arXiv subjects

Changhan Zou

Publications and source records attributed to Changhan Zou.

5 recordsLinked to original sources

arXiv2026

Local Bousfield classes via homological support

Given an object $A$ in a big tensor-triangulated category, we study the homological and cohomological Bousfield classes of the associated localization: the tensor-triangulated category of $A$-local objects. We show that the homological support classifies the homological Bousfield classes of the $A$-local category precisely when an $A$-relative form of the homological detection property holds. Moreover, we prove that this holds if and only if $A$ is Bousfield equivalent to a coproduct of homological residue fields. The analogous classification of cohomological Bousfield classes by homological cosupport is strictly stronger: it is equivalent to an $A$-relative form of homological stratification. This equivalence between stratification and the classification of cohomological Bousfield classes is new even in the absolute case. A further surprise is that stratification is also equivalent to the classification of homological Bousfield classes together with the statement that every cohomological Bousfield class is homological. Applied to chromatic homotopy theory, these results classify the homological Bousfield classes of any localization of spectra with respect to a coproduct of Morava $K$-theories. This covers many localizations of interest. We also completely characterize when such chromatic localizations are relatively homologically stratified. This yields new examples of cohomological Bousfield classes that are not homological. In particular, it answers a question of Wolcott concerning the category of harmonic spectra. Our examples are produced by exhibiting local spectra with empty homological cosupport.

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arXiv2026

Smashing, Balmer, Zariski spectra: an ideal approach

We introduce the Zariski frame of any presentably symmetric monoidal $\infty$-category. This allows us to unify several spectral theories arising in higher algebra. The Zariski frame is coherent whenever the category is compactly generated, and the associated spectral space recovers both the classical Zariski spectrum of a commutative ring and the Hochster dual of the Balmer spectrum of a commutative $2$-ring. Moreover, the smashing frame of any stable presentably symmetric monoidal $\infty$-category can be identified with the Zariski frame of its category of dualizable modules. This construction is based on the principle that ideals in a symmetric monoidal $\infty$-category should be understood as monomorphisms into the unit object. In suitable contexts, this notion recovers the kinds of ideals appearing in the preceding examples, including thick ideals and smashing ideals, and it also accommodates the smashing ideals of non-stable $\infty$-categories. We also study the problem of forming quotients by ideals, which is subtle in the setting of higher algebra. To address this, we introduce two properties of pointed $\infty$-categories, called $\Sigma$-triviality and $\Sigma$-exactness. These conditions ensure that quotienting by ideals behaves well. As an application, we construct quotients of $\mathbb{E}_\infty$-semirings.

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arXiv2025

Convexity in tensor triangular geometry

We classify the dualizable localizing ideals of rigidly-compactly generated tt-$\infty$-categories that are cohomologically stratified. By definition, these are the localizing ideals that are dualizable with respect to the Lurie tensor product. We prove that these ideals correspond to the convex subsets of the Balmer spectrum. More generally, we establish this classification for categories which are locally cohomologically stratified and whose Balmer spectrum is noetherian. The classification thus applies to many categories arising in algebra and topology, including derived categories of noetherian schemes. Our result generalizes, and is motivated by, a recent theorem of Efimov which establishes this classification for derived categories of commutative noetherian rings.

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arXiv2024

Homological stratification and descent

We introduce a notion of stratification for rigidly-compactly generated tensor-triangulated categories relative to the homological spectrum and develop the fundamental features of this theory. In particular, we demonstrate that it exhibits excellent descent properties. In conjunction with Balmer's Nerves of Steel conjecture, we conclude that stratification admits a general form of descent. This gives a uniform treatment of several recent stratification results and provides a complete answer to the question: When does stratification descend? As a new application, we extend earlier work on the tensor triangular geometry of equivariant module spectra from finite groups to compact Lie groups.

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arXiv2023

Support theories for non-Noetherian tensor triangulated categories

We extend the support theory of Benson--Iyengar--Krause to the non-Noetherian setting by introducing a new notion of small support for modules. This enables us to prove that the stable module category of a finite group is canonically stratified by the action of the Tate cohomology ring, despite the fact that this ring is rarely Noetherian. In the tensor triangular context, we compare the support theory proposed by W. Sanders (which extends the Balmer--Favi support theory beyond the weakly Noetherian setting) with our generalized BIK support theory. When the Balmer spectrum is homeomorphic to the Zariski spectrum of the endomorphism ring of the unit, the two support theories coincide as do their associated theories of stratification. We also prove a negative result which states that the Balmer--Favi--Sanders support theory can only stratify categories whose spectra are weakly Noetherian. This provides additional justification for the weakly Noetherian hypothesis in the work of Barthel, Heard and B. Sanders. On the other hand, the detection property and the local-to-global principle remain interesting in the general setting.

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