Inspired in part by recent work of Šaroch and Šťov\'ıček in the setting of Gorenstein homological algebra, we extend the notion of Foxby-Golod ${\rm G_C}$-dimension of finitely generated modules with respect to a semidualizing module $C$ to arbitrary modules over arbitrary rings, with respect to a module $C$ that is not necessarily semidualizing. We call this dimension ${\rm PG_CF}$ dimension and show that it can serve as an alternative definition of the ${\rm G_C}$-projective dimension introduced by Holm and Jørgensen. Modules with ${\rm PG_CF}$ dimension zero are called ${\rm PG_CF}$ modules. When the module $C$ is nice enough, we show that the class ${\rm PG_CF}(R)$ of these modules is projectively resolving. This enables us to obtain good homological properties of this new dimension. We also show that ${\rm PG_CF}(R)$ is the left-hand side of a complete hereditary cotorsion pair. This yields, from a homotopical perspective, a hereditary Hovey triple where the cofibrant objects coincide with the ${\rm PG_CF}$ modules and the fibrant objects coincide with the modules in the well-known Bass class $\mathcal{B}_C(R)$.
We give necessary and sufficient conditions in order for the class of projectively coresolved Gorenstein flat modules, $\mathcal{PGF}$, (respectively that of projectively coresolved Gorenstein $\mathcal{B}$ flat modules, $\mathcal{PGF}_{\mathcal{B}}$) to coincide with the class of Ding projective modules ($\mathcal{DP})$. We show that $\mathcal{PGF} = \mathcal{DP}$ if and only if every Ding projective module is Gorenstein flat. This is the case if the ring $R$ is coherent for example. We include an example to show that the coherence is a sufficient, but not a necessary condition in order to ha…
In this paper, we study the relative homological dimension based on the class of projectively coresolved Gorenstein flat modules (PGF-modules), that were introduced by Saroch and Stovicek. The resulting PGF-dimension of modules has several properties in common with the Gorenstein projective dimension, the relative homological theory based on the class of Gorenstein projective modules. In particular, there is a hereditary Hovey triple in the category of modules of finite PGF-dimension, whose associated homotopy category is triangulated equivalent to the stable category of PGF-modules. Studying …
We introduce the notion of relative singularity category with respect to any self-orthogonal subcategory $ω$ of an abelian category. We introduce the Frobenius category of $ω$-Cohen-Macaulay objects, and under some reasonable conditions, we show that the stable category of $ω$-Cohen-Macaulay objects is triangle-equivalent to the relative singularity category. As applications, we relate the stable category of (unnecessarily finitely-generated) Gorenstein-projective modules with singularity categories of rings. We prove that for a Gorenstein ring, the stable category of Gorenstein-projective mod…
We revisit faithfully balanced modules. These are faithful modules having the double centralizer property. For finite-dimensional algebras our main tool is the category ${\rm cogen}^1(M)$ of modules with a copresentation by summands of finite sums of $M$ on which ${\rm Hom}(-,M)$ is exact. For a faithfully balanced module $M$ the functor ${\rm Hom}(-,M)$ is a duality on these categories - for cotilting modules this is the Brenner-Butler theorem. We also study new classes of faithfully balanced modules combining cogenerators and cotilting modules. Then we turn to relative homological algebra in…
Let $\mathcal{T}$ be a Krull-Schmidt, Hom-finite triangulated category with suspension functor $[1]$. Let $R$ be a basic rigid object, $Γ$ the endomorphism algebra of $R$, and $\operatorname{\mathsf{pr}}(R)\subseteq \mathcal{T}$ the subcategory of objects finitely presented by $R$. We investigate the relative rigid objects, \ie $R[1]$-rigid objects of $\mathcal{T}$. Our main results show that the $R[1]$-rigid objects in $\operatorname{\mathsf{pr}}(R)$ are in bijection with $τ$-rigid $Γ$-modules, and the maximal $R[1]$-rigid objects with respect to $\operatorname{\mathsf{pr}}(R)$ are in bijecti…
For a Noetherian ring $R$ and a cotilting $R$-module $T$ of injective dimension at least $1$, we prove that the derived dimension of $R$ with respect to the category $\mathcal{X}_T$ is precisely the injective dimension of $T$ by applying Auslander-Buchweitz theory and Ghost Lemma. In particular, when $R$ is a commutative Noetherian local ring with a canonical module $ω_R$ and $\dim R\ge1$, the derived dimension of R with respect to the category of maximal Cohen-Macaulay modules is precisely $\dim R$.
Let $R$ be any ring. We prove that all direct products of flat right $R$-modules have finite flat dimension if and only if each finitely generated left ideal of $R$ has finite projective dimension relative to the class of all $\mathcal F$-Mittag-Leffler left $R$-modules, where $\mathcal F$ is the class of all flat right $R$-modules. In order to prove this theorem, we obtain a general result concerning global relative dimension. Namely, if $\mathcal X$ is any class of left $R$-modules closed under filtrations that contains all projective modules, then $R$ has finite left global projective dimen…
We investigate purities determined by classes of finitely presented modules including the correspondence between purities for left and right modules. We show some cases where purities determined by matrices of given sizes are different. Then we consider purities over finite-dimensional algebras, giving a general description of the relative pure injectives which we make completely explicit in the case of tame hereditary algebras.
In this paper, we prove a Maschke type theorem for the category of relative Hom-Hopf modules. In fact, we give necessary and sufficient conditions for the functor that forgets the $(H, \a)$-coaction to be separable. This leads to a generalized notion of integrals.
Let $H$ be a Hopf algebra over a field $k$, and $A$ an $H$-comodule algebra. The categories of comodules and relative Hopf modules are then Grothendieck categories with enough injectives. We study the derived functors of the associated Hom functors, and of the coinvariants functor, and discuss spectral sequences that connect them. We also discuss when the coinvariants functor preserves injectives.
Let $R$ be an arbitrary ring with identity and $M$ a right $R$-module with $S=$ End$_R(M)$. Let $Z_2(M)$ be the second singular submodule of $M$. In this paper, we define Goldie Rickart modules by utilizing the endomorphisms of a module. The module $M$ is called Goldie Rickart if for any $f\in S$, $f^{-1}(Z_2(M))$ is a direct summand of $M$. We provide several characterizations of Goldie Rickart modules and study their properties. Also we present that semisimple rings and right $Σ$-$t$-extending rings admit some characterizations in terms of Goldie Rickart modules.
In this work we introduce a new concept, namely, $τ_{s}$-extending modules (rings) which is torsion-theoretic analogues of extending modules and then we extend many results from extending modules to this new concept. For instance we show that for any ring $R$ with unit, if $R$ is purely $τ_{s}$-extending then every cyclic $τ$-nonsingular $R$-module is flat and we show that this fact is true over a principal ideal domain as well. Also, we make a classification for the direct sums of the rings to be purely $τ_{s}$-extending.
Let $k$ be a commutative ring, $H$ a faithfully flat Hopf algebra with bijective antipode, $A$ a $k$-flat right $H$-comodule algebra. We investigate when a relative Hopf module is projective over the subring of coinvariants $B=A^{{\rm co}H}$, and we study the semisimplicity of the category of relative Hopf modules.
Let $S$ and $R$ be rings and $_SC_R$ a (faithfully) semidualizing bimodule. We introduce and study $C$-weak flat and $C$-weak injective modules as a generalization of $C$-flat and $C$-injective modules (J. Math. Kyoto Univ. 47(2007), 781-808) respectively, and use them to provide additional information concerning the important Foxby equivalence between the subclasses of the Auslander class $\mathcal{A}_C(R)$ and that of the Bass class $\mathcal{B}_C(S)$. Then we study the stability of Auslander and Bass classes, which enables us to give some alternative characterizations of the modules in $\ma…
In this article, we present actions by central elements on Hochschild cohomology groups with arbitrary bimodule coefficients, as well as an interpretation of these actions in terms of exact sequences. Since our construction utilises the monoidal structure that the category of bimodules possesses, we will further recognise that these actions are compatible with monoidal functors and thus, as a consequence, are invariant under Morita equivalences. By specialising the bimodule coefficients to the underlying algebra itself, our efforts in particular yield a description of the degree-$(n,0)$-part o…
We use the concept of a regular object with respect to another object in an arbitrary category, defined in \cite{dntd}, in order to obtain the transfer of regularity in the sense of Zelmanowitz between the categories $R-$mod and $S-$mod, when $S$ is an excellent extension of the ring $R$. Consequently, we obtain a result of \cite{ps}: if $S$ is an excellent extension of the ring $R$, then $S$ is von Neumann regular ring if and only if $R$ is also von Neumann regular ring. In the second part, using relative regular modules, we give a new proof of a classical result: the von Neumann regular prop…
In their investigation of horizontal linkage of modules of finite Gorenstein dimension over a commutative, Noetherian, semiperfect (e.g., local) ring, Dibaei and Sadeghi introduced the class of reduced G-perfect modules, making use of Bass' concept of reduced grade. A few years later, the same authors extended this class by considering the relative property of reduced G$_C$-perfection, where $C$ is a semidualizing module, and studied linkage even further. In the present paper, we contribute to their theory and also generalize results of Auslander and Bridger as well as of Martsinkovsky and Str…
A recent result by J. Šaroch and J. Šťov\'ıček asserts that there is a unique abelian model structure on the category of left $R$-modules, for any associative ring $R$ with identity, whose (trivially) cofibrant and (trivially) fibrant objects are given by the classes of Gorenstein flat (resp., flat) and cotorsion (resp., Gorenstein cotorsion) modules. In this paper, we generalise this result to a certain relativisation of Gorenstein flat modules, which we call Gorenstein $\mathcal{B}$-flat modules, where $\mathcal{B}$ is a class of right $R$-modules. Using some of the techniques considered by …
Let $\mathcal{C}$ be a triangulated category with shift functor $[1]$ and $\mathcal{R}$ a rigid subcategory of $\mathcal{C}$. We introduce the notions of two-term $\mathcal{R}[1]$-rigid subcategories, two-term (weak) $\mathcal{R}[1]$-cluster tilting subcategories and two-term maximal $\mathcal{R}[1]$-rigid subcategories, and discuss relationship between them. Our main result shows that there exists a bijection between the set of two-term $\mathcal{R}[1]$-rigid subcategories of $\mathcal{C}$ and the set of $τ$-rigid subcategories of $\mod\mathcal{R}$, which induces a one-to-one correspondence b…
Let (N, G), where N is a normal subgroup of G<SL_n(C), be a pair of finite groups and V a finite-dimensional fundamental G-module. We study the G-invariants in the symmetric algebra S(V) by giving explicit formulas of the Poincaré series for the induced modules and restriction modules. In particular, this provides a uniform formula of the Poincaré series for the symmetric invariants in terms of the McKay-Slodowy correspondence. Moreover, we also derive a global version of the Poincaré series in terms of Tchebychev polynomials in the sense that one needs only the dimensions of the subgroups and…
Let $(H, \a)$ be a monoidal Hom-Hopf algebra and $(A, \b)$ a right $(H, \a)$-Hom-comodule algebra. We first investigate the criterion for the existence of a total integral of $(A, \b)$ in the setting of monoidal Hom-Hopf algebras. Also we prove that there exists a total integral $ϕ: (H, \a)\rightarrow (A, \b)$ if and only if any representation of the pair $(H,A)$ is injective in a functorial way, as a corepresentation of $(H, \a)$, which generalizes Doi's result. Finally, we define a total quantum integral $\g: H\rightarrow Hom(H, A)$ and prove the following affineness criterion: if there exis…
Similar to the idea of relative projectivity, we introduce the notion of relative subprojectivity, which is an alternative way to measure the projectivity of a module. Given modules $M$ and $N$, $M$ is said to be {\em $N$-subprojective} if for every epimorphism $g:B \rightarrow N$ and homomorphism $f:M \rightarrow N$, then there exists a homomorphism $h:M \rightarrow B$ such that $gh=f$. For a module $M$, the {\em subprojectivity domain of $M$} is defined to be the collection of all modules $N$ such that $M$ is $N$-subprojective. A module is projective if and only if its subprojectivity domain…
In this paper, the notion of strongly G_C-projective and injective modules is introduced, where C is a semidualizing module. Using these modules we can obtain a new characterization of G_C-projective and injective modules, similar to the one of projective modules by the free modules. We then define and study the global dimensions of rings relative to a semidualizing module C, and prove that the global G_C-projective dimension of a ring R is equal to the global G_C-injective dimension of R.
We study Martsinkovsky-Russell torsion modules [MaRu20] with pure embeddings as an abstract elementary class. We give a model-theoretic characterization of the pure-injective and the $Σ$-pure-injective modules relative to the class of torsion modules assuming that the torsion submodule is a pure submodule. Our characterization of relative $Σ$-pure-injective modules strictly extends the classical charactetization of [GrJe76] and [Zim, 3.6]. We study the limit models of the class and determine when the class is superstable assuming that the torsion submodule is a pure submodule. As a corollary, …
We extend Morita theory to abelian categories by using wide Morita contexts. Several equivalence results are given for wide Morita contexts between abelian categories, widely extending equivalence theorems for categories of modules and comodules due to Kato, Muller and Berbec. In the case of Grothendieck categories we derive equivalence results by using quotient categories. We apply the general equivalence results to rings with identity, rings with local units, graded rings, Doi-Hopf modules and coalgebras.
We study (relative) $\mathcal K$-Mittag-Leffler modules as was done in the author's habilitation thesis, rephrase old, unpublished results in terms of definable subcategories, and present newer ones, culminating in a characterization of countably generated $\cal K$-Mittag-Leffler modules.
We investigate relative CS-Baer objects in abelian categories in relationship with other relevant classes of objects such as relative Baer objects, extending objects, objects having certain summand intersection properties and relative CS-Rickart objects. Dual results are automatically obtained by applying the duality principle in abelian categories. We also study direct sums of relative CS-Baer objects, and we determine the complete structure of dual self-CS-Baer modules over Dedekind domains. Further applications are given to module categories.
In the last chapter of his book "The Algebraic Theory of Modular Systems " published in 1916, F. S. Macaulay developped specific techniques for dealing with " unmixed polynomial ideals " by introducing what he called " inverse systems ". The purpose of this paper is to extend such a point of view to differential modules defined by linear multidimensional systems, that is by linear systems of ordinary differential (OD) or partial differential (PD) equations of any order, with any number of independent variables, any number of unknowns and even with variable coefficients in a differential field.…
