An equivalence between derived categories (typically bounded or unbounded derived categories of abelian or differential graded contexts) that identifies objects and morphisms up to quasi-isomorphism and preserves the triangulated structure, thus matching homological behavior across different settings.

Principle

Derived equivalence asserts that two contexts share the same derived (homological) information: complexes, cohomology, and distinguished triangles correspond under an exact equivalence of triangulated categories (often induced by a tilting object or complex).

Demonstration

Example: two algebras A and B are derived equivalent when there exists a tilting complex T over A whose endomorphism algebra is quasi-isomorphic to B, inducing an equivalence D^b(A-mod) ≅ D^b(B-mod) that preserves Ext-groups and derived invariants.

Misapplication

Confusing derived equivalence with Morita equivalence of module categories (which is stronger) or assuming that an equivalence of abelian categories implies derived equivalence without checking derived-level behavior; ignoring needed dg- or enhancement data for a well-behaved equivalence.

Consequence

Derived equivalence preserves a wide range of homological invariants (Ext-algebras, Hochschild cohomology up to subtle transformations, many numerical invariants) and allows transfer of deformation and obstruction theories between contexts.

Reversal

The reversal contrasts derived equivalence with mere similarity of individual cohomology groups: matching cohomology groups in degrees does not suffice for derived equivalence if the extension and triangulated structures disagree.

Boundary

Derived equivalence is a statement about derived (triangulated or enhanced) categories and requires control of quasi-isomorphisms and enhancements; it excludes raw abelian-level coincidences and may fail when dg-structures or infinity-enhancements differ.

Semantic Tension

Derived equivalence competes with Morita equivalence and with simpler invariants: Morita equivalence implies equivalence of module categories but not always derived equivalence in the same sense, while derived equivalence is weaker than isomorphism of algebras but stronger than matching cohomology groups.

Synthesis

A derived equivalence identifies two mathematical contexts at the homological level: an exact equivalence of derived categories aligns complexes, triangles, and Ext-structures so that homological computations and deformation-theoretic phenomena correspond across the equivalence.