ABSTRACT
We define the associated chain complex (Xn, ∂) by setting
Xn = ∑
p+q=n Xp,q, ∂n =
∑
We call ∂ the total boundary operator, and ∂ ′, ∂ ′′ the partial boundary operators.
partial derived functor SupposeF is a functor of n variables. If S is a subset of {1, . . . , n}, we consider the variables whose indices are in S as active and those whose indices are in {1, . . . n}\S as passive. By fixing all the passive variables, we obtain a functor FS in the active variables. The partial derived functors are then defined as the derived functors RkFS . See also functor, derived functor.