Another version of cosupport for complexes

The goal of the article is to get a satisfactory theory of cosupport in the derived category $\mathrm{D}(R)$, this is done by introducing another versions of the "big" and "small" cosupport for complexes. We provide some properties for cosupport that are similar--or rather dual--to those of support for complexes. By examples we show that these versions are differ from the cosupport in [J. Reine Angew. Math. 673 (2012) 161--207]. We also study some relations between the "big" and "small" cosupport and give some computations and comparisons of the "small" support and "small" cosupport.