Related papers: Cotorsion pairs associated with Auslander categori…
Let $R$ and $S$ be any rings and $_RC_S$ a semidualizing bimodule, and let $\mathcal{A}_C(R^{op})$ and $\mathcal{B}_C(R)$ be the Auslander and Bass classes respectively. Then both the pairs $$(\mathcal{A}_C(R^{op}),\mathcal{B}_C(R))\ {\rm…
We introduce the notion of a duality pair and demonstrate how the left half of such a pair is often covering and preenveloping. As an application, we generalize a result by Enochs et al. on Auslander and Bass classes, and we prove that the…
As a dual of the Auslander transpose of modules, we introduce and study the cotranspose of modules with respect to a semidualizing module $C$. Then using it we introduce $n$-$C$-cotorsionfree modules, and show that $n$-$C$-cotorsionfree…
We extend the definition of a semidualizing module to associative rings. This enables us to define and study Auslander and Bass classes with respect to a semidualizing bimodule C. We then study the classes of C-flats, C-projectives, and…
We study homological behavior of modules satisfying the Auslander condition. Assume that $\mathcal{AC}$ is the class of left $R$-modules satisfying the Auslander condition. It is proved that each cycle of an exact complex with each term in…
We continue our work on adic semidualizing complexes over a commutative noetherian ring $R$ by investigating the associated Auslander and Bass classes (collectively known as Foxby classes), following Foxby and Christensen. Fundamental…
For a semidualizing module $C$ over a ring $R$, we study the following classes modulo exact zero divisors: $\g_C$--projectives, $\mathcal G_C$; the Auslander class $\mathcal A_C$; the Bass class $\mathcal B_C$; $\mathcal{P}_C$--projective;…
Let $R$ and $S$ be rings and $_R\omega_S$ a semidualizing bimodule. We prove that there exists a Morita equivalence between the class of $\infty$-$\omega$-cotorsionfree modules and a subclass of the class of $\omega$-adstatic modules. Also…
We define the symmetric Auslander category A^s(R) to consist of complexes of projective modules whose left- and right-tails are equal to the left- and right tails of totally acyclic complexes of projective modules. The symmetric Auslander…
For a dualizing module $D$ over a commutative Noetherian ring $R$ with identity, it is known that its Auslander class $\mathscr{A}_D\left(R\right)$ (respectively, Bass class $\mathscr{B}_D\left(R\right)$) is characterized as those…
We use the theory of Auslander Buchweitz approximations to classify certain resolving subcategories containing a semidualizing or a dualizing module. In particular, we show that if the ring has a dualizing module, then the resolving…
We introduce and study "quasidualizing" modules. An artinian R-module T is quasidualizing if the homothety map \hat R\rightarrow Hom(T,T) is an isomorphism and Ext_R^i(T,T)=0 for each integer i>0. Quasidualizing modules are associated to…
We study the properties of rings satisfying Auslander-type conditions. If an artin algebra $\Lambda$ satisfies the Auslander condition (that is, $\Lambda$ is an $\infty$-Gorenstein artin algebra), then we construct two kinds of…
Auslander-Reiten duality for module categories is generalised to Grothendieck abelian categories that have a sufficient supply of finitely presented objects. It is shown that Auslander-Reiten duality amounts to the fact that the functor…
We provide a complete classification of all tilting modules and tilting classes over almost perfect domains, which generalizes the classifications of tilting modules and tilting classes over Dedekind and 1-Gorenstein domains. Assuming the…
We show that if a class of modules is closed under pure quotients, then it is precovering if and only if it is covering, and this happens if and only if it is closed under direct sums. This is inspired by a dual result by Rada and…
Torsion pairs in the category of finitely presented modules over a noetherian ring can be parametrised by the class of cosilting modules. In this paper, we characterise such modules in terms of their indecomposable summands, providing a new…
In this paper, we introduce and study relative Auslander--Gorenstein pairs. This consists of a finite-dimensional Gorenstein algebra together with a self-orthogonal module that provides a further homological feature of the algebra in terms…
We 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…
As a special case of Bass' theory of perfect rings, one obtains the assertion that, over a finite-dimensional associative algebra over a field, all flat modules are projective. In this paper we prove the following relative version of this…