Cotorsion Dimension Research Articles

Gorenstein (ℒ,𝒜)-flat dimension of complexes and relative singularity categories

Assume that the class of all Gorenstein [Formula: see text]-flat modules is closed under extensions. We define a notion of Gorenstein [Formula: see text]-flat dimension for complexes and consider equivalent characterizations of the finiteness of Gorenstein [Formula: see text]-flat dimension of complexes. Furthermore, in addition, assume that [Formula: see text] is a right coherent ring. We construct the relative singularity category with respect to Gorenstein [Formula: see text]-flat modules, as the triangulated quotient of the triangulated subcategory of [Formula: see text] consisting of all complexes with both finite Gorenstein [Formula: see text]-flat dimension and cotorsion dimension by the bounded homotopy category of flat-cotorsion modules, and show that the relative singularity category is triangulated equivalent to the stable category of the Frobenius category consisting of all Gorenstein [Formula: see text]-flat and cotorsion modules.

$S$-cotorsion modules and dimensions

Let $R$ be a ring, $S$ a multiplicative subset of $R$. An $R$-module $M$ is said to be $u$-$S$-flat ($u$- always abbreviates uniformly) if ${\rm Tor}^R_1 (M, N)$ is $u$-$S$-torsion $R$-module for all $R$-modules $N$. In this paper, we introduce and study the concept of $S$-cotorsion module which is in some way a generalization of the notion of cotorsion module. An $R$-module $M$ is said to be $S$-cotorsion if ${\rm Ext}^1_R(F,M)=0$ for any $u$-$S$-flat module $F$. This new class of modules will be used to characterize $u$-$S$-von Neumann regular rings. Hence, we introduce the $S$-cotorsion dimensions of modules and rings. The relations between the introduced dimensions and other (classical) homological dimensions are discussed. As applications, we give a new upper bound on the global dimension of rings.

The Recollements of Abelian Categories: Cotorsion Dimensions and Cotorsion Triples

The Recollements of Abelian Categories: Cotorsion Dimensions and Cotorsion Triples

Homological Dimensions of Skew Group Rings

Let R be a ring and let H be a subgroup of a finite group G. We consider the weak global dimension, cotorsion dimension and weak Gorenstein global dimension of the skew group ring RσG and its coefficient ring R. Under the assumption that RσG is a separable extension over RσH, it is shown that RσG and RσH share the same homological dimensions. Several known results are then obtained as corollaries. Moreover, we investigate the relationships between the homological dimensions of RσG and the homological dimensions of a commutative ring R, using the trivial RσG-module.

Flat model structures under excellent extensions

We prove that the weak equivalences, cofibrations and fibrations in Gillespie’s flat model structure are invariant under excellent extensions of rings. As applications, the flat and cotorsion dimensions of unbounded complexes along excellent extensions are investigated, and moreover, some homological invariant properties under excellent extensions are studied in the present framework of complexes.

On copure projective and cotorsion modules

Abstract Let R be an IF ring, or be a ring such that each right R-module has a monomorphic flat envelope and the class of flat modules is coresolving. We firstly give a characterization of copure projective and cotorsion modules by lifting and extension diagrams, which implies that the classes of copure projective and cotorsion modules have some balanced properties. Then, a relative right derived functor is introduced to investigate copure projective and cotorsion dimensions of modules. As applications, some new characterizations of QF rings, perfect rings and noetherian rings are given.

Weak Gorenstein Cotorsion Modules

Let R be a ring and let 𝒮𝒢ℱ be the class of strongly Gorenstein at right R-modules. We call a right R-module M a weak Gorenstein cotorsion module if M is in the class 𝒮𝒢ℱ⊥. Properties of weak Gorenstein cotorsion modules are investigated. It is shown that weak Gorenstein cotorsion R-modules over coherent rings are indeed weaker than Gorenstein cotorsion R-modules. Weak Gorenstein cotorsion dimension for modules and rings are also studied.

Relative singularity categories with respect to gorenstein flat modules

Let R be a right coherent ring and D b (R-Mod) the bounded derived category of left R-modules. Denote by \({D^b}{\left( {R - Mod} \right)_{\widehat {\left[ {GF,C} \right]}}}\) the subcategory of D b (R-Mod) consisting of all complexes with both finite Gorenstein flat dimension and cotorsion dimension and K b (F ∩ C) the bounded homotopy category of flat cotorsion left R-modules. We prove that the quotient triangulated category \({D^b}{\left( {R - Mod} \right)_{\widehat {\left[ {GF,C} \right]}}}\)/K b (F ∩ C) is triangle-equivalent to the stable category GF ∩ C of the Frobenius category of all Gorenstein flat and cotorsion left R-modules.

On homological dimensions in some functor categories

In this paper, we investigate the homological properties of the functor categories (mod−R, Ab) and ((mod−R)op, Ab). Some new homological dimensions in these functor categories such as FP-projecive dimensions and cotorsion dimensions for functors and functor categories are introduced and studied. We also characterize functor categories of homological dimensions zero and explore the connections among some different homological dimensions.

Flat quasi-coherent sheaves of finite cotorsion dimension

Let [Formula: see text] be a quasi-compact and semi-separated scheme. If every flat quasi-coherent sheaf has finite cotorsion dimension, we prove that [Formula: see text] is [Formula: see text]-perfect for some [Formula: see text]. If [Formula: see text] is coherent and [Formula: see text]-perfect (not necessarily of finite Krull dimension), we prove that every flat quasi-coherent sheaf has finite pure injective dimension. Also, we show that there is an equivalence [Formula: see text] of homotopy categories, whenever [Formula: see text] is the homotopy category of pure injective flat quasi-coherent sheaves and [Formula: see text] is the pure derived category of flat quasi-coherent sheaves.

Bounded complexes of cotorsion sheaves

ABSTRACTWe provide a necessary and sufficient condition which ensure that every flat quasi-coherent sheaf has finite cotorsion dimension. Also, we will show that every locally noetherian scheme with a dualizing complex has this requirement.

Cotorsion dimensions relative to semidualizing modules

Let [Formula: see text] be a semidualizing [Formula: see text]-module, where [Formula: see text] is a commutative ring. We first introduce the definition of [Formula: see text]-cotorsion modules, and obtain the properties of [Formula: see text]-cotorsion modules. As applications, we give some new characterizations for perfect rings. Second, we study the Foxby equivalences between the subclasses of the Auslander class and that of the Bass class with respect to [Formula: see text]. Finally, we discuss [Formula: see text]-cotorsion dimensions and investigate the transfer properties of strongly [Formula: see text]-cotorsion dimensions under almost excellent extensions.

Notes on cotorsion dimension of Hopf--Galois extensions

Let H be anite dimensional Hopf algebra over aeld k and A=B be a right H -Galois extension. In this note the relationship of cotorsion dimensions between A and B will be studied. We prove that r.cot.D( A) � r.cot.D( B) + l.D( H) . Moreover, we give some sufficient conditions for which r.cot.D( A) = r.cot.D( B) . As applications, we obtain some results about cotorsion dimension of the smash product.

Notes on Gorenstein cotorsion modules

Let R be a ring. An R-module N is called Gorenstein cotorsion module if ExtR1 (F, N) = 0 for any Gorenstein flat R-module F. We investigate some properties of Gorenstein cotorsion envelopes and introduce Gorenstein cotorsion dimension. Some characterizations of Gorenstein cotorsion dimension of modules and rings are given.

ON GORENSTEIN COTORSION DIMENSION OVER GF-CLOSED RINGS

In this article, we introduce and study the Gorenstein cotorsion dimension of modules and rings. It is shown that this dimension has nice properties when the ring in question is left GF-closed. The relations between the Gorenstein cotorsion dimension and other homological dimensions are discussed. Finally, we give some new characterizations of weak Gorenstein global dimension of coherent rings in terms of Gorenstein cotorsion modules.

Cotorsion Dimension of Unbounded Complexes

We extend the cotorsion dimension of R-modules to unbounded R-complexes by applying the flat model structure on Ch(R) proposed by J. Gillespie. This is not natural because there has been no sufficiently general result available for the existence of proper “cotorsion” resolutions of unbounded complexes, for which one would be able to define the derived functors. The global cotorsion dimension of ring is discussed in our present framework, and the relations between it and other dimensions are investigated as well. Some rings are characterized and some known results are extended.

Gorenstein Flat and Cotorsion Dimensions of Unbounded Complexes

In this article, we define and study the Gorenstein flat dimension and Gorenstein cotorsion dimension for unbounded complexes over GF-closed rings by constructions of resolutions of unbounded complexes. The behavior of the dimensions under change of rings is investigated.

Cotorsion dimensions and Hopf algebra actions

Let H be a finite-dimensional Hopf algebra over a field k, and let A be an H-module algebra. In this paper, we discuss the cotorsion dimension of the smash product A # H. We prove that $$l.\cot .D\left( {A\# H} \right) \leqslant l.\cot .D\left( A \right) + r.D\left( H \right),$$ which generalizes the result of group rings. Moreover, we give some sufficient conditions for which $$l.\cot .D\left( {A\# H} \right) = l.\cot .D\left( A \right).$$ As applications, we study the invariants of IF properties and Gorenstein global dimensions.

Excellent Extensions and Global Cotorsion Dimensions

In this paper, we mainly investigate the global cotorsion dimension under the excellent extension of rings. We show that if $S$ is an excellent extension of $R$, then $\mathrm{cot.D}(S)=\mathrm{cot.D}(R)$. Furthermore, some known results, such as Corollary 3.8 and 3.12, can be also obtained as direct corollaries of our theorem.

STRONGLY COTORSION (TORSION-FREE) MODULES AND COTORSION PAIRS

In this paper, strongly cotorsion (torsion-free) modules are studied and strongly cotorsion (torsion-free) dimension is introduced. It is shown that every module has a special <TEX>$\mathcal{SC}_n$</TEX>-preenvelope and an ST <TEX>$\mathcal{F}_n$</TEX>-cover for any <TEX>$n\;{\in}\;\mathbb{N}$</TEX> based on some results of cotorsion pairs from [9]. Some characterizations of strongly cotorsion (torsion-free) dimension of a module are given.