Spectra, Growth, and Localization in Half-Bounded Tensor Triangular Geometry

Source abstract

Over a noetherian scheme, classical cohomological support determines thick tensor ideals of perfect complexes. In half-bounded derived categories, complexes with the same support can generate different radical thick tensor ideals. We study how these distinctions affect the Balmer spectrum and localization. For a commutative noetherian ring $R$, we prove that $\operatorname{Spc}(\mathbf D^-_{\mathrm{fg}}(R))$ is noetherian if and only if $R$ is artinian. For a nonempty noetherian scheme $X$, the corresponding condition is $\dim X=0$. For such a ring $R$ and a finite group $G$, $\operatorname{Spc}(\mathbf D^-_{\mathrm{fg}}(RG))$ is noetherian if and only if $R$ is artinian and $|G|$ is invertible in $R$. Using the Sanders--Zhang classification over a discrete valuation ring, we compare objects whose annihilator exponents grow at different rates in different degrees. We lift the relevant generation relations before changing coefficients and obtain spectral retractions. These comparisons persist after thick quotients and algebraic localization under explicit uniform annihilation and nonvanishing conditions. A second argument uses nonnilpotent classes of positive degree and passes from cyclic subgroups to finite groups. Applications include relative results for schemes and bounded-below spectra with degreewise finitely generated homotopy. We also describe the cohomological information shared by the resulting ideals. The localization comparisons for bounded-below complexes of finitely generated modules and for bounded-below derived objects with finitely generated cohomology are established separately.