Homological invariants in the representation theory of finite dimensional algebras

A lattice-valued version of complexity is introduced for finitely generated representations of finite dimensional algebras which measures the dimension growth of their syzygies. The spectrum of complexities over a finite dimensional algebra is shown to be invariant under derived equivalences in many instances. Directed graphs called "syzygy quivers" are used to explicitly compute complexity for representations of monomial algebras and to confirm that the spectrum of complexities of a monomial algebra is a derived invariant. "Relative complexity spectra" are used to show that this complexity spectrum is a derived invariant for any finite dimensional algebra which satisfies certain "translation invariance" properties. Examples of such algebras include Gorenstein algebras, local algebras, and commutative algebras; no example is known of an algebra which lacks this property.