  In topological data analysis, two-parameter persistence can be studied using
the representation theory of the 2d commutative grid, the tensor product of two
Dynkin quivers of type A. In a previous work, we defined interval
approximations using restrictions to essential vertices of intervals together
with Mobius inversion. In this work, we consider homological approximations
using interval resolutions, and show that the interval resolution global
dimension is finite for finite posets and that it is equal to the maximum of
the interval dimensions of the Auslander-Reiten translates of the interval
representations. In fact, for the latter equality, we obtained a general
formula in the setting of finite-dimensional algebras and resolutions relative
to a generator-cogenerator. Furthermore, in the commutative ladder case, by a
suitable modification of our interval approximation, we provide a formula
linking the two conceptions of approximation.
