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Derived equivalences for the derived discrete algebras are standard

Grzegorz Bobinski, Tomasz Ciborski

Abstract

We prove that any derived equivalence between derived discrete algebras is standard, i.e.\ is isomorphic to the derived tensor product by a two-sided tilting complex.

Derived equivalences for the derived discrete algebras are standard

Abstract

We prove that any derived equivalence between derived discrete algebras is standard, i.e.\ is isomorphic to the derived tensor product by a two-sided tilting complex.
Paper Structure (14 sections, 23 theorems, 56 equations)

This paper contains 14 sections, 23 theorems, 56 equations.

Table of Contents

  1. Introduction and the main result
  2. Preliminaries
  3. Objects and morphisms in the homotopy category of $\mathop{\mathrm{proj}}\nolimits \Lambda (n, m)$
  4. Objects
  5. Morphisms
  6. A model of the homotopy category of $\mathop{\mathrm{proj}}\nolimits \Lambda (n, m)$
  7. The category
  8. Irreducible morphisms
  9. Proof of the main result
  10. Construction of the isomorphism
  11. Verification
  12. Connecting isomorphism
  13. Summary
  14. Proof of Theorem \ref{['theo:main_special']}

Key Result

Theorem 1.1

If $A$ and $B$ are derived discrete $\Bbbk$-algebras, then every derived equivalence between $A$ and $B$ is standard.

Theorems & Definitions (38)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Corollary 3.3
  • ...and 28 more