arXiv subjects

Rachid El Maaouy

Publications and source records attributed to Rachid El Maaouy.

5 recordsLinked to original sources

arXiv2026

Approximations and Hovey triples by objects of finite homological dimensions: Applications to sheaves

Let $\mathcal{Q}$ be a class of objects in an abelian category $\mathcal{A}$ which need not have enough projective or injective objects. In this paper, we prove that if $\mathcal{Q}$ is the first class of a Hovey triple $(\mathcal{Q},\mathcal{W},\mathcal{R})$ in $\mathcal{A}$ satisfying certain assumptions-weaker than those required in the recent literature-then $\mathcal{Q}_n$, the class of objects with $\mathcal{Q}$-resolution dimension at most an integer $n\ge 0$, forms the first class of a hereditary Hovey triple $\mathcal{M}_n=(\mathcal{Q}_n,\mathcal{W}_{\mathcal{Q},n},\mathcal{R}_{\mathcal{Q},n})$, where $\mathcal{W}_{\mathcal{Q},n}$ and $\mathcal{R}_{\mathcal{Q},n}$ are described explicitly. Consequently, $\mathcal{Q}_n$ is the left-hand side of a complete hereditary cotorsion pair and hence a special precovering class. The dual statement is also established. As a main application, we construct an abelian model structure on $Qcoh(X)$, the category of quasi-coherent sheaves over a semi-separated Noetherian scheme $X$, in which the cofibrant (resp. fibrant) objects are precisely the sheaves with Gorenstein flat (resp. Gorenstein injective) dimension at most $n$.

math.AG↗

arXiv2024

Model structures, n-Gorenstein flat modules and PGF dimensions

Given a non-negative integer $n$ and a ring $R$ with identity, we construct an abelian model structure on the category of left $R$-modules where the class of cofibrant objects coincides with $\mathcal{GF}_n(R)$ the class of left $R$-modules with Gorenstein flat dimension less than $n$, the class of fibrant objects coincides with $\mathcal{F}_n(R)^\perp$ the right ${\rm Ext}$-orthogonal class of left $R$-modules with flat dimension less than $n$, and the class of trivial objects coincides with $\mathcal{PGF}(R)^\perp$ the right ${\rm Ext}$-orthogonal class of PGF left $R$-modules recently introduced by Šaroch and Šťov\'ıček. The homotopy category of this model structure is triangulated equivalent to the stable category $\underline{\mathcal{GF}(R)\cap\mathcal{C}(R)}$ modulo flat-cotorsion modules and it is compactly generated when $R$ has finite global Gorenstein AC-projective dimension. The second part of this paper deals with the PGF dimension of modules and rings. Our results suggest that this dimension could serve as an alternative definition of the Gorenstein projective dimension. We show, among other things, that ($n$-)perfect rings can be characterized in terms of Gorenstein homological dimensions, similar to the classical ones, and the global Gorenstein projective dimension coincides with the global PGF dimension.

math.RA↗

arXiv2024

Relative PGF modules and dimensions

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 $\mathcal{B}_C(R)$.

math.RA↗

arXiv2024

Relative Gorenstein flat modules and Foxby classes and their model structures

A model structure on a category is a formal way of introducing a homotopy theory on that category, and if the model structure is abelian and hereditary, its homotopy category is known to be triangulated. So a good way to both build and model a triangulated category is to build a hereditary abelian model structure. Given a ring $R$ and a (non necessarily semidualizing) left $R$-module $C$, we introduce and study new concepts of relative Gorenstein cotorsion and cotorsion modules: $\rm G_C$-cotorsion and (strongly) $\mathcal{C}_C$-cotorsion. As an application, we prove that there is a unique hereditary abelian model structure on the category of left $R$-modules, in which the cofibrations are the monomorphisms with $\rm G_C$-flat cokernel and the fibrations are the epimorphisms with $\mathcal{C}_C$-cotorsion kernel belonging to the Bass class $\mathcal{B}_C(R)$. In the second part, when $C$ is a semidualizing $(R,S)$-bimodule, we investigate the existence of abelian model structures on the category of left (resp., right) $R$-modules where the cofibrations are the epimorphisms (resp., monomorphisms) with kernel (resp., cokernel) belonging to the Bass (resp., Auslander) class $\mathcal{B}_C(R)$ (resp., $\mathcal{A}_C(R)$). We also study the class of $\rm G_C$-flat modules and the Bass class from the Auslander-Buchweitz approximation theory point of view. We show that they are part of weak AB-contexts. As the concept of weak AB-context can be dualized, we also give dual results that involve the class of $\rm G_C$-cotorsion modules and the Auslander class.

math.RA↗

arXiv2021

Relative Gorenstein dimensions over triangular matrix rings

Let $A$ and $B$ be rings, $U$ a $(B,A)$-bimodule and $T=\begin{pmatrix} A&0\\U&B \end{pmatrix}$ the triangular matrix ring. In this paper, several notions in relative Gorenstein algebra over a triangular matrix ring are investigated. We first study how to construct w-tilting (tilting, semidualizing) over $T$ using the corresponding ones over $A$ and $B$. We show that when $U$ is relative (weakly) compatible we are able to describe the structure of $G_C$-projective modules over $T$. As an application, we study when a morphism in $T$-Mod has a special $G_CP(T)$-precover and when the class $G_CP(T)$ is a special precovering class. In addition, we study the relative global dimension of $T$. In some cases, we show that it can be computed from the relative global dimensions of $A$ and $B$. We end the paper with a counterexample to a result that characterizes when a $T$-module has a finite projective dimension.

math.RA↗