  We introduce notions such as cdp presheaf, cds precosheaf, Mayer-Vietoris
system, and investigate their properties. As applications, we study
cohomologies with values in local systems on smooth manifolds and Dolbeault
cohomologies with values in locally free sheaves on complex manifolds, where
the compactness is not necessary for both cases. In particular, we write out
explicit blow-up formulas for these cohomologies. Moreover, we compare the
blow-up formula given by Rao, S., Yang, S. and Yang, X.-D. with ours, and then
deduce that their formula is still an isomorphism in the noncompact case.
