Support and vanishing for non-Noetherian rings and tensor triangulated categories

We define and characterise small support for complexes over non-Noetherian rings and in this context prove a vanishing theorem for modules. Our definition of support makes sense for any rigidly compactly generated tensor triangulated category. Working in this generality, we establish basic properties of support and investigate when it detects vanishing. We use pointless topology to relate support, the topology of the Balmer spectrum, and the structure of the idempotent Bousfield lattice.