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Greg Stevenson

Publications and source records attributed to Greg Stevenson.

At least 19 recordsLinked to original sources
arXiv2026

The importance of being isolated

We give both a sufficient condition for and an obstruction to the derived category of a commutative ring being generated by its residue fields. As an illustration, we exhibit a ring for which Foxby's small support classifies localizing subcategories despite the failure of the local-to-global principle. We also conclude that the residue fields do not generate for a polynomial ring in infinitely many variables. Finally, we give a necessary and sufficient criterion for the derived category to satisfy the local-to-global principle; it turns out to depend solely on the topology of the spectrum.

math.AC↗
arXiv2026

Ore's theorem for thick subcategories

We characterize those finite groups for which the bounded derived category of finite dimensional representations over an algebraically closed field of characteristic $p$ has distributive lattice of thick subcategories: they are precisely the $p$-nilpotent groups. Along the way we give necessary and sufficient criteria for the bounded derived category and perfect complexes of a finite dimensional $k$-algebra to have distributive lattices of thick subcategories.

math.RT↗
arXiv2026

Distributivity, affineness, and the structure sheaf

We observe that for a quasi-compact and quasi-separated scheme the structure sheaf generates the perfect complexes if and only if the lattice of thick subcategories is distributive if and only if the affinization map is 0-affine. Examples are discussed, including an example of a quasi-projective scheme which is not quasi-affine but for which these equivalent conditions hold.

math.AG↗
arXiv2026

Some notes on tensor triangular geometry

These are notes from the lectures I gave at the Oberwolfach seminar `Tensor Triangular Geometry and Interactions' which was held in October 2025. The aim of these notes is to give an introduction to tensor triangular geometry, for both small and large categories, through the lens of lattice theory. We do not try to be exhaustive and this is reflected in both the content and the bibliography. For instance we are quite light on triangulated preliminaries, especially for compactly generated categories. The first three sections treat the essentially small case and conclude with a tensor triangular proof of Thomason's theorem computing the spectrum of the perfect complexes on a quasi-compact and quasi-separated scheme. The last section treats the compactly generated case. This final section is somewhat experimental and contains some new thoughts.

math.CT↗
arXiv2026

Geometric purity and the frame of smashing ideals

We introduce the notion of geometric purity in rigidly-compactly generated tt-categories by considering exact triangles that are pure at each tt-stalk. We develop a systematic study of this concept, including examples and applications. In particular, we show that geometric purity is, in general, strictly stronger than ordinary purity, and that it naturally leads to the notion of geometrically pure-injective objects. We prove that such objects arise as pushforwards of pure-injective objects from suitable tt-stalks. Moreover, we give a detailed analysis of indecomposable geometrically pure-injective objects in the derived category of the projective line. Under mild additional assumptions, we identify the geometric part of the Ziegler spectrum as a closed subset. As an application, we demonstrate that this new notion of purity can be used to tackle the problem of spatiality of the frame of smashing ideals via the geometric Ziegler spectrum. In particular, we show that our approach rules out the counterexamples of Balchin and Stevenson to existing methods.

math.RT↗
arXiv2025

Higher Zariski Geometry

We revisit the classical constructions of tensor-triangular geometry in the setting of stably symmetric monoidal idempotent-complete $\infty$-categories, henceforth referred to as 2-rings. In this setting, we produce a Zariski topology, a Zariski spectrum, a category of locally 2-ringed spaces (more generally $\infty$-topoi), and an affine spectrum-global sections adjunction, based on the framework of ``$\infty$-topoi with geometric structure'' as developed by Lurie in \cite{LurieDAG5}. Using work of Kock and Pitsch, we compute that the underlying space of the Zariski spectrum of a 2-ring recovers the Balmer spectrum of its homotopy category. These constructions mirror the analogous structures in the classical Zariski geometry of commutative rings (and commutative ring spectra), and we also demonstrate additional compatibility between classical Zariski and higher Zariski geometry. For rigid 2-rings, we show that the descent results of Balmer and Favi admit coherent enhancements. As a corollary, we obtain that the Zariski spectrum fully faithfully embeds rigid 2-rings into locally 2-ringed $\infty$-topoi. In an appendix, we prove a ``stalk-locality principle'' for the telescope conjecture in the rigid setting, extending earlier work of Hrbek.

math.AG↗
arXiv2024

Tilting in $Q$-shaped derived categories

The main result of this paper is that there is sometimes a triangulated equivalence between $D_Q( A )$, the $Q$-shaped derived category of an algebra $A$, and $D( B )$, the classic derived category of a different algebra $B$. By construction, $D_Q( A )$ consists of $Q$-shaped diagrams of $A$-modules for a suitable small category $Q$. Our result concerns the case where $Q$ consists of shifts of indecomposable projective modules over a self-injective $\mathbb{Z}$-graded algebra $\Lambda$. A notable special case is the result by Iyama, Kato, and Miyachi that $D_N( A )$, the $N$-derived category of $A$, is triangulated equivalent to $D( T_{ N-1 }A )$, the classic derived category of $T_{ N-1 }( A )$, which denotes upper diagonal $( N-1 ) \times ( N-1 )$-matrices over $A$. Several other special cases will also be discussed.

math.RT↗
arXiv2024

Approximable Triangulated Categories and Reflexive DG-categories

We use the theory of approximable triangulated categories to give a condition for a proper DG-category to be reflexive in the sense of Kuznetsov and Shinder. To do this we provide another description of the completion of an approximable triangulated category under a properness assumption. We apply our results to proper schemes, proper connective DG-algebras and Azumaya algebras over proper schemes. We include an appendix by Raedschelders and Stevenson showing that proper connective DG-algebras admit finite dimensional models over any field.

math.AG↗
arXiv2024

Proxy-small objects present compactly generated categories

We develop a correspondence between presentations of compactly generated triangulated categories as localizations of derived categories of ring spectra and proxy-small objects, and explore some consequences. In addition, we give a characterization of proxy-smallness in terms of coproduct preservation of the associated corepresentable functor `up to base change'.

math.CT↗
arXiv2022

Approximating triangulated categories by spaces

We initiate a systematic study of lattices of thick subcategories for arbitrary essentially small triangulated categories. To this end we give several examples illustrating the various properties these lattices may, or may not, have and show that as soon as a lattice of thick subcategories is distributive it is automatically a spatial frame. We then construct two non-commutative spectra, one functorial and one with restricted functoriality, that give universal approximations of these lattices by spaces.

math.CT↗
arXiv2021

Big categories, big spectra

We introduce a new topological invariant of a rigidly-compactly generated tensor-triangulated category and two new notions of support. The first is based on smashing subcategories: it is unknown whether the frame of smashing subcategories is spatial in general, but supposing it is as in the examples we understand, we show it is equipped with a morphism to the Balmer spectrum which detects the failure of the telescope conjecture and we develop the corresponding support theory. The new invariant, the big spectrum, results from taking the entire collection of localizing ideals seriously and considering prime localizing ideals. Although there are, in principle, a proper class of localizing ideals, we are able to prove the existence of at least one big prime lying over every Balmer prime. We conclude with a pair of examples illustrating our constructions.

math.CT↗
arXiv2021

Homological residue fields as comodules over coalgebras

We explicitly present homological residue fields for tensor triangulated categories as categories of comodules in a number of examples across algebra, geometry, and topology. Our results indicate that, despite their abstract nature, they are very natural objects and encode tangent data at the corresponding point on the spectrum.

math.CT↗
arXiv2020

Relative stable categories and birationality

We propose a general method to construct new triangulated categories, relative stable categories, as additive quotients of a given one. This construction enhances results of Beligiannis, particularly in the tensor-triangular setting. We prove a birationality result showing that the original category and its relative stable quotient are equivalent on some open piece of their spectrum.

math.CT↗
arXiv2019

Proper connective differential graded algebras and their geometric realizations

We prove that every proper connective DG-algebra $A$ admits a geometric realization (as defined by Orlov) by a smooth projective scheme with a full exceptional collection. As a corollary we obtain that $A$ is quasi-isomorphic to a finite dimensional DG-algebra and in the smooth case we compute the noncommutative Chow motive of $A$. We go on to analyse the relationship between smoothness and regularity in more detail as well as commenting on smoothness of the degree zero cohomology for smooth proper connective DG-algebras.

math.KT↗
arXiv2018

Homotopy invariants of singularity categories

We present a method for computing $\mathbb{A}^1$-homotopy invariants of singularity categories of rings admitting suitable gradings. Using this we describe any such invariant, e.g. homotopy K-theory, for the stable categories of self-injective algebras admitting a connected grading. A remark is also made concerning the vanishing of all such invariants for cluster categories of type $A_{2n}$ quivers.

math.KT↗
arXiv2017

The derived category of the projective line

We examine the localizing subcategories of the derived category of quasi-coherent sheaves on the projective line over a field. We provide a complete classification of all such subcategories which arise as the kernel of a cohomological functor to a Grothendieck category.

math.CT↗