A finite type algebra $A$ has many special features such as its irreducible representations can be parameterized by its primitive ideal spectrum $Prim(A)$. The homology groups $HH_*(A)$ and $HP_*(A)$ describe the topology of the space Prim(A). In this talk I will consider twisted versions of these homologies for a finite type algebra in presence of topology provided by powers of an ideal $I$. An application is the computation of twistd homologies for algebras of regular functions on possibly nonsmooth affine complex algebraic variety. This is joint work with V. Nistor and J. Brodzki.