The phantom cover of a module

A morphism of left R-modules is a phantom morphism if for any morphism , with A finitely presented, the composition fg factors through a projective module. Equivalently, Tor₁(X,f)=0 for every right R-module X. It is proved that every R-module possesses a phantom cover, whose kernel is pure injective.If is the category of finitely presented right R-modules modulo projectives, then the association M?Tor₁(−,M) is a functor from the category of left R-modules to that of the flat functors on . The phantom cover is used to characterize when this functor is faithful or full. It is faithful if and only if the flat cover of every module has a pure injective kernel; this is equivalent to the flat cover being the phantom cover. The question of fullness is only reasonable when the functor is restricted to the subcategory of cotorsion modules. This restriction is full if and only if every phantom cover of a cotorsion module is pure injective.     

Gorenstein injective and strongly cotorsion modules

By investigating the properties of some special covers and envelopes of modules, we prove that if R is a Gorenstein ring with the injective envelope of _R R flat, then a left R-module is Gorenstein injective if and only if it is strongly cotorsion, and a right R-module is Gorenstein flat if and only if it is strongly torsionfree. As a consequence, we get that for an Auslander-Gorenstein ring R, a left R-module is Gorenstein injective (resp. flat) if and only if it is strongly cotorsion (resp. torsionfree).     

Flat and Cotorsion Quasi-Coherent Sheaves. Applications

The study of flat covers and cotorsion envelopes has turned out to be very useful since their existence was proved in [3] for the category of R-modules. The problem is even more interesting in categories of sheaves on a topological space, because these categories do not have enough projectives. But the existence of flat covers and cotorsion envelopes allow us to form flat and cotorsion resolutions to compute cohomology.     

Reflexive modules on normal surface singularities and representations of the local fundamental group

We consider the Riemann-Hilbert correspondence on the complement of a normal surface singularity (X,x). Through a closure operation we obtain a correspondence between the category of finite dimensional representations of the local fundamental group and the category of left D_X,x-modules that are reflexive as O_X,x-modules. We show that under this correspondence profinite representations correspond to invariant modules and that these admit a canonical structure as left D_X,x-modules. We prove that the fundamental module is an invariant module if and only if (X,x) is a quotient singularity. Finally we investigate some algebraisation aspects.     

Injective Envelopes and (Gorenstein) Flat Covers

We characterize left Noetherian rings in terms of the duality property of injective preenvelopes and flat precovers. For a left and right Noetherian ring R, we prove that the flat dimension of the injective envelope of any (Gorenstein) flat left R-module is at most the flat dimension of the injective envelope of _R R. Then we get that the injective envelope of _R R is (Gorenstein) flat if and only if the injective envelope of every Gorenstein flat left R-module is (Gorenstein) flat, if and only if the injective envelope of every flat left R-module is (Gorenstein) flat, if and only if the (Gorenstein) flat cover of every injective left R-module is injective, and if and only if the opposite version of one of these conditions is satisfied.     

The derived category of quasi-coherent sheaves and axiomatic stable homotopy

We prove in this paper that for a quasi-compact and semi-separated (nonnecessarily noetherian) scheme X, the derived category of quasi-coherent sheaves over X, D(Aqc(X)), is a stable homotopy category in the sense of Hovey, Palmieri and Strickland, answering a question posed by Strickland. Moreover we show that it is unital and algebraic. We also prove that for a noetherian semi-separated formal scheme X, its derived category of sheaves of modules with quasi-coherent torsion homologies Dqct(X) is a stable homotopy category. It is algebraic but if the formal scheme is not a usual scheme, it is not unital, therefore its abstract nature differs essentially from that of the derived category Dqc(X) (which is equivalent to D(Aqc(X))) in the case of a usual scheme.     

On Strongly Copure Flat Modules and Copure Flat Dimensions

Let R be a left coherent ring. We first prove that a right R-module M is strongly copure flat if and only if Ext ⁱ (M, C) = 0 for all flat cotorsion right R-modules C and i ≥ 1. Then we define and investigate copure flat dimensions of left coherent rings. Finally, we give some new characterizations of n-FC rings.     

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(R-Mod)_([G F,C]) 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(R-Mod)_([G F,C])/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.     

COTORSION MODULES AND PROJECTIVE DIMENSION ONE

If T is a perfect torsion theory for a category of modules over a commutative ring R, a module C is called T—cotorsion provided Hom_R(Q_T,C) = 0 = Ext_R (Q_T,C) where Q_T denotes the T-injective hull of R. Motivated by the now classical results of D. K. Harrison for abelian groups and of E. Matlis for modules over a domain, the theory of T—cotorsion modules is extended. For example, a category equivalence is obtained between the category of T—compact T-cotorsion modules and the category of T-torsion T-reduced modules. The class of T-divisible modules (homomorphic images of T-injective modules) is shown to be closed under formation of extensions if and only if pd_RQ_T ≤ 1, in the case that Q_T is T—cocritical.     

IFP-Flat Modules and IFP-Injective Modules

In this article, we introduce the concept of IFP-flat (resp., IFP-injective) modules as nontrivial generalization of flat (resp., injective) modules. We investigate the properties of these modules in various ways. For example, we show that the class of IFP-flat (resp., IFP-injective) modules is closed under direct products and direct sums. Therefore, the direct product of flat modules is not flat in general; however, the direct product of flat modules is IFP-flat over any ring. We prove that (^⊥??, ??) is a complete cotorsion theory and (??, ??^⊥) is a perfect cotorsion theory, where ?? stands for the class of all IFP-injective left R-modules, and ?? denotes the class of all IFP-flat right R-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.     

Characterization of Quasi-Coherent Modules that are Module Schemes

Let R be a commutative ring with unit, and let E be an R-module. We say the functor of R-modules E, defined by E(B) = E ? _R B, is a quasi-coherent R-module, and its dual E* is an R-module scheme. Both types of R-module functors are essential for the development of the theory of the linear representations of an affine R-group. We prove that a quasi-coherent R-module E is an R-module scheme if and only if E is a projective R-module of finite type, and, as a consequence, we also characterize finitely generated projective R-modules.     

Cofiniteness conditions, projective covers and the logarithmic tensor product theory

We construct projective covers of irreducible V-modules in the category of grading-restricted generalized V-modules when V is a vertex operator algebra satisfying the following conditions: 1. V is C₁-cofinite in the sense of Li. 2. There exists a positive integer N such that the differences between the real parts of the lowest conformal weights of irreducible V-modules are bounded by N and such that the associative algebra A_N(V) is finite dimensional. This result shows that the category of grading-restricted generalized V-modules is a finite abelian category over C. Using the existence of projective covers, we prove that if such a vertex operator algebra V also satisfies Condition 3 that irreducible V-modules are R-graded and C₁-cofinite in the sense of the author, then the category of grading-restricted generalized V-modules is closed under operations ?_P(z) for z∈C^×. We also prove that other conditions for applying the logarithmic tensor product theory developed by Lepowsky, Zhang and the author hold. Consequently, for such V, this category has a natural structure of braided tensor category. In particular, when V is of positive energy and C₂-cofinite, Conditions 1-3 are satisfied and thus all the conclusions hold.     

FP-Injective Complexes

In this article, FP-injective complexes are introduced and studied. We prove that (^⊥??, ??) is a hereditary cotorsion theory if and only if R is a left coherent ring, where ?? denotes the class of all FP-injective complexes of left R-modules. Simultaneously, we study the existence of FP-injective preenvelopes and FP-injective covers.     

Brauer–Thrall for Totally Reflexive Modules over Local Rings of Higher Dimension

Let R be a commutative Noetherian local ring. Assume that R has a pair {x,y} of exact zerodivisors such that dim R/(x,y)?≥?2 and all totally reflexive R/(x)-modules are free. We show that the first and second Brauer–Thrall type theorems hold for the category of totally reflexive R-modules. More precisely, we prove that, for infinitely many integers n, there exists an indecomposable totally reflexive R-module of multiplicity n. Moreover, if the residue field of R is infinite, we prove that there exist infinitely many isomorphism classes of indecomposable totally reflexive R-modules of multiplicity n.     

Cotorsion Theories Cogenerated by a Torsionfree Class

Let R be a right perfect ring, and let (?, 𝒞) be a cotorsion theory in the category of right R-modules ? _R . In this article, it is shown that every right R-module has a superfluous ?-cover if and only if there exists a torsion theory (𝒜, ?) such that (?, 𝒞) is cogenerated by ?. It is also proved that if (𝒜, ?) is a cosplitting torsion theory, then (^⊥?, (^⊥?)^⊥) is a hereditary and complete cotorsion theory, and if (𝒜, ?) is a centrally splitting torsion theory, then (^⊥?, (^⊥?)^⊥) is a hereditary and perfect cotorsion theory.     

Injective dimension of sheaves of rational vector spaces

The Cantor–Bendixson rank of a topological space X is a measure of the complexity of the topology of X. We will be interested primarily in the case that the space is profinite: Hausdorff, compact and totally disconnected. In this paper, we prove that the injective dimension of the abelian category of sheaves of $\mathbb{Q}$-modules over a profinite space X is determined by the Cantor–Bendixson rank of X.     

Applications of balanced pairs

Let (X, Y) be a balanced pair in an abelian category. We first introduce the notion of cotorsion pairs relative to (X, Y), and then give some equivalent characterizations when a relative cotorsion pair is hereditary or perfect. We prove that if the X-resolution dimension of Y (resp. Y-coresolution dimension of X) is finite, then the bounded homotopy category of Y (resp. X) is contained in that of X (resp. Y). As a consequence, we get that the right X-singularity category coincides with the left Y-singularity category if the X-resolution dimension of Y and the Y-coresolution dimension of X are finite.     

When the heart of a faithful torsion pair is a module category

An abelian category with arbitrary coproducts and a small projective generator is equivalent to a module category (Mitchell (1964) [17]). A tilting object in an abelian category is a natural generalization of a small projective generator. Moreover, any abelian category with a tilting object admits arbitrary coproducts (Colpi et al. (2007) [8]). It naturally arises the question when an abelian category with a tilting object is equivalent to a module category. By Colpi et al. (2007) [8], the problem simplifies in understanding when, given an associative ring R and a faithful torsion pair (X,Y) in the category of right R-modules, the heartH(X,Y)of the t-structure associated with (X,Y) is equivalent to a category of modules. In this paper, we give a complete answer to this question, proving necessary and sufficient conditions on (X,Y) for H(X,Y) to be equivalent to a module category. We analyze in detail the case when R is right artinian.