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We study cluster tilting modules in mesh algebras of Dynkin type as defined in [12] , providing a new proof for their existence. Except for type G 2 , we show that these are precisely the maximal rigid modules, and that they are equivariant for a certain automorphism. We further study their mutation, providing an example of mutation in an abelian category which is not stably 2-Calabi-Yau, and explicitly describe the combinatorics.

In category-theoretic models for the anyon systems proposed for topological quantum computing, the essential ingredients are two monoidal structures, $\oplus$ and $\otimes$. The former is symmetric but the latter is only braided, and $\otimes$ is required to distribute over $\oplus$. What are the appropriate coherence conditions for the distributivity isomorphisms? We came to this question working on a simplification of the category-theoretical foundation of topological quantum computing, which is the intended application of the research reported here. This question was answered by Laplaza when both monoidal structures are symmetric, but topological quantum computation depends crucially on $\otimes$ being only braided, not symmetric. We propose coherence conditions for distributivity in this situation, and we prove that our conditions are (a) strong enough to imply Laplaza's when the latter are suitably formulated, and (b) weak enough to hold when --- as in the categories used to model anyons --- the additive structure is that of an abelian category and the braided $\otimes$ is additive. Working on these results, we found a new redundancy in Laplaza's conditions., This is a companion paper for article "Witness algebra and anyon braiding," arXiv:1807.10414, proving results used there

For an abelian category A , we define the category PEx( A ) of pullback diagrams of short exact sequences in A , as a subcategory of the functor category Fun( Δ , A ) for a fixed diagram category Δ. For any object M in PEx ( A ) , we prove the existence of a short exact sequence 0 → K → P → M → 0 of functors, where the objects are in PEx( A ) and P ( i ) ∈ Proj ( A ) for any i ∈ Δ . As an application, we prove that if ( C , D , E ) is a triple of syzygy finite classes of objects in mod Λ satisfying some special conditions, then Λ is an Igusa–Todorov algebra. Finally, we study lower triangular matrix Artin algebras and determine in terms of their components, under reasonable hypothesis, when these algebras are syzygy finite or Igusa–Todorov.

A recollement is a decomposition of a given category (abelian or triangulated) into two subcategories with functorial data that enables the glueing of structural information. This paper is dedicated to investigating the behaviour under glueing of some basic properties of abelian categories (well-poweredness, Grothendieck's axioms AB3, AB4 and AB5, existence of a generator) in the presence of a recollement. In particular, we observe that in a recollement of a Grothendieck abelian category the other two categories involved are also Grothendieck abelian and, more significantly, we provide an example where the converse does not hold and explore multiple sufficient conditions for it to hold.

Let A be a finitary hereditary abelian category. In this note, we show that Green's formula on Hall numbers of A can be deduced from the associativity of the derived Hall algebra of A whose multiplication structure constants are given by the so-called Toen's formula.

Let $\C$ be an $(n+2)$-angulated category with shift functor $\Sigma$ and $\X$ be a cluster-tilting subcategory of $\C$. Then we show that the quotient category $\C/\X$ is an $n$-abelian category. If $\C$ has a Serre functor, then $\C/\X$ is equivalent to an $n$-cluster tilting subcategory of an abelian category $\textrm{mod}(\Sigma^{-1}\X)$. Moreover, we also prove that $\textrm{mod}(\Sigma^{-1}\X)$ is Gorenstein of Gorenstein dimension at most $n$. As an application, we generalize recent results of Jacobsen-J{\o}rgensen and Koenig-Zhu., Comment: 14 pages

The Frobenius-Perron dimension for an abelian category was recently introduced. We apply this theory to the category of representations of the finite-dimensional radical squared zero algebras associated to certain modified ADE graphs. In particular, we take an ADE quiver with arrows in a certain orientation and an arbitrary number of loops at each vertex. We show that the Frobenius-Perron dimension of this category is equal to the maximum number of loops at a vertex. Along the way, we introduce a result which can be applied in general to calculate the Frobenius-Perron dimension of a radical square zero bound quiver algebra. We use this result to introduce a family of abelian categories which produce arbitrarily large irrational Frobenius-Perron dimensions., Comment: typos corrected

In this paper we define the modified Ringel–Hall algebra M H ( A ) of a hereditary abelian category A from the category C b ( A ) of bounded Z -graded complexes. Two main results are obtained. One is to give a new proof of Green's formula on Ringel–Hall numbers by using the associative multiplication of the modified Ringel–Hall algebra. The other is to show that in certain twisted cases the derived Hall algebra can be embedded in the modified Ringel–Hall algebra. As a consequence of the second result, we get that in certain twisted cases the modified Ringel–Hall algebra is isomorphic to the tensor algebra of the derived Hall algebra and the torus of acyclic complexes and therefore the modified Ringel–Hall algebra is invariant under derived equivalences.

We investigate how to characterize subcategories of abelian categories in terms of intrinsic axioms. In particular, we find intrinsic axioms which characterize generating cogenerating functorially finite subcategories, precluster tilting subcategories, and cluster tilting subcategories of abelian categories. As a consequence we prove that any $d$-abelian category is equivalent to a $d$-cluster tilting subcategory of an abelian category, without any assumption on the categories being projectively generated., Comment: 29 pages. Accepted for publication in Journal of Pure and Applied Algebra

Given two complete hereditary cotorsion pairs ( Q , R ) and ( Q ′ , R ′ ) in a bicomplete abelian category G such that Q ′ ⊆ Q and Q ∩ R = Q ′ ∩ R ′ , Becker showed that there exists a hereditary abelian model structure M = ( Q , W , R ′ ) on G , where W is a thick subcategory of G . We prove that the homotopy category Ho( M ) of M is triangulated equivalent to the triangulated quotient category D b ( G ) [ Q , R ′ ] ˆ / K b ( Q ′ ∩ R ′ ) , where D b ( G ) [ Q , R ′ ] ˆ is the subcategory of D b ( G ) consisting of all homology bounded complexes with both finite Q dimension and R ′ dimension and K b ( Q ′ ∩ R ′ ) is the bounded homotopy category of Q ′ ∩ R ′ (core) objects. Applications are given in the category of modules. It is shown that the homotopy category of the Gorenstein flat (resp., Ding projective and Gorenstein AC-projective) model structure on the category of modules established by Gillespie and his coauthors can be realized as a certain triangulated quotient category.

We introduce the notions of a $\mathbf{D}$-standard abelian category and a $\mathbf{K}$-standard additive category. We prove that for a finite dimensional algebra $A$, its module category is $\mathbf{D}$-standard if and only if any derived autoequivalence on $A$ is standard, that is, given by a two-sided tilting complex. We prove that if the subcategory of projective $A$-modules is $\mathbf{K}$-standard, then the module category is $\mathbf{D}$-standard. We provide new examples of $\mathbf{D}$-standard module categories., revised, comments are welcome

For an abelian category $\mathcal{A}$ we investigate when the stable categories $\underline{\mathrm{GPro}}\mathrm{j}(\mathcal{A})$ and $\underline{\mathrm{GIn}}\mathrm{j}(\mathcal{A})$ are triangulated equivalent. To this end, we realize these stable categories as homotopy categories of certain (non-trivial) model categories and give conditions on $\mathcal{A}$ that ensure the existence of a Quillen equivalence between the model categories in question. We also study when such a Quillen equivalence transfers from $\mathcal{A}$ to categories naturally associated to $\mathcal{A}$, such as $\mathrm{Ch}(\mathcal{A})$, the category of chain complexes in $\mathcal{A}$, or $\mathrm{Rep}(Q,\mathcal{A})$, the category of $\mathcal{A}$-valued representations of a quiver $Q$., Comment: Revised version; now 16 pages

Alexander L. Rosenberg has constructed a spectrum for abelian categories which is able to reconstruct a quasi-separated scheme from its category of quasi-coherent sheaves. In this note we present a detailed proof of this result which is due to Ofer Gabber. Moreover, we determine the automorphism class group of the category of quasi-coherent sheaves.

Let R be a commutative Noetherian ring, I ⊆ J be ideals of R and M be a finitely generated R-module. In this paper it is shown that q ( J , M ) ≤ q ( I , M ) + cd ( J , M / I M ) . Furthermore, it is shown that, for any ideal I of R and any finitely generated R-module M with q ( I , M ) ≤ 1 , the local cohomology modules H I i ( M ) are I-cofinite for all integers i ≥ 0 . As a consequence of this result it is shown that, if q ( I , R ) ≤ 1 , then for any finitely generated R-module M, the local cohomology modules H I i ( M ) are I-cofinite for all integers i ≥ 0 . Finally, it is shown that the category of all I-cofinite R-modules C ( R , I ) c o f is an Abelian subcategory of the category of all R-modules, whenever ( R , m ) is a complete Noetherian local ring and I is an ideal of R with q ( I , R ) ≤ 1 . These assertions answer affirmatively two questions raised by R. Hartshorne in [16] , in the some special cases.

Let A be a homomorphic image of a Gorenstein ring of finite Krull dimension, J an ideal of A of dimension one, and N • a bounded-below complex of A-modules. Suppose that A is complete with respect to a J-adic topology. In this paper, we prove that N • is a J-cofinite complex if and only if H i ( N • ) is a J-cofinite module for all i. The same result is also proved for principal ideals J. Consequently, for the fourth question given by R. Hartshorne (cf. [6, Fourth Question, p. 149] ), we obtain an answer over the ring, on affine curves and hypersurfaces.

Generalizing our previous results on Walsh system, we consider the system of characters of zero-dimensional compact abelian group with the second axiom of countability and give for the sets of uniqueness with respect to this system a solution of category problem. We also obtain a generalization of a theorem on existence for this system a perfect M 0 -set whose Hausdorff h -measure equals zero.

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., As accepted by Advances in Mathematics; some differences in label and page numbering may occur between this version and the published version

In this note, we prove that the category of unipotent torsion Breuil modules is an abelian category, under the condition e r = p − 1 , r p − 1 .

We describe a general correspondence between injective (resp. projective) recollements of triangulated categories and injective (resp. projective) cotorsion pairs. This provides a model category description of these recollement situations. Our applications focus on displaying several recollements that glue together various full subcategories of K(R), the homotopy category of chain complexes of modules over a general ring R. When R is (left) Noetherian ring, these recollements involve complexes built from the Gorenstein injective modules. When R is a (left) coherent ring for which all flat modules have finite projective dimension we obtain the duals. These results extend to a general ring R by replacing the Gorenstein modules with the Gorenstein AC-modules introduced recently in the work of Bravo-Gillespie-Hovey. We also see that in any abelian category with enough injectives, the Gorenstein injective objects enjoy a maximality property in that they contain every other class making up the right half of an injective cotorsion pair., 45 pages. Revisions from previous version include: 0) A major rewrite resulting in a completely different numbering of Lemmas, Propositions, etc. 1) removal of a false claim in Example 4.8 on extending the recollement of Krause. 2) A converse to the main theorem. 3) An extension of the Gorenstein applications to arbitrary rings using the Gorenstein AC-modules

The paper exhibits three constitutive exact subcategories of Adelman's free abelian category Ab ( A ) over a Quillen exact category A , which provide an intrinsic description of Ab ( A ) and encapsulate the mechanism of (finite or infinite) tilting. In general, two of these subcategories are left and right abelian, respectively, and a tilting adjunction takes place between them if they are abelian. The third category Im ( A ) contains the acyclic closure T ( A ) of A as an exact subcategory of those objects which arise as images of the morphisms in an exact complex over A . If the exact structure of A is trivial, T ( A ) consists of the Gorenstein projectives over A , a category that does not depend on an embedding of A into some ambient abelian category. The construction of Ab ( A ) and its intrinsic description sheds some light upon old and new concepts and results concerning Gruson-Jensen duality, resolving subcategories, representations of Cohen-Macaulay orders, Gorenstein projectivity, non-commutative resolutions, and representation dimension. Applications to infinite tilting and existence of derived equivalences will be given in a forthcoming publication.

We study homological properties of contravariantly finite rigid subcategories of an arbitrary triangulated or abelian category, concentrating at the class of higher cluster tilting subcategories. In both the triangulated and abelian context we present several new characterizations of such subcategories and we give conditions ensuring that the associated cluster tilted category is Gorenstein and/or stably Calabi–Yau. Thus we generalize and improve on several recent results of the literature. In the context of abelian categories, we prove a categorified version of higher Auslander correspondence, as developed by Iyama in the case of module categories. In this connection we investigate in detail homological properties of (stable) Auslander categories associated to higher cluster tilting subcategories in an abelian category, and we present applications to (possibly infinitely generated) cluster tilting modules over an arbitrary ring. Finally we give bounds on the global and representation dimension of certain categories of coherent functors associated to Cohen–Macaulay objects.

We show that every exact fully faithful functor from the category of perfect complexes on the spectrum of dual numbers to the bounded derived category of a noetherian separated scheme is of Fourier-Mukai type. The kernel turns out to be an object of the bounded derived category of coherent complexes on the product of the two schemes. We also study the space of stability conditions on the derived category of the spectrum of dual numbers., 23 pages, Final version to appear in J. Algebra

We study the properties of the relative derived category $D_{\mathscr{C}}^{b}$($\mathscr{A}$) of an abelian category $\mathscr{A}$ relative to a full and additive subcategory $\mathscr{C}$. In particular, when $\mathscr{A}=A{\text -}\mod$ for a finite-dimensional algebra $A$ over a field and $\mathscr{C}$ is a contravariantly finite subcategory of $A$-$\mod$ which is admissible and closed under direct summands, the $\mathscr{C}$-singularity category $D_{\mathscr{C}{\text sg}}$($\mathscr{A}$)=$D_{\mathscr{C}}^{b}$($\mathscr{A}$)/$K^{b}(\mathscr{C})$ is studied. We give a sufficient condition when this category is triangulated equivalent to the stable category of the Gorenstein category $\mathscr{G}(\mathscr{C})$ of $\mathscr{C}$., 17 pages, to appear in Journal of Pure and Applied Algebra

This paper is devoted to investigate the tilting bundles, i.e. tilting sheaves that are vector bundles, in the category of coherent sheaves on a weighted projective line of type (2,2,n)(2,2,n). We classify all the tilting bundles into two classes, one containing the tilting bundles that are consisting of line bundles, and the other one containing tilting bundles with indecomposable summands that are not line bundles. Moreover, for each tilting bundle T (with endomorphism algebra Λ) we prove that the missing part, from the category of coherent sheaves to the category of finitely generated right Λ-modules, carries the structure of an abelian category.

The aim of this paper is to introduce Gorenstein singularity category D g p s g b ( A ) , as the Verdier quotient of the Gorenstein derived category D g p b ( A ) by the triangulated subcategory K b ( GP ( A ) ) . The main result shows that, if A is CM-contravariantly finite abelian category, then the Gorenstein singularity category D g p s g b ( A ) is triangle-equivalent to the Gorenstein defect category D defect b ( A ) in the sense of [7] . We point out several classes of algebras with non-zero singularity categories, but their Gorenstein singularity categories become zero. We also prove that a Gorenstein derived equivalence between CM-contravariantly finite abelian categories with infinite direct sums induces a Gorenstein singular equivalence.

We prove that a Hom-finite additive category having determined morphisms on both sides is a dualizing variety. This complements a result by Krause. We prove that in a Hom-finite abelian category having Serre duality, a morphism is right determined by some object if and only if it is an epimorphism. We give a characterization to abelian categories having Serre duality via determined morphisms.

In this paper, we introduce the concepts of category of lattice-valued finite state machines, category of quasi-lattice-valued transformation semigroups, category of quasi-lattice-valued transformation monoids, category of lattice-valued transformation semigroups, category of lattice-valued transformation monoids and six pairs of functors, each pair of functors have the adjoint property, we discuss the relation between the categories with functors and give the adjoint theorems and relation theorems.

As a generalization of a resolving subcategory, which was introduced by Auslander and Bridger in [2] , the notion of a quasi-resolving subcategory of an abelian category is introduced. The homological theory on quasi-resolving subcategories is studied. It is shown that many results in [1] , [2] , [4] , [5] , [6] , [8] , [13] , [14] , [17] , [18] , [22] , [25] , [26] , [27] are unified and naturally generated.

In this article we study the appearance of abelian borders in binary words, a notion closely related to the abelian period of a word. We show how many binary words have shortest border of a given length by identifying relations with Dyck words. Furthermore, we give some bounds on the number of abelian border-free words of a given length and on the number of abelian words of a given length that have at least one abelian border. Finally, using some techniques employed in a recent paper by Christodoulakis et al. (2013), we show that there exists an algorithm that finds the shortest abelian border of a binary word that is not abelian border-free in Θ ( n ) time on average.

The main objective of this paper is to study the relative derived categories from various points of view. Let A be an abelian category and C be a contravariantly finite subcategory of A . One can define C -relative derived category of A , denoted by D C ⁎ ( A ) . The interesting case for us is when A has enough projective objects and C = GP - A is the class of Gorenstein projective objects, where D C ⁎ ( A ) is known as the Gorenstein derived category of A . We explicitly study the relative derived categories, specially over artin algebras, present a relative version of Rickardʼs theorem, specially for Gorenstein derived categories, provide some invariants under Gorenstein derived equivalences and finally study the relationships between relative and (absolute) derived categories.

This paper introduces a concept of dimension of a triangulated category with respect to a fixed full subcategory. For the bounded derived category of an abelian category, upper bounds of the dimension with respect to a contravariantly finite subcategory and a resolving subcategory are given. Our methods not only recover some known results on the dimensions of derived categories in the sense of Rouquier, but also apply to various commutative and non-commutative noetherian rings.

Assume that H is a Hilbert space of dimension greater than two. We prove that an abelian Kadison–Singer algebra acting on H cannot contain any non-trivial idempotent. Based on this, we show that an abelian KS-algebra in matrix algebra M n ( C ) ( n ⩾ 3 ) cannot be generated by a single element. As a corollary, it is also proved that the lattice of an abelian KS-algebra cannot be completely distributive.

Let A be an abelian category and B be the Happel-Reiten-Smalo tilt of A with respect to a torsion pair. We give necessary and sufficient conditions for the existence of a derived equivalence between A and B , which is compatible with the inclusion of B into the derived category of A . As applications, we obtain new derived equivalences related to splitting torsion pairs, TTF-triples and two-term silting subcategories, respectively. We prove that for the realization functor of any bounded t-structure, its denseness implies its fully-faithfulness.

We prove that some subquotient categories of exact categories are abelian. This generalizes a result by Koenig-Zhu in the case of (algebraic) triangulated categories. As a particular case, if an exact category B with enough projectives and injectives has a cluster tilting subcategory M, then B/M is abelian. More precisely, it is equivalent to the category of finitely presented modules over the stable category of M., Comment: 21 pages. Slight modifications after referring. Accepted for publication in Journal of Pure and Applied Algebra

Given a torsion theory (Y,X) in an abelian category C, the reflector I from C to the torsion-free subcategory X induces a reflective factorisation system (E, M) on C. It was shown by A. Carboni, G.M. Kelly, G. Janelidze and R. Par\'e that (E, M) induces a monotone-light factorisation system (E',M*) by simultaneously stabilising E and localising M, whenever the torsion theory is hereditary and any object in C is a quotient of an object in X. We extend this result to arbitrary normal categories, and improve it also in the abelian case, where the heredity assumption on the torsion theory turns out to be redundant. Several new examples of torsion theories where this result applies are then considered in the categories of abelian groups, groups, topological groups, commutative rings, and crossed modules., Comment: 12 pages

In the papers of Nakaoka, he introduced the notion of hearts of (twin) cotorsion pairs on triangulated categories and showed that they have structures of (semi-) abelian categories. We study in this article a twin cotorsion pair (S,T),(U,V) on an exact category B with enough projectives and injectives and introduce a notion of the heart. First we show that its heart is preabelian. Moreover we show the heart of a single cotorsion pair is abelian. These results are analog of Nakaoka's results in triangulated categories. We also consider special cases where the heart has nicer structure. By our results, the heart of a special twin cotorsion pair (S,T),(T,V), is integral and almost abelian. Finally we show that the Gabriel-Zisman localisation of the heart at the class of regular morphisms is abelian, and moreover it is equivalent to the category of finitely presented modules over a stable subcategory of B., 31pages

The notion of a ( φ , G ˆ ) -module is defined by Tong Liu in 2010 to classify lattices in semi-stable representations. In this paper, we study torsion ( φ , G ˆ ) -modules, and torsion p-adic representations associated with them, including the case where p = 2 . First we prove that the category of torsion p-adic representations arising from torsion ( φ , G ˆ ) -modules is an abelian category. Secondly, we construct a maximal (minimal) theory for ( φ , G ˆ ) -modules by using the theory of etale ( φ , G ˆ ) -modules, essentially proved by Xavier Caruso, which is an analogue of Fontaineʼs theory of etale ( φ , Γ ) -modules. Non-isomorphic two maximal (minimal) objects give non-isomorphic two torsion p-adic representations.

This article proposes an effective criterion for lifting automorphisms along regular coverings of graphs, with the covering transformation group being any finite abelian group., Comment: 10 pages

In this article we complete the work of enumerating typical abelian coverings of Cayley graphs, by reducing the problem to enumerating certain subgroups of finite abelian groups., 15pages

One-sided exact categories appear naturally as instances of Grothendieck pretopologies. In an additive setting they are given by considering the one-sided part of Keller’s axioms defining Quillen’s exact categories. We study one-sided exact additive categories and a stronger version defined by adding the one-sided part of Quillen’s “obscure axiom”. We show that some homological results, such as the Short Five Lemma and the 3×3 Lemma, can be proved in our context. We also note that the derived category of a one-sided exact additive category can be constructed.

In our previous article, we constructed an abelian category from any torsion pair on a triangulated category. This generalizes the heart of a $t$-structure and the ideal quotient by a cluster tilting subcategory. Recently, generalizing the quotient by a cluster tilting subcategory, Buan and Marsh showed that an integral preabelian category can be constructed as a quotient, from a rigid object in a triangulated category with some conditions. In this article, by considering a pair of torsion pairs, we make a simultaneous genralization of these two constructions., Comment: 18 pages

It is well known that diagram lemmas for abelian groups (and more generally in abelian categories) used in algebraic topology, can be suitably extended to “non-abelian” structures such as groups, rings, loops, etc. Moreover, they are equivalent to properties which arise in the axiomatic study of these structures. For the five lemma this is well known, and in the present paper we establish this for the snake lemma and the 3 × 3 lemma, which, when suitably formulated, turn out to be equivalent to each other for all (pointed) algebraic structures, and also in general categories of a special type. In particular, we show that among varieties of universal algebras, they are satisfied precisely in the so-called (pointed) ideal determined varieties.

For an arbitrary commutative ring k and t in k, we construct a 2-functor S_t which sends a tensor category to a new tensor category. By applying it to the representation category of a bialgebra we obtain a family of categories which interpolates the representation categories of the wreath products of the bialgebra. This generalizes the construction of Deligne's category Rep(S_t,k) for representation categories of symmetric groups., Comment: [v3] 41 pages, appendix added; final version [v2] 39 pages, title changed [v1] 32 pages

By a multi-pointed category we mean a category C equipped with an ideal of null morphisms, i.e. a class N of morphisms satisfying f ∈ N ∨ g ∈ N ⇒ f g ∈ N for any composable pair f , g of morphisms. Such categories are precisely the categories enriched in the category of pairs X = ( X , N ) where X is a set and N is a subset of X , whereas a pointed category has the same enrichment, but restricted to those pairs X = ( X , N ) where N is a singleton. We extend the notion of an “ideal” from regular pointed categories to regular multi-pointed categories, and having “a good theory of ideals” will mean that there is a bijection between ideals and kernel pairs, which in the pointed case is the main property of ideal determined categories. The study of general categories with a good theory of ideals allows in fact a simultaneous treatment of ideal determined and Barr exact Goursat categories: we prove that in the case when all morphisms are chosen as null morphisms, the presence of a good theory of ideals becomes precisely the property for a regular category to be a Barr exact Goursat category. Among other things, this allows to obtain a unified proof of the fact that lattices of effective equivalence relations are modular both in the case of Barr exact Goursat categories and in the case of ideal determined categories.

We define and study the category C o h n ( P 1 ) of normal coherent sheaves on the monoid scheme P 1 (equivalently, the M 0 -scheme P 1 / F 1 in the sense of Connes–Consani–Marcolli, Connes (2009) [2] ). This category resembles in most ways a finitary abelian category, but is not additive. As an application, we define and study the Hall algebra of C o h n ( P 1 ) . We show that it is isomorphic as a Hopf algebra to the enveloping algebra of the product of a non-standard Borel in the loop algebra L g l 2 and an abelian Lie algebra on infinitely many generators. This should be viewed as a ( q = 1 ) version of Kapranov’s result relating (a certain subalgebra of) the Ringel–Hall algebra of P 1 over F q to a non-standard quantum Borel inside the quantum loop algebra U ν ( s l 2 ) , where ν 2 = q .

An abelian category with arbitrary coproducts and a small projective generator is equivalent to a module category (Mitchell (1964) [17]). A tilting object in an abelian category is a natural generalization of a small projective generator. Moreover, any abelian category with a tilting object admits arbitrary coproducts (Colpi et al. (2007) [8]). It naturally arises the question when an abelian category with a tilting object is equivalent to a module category. By Colpi et al. (2007) [8], the problem simplifies in understanding when, given an associative ring R and a faithful torsion pair (X,Y) in the category of right R-modules, the heart H(X,Y) of the t-structure associated with (X,Y) is equivalent to a category of modules. In this paper, we give a complete answer to this question, proving necessary and sufficient conditions on (X,Y) for H(X,Y) to be equivalent to a module category. We analyze in detail the case when R is right artinian.

Expansions of abelian categories are introduced. These are certain functors between abelian categories and provide a tool for induction/reduction arguments. Expansions arise naturally in the study of coherent sheaves on weighted projective lines; this is illustrated by various applications., 14 pages

We use t-structures on the homotopy category Kb(R-mod) for an artin algebra R and Wattsʼ representability theorem to give an existence proof for Auslander–Reiten sequences of R-modules. This framework naturally leads to a notion of generalized (or higher) Auslander–Reiten sequences.

We prove a Dold–Kan type correspondence between the category of planar dendroidal abelian groups and a suitably constructed category of planar dendroidal chain complexes. Our result naturally extends the classical Dold–Kan correspondence between the category of simplicial abelian groups and the category of non-negatively graded chain complexes.