Relative Homological Algebra Research Articles

Koszul Complexes and Relative Homological Algebra of Functors Over Posets

AbstractUnder certain conditions, Koszul complexes can be used to calculate relative Betti diagrams of vector space-valued functors indexed by a poset, without the explicit computation of global minimal relative resolutions. In relative homological algebra of such functors, free functors are replaced by an arbitrary family of functors. Relative Betti diagrams encode the multiplicities of these functors in minimal relative resolutions. In this article we provide conditions under which grading the chosen family of functors leads to explicit Koszul complexes whose homology dimensions are the relative Betti diagrams, thus giving a scheme for the computation of these numerical descriptors.

Cut cotorsion pairs

AbstractWe present the concept of cotorsion pairs cut along subcategories of an abelian category. This provides a generalization of complete cotorsion pairs, and represents a general framework to find approximations restricted to certain subcategories. We also exhibit some connections between cut cotorsion pairs and Auslander–Buchweitz approximation theory, by considering relative analogs for Frobenius pairs and Auslander–Buchweitz contexts. Several applications are given in the settings of relative Gorenstein homological algebra, chain complexes, and quasi-coherent sheaves, as well as to characterize some important results on the Finitistic Dimension Conjecture, the existence of right adjoints of quotient functors by Serre subcategories, and the description of cotorsion pairs in triangulated categories as co-t-structures.

Gorenstein-Projective Modules over Upper Triangular Matrix Artin Algebras

Gorenstein-projective module is an important research topic in relative homological algebra, representation theory of algebras, triangulated categories, and algebraic geometry (especially in singularity theory). For a given algebra A , how to construct all the Gorenstein-projective A -modules is a fundamental problem in Gorenstein homological algebra. In this paper, we describe all complete projective resolutions over an upper triangular Artin algebra Λ = A M B A 0 B . We also give a necessary and sufficient condition for all finitely generated Gorenstein-projective modules over Λ = A M B A 0 B .

An integral theory of dominant dimension of Noetherian algebras

Dominant dimension is introduced into integral representation theory, extending the classical theory of dominant dimension of Artinian algebras to projective Noetherian algebras (that is, algebras which are finitely generated projective as modules over a commutative Noetherian ring). This new homological invariant is based on relative homological algebra introduced by Hochschild in the 1950s. Amongst the properties established here are a relative version of the Morita-Tachikawa correspondence and a relative version of Mueller's characterization of dominant dimension. The behaviour of relative dominant dimension of projective Noetherian algebras under change of ground ring is clarified and we explain how to use this property to determine the relative dominant dimension of projective Noetherian algebras. In particular, we determine the relative dominant dimension of Schur algebras and quantized Schur algebras.

N-Cotorsion pairs

Motivated by some properties satisfied by Gorenstein projective and Gorenstein injective modules over an Iwanaga-Gorenstein ring, we present the concept of left and right $n$-cotorsion pairs in an abelian category $\mathcal{C}$. Two classes $\mathcal{A}$ and $\mathcal{B}$ of objects of $\mathcal{C}$ form a left $n$-cotorsion pair $(\mathcal{A,B})$ in $\mathcal{C}$ if the orthogonality relation $\mathsf{Ext}^i_{\mathcal{C}}(\mathcal{A,B}) = 0$ is satisfied for indexes $1 \leq i \leq n$, and if every object of $\mathcal{C}$ has a resolution by objects in $\mathcal{A}$ whose syzygies have $\mathcal{B}$-resolution dimension at most $n-1$. This concept and its dual generalise the notion of complete cotorsion pairs, and has an appealing relation with left and right approximations, especially with those having the so called unique mapping property. The main purpose of this paper is to describe several properties of $n$-cotorsion pairs and to establish a relation with complete cotorsion pairs. We also give some applications in relative homological algebra, that will cover the study of approximations associated to Gorenstein projective, Gorenstein injective and Gorenstein flat modules and chain complexes, as well as $m$-cluster tilting subcategories.

On faithfully balanced modules, F-cotilting and F-Auslander algebras

The aim of this paper is to give a unification of the theory of (co)tilting modules and (higher) Auslander correspondence by using faithfully balanced modules (i.e., faithful modules with the double centralizer property). In the language of relative homological algebra following Auslander-Solberg, a relative version of faithfully balancedness is given, which will be called 1-F-faithful. The relative versions of (higher) Auslander correspondence and Brenner-Butler theorem are also developed.

Infinitely generated pseudocompact modules for finite groups and Weiss' Theorem

One of the most beautiful results in the integral representation theory of finite groups is a theorem of A. Weiss that detects a permutation R-lattice for the finite p-group G in terms of the restriction to a normal subgroup N and the N-fixed points of the lattice, where R is a finite extension of the p-adic integers. Using techniques from relative homological algebra, we generalize Weiss' Theorem to the class of infinitely generated pseudocompact lattices for a finite p-group, allowing R to be any complete discrete valuation ring in mixed characteristic. A related theorem of Cliff and Weiss is also generalized to this class of modules. The existence of the permutation cover of a pseudocompact module is proved as a special case of a more general result. The permutation cover is explicitly described.

The lower extension groups and quotient categories

For certain full additive subcategories X of an additive category A, one defines the lower extension groups in relative homological algebra. We show that these groups are isomorphic to the suspended Hom groups in the Verdier quotient category of the bounded homotopy category of A by that of X. Alternatively, these groups are isomorphic to the negative cohomology groups of the Hom complexes in the dg quotient category A/X, where both A and X are viewed as dg categories concentrated in degree zero.

Tilting objects in triangulated categories

Based on Beligiannis’s theory in [Beligiannis, A. (2000). Relative homological algebra and purity in triangulated categories. J. Algebra 227(1):268–361], we introduce and study -tilting objects in a triangulated category, where is a proper class of triangles. We show that each -tilting object cogenerates an -cotorsion pair. Meanwhile, we also achieve some nice characterizations with respect to the -tilting object. As an application, we provide a necessary and sufficient condition for a triangulated category to be -1-Gorenstein. Finally, we give a one to one correspondence between the class of -tilting objects and the class of tilting subcategories in a suitable functor category.

Balanced pairs, cotorsion triplets and quiver representations

AbstractBalanced pairs appear naturally in the realm of relative homological algebra associated with the balance of right-derived functors of the Hom functor. Cotorsion triplets are a natural source of such pairs. In this paper, we study the connection between balanced pairs and cotorsion triplets by using recent quiver representation techniques. In doing so, we find a new characterization of abelian categories that have enough projectives and injectives in terms of the existence of complete hereditary cotorsion triplets. We also provide a short proof of the lack of balance for derived functors of Hom computed using flat resolutions, which extends the one given by Enochs in the commutative case.

Balance for Relative (Co)Homology in Abelian Categories

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.

Duality pairs and stable module categories

Let R be a commutative ring. We show that any complete duality pair gives rise to a theory of relative homological algebra, analogous to Gorenstein homological algebra. Indeed Gorenstein homological algebra over a commutative Noetherian ring of finite Krull dimension can be recovered from the duality pair (F,I) where F is the class of flat R-modules and I is the class of injective R-modules. For a general R, the AC-Gorenstein homological algebra of Bravo–Gillespie–Hovey is the one coming from the duality pair (L,A) where L is the class of level R-modules and A is class of absolutely clean R-modules. Indeed we show here that the work in [1] can be extended to obtain similar abelian model structures on R-Mod from any a complete duality pair (L,A). It applies in particular to the original duality pairs constructed by Holm–Jørgensen.

STRATIFYING SYSTEMS FOR EXACT CATEGORIES

AbstractIn this paper, we develop the theory of stratifying systems in the context of exact categories as a generalisation of the notion of stratifying systems in module categories, which have been studied by different authors. We prove that attached to a stratifying system in an exact category $(\mathcal{A},\mathcal{E})$ there is an standardly stratified algebra B such that the category $\mathscr{F}$F(Θ), of F-filtered objects in the exact category $(\mathcal{A},\mathcal{E})$ is equivalent to the category $\mathscr{F}$(Δ) of Δ-good modules associated to B. The theory we develop in exact categories, give us a way to produce standardly stratified algebras from module categories by just changing the exact structure on it. In this way, we can construct exact categories whose bounded derived category is equivalent to the bounded derived category of an standardly stratified algebra. Finally, applying the relative homological algebra developed by Auslander–Solberg, we can construct examples of stratifying systems that are not a stratifying system in the classical sense, so our approach really produces new stratifying systems.

A categorical perspective on the Atiyah–Segal completion theorem in KK-theory

We investigate the homological ideal \mathcal J _G^H , the kernel of the restriction functors in compact Lie group equivariant Kasparov categories. Applying the relative homological algebra developed by Meyer and Nest, we relate the Atiyah–Segal completion theorem with the comparison of \mathcal J_G^H with the augmentation ideal of the representation ring. In relation to it, we study on the Atiyah–Segal completion theorem for groupoid equivariant KK-theory, McClure's restriction map theorem and permanence property of the Baum–Connes conjecture under extensions of groups.

Auslander–Gorenstein algebras and precluster tilting

We generalize the notions of n-cluster tilting subcategories and τ-selfinjective algebras into n-precluster tilting subcategories and τn-selfinjective algebras, where we show that a subcategory naturally associated to n-precluster tilting subcategories has a higher Auslander–Reiten theory. Furthermore, we give a bijection between n-precluster tilting subcategories and n-minimal Auslander–Gorenstein algebras, which is a higher dimensional analog of Auslander–Solberg correspondence [8] as well as a Gorenstein analog of n-Auslander correspondence [22]. The Auslander–Reiten theory associated to an n-precluster tilting subcategory is used to classify the n-minimal Auslander–Gorenstein algebras into four disjoint classes. Our method is based on relative homological algebra due to Auslander–Solberg.

Relative homological algebra and Waldhausen $K$-theory

Relative homological algebra and Waldhausen $K$-theory

Relative Homological Algebra via Truncations

To do homological algebra with unbounded chain complexes one needs to first find a way of constructing resolutions. Spaltenstein solved this problem for chain complexes of R -modules by truncating further and further to the left, resolving the pieces, and gluing back the partial resolutions. Our aim is to give a homotopy theoretical interpretation of this procedure, which may be extended to a relative setting. We work in an arbitrary abelian category \mathcal{A} and fix a class of “injective objects” \mathcal{I} . We show that Spaltenstein's construction can be captured by a pair of adjoint functors between unbounded chain complexes and towers of non-positively graded ones. This pair of adjoint functors forms what we call a Quillen pair and the above process of truncations, partial resolutions, and gluing, gives a meaningful way to resolve complexes in a relative setting up to a split error term. In order to do homotopy theory, and in particular to construct a well behaved relative derived category D(\mathcal{A};\mathcal{I}) , we need more: the split error term must vanish. This is the case when \mathcal I is the class of all injective R -modules but not in general, not even for certain classes of injectives modules over a Noetherian ring. The key property is a relative analogue of Roos's AB4 ^\ast - n axiom for abelian categories. Various concrete examples such as Gorenstein homological algebra and purity are also discussed.

Gorenstein FI-flat dimension and relative homology

In this paper, we study some properties and behavior of finite Gorenstein FI-flat dimension through the methods of relative homological algebra.

Finiteness conditions and cotorsion pairs

We study the interplay between the notions of n-coherent rings and finitely n-presented modules, and also study the relative homological algebra associated to them. We show that the n-coherency of a ring is equivalent to the thickness of the class of finitely n-presented modules. The relative homological algebra part comes from the study of orthogonal complements to this class of modules with respect to the Ext and Tor functors. We also construct cotorsion pairs from these orthogonal complements, allowing us to provide further characterizations of n-coherent rings.

Gorenstein Cotilting and Tilting Modules

Auslander and Solberg introduced the concepts of finitely generated cotilting and tilting modules in relative homological algebra considering subfunctors of the Ext-functor. In this article we generalize Auslander–Solberg relative notions by giving the definitions of infinitely generated Gorenstein cotilting and tilting modules by means of Gorenstein exact sequences. Using the theory developed by Enochs on the existence of Gorenstein preenvelopes and precovers, we prove a characterization of relative Gorenstein cotilting and tilting modules, which is a generalization of the beautiful characterization of relative cotilting and tilting modules given by Bazzoni.