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Amnon Neeman

Publications and source records attributed to Amnon Neeman.

At least 19 recordsLinked to original sources
arXiv2026

Metrics on triangulated categories and their enhancements

In this paper we investigate the uniqueness of enhancements of the natural subcategories of weakly approximable triangulated categories. The main idea is to enhance at the level of $\infty$-categories the recently developed theory of excellent metrics. The applications of our results include a vast generalization of the known results about the (strong) uniqueness of enhancements in the linear and nonlinear setting, providing positive answers to some open questions. In addition we prove that, under some natural assumptions, the equivalences between such subcategories can be lifted through their natural inclusions. This completes the picture started in our previous paper arXiv:2505.10374 and extends the known results about Margolis Uniqueness Conjecture for the homotopy category of spectra.

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arXiv2025

Strong uniqueness of enhancements for the dual numbers: a case study

We prove that the bounded and bounded below derived categories of (all) modules over the dual numbers have strongly unique (dg) enhancements. To this end we relate those categories to the category of sequences of vector spaces, which allows a complete classification of indecomposable objects. Along the way we also prove that all the derived categories of any hereditary category have strongly unique enhancements.

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arXiv2025

Excellent metrics on triangulated categories, and the involutivity of the map taking $\mathcal{S}$ to $\mathfrak{S}({\mathcal{S})^{\mathrm{op}}}$

In the article arXiv:1806.06471 we defined good metrics on triangulated categories, and then studied the construction, that began with a triangulated category $\mathcal{S}$ together with a good metric $\{\mathcal{M}_i,\,i\in\mathbb{N}\}$, and out of it cooked up another triangulated category $\mathfrak(\mathcal{S})$. We went on to study examples, and produced many for which the construction is involutive. By this we mean that, if you let $\mathcal{T}=\mathfrak{S}(\mathcal{S})^{\mathrm{op}}$, then there is a choice of metric on $\mathcal{T}$ for which $\mathcal{S}=\mathfrak{S}(\mathcal{T})^{\mathrm{op}}$. In this article we study this phenomenon much more carefully, with the focus being on understanding the metrics for which involutivity occurs. As it turns out there is a large class of them, the excellent metrics on triangulated categories. At the end we will produce a few new examples of excellent metrics. And our reason for going to all this trouble is that the results of this article will permit us to prove new and surprising statements about uniqueness of enhancements. Those results will come in a sequel to this article, which is joint with Canonaco and Stellari.

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arXiv2024

Weakly approximable triangulated categories and enhancements: a survey

This paper surveys some recent results, concerning the intrinsicness of natural subcategories of weakly approximable triangulated categories. We also review the results about uniqueness of enhancements of triangulated categories, with the aim of showing the fruitful interplay. In particular, we show how this leads to a vast generalization of a result by Rickard about derived invariance for schemes and rings.

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arXiv2024

The passage among the subcategories of weakly approximable triangulated categories

In this article we prove that all the inclusions between the 'classical' and naturally defined full triangulated subcategories of a weakly approximable triangulated category are intrinsic (in one case under a technical condition). This extends all the existing results about subcategories of weakly approximable triangulated categories. Together with a forthcoming paper about uniqueness of enhancements, our result allows us to generalize a celebrated theorem by Rickard which asserts that if $R$ and $S$ are left coherent rings, then a derived equivalence of $R$ and $S$ is "independent of the decorations". That is, if $D^?(R\text{-}\square)$ and $D^?(S\text{-}\square)$ are equivalent as triangulated categories for some choice of decorations $?$ and $\square$, then they are equivalent for every choice of decorations. But our theorem is much more general, and applies also to quasi-compact and quasi-separated schemes -- even to the relative version, in which the derived categories consist of complexes with cohomology supported on a given closed subscheme with quasi-compact complement.

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arXiv2022

Finite approximations as a tool for studying triangulated categories

Small, finite entities are easier and simpler to manipulate than gigantic, infinite ones. Consequently huge chunks of mathematics are devoted to methods reducing the study of big, cumbersome objects to an analysis of their finite building blocks. The manifestation of this general pattern, in the study of derived and triangulated categories, dates back almost to the beginnings of the subject -- more precisely to articles by Illusie in SGA6, way back in the early 1970s. What's new, at least new in the world of derived and triangulated categories, is that one gets extra mileage from analysing more carefully and quantifying more precisely just how efficiently one can estimate infinite objects by finite ones. This leads one to the study of metrics on triangulated categories, and of how accurately an object can be approximated by finite objects of bounded size.

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arXiv2022

Obstructions to the existence of Bounded {\it t}--structures

In a striking 2019 article, Antieau, Gepner and Heller found {\it K--}theoretic obstructions to bounded t-structures. We will survey their work, as well as some progress since. The focus will be on the open problems that arise from this.

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arXiv2022

Bounded t-structures on the category of perfect complexes

Let $X$ be a finite-dimensional, noetherian scheme. Antieau, Gepner and Heller conjectured that its derived category of perfect complexes has a bounded t-structure if and only if $X$ is regular. We prove a generalization, and to do so we sharpen some of the techniques so far obtained in the theory of approximable triangulated categories.

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arXiv2021

Uniqueness of enhancements for derived and geometric categories

We prove that the derived categories of abelian categories have unique enhancements -- all of them, the unbounded, bounded, bounded above and bounded below derived categories. The unseparated and left completed derived categories of a Grothendieck abelian category are also shown to have unique enhancements. Finally we show that the derived category of complexes with quasi-coherent cohomology and the category of perfect complexes have unique enhancements for quasi-compact and quasi-separated schemes.

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arXiv2020

A counterexample to vanishing conjectures for negative $K$-theory

In a 2006 article Schlichting conjectured that the negative {\it K--}theory of any abelian category must vanish. This conjecture was generalized in a 2019 article by Antieau, Gepner and Heller, who hypothesized that the negative {\it K--}theory of any category with a bounded {\it t--}structure must vanish. Both conjectures will be shown to be false.

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arXiv2019

Metrics on triangulated categories

In this survey we explain the results of the recent article arXiv:1806.06471. Following a 1973 article by Lawvere one can define metrics on categories, and following Kelly's 1982 book one can complete a category with respect to its metric. We specialize these general constructions to triangulated categories, and restrict our attention to "good metrics". And the remarkable new theorem is that, when we start with a triangulated category $\mathcal S$ with a good metric, its completion $\mathfrak{L}(\mathcal{S})$ contains an interesting subcategory $\mathfrak{S}(\mathcal{S})$ which is always triangulated. As special cases we obtain $\mathcal{H}^0(\mathrm{Perf}(X))$ and $D^b_{\mathrm{coh}}(X)$ from each other. We also give a couple of other examples.

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arXiv2018

The t-structures generated by objects

Let $\mathcal T$ be a well generated triangulated category, and let $S\subset\mathcal T$ be a set of objects. We prove that there is a t-structure on $\mathcal T$ with ${\mathcal T}^{\leq0}=\overline{\langle S\rangle}^{(-\infty,0]}$. This article is an improvement on the main result of a 2003 article by Alonso, Jeremias and Souto---in that article the theorem was proved under the assumption that $\mathcal T$ has a nice enough model. It should be mentioned that the theorem of Alonso, Jeremias and Souto has been influential---it turns out to be interesting to study all of these t-structures.

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arXiv2018

Approximable triangulated categories

In this survey we present the relatively new concept of \emph{approximable triangulated categories.} We will show that the definition is natural, that it leads to powerful new results, and that it throws new light on old, familiar objects. In particular: a recent theorem says that the category $D_{\text{qc}}(X)$ is approximable whenever $X$ is a quasicompact separated scheme. As corollaries of this (seemingly technical) statement one can prove striking improvements on old theorems by Bondal, Rickard, Rouquier and Van den Bergh, about the (much smaller) categories $D^{\text{perf}}(X)$ and $D^b_{\text{coh}}(X)$.

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arXiv2018

The categories ${\mathcal T}^c$ and ${\mathcal T}^b_c$ determine each other

Given an essentially small triangulated category it is possible to give a metric on it, to complete it with respect to the metric, and to look at the subcategory of objects in the completion which are compactly supported with respect to the metric. The main theorem says that this procedure produces a new triangulated category. And then we give examples: for example we learn that it is possible, for suitable choices of metrics, to produce the categories $D^b(R-\text{mod})$ and $K^b(R-\text{proj})$ out of each other.

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arXiv2018

The category $\big[{\mathcal T}^c\big]^{\text{op}}$ as functors on ${\mathcal T}^b_c$

We revisit an old assertion due to Rouquier, characterizing the perfect complexes as bounded homological functors on the bounded complexes of coherent sheaves. The new results vastly generalize the old statement---first of all the ground ring is not restricted to be a field, any commutative, noetherian ring will do. But the generalization goes further, to the abstract world of approximable triangulated categories.

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arXiv2018

Gluing Approximable Triangulated Categories

Given a bounded-above cochain complex of modules over a ring, it is standard to replace it by a projective resolution, and it is classical that doing so can be very useful. Recently, a modified version of this was introduced in triangulated categories other than the derived category of a ring. A triangulated category is approximable if this modified procedure is possible. Not surprisingly this has proved a powerful tool. For example: the fact that the derived category of a quasi compact, separated scheme is approximable has led to major improvements on old theorems due to Bondal, Van den Bergh and Rouquier. In this article we prove that, under weak hypotheses, the recollement of two approximable triangulated categories is approximable. In particular, this shows many of the triangulated categories that arise in noncommutative algebraic geometry are approximable. Furthermore, the lemmas and techniques developed in this article form a powerful toolbox which has interesting applications in existing and forthcoming work by the authors.

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arXiv2018

Grothendieck duality made simple

It has long been accepted that the foundations of Grothendieck duality are complicated. This has changed recently. By "Grothendieck duality" we mean what, in the old literature, used to go by the name "coherent duality". This isn't to be confused with what is nowadays called "Verdier duality", and used to pass as "$\ell$-adic duality".

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