For a finite dimensional algebra A of finite global dimension the bounded derived category of finite dimensional A-modules admits Auslander–Reiten triangles such that the Auslander–Reiten translation τ is an equivalence. On the level of the Grothendieck group τ induces the Coxeter transformation ΦA. More generally this extends to a homologically finite triangulated category T admitting Serre duality. In both cases the Coxeter polynomial, that is, the characteristic polynomial of the Coxeter transformation, yields an important homological invariant of A or T. Spectral analysis is the study of this interplay, it often reveals unexpected links between apparently different subjects.
This paper gives a summary on spectral techniques and studies the links to singularity theory. In particular, it offers a contribution to the categorification of the Milnor lattice through triangulated categories which are naturally attached to a weighted projective line.