Torsion models for tensor-triangulated categories

TL;DR

This work develops a torsion model for rigidly-compactly generated tensor-triangulated categories with finite dimensional Noetherian Balmer spectra by building an adelic framework assembled from single-prime pieces. Concretely, objects are reconstructed from pieces Gamma_p L_p X via assembly data and iterated cofibres that glue to form a global diagram, yielding an equivalence with the original category. The construction generalizes the one-dimensional adelic model, connects to Cousin complexes and monochromatic layers in chromatic stable homotopy theory, and provides explicit algebraic/topological examples including derived categories, chromatic theory, and rational torus-equivariant spectra. The framework hinges on cubes and punctured cubes, cofibre functors, and assembly data to organize local data into a global torsion model, offering a versatile approach to categorifying local-to-global reconstruction across contexts.

Abstract

Given a rigidly-compactly generated tensor-triangulated category whose Balmer spectrum is finite dimensional and Noetherian, we construct a torsion model for it, which is equivalent to the original tensor-triangulated category. The torsion model is determined in an adelic fashion by objects with singleton supports. This categorifies the Cousin complex from algebra, and the process of reconstructing a spectrum from its monochromatic layers in chromatic stable homotopy theory. This model is inspired by work of the second author in rational equivariant stable homotopy theory, and extends previous work of the authors from the one-dimensional setting.

Figures (4)

• Figure 1: The poset arising from the Balmer spectrum of $L_{\mathbb{Q}} \mathsf{Sp}_{\mathbb{T}^2}$.
• Figure 2: The assembly data given by applying the function $\mathrm{conn}$ to \ref{['fig:figure1']}.
• Figure 3: The coarsest assembly data for $L_{\mathbb{Q}} \mathsf{Sp}_{\mathbb{T}^2}$.
• Figure 4: The adelic rings associated to the finest and coarsest assembly data on $\mathsf{D}(R)$ for a local Noetherian domain $R$ of Krull dimension $2$. For an $R$-module $M$, the chain complex $M[\mathfrak{m}^{-1}]$ has homology given by the Čech cohomology of punctured affine space, and $\Lambda_{\leqslant 1}R$ can be described as the cofibre of $\operatorname{Hom}_R(R_\mathfrak{g}, R) \to R$ (as an object of $\mathsf{D}(R)$ rather than as a commutative algebra object).

Theorems & Definitions (153)

• Theorem : Informal version
• Theorem : \ref{['torsionmodel']}
• Theorem 2.2
• proof
• Lemma 2.3: BalmerFavi11
• Proposition 2.4
• proof
• Definition 2.6
• Lemma 2.7
• proof