  We define support varieties in an axiomatic setting using the prime spectrum
of a lattice of ideals. A key observation is the functoriality of the spectrum
and that this functor admits an adjoint. We assign to each ideal its support
and can classify ideals in terms of their support. Applications arise from
studying abelian or triangulated tensor categories. Specific examples from
algebraic geometry and modular representation theory are discussed,
illustrating the power of this approach which is inspired by recent work of
Balmer.
