UC Santa Cruz

For a rigidly-compactly generated tensor triangulated category, we construct a support theory valued in the homogeneous Zariski spectrum of the graded endomorphism ring of the unit object, extending earlier work of Benson–Iyengar–Krause by removing the Noetherian hypothesis on the endomorphism ring. This allows us to define a non-Noetherian stratification theory. As an example, we show that the stable module category of a finite group is stratified by the canonical action of the Tate cohomology ring, which is typically non-noetherian. We also compare this support theory with the extension of the Balmer–Favi support by Sanders, which takes values in the Balmer spectrum of the category. When the comparison map from the Balmer spectrum to the homogeneous Zariski spectrum of the endomorphism ring is a homeomorphism, the two support theories agree, as do their corresponding stratification theories. As an application of this comparison, we show that the dualizable localizing ideals of a rigidly-compactly generated tt-∞-category are classified by the convex subsets of the Balmer spectrum of the category, assuming that the category is locally cohomologically stratified. This generalizes a recent classification theorem of Efimov.