Realizing modules over the homology of a DGA

Let A be a DGA over a field and X a module over H_*(A). Fix an \(A_\infty\)-structure on H_*(A) making it quasi-isomorphic to A. We construct an equivalence of categories between A_{n+1}-module structures on X and length n Postnikov systems in the derived category of A-modules based on the bar resol...

Full description

Saved in:
Bibliographic Details
Published inarXiv.org
Main Authors Granja, Gustavo, Hollander, Sharon
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 16.08.2007
Subjects
Codes
Equivalence
Homology
Isomorphism
Modules
Online AccessGet full text

Cover

More Information
Summary:Let A be a DGA over a field and X a module over H_*(A). Fix an \(A_\infty\)-structure on H_*(A) making it quasi-isomorphic to A. We construct an equivalence of categories between A_{n+1}-module structures on X and length n Postnikov systems in the derived category of A-modules based on the bar resolution of X. This implies that quasi-isomorphism classes of A_n-structures on X are in bijective correspondence with weak equivalence classes of rigidifications of the first n terms of the bar resolution of X to a complex of A-modules. The above equivalences of categories are compatible for different values of n. This implies that two obstruction theories for realizing X as the homology of an A-module coincide.
ISSN:2331-8422