11. Homological Lemma
  12. Stability lemmas

Key Result

Theorem 1.1

ReitenvDB*Thm. V.1.2 Let $\cA$ be a Noetherian, smooth, and connected non-commutative curve, then $\cA$ is equivalent to either

Figures (1)

  • Figure 1: Above are the various dimensions of $\mathrm{D}^{\mathrm{b}}(\cA)$ for Abelian $\cA$. Note that here $Q$ denotes a finite acyclic quiver and $\mathrm{rep}(Q)$ the category of its finite-dimensional representations. $Q_{\rm{ADE}}$ denotes a quiver of ADE (Dynkin) type (besides $A_1$). The boldfaced values are results obtained in the present work, while the others follow from the results cited in the previous sections.

Theorems & Definitions (75)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem A: =\ref{['T:Orlovtheorem']}
  • Theorem B: =\ref{['T:diagDimOrbCurve']}
  • Theorem C: =\ref{['T:stabcond']}, \ref{['C:orbifoldcurvegldim']}
  • Definition 2.1
  • Proposition 2.2: Taams*Thm. 1.1.7
  • Remark 2.3
  • Definition 2.4
  • Lemma 2.5: VoightZB*Lem. 5.4.5
  • ...and 65 more