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Clausen predicted that Chevalley's id\`{e}le class group of a number field $F$ appears as the first $K$-group of the category of locally compact $F$-vector spaces. This has turned out to be true, and even generalizes to the higher $K$-groups in a suitable sense. We replace $F$ by a semisimple $\mathbb{Q}$-algebra, and obtain Fr\"{o}hlich's non-commutative id\`{e}le class group in an analogous fashion, modulo the reduced norm one elements. Even in the number field case our proof is simpler than the existing one, and based on the localization theorem for percolating subcategories. Finally, using class field theory as input, we interpret Hilbert's reciprocity law (as well as a noncommutative variant) in terms of our results., Comment: 36 pages. Comments welcome

Quasi-abelian categories are abundant in functional analysis and representation theory. It is known that a quasi-abelian category $\mathcal{E}$ is a cotilting torsionfree class of an abelian category. In fact, this property characterizes quasi-abelian categories. This ambient abelian category is derived equivalent to the category $\mathcal{E}$, and can be constructed as the heart $\mathcal{LH}(\mathcal{E})$ of a $\operatorname{t}$-structure on the bounded derived category $\operatorname{D^b}(\mathcal{E})$ or as the localization of the category of monomorphisms in $\mathcal{E}.$ However, there are natural examples of categories in functional analysis which are not quasi-abelian, but merely one-sided quasi-abelian or even weaker. Examples are the category of $\operatorname{LB}$-spaces or the category of complete Hausdorff locally convex spaces. In this paper, we consider additive regular categories as a generalization of quasi-abelian categories that covers the aforementioned examples. Additive regular categories can be characterized as those subcategories of abelian categories which are closed under subobjects. As for quasi-abelian categories, we show that such an ambient abelian category of an additive regular category $\mathcal{E}$ can be found as the heart of a $\operatorname{t}$-structure on the bounded derived category $\operatorname{D^b}(\mathcal{E})$, or as the localization of the category of monomorphisms of $\mathcal{E}$. In our proof of this last construction, we formulate and prove a version of Auslander's formula for additive regular categories. Whereas a quasi-abelian category is an exact category in a natural way, an additive regular category has a natural one-sided exact structure. Such a one-sided exact category can be 2-universally embedded into its exact hull. We show that the exact hull of an additive regular category is again an additive regular category.

The Auslander correspondence is a fundamental result in Auslander-Reiten theory. In this paper we introduce the category $\operatorname{mod_{\mathsf{adm}}}(\mathcal{E})$ of admissibly finitely presented functors and use it to give a version of Auslander correspondence for any exact category $\mathcal{E}$. An important ingredient in the proof is the localization theory of exact categories. We also investigate how properties of $\mathcal{E}$ are reflected in $\operatorname{mod_{\mathsf{adm}}}(\mathcal{E})$, for example being (weakly) idempotent complete or having enough projectives or injectives. Furthermore, we describe $\operatorname{mod_{\mathsf{adm}}}(\mathcal{E})$ as a subcategory of $\operatorname{mod}(\mathcal{E})$ when $\mathcal{E}$ is a resolving subcategory of an abelian category. This includes the category of Gorenstein projective modules and the category of maximal Cohen-Macaulay modules as special cases. Finally, we use $\operatorname{mod_{\mathsf{adm}}}(\mathcal{E})$ to give a bijection between exact structures on an idempotent complete additive category $\mathcal{C}$ and certain resolving subcategories of $\operatorname{mod}(\mathcal{C})$., Comment: Accepted in Advances in Mathematics

It is well known that a resolving subcategory $\mathcal{A}$ of an abelian subcategory $\mathcal{E}$ induces several derived equivalences: a triangle equivalence $\mathbf{D}^-(\mathcal{A})\to \mathbf{D}^-(\mathcal{E})$ exists in general and furthermore restricts to a triangle equivalence $\mathbf{D}^{\mathsf{b}}(\mathcal{A})\to \mathbf{D}^{\mathsf{b}}(\mathcal{E})$ if $\operatorname{res.dim}_{\mathcal{A}}(E)<\infty$ for any object $E\in \mathcal{E}$. If the category $\mathcal{E}$ is uniformly bounded, i.e. $\operatorname{res.dim}_{\mathcal{A}}(\mathcal{E})<\infty$, one obtains a triangle equivalence $\mathbf{D}(\mathcal{A})\to \mathbf{D}(\mathcal{E})$. In this paper, we show that all of the above statements hold for preresolving subcategories of (one-sided) exact categories. By passing to a one-sided language, one can remove the assumption that $\mathcal{A}\subseteq \mathcal{E}$ is extension-closed completely from the classical setting, yielding easier criteria and more examples. To illustrate this point, we consider the Isbell category $\mathcal{I}$ and show that $\mathcal{I}\subseteq \mathsf{Ab}$ is preresolving but $\mathcal{I}$ cannot be realized as an extension-closed subcategory of an exact category. We also consider a criterion given by Keller to produce derived equivalences of fully exact subcategories. We show that this criterion fits into the framework of preresolving subcategories by considering the relative weak idempotent completion of said subcategory., Comment: 14 pages, comments welcome

One-sided exact categories are obtained via a weakening of a Quillen exact category. Such one-sided exact categories are homologically similar to Quillen exact categories: a one-sided exact category $\mathcal{E}$ can be (essentially uniquely) embedded into its exact hull ${\mathcal{E}}^{\textrm{ex}}$; this embedding induces a derived equivalence $\textbf{D}^b(\mathcal{E}) \to \textbf{D}^b({\mathcal{E}}^{\textrm{ex}})$. Whereas it is well known that Quillen's obscure axioms are redundant for exact categories, some one-sided exact categories are known to not satisfy the corresponding obscure axiom. In fact, we show that the failure of the obscure axiom is controlled by the embedding of $\mathcal{E}$ into its exact hull ${\mathcal{E}}^{\textrm{ex}}.$ In this paper, we introduce three versions of the obscure axiom (these versions coincide when the category is weakly idempotent complete) and establish equivalent homological properties, such as the snake lemma and the nine lemma. We show that a one-sided exact category admits a closure under each of these obscure axioms, each of which preserves the bounded derived category up to triangle equivalence., Comment: 27 pages, comments welcome

We present a quick approach to computing the $K$-theory of the category of locally compact modules over any order in a semisimple $\mathbb{Q}$-algebra. We obtain the $K$-theory by first quotienting out the compact modules and subsequently the vector modules. Our proof exploits the fact that the pair (vector modules plus compact modules, discrete modules) becomes a torsion theory after we quotient out the finite modules. Treating these quotients as exact categories is possible due to a recent localization formalism., Comment: 8 pages. Comments welcome

Fragment and glider representations (introduced by F. Caenepeel, S. Nawal, and F. Van Oystaeyen) form a generalization of filtered modules over a filtered ring. Given a $\Gamma$-filtered ring $FR$ and a subset $\Lambda \subseteq \Gamma$, we provide a category $\operatorname{Glid}_\Lambda FR$ of glider representations, and show that it is a complete and cocomplete deflation quasi-abelian category. We discuss its derived category, and its subcategories of natural gliders and Noetherian gliders. If $R$ is a bialgebra over a field $k$ and $FR$ is a filtration by bialgebras, we show that $\operatorname{Glid}_\Lambda FR$ is a monoidal category which is derived equivalent to the category of representations of a semi-Hopf category (in the sense of E. Batista, S. Caenepeel, and J. Vercruysse). We show that the monoidal category of glider representations associated to the one-step filtration $k \cdot 1 \subseteq R$ of a bialgebra $R$ is sufficient to recover the bialgebra $R$ by recovering the usual fiber functor from $\operatorname{Glid}_\Lambda FR.$ When applied to a group algebra $kG$, this shows that the monoidal category $\operatorname{Glid}_\Lambda F(kG)$ alone is sufficient to distinguish even isocategorical groups., Comment: 33 pages, comments welcome

We consider the quotient of an exact or one-sided exact category $\mathcal{E}$ by a so-called percolating subcategory $\mathcal{A}$. For exact categories, such a quotient is constructed in two steps. Firstly, one localizes $\mathcal{E}$ at a suitable class $S_\mathcal{A} \subseteq \operatorname{Mor}(\mathcal{E})$ of morphisms. The localization $\mathcal{E}[S_\mathcal{A}^{-1}]$ need not be an exact category, but will be a one-sided exact category. Secondly, one constructs the exact hull $\mathcal{E}{/\mkern-6mu/} \mathcal{A}$ of $\mathcal{E}[S_\mathcal{A}^{-1}]$ and shows that this satisfies the 2-universal property of a quotient amongst exact categories. In this paper, we show that this quotient $\mathcal{E} \to \mathcal{E} {/\mkern-6mu/} \mathcal{A}$ induces a Verdier localization $\mathbf{D}^b(\mathcal{E}) \to \mathbf{D}^b(\mathcal{E} {/\mkern-6mu/} \mathcal{A})$ of bounded derived categories. Specifically, (i) we study the derived category of a one-sided exact category, (ii) we show that the localization $\mathcal{E} \to \mathcal{E}[S_\mathcal{A}^{-1}]$ induces a Verdier quotient $\mathbf{D}^b(\mathcal{E}) \to \mathbf{D}^b(\mathcal{E}[S^{-1}_\mathcal{A}])$, and (iii) we show that the natural embedding of a one-sided exact category $\mathcal{F}$ into its exact hull $\overline{\mathcal{F}}$ lifts to a derived equivalence $\mathbf{D}^b(\mathcal{F}) \to \mathbf{D}^b(\overline{\mathcal{F}})$. We furthermore show that the Verdier localization is compatible with several enhancements of the bounded derived category, so that the above Verdier localization can be used in the study of localizing invariants, such as non-connective $K$-theory., Comment: 37 pages. Improved exposition and the formulation of the main theorem. Added section 8. Comments welcome

In this paper, we introduce quotients of exact categories by percolating subcategories. This approach extends earlier localization theories by Cardenas and Schlichting for exact categories, allowing new examples. Let $\mathcal{A}$ be a percolating subcategory of an exact category $\mathcal{E}$, the quotient $\mathcal{E} {/\mkern-6mu/} \mathcal{A}$ is constructed in two steps. In the first step, we associate a set $S_\mathcal{A} \subseteq \operatorname{Mor}(\mathcal{E})$ to $\mathcal{A}$ and consider the localization $\mathcal{E}[S^{-1}_\mathcal{A}]$. In general, $\mathcal{E}[S_\mathcal{A}^{-1}]$ need not be an exact category, but will be a one-sided exact category. In the second step, we take the exact hull $\mathcal{E} {/\mkern-6mu/} \mathcal{A}$ of $\mathcal{E}[S_\mathcal{E}^{-1}]$. The composition $\mathcal{E} \rightarrow \mathcal{E}[S_\mathcal{A}^{-1}] \rightarrow \mathcal{E} {/\mkern-6mu/} \mathcal{A}$ satisfies the 2-universal property of a quotient in the 2-category of exact categories. We formulate our results in slightly more generality, allowing to start from a one-sided exact category. Additionally, we consider a type of percolating subcategories which guarantee that the morphisms of the set $S_\mathcal{A}$ are admissible. In upcoming work, we show that these localizations induce Verdier localizations on the level of the bounded derived category., Comment: 43 pages. Weakened the conditions of a percolating subcategory. Comments welcome

Using the framework of noncommutative Kahler structures, we generalise to the noncommutative setting the celebrated vanishing theorem of Kodaira for positive line bundles. The result is established under the assumption that the associated Dirac-Dolbeault operator of the line bundle is diagonalisable, an assumption that is shown to always hold in the quantum homogeneous space case. The general theory is then applied to the covariant Kahler structure of the Heckenberger-Kolb calculus of the quantum Grassmannians allowing us to prove a direct q-deformation of the classical Grassmannian Bott-Borel-Weil theorem for positive line bundles., Comment: 54 pages

As a generalization of acyclic 2-Calabi-Yau categories, we consider 2-Calabi-Yau categories with a directed cluster-tilting subcategory; we study their cluster-tilting subcategories and the cluster combinatorics that they encode. We show that such categories have a cluster structure. Triangulated 2-Calabi-Yau categories with a directed cluster-tilting subcategory are closely related to representations of certain semi-hereditary categories, more specifically to representations of thread quivers. Thread quivers are a tool to classify and study certain semi-hereditary categories using both quivers and linearly ordered sets (threads). We study the case where the thread quiver consists of a single thread (so that representations of this thread quiver correspond to representations of some linearly ordered set), and show that, similar to the case of a Dynkin quiver of type $A$, the cluster-tilting subcategories can be understood via triangulations of an associated cyclically ordered set. In this way, we gain insight into the structure of the cluster-tilting subcategories of 2-Calabi-Yau categories with a directed cluster-tilting subcategory. As an application, we show that every 2-Calabi-Yau category which admits a directed cluster-tilting subcategory with countably many isomorphism classes of indecomposable objects has a cluster-tilting subcategory $\mathcal{V}$ with the following property: any rigid object in the cluster category can be reached from $\mathcal{V}$ by finitely many mutations. This implies that there is a cluster map which is defined on all rigid objects, and thus that there is a cluster algebra whose cluster variables are exactly given by the rigid indecomposable objects., Comment: 61 pages

We study the role of the Serre functor in the theory of derived equivalences. Let $\mathcal{A}$ be an abelian category and let $(\mathcal{U}, \mathcal{V})$ be a $t$-structure on the bounded derived category $D^b \mathcal{A}$ with heart $\mathcal{H}$. We investigate when the natural embedding $\mathcal{H} \to D^b \mathcal{A}$ can be extended to a triangle equivalence $D^b \mathcal{H} \to D^b \mathcal{A}$. Our focus of study is the case where $\mathcal{A}$ is the category of finite-dimensional modules over a finite-dimensional hereditary algebra. In this case, we prove that such an extension exists if and only if the $t$-structure is bounded and the aisle $\mathcal{U}$ of the $t$-structure is closed under the Serre functor., Comment: As accepted by Advances in Mathematics; some differences in label and page numbering may occur between this version and the published version

Let A be an abelian hereditary category with Serre duality. We provide a classification of such categories up to derived equivalence under the additional condition that the Grothendieck group modulo the radical of the Euler form is a free abelian group of finite rank. Such categories are called numerically finite, and this condition is satisfied by the category of coherent sheaves on a smooth projective variety., Comment: 41 pages, as accepted by the Transactions of the American Mathematical Society

We study aisles in the derived category of a hereditary abelian category. Given an aisle, we associate a sequence of subcategories of the abelian category by considering the different homologies of the aisle. We then obtain a sequence, called a narrow sequence. We then prove that a narrow sequence in a hereditary abelian category consists of a nondecreasing sequence of wide subcategories, together with a tilting torsion class in each of these wide subcategories. Furthermore, there are relations these torsion classes have to satisfy. These results are sufficient to recover known classifications of t-structures for smooth projective curves, and for finitely generated modules over a Dedekind ring. In some special cases, including the case of finite dimensional modules over a finite dimensional hereditary algebra, we can reduce even further, effectively decoupling the different tilting torsion theories one chooses in the wide subcategories., Comment: 28 pages

An abelian Krull-Schmidt category is said to be uniserial if the isomorphism classes of subobjects of a given indecomposable object form a linearly ordered poset. In this paper, we classify the hereditary uniserial categories with Serre duality. They fall into two types: the first type is given by the representations of the quiver A_n with linear orientation (and infinite variants thereof), the second type by tubes (and an infinite variant). These last categories give a new class of hereditary categories with Serre duality, called big tubes., Comment: 24 pages, 2 figures

As a generalization of a Calabi-Yau category, we will say a k-linear Hom-finite triangulated category is fractionally Calabi-Yau if it admits a Serre functor S and there is an n > 0 with S^n = [m]. An abelian category will be called fractionally Calabi-Yau is its bounded derived category is. We provide a classification up to derived equivalence of abelian hereditary fractionally Calabi-Yau categories (for algebraically closed k). They are: the category of finite dimensional representations of a Dynkin quiver, the category of finite dimensional nilpotent representations of a cycle, and the category of coherent sheaves on an elliptic curve or a weighted projective line of tubular type. To obtain this classification, we introduce generalized 1-spherical objects and use them to obtain results about tubes in hereditary categories (which are not necessarily fractionally Calabi-Yau)., Comment: 23 pages, 1 figure, simplified the proof of the main theorem by adding Proposition 6.2

We show that every k-linear abelian Ext-finite hereditary category with Serre duality which is generated by preprojective objects is derived equivalent to the category of representations of a strongly locally finite thread quiver., Comment: 32 pages, 12 pictures

We introduce thread quivers as an (infinite) generalization of quivers, and show that every k-linear (k algebraically closed) hereditary category with Serre duality and enough projectives is equivalent to the category of finitely presented representations of a thread quiver. In this way, we obtain an explicit construction of a new class of hereditary categories with Serre duality., Comment: As accepted by the Proceedings of the London Mathematical Society; some differences in label and page numbering may occur between this version and the published version

Let k be an algebraically closed field and A a k-linear hereditary category satisfying Serre duality with no infinite radicals between the preprojective objects. If A is generated by the preprojective objects, then we show that A is derived equivalent to rep_k Q for a so called strongly locally finite quiver Q. To this end, we introduce light cone distances and round trip distances on quivers which will be used to investigate sections in stable translation quivers of the form \mathbb{Z} Q., Comment: 16 pages, as accepted by Journal of Pure and Applied Algebra

In this paper, we show all k-linear abelian 1-Calabi-Yau categories over an algebraically closed field k are derived equivalent to either the category of coherent sheaves on an elliptic curve, or to the finite dimensional representations of k[[t]]. Since all abelian categories derived equivalent with these two are known, we obtain a classification of all k-linear abelian 1-Calabi-Yau categories up to equivalence., Comment: 12 pages, as accepted by the International Mathematics Research Notices

In an ongoing project to classify all hereditary abelian categories, we provide a classification of Ext-finite directed hereditary abelian categories satisfying Serre duality up to derived equivalence. In order to prove the classification, we will study the shapes of the Auslander-Reiten components extensively and use appropriate generalisations of tilting objects and coordinates, namely partial tilting sets and probing of objects by quasi-simples., Comment: 30 pages with 12 figures; incorporated referee's comments