Liftable derived equivalences and objective categories
| Autor: | Xiao-Wu Chen, Xiaofa Chen |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2019 |
| Předmět: |
Derived category
Pure mathematics Functor General Mathematics 010102 general mathematics Mathematics - Rings and Algebras 01 natural sciences Coherent sheaf Mathematics - Algebraic Geometry Rings and Algebras (math.RA) Tensor (intrinsic definition) Scheme (mathematics) Mathematics::Category Theory FOS: Mathematics Isomorphism Abelian category 0101 mathematics Representation Theory (math.RT) Equivalence (measure theory) Algebraic Geometry (math.AG) Mathematics - Representation Theory Mathematics |
| Popis: | We give two proofs to the following theorem and its generalization: if a finite dimensional algebra $A$ is derived equivalent to a smooth projective scheme, then any derived equivalence between $A$ and another algebra $B$ is standard, that is, isomorphic to the derived tensor functor by a two-sided tilting complex. The main ingredients of the proofs are as follows: (1) between the derived categories of two module categories, liftable functors coincide with standard functors; (2) any derived equivalence between a module category and an abelian category is uniquely factorized as the composition of a pseudo-identity and a liftable derived equivalence; (3) the derived category of coherent sheaves on a certain projective scheme is triangle-objective, that is, any triangle autoequivalence on it, which preserves the the isomorphism classes of complexes, is necessarily isomorphic to the identity functor. |
| Databáze: | OpenAIRE |
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