Obstruction Theory for Objects in Abelian and Derived Categories

ABSTRACT

In this article, we develop the obstruction theory for lifting complexes, up to quasi-isomorphism, to derived categories of flat nilpotent deformations of abelian categories. As a particular case we also obtain the corresponding obstruction theory for lifting of objects in terms of Yoneda Ext-groups. In an appendix we prove the existence of miniversal derived deformations of complexes.

Keywords:

• Abelian categories
• Deformation theory
• Derived categories
• Obstruction theory

Mathematics Subject Classification:

• 13D10
• 18E10
• 18E15
• 18G10
• 18G35

ACKNOWLEDGMENTS

The author wishes to thank Michel Van den Bergh for suggesting the use of injective resolutions and for several interesting discussions.

Notes

^aIn applications R and R ₀ will probably be artinian local rings but the added generality we allow incurs very little cost.

^#Communicated by J. Alev. *Aspirant at the FWO.