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We introduce and study strongly Gorenstein subcategory [Formula: see text], relative to an additive full subcategory [Formula: see text] of an abelian category [Formula: see text]. When [Formula: see text] is self-orthogonal, we give some sufficient conditions under which the property of an object in [Formula: see text] can be inherited by its subobjects and quotient objects. Then, we introduce the notions of one-sided (strongly) Gorenstein subcategories. Under the assumption that [Formula: see text] is closed under countable direct sums (respectively, direct products), we prove that an object is in right (respectively, left) Gorenstein category [Formula: see text] (respectively, [Formula: see text]) if and only if it is a direct summand of an object in right (respectively, left) strongly Gorenstein subcategory [Formula: see text] (respectively, [Formula: see text]). As applications, some known results are obtained as corollaries.

Let $\mathcal{A}$ be an abelian category. In this paper, we investigate the global $(\mathcal{X} , \mathcal{Y})$-Gorenstein projective dimension $\mathrm{gl.GPD}(\mathcal{X} ,\mathcal{Y})(\mathcal{A})$, associated to a GP-admissible pair $(\mathcal{X} , \mathcal{Y} )$. We give homological conditions over $(\mathcal{X} , \mathcal{Y})$ that characterize it. Moreover, given a GI-admisible pair $(\mathcal{Z} , \mathcal{W} )$, we study conditions under which $\mathrm{gl.GID}(\mathcal{Z},\mathcal{W})(\mathcal{A})$ and $\mathrm{gl.GPD}(\mathcal{X},\mathcal{Y})(\mathcal{A})$ are the same.

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.

We define the notion of an infinitely generated tilting object of infinite homological dimension in an abelian category. A one-to-one correspondence between $\infty$-tilting objects in complete, cocomplete abelian categories with an injective cogenerator and $\infty$-cotilting objects in complete, cocomplete abelian categories with a projective generator is constructed. We also introduce $\infty$-tilting pairs, consisting of an $\infty$-tilting object and its $\infty$-tilting class, and obtain a bijective correspondence between $\infty$-tilting and $\infty$-cotilting pairs. Finally, we discuss the related derived equivalences and t-structures.

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 R be a commutative Noetherian ring and I be an ideal of R. In this article, we answer affirmatively a question raised by the present author. Also, as a consequence, it is shown that the categor...

This report is the announcement of the result on the characterization of cofinite complexes (the first one is in the talk at the RIMS conferencet). Let A be a homomorphic image of a Gorenstein ring of finite Krull dimension, J an ideal of A, and N^{cdot} a bounded-below complex of A-modules. Suppose that A is complete with respect to a J-adic topology. In this report, we introduce that N^{cdot} is a J-cofinite complex if and only if the support of H^{i}(N^{cdot}) is in V(J) for all i, and Ext^{j}(A/J, N^{cdot}) is of finite type for all j.

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 the previous article "Hearts of twin cotorsion pairs on exact categories", we introduced the notion of the heart for any cotorsion pair on an exact category with enough projectives and injectives, and showed that it is an abelian category. In this paper, we construct a half exact functor from the exact category to the heart. This is analog of the construction of Abe and Nakaoka for triangulated categories. We will also use this half exact functor to find out a sufficient condition when two different hearts are equivalent., 26 pages

Let (X,Y) be a complete and hereditary cotorsion pair in an abelian category A. We show that if Y is closed under kernels of epimorphisms, then (G(X)∩Y,X∩Y) is a strong left Frobenius pair,...

Let $R\to U$ be an associative ring epimorphism such that $U$ is a flat left $R$-module. Assume that the related Gabriel topology $\mathbb G$ of right ideals in $R$ has a countable base. Then we show that the left $R$-module $U$ has projective dimension at most $1$. Furthermore, the abelian category of left contramodules over the completion of $R$ at $\mathbb G$ fully faithfully embeds into the Geigle-Lenzing right perpendicular subcategory to $U$ in the category of left $R$-modules, and every object of the latter abelian category is an extension of two objects of the former one. We discuss conditions under which the two abelian categories are equivalent. Given a right linear topology on an assocative ring $R$, we consider the induced topology on every left $R$-module, and for a perfect Gabriel topology $\mathbb G$ compare the completion of a module with an appropriate Ext module. Finally, we characterize the $U$-strongly flat left $R$-modules by the two conditions of left positive-degree Ext-orthogonality to all left $U$-modules and all $\mathbb G$-separated $\mathbb G$-complete left $R$-modules., Comment: LaTeX 2e with pb-diagram and xy-pic, 64 pages, 6 commutative diagrams + Corrigenda, LaTeX 2e with ulem.sty, 10 pages; v.6: corrigenda added (two mistakes, one in Remark 3.3 and the other one in Section 5); v.7: third section added to corrigenda (confusion in Remark 11.3); v.8: fourth section added to corrigenda (about an unjustified assertion in the preliminaries), main results unaffected

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.

Let $${\mathcal {A}}$$ be an abelian category, and $${\mathcal {X}}$$ and $${\mathcal {Y}}$$ subcategories. We derive in this paper an additive functor $$T(-, -)$$ using proper $${\mathcal {X}}$$ -resolutions (respectively, proper $${\mathcal {Y}}$$ -coresolutions). Under certain conditions, we establish balance results for such relative homology (respectively, cohomology) over $${\mathcal {A}}$$ . Our main theorem simultaneously recovers theorems of Sather-Wagstaff et al. (J Math Kyoto Univ 48(3):571–596, 2008), Enochs et al. (Relative homological algebra. De Gruyter, Berlin 2000) and Di et al. (Bull Korean Math Soc 52(1):137–147, 2015) as special cases.

Quasi-resolving subcategory, as a generalization of resolving subcategory, was introduced in Zhu (J Algebra 414:6–40, 2014). Let $${\mathcal {A}}$$ be an abelian category, $${\mathcal {X}}$$ a quasi-resolving subcategory of $${\mathcal {A}}$$ and $${\mathcal {H}}$$ an Ext-injective cogenerator for $${\mathcal {X}}$$ . We study the subcategory consisting of all the objects with finite $${\mathcal {X}}$$ -resolution dimension. Furthermore, we investigate the objects with approximations by objects which are either with finite $${\mathcal {H}}$$ -resolution dimension or else in $${\mathcal {X}}$$ . As applications, we give the finiteness criteria of the homological dimensions in Gorenstein categories of modules and complexes.

Giving a cotorsion pair in an abelian category $${\mathscr {C}}$$ , we have a sequence of exact functors between triangulated categories with respect to the pair, and construct right (left) adjoints of the exact functors such that the sequence is a (co)localization sequence. Further, for some especial cotorsion pairs, we gain a recollement and triangle-equivalences between corresponding triangulated categories. In particular, let ( $${\mathcal {A}}, {\mathcal {Z}}, {\mathcal {B}}$$ ) and ( $${\mathcal {A}}_{1}, {\mathcal {Z}}_{1}, {\mathcal {B}}_{1}$$ ) be two complete and hereditary cotorsion triples in $${\mathscr {C}}$$ with $${\mathcal {A}}_{1}\subseteq {\mathcal {A}}$$ . We obtain a triangle-equivalence $$K({\mathcal {A}})\simeq K({\mathcal {B}})$$ , which restricts to an equivalence $$K({\mathcal {A}}_{1})\simeq K({\mathcal {B}}_{1})$$ .

For a given hereditary abelian category satisfying some finiteness conditions, in certain twisted cases it is shown that the modified Ringel-Hall algebra is isomorphic to the naive lattice algebra and there exists an epimorphism from the modified Ringel-Hall algebra to the lattice algebra. Furthermore, the kernel of this epimorphism is described explicitly. Finally, we show that the naive lattice algebra is invariant under the derived equivalences of hereditary abelian categories.