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On $τ$-tilting theory

Takahide Adachi, Osamu Iyama, Idun Reiten

Abstract

We give a brief introduction to $τ$-tilting theory [AIR]. In particular, we will see how our theory unifies two different branches of tilting theory, namely, silting theory and cluster tilting theory. We also introduce the history and recent developments.

On $τ$-tilting theory

Abstract

We give a brief introduction to -tilting theory [AIR]. In particular, we will see how our theory unifies two different branches of tilting theory, namely, silting theory and cluster tilting theory. We also introduce the history and recent developments.

Paper Structure

This paper contains 12 sections, 15 theorems, 9 equations.

Table of Contents

  1. History
  2. Tilting theory
  3. Silting theory
  4. Cluster tilting theory
  5. $\tau$-tilting theory
  6. Definition
  7. Unification
  8. Completion and mutation
  9. Application
  10. Further results
  11. The correspondences in module/derived categories
  12. Recent developments

Key Result

Theorem 1.2

R Two rings $A$ and $B$ are derived equivalent if and only if there exists a tilting complex $T$ of $A$ such that the ring $\operatorname{End}\nolimits_{\mathsf{D}(A)}(T)$ is isomorphic to $B$.

Theorems & Definitions (24)

  • Definition 1.1
  • Theorem 1.2
  • Definition 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Definition 1.7
  • Theorem 1.8
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • ...and 14 more