可对偶范畴的色纯度
Chromatic Purity of Dualizable Categories
- School of Mathematical Sciences, Beijing Normal University（北京师范大学数学科学学院）
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AI总结：
本文发展可对偶稳定范畴的色理论，推广色纯度定理至连续K-理论，构造色断裂方块并建立色下降，从而将红移界、Tate消失和蓝移从谱提升到范畴，并证明核固体与核气体模范畴满足色红移。
AI中文摘要：
我们发展了可对偶稳定范畴的色理论，其核心是连续$K$-理论的色纯度。我们将代数$K$-理论的色纯度定理推广到可对偶稳定范畴的连续$K$-理论，并将其重新表述为$H$-幺环的纯度。利用范畴完备化理论，我们构造了色断裂方块，并获得了色纯度的精细化：对于每个可对偶稳定范畴$\mathcal{C}$，$T(n)\oplus T(n-1)$-完备化映射诱导等价$$ K_{T(n)}^{\mathrm{cont}}(\mathcal{C})\xrightarrow{\simeq}K_{T(n)}^{\mathrm{cont}}(\mathrm{Nuc}_{T(n)\oplus T(n-1)}(\mathcal{C})). $$我们还建立了可对偶同伦不动点的连续$K$-理论的色下降，并证明了下降的范畴核精细化。这一框架使我们能够将红移界、Tate消失和蓝移从谱提升到可对偶稳定范畴。我们证明，刚性对称幺半稳定范畴的$T(n)$-完备化等价于类型$n$广义Moore谱塔上的模范畴的可对偶极限。作为应用，我们证明了核固体和核气体模范畴都满足色红移。
英文摘要：
We develop the chromatic theory of dualizable stable categories, with chromatic purity for continuous $K$-theory at its center. We generalize the chromatic purity theorem for algebraic $K$-theory to continuous $K$-theory of dualizable stable categories and reformulate it as the purity of $H$-unital rings. Using categorical completion theory, we construct chromatic fracture squares and obtain a refinement of chromatic purity: for every dualizable stable category $\mathcal{C}$, the $T(n)\oplus T(n-1)$-completion map induces an equivalence $$ K_{T(n)}^{\mathrm{cont}}(\mathcal{C})\xrightarrow{\simeq}K_{T(n)}^{\mathrm{cont}}(\mathrm{Nuc}_{T(n)\oplus T(n-1)}(\mathcal{C})). $$ We also establish chromatic descent for continuous $K$-theory of dualizable homotopy fixed points and prove a categorical nuclear refinement of descent. This framework allows us to lift redshift bounds, Tate vanishing, and blueshift from spectra to dualizable stable categories. We show that the $T(n)$-completion of a rigid symmetric monoidal stable category is equivalent to the dualizable limit of module categories over a tower of type $n$ generalized Moore spectra. As applications, we prove that both nuclear solid and nuclear gaseous module categories satisfy chromatic redshift.
