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.     

Torsion-free covers II

This paper continues the study of the existence of torsion-free covers with respect to a faithful hereditary torsion theory (ℑ,F) of left modules over a ringR with unity. If the filter of left ideals associated with (ℑ,F) has a cofinal subset of finitely generated left ideals, then every leftR-module has a torsion-free cover. An example is given to illustrate how this result generalizes all previously known existence theorems for torsion-free covers.     

Relative Flatness,Mittag–Leffler Modules,and Endocoherence

Let M _R be a right R-module over a ring R with S = End(M _R ). We study the coherence of the left S-module _S M relative to a hereditary torsion theory for the category of right R-modules. Various results are developed, many extending known results.     

Preservation of quasi-isomorphisms of complexes

We consider the preservation property of the homomorphism and tensor product functors for quasi-isomorphisms and equivalences of complexes.Let X and Y be two classes of R-modules with Ext ≥1(X,Y) = 0 for each object X ∈X and each object Y ∈Y.We show that if A,B ∈C■(R) are X-complexes and U,V ∈ C■(R) are Y-complexes,then U■V■Hom(A,U)■Hom(A,V);A■B■Hom(B,U)■Hom(A,U).As an application,we give a sufficient condition for the Hom evaluation morphism being invertible.     

Adjoint functors and equivalences of subcategories

For any left R-module P with endomorphism ring S, the adjoint pair of functors P⊗_S− and Hom_R(P,−) induce an equivalence between the categories of P-static R-modules and P-adstatic S-modules. In particular, this setting subsumes the Morita theory of equivalences between module categories and the theory of tilting modules. In this paper we consider, more generally, any adjoint pair of covariant functors between complete and cocomplete Abelian categories and describe equivalences induced by them. Our results subsume the situations mentioned above but also equivalences between categories of comodules.     

On Gorenstein Projective Dimensions of Unbounded Complexes

Let R → S be a ring homomorphism and X be a complex of R-modules. Then the complex of S-modules S?_R~L X in the derived category D(S) is constructed in the natural way. This paper is devoted to dealing with the relationships of the Gorenstein projective dimension of an R-complex X(possibly unbounded) with those of the S-complex S?_R~L X.It is shown that if R is a Noetherian ring of finite Krull dimension and φ : R → S is a faithfully flat ring homomorphism, then for any homologically degree-wise finite complex X, there is an equality Gpd_RX = GpdS(S?_R~L X). Similar result is obtained for Ding projective dimension of the S-complex S?_R~L X.     

Semisimple quotient rings and morita context

Let (U, V; I, J) be a Morita context with the trace ideals I and J, τ a Gabriel topology containing I on R-Mod, and τ′ the corresponding Gabriel topology containing J on S-Mod if τ - d( _RU) < ∞ and R is semisimple, simple respectively, then s is semisimple, simple respectively, moreover they are Morita equivalent, where R S are the rings of quotients of R with respect to τ, resp., and of quotients of s with respect of τ′.     

Torsion-free covers

This paper studies the existence and properties of a torsion-free cover with respect to a faithful hereditary torsion theory (T, F) of modules over a ring with unity. A direct sum of a finite number of torsion-free covers of modules is the torsion-free cover of the direct sum of the modules. The concept of aT-near homomorphism, which generalizes Enochs’ definition of a neat submodule, is introduced and studied. This allows the generalization of a result of Enochs on liftings of homomorphisms. Hereditary torsion theories for which every module has a torsion-free cover are called universally covering. If the inclusion map ofR into the appropriate quotient ringQ is a left localization in the sense of Silver, the problem of the existence of universally-covering torsion theories can be reduced to the caseR=Q. As a consequence, many sufficient conditions for a hereditary torsion theory to be universally covering are obtained. For a universally-covering hereditary torsion theory (T, F), the following conditions are equivalent: (1) the product ofF-neat homomorphisms is alwaysT-neat; (2) the product of torsion-free covers is alwaysT-neat; (3) every nonzero module inT has a nonzero socle.     

Torsion Zero-Cycles on a Product of Canonical Lifts of Elliptic Curves

Let X be a surface over a p-adic field with good reduction and let Y be its special fiber. We write T(X) and T(Y) for the kernels of the Albanese maps of X and Y, respectively. Then, F(X) = T(X)/T(X)_div is conjectured to be finite, where T(X)_div is the maximal divisible subgroup of T(X). Furthermore, F(X) is conjectured to be isomorphic to T(Y) modulo p-primary torsion. We show that the p-primary torsion subgroup of F(X) can be arbitrary large even though we fix the special fiber Y.     

Gorenstein projective dimensions of complexes

We show that over a right coherent left perfect ring R, a complex C of left R-modules is Gorenstein projective if and only if C ^m is Gorenstein projective in R-Mod for all m ∈ ℤ. Basing on this we show that if R is a right coherent left perfect ring then Gpd(C) = sup{Gpd(C ^m )|m ∈ ℤ} where Gpd(−) denotes Gorenstein projective dimension.     

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.     

A module as a torsion-free cover

LetR be a ring, and let (ℐ, ℱ) be an hereditary torsion theory of leftR-modules. An epimorphism ψ:M→X is called a torsion-free cover ofX if (1)M∈ ℱ, (2) every homomorphism from a torsion-free module intoX can be factored throughM, and (3) ker ψ contains no nonzero ℐ -closed submodules ofM. Conditions onM andN are studied to determine when the natural mapsM→M/N andQ(M)→Q(M)/N are torsion-free covers, whenQ(M) is the localization ofM with respect to (ℐ, ℱ). IfM→M/N is a torsion-free cover andM is projective, thenN⊆radM. Consequently, the concepts of projective cover and torsion-free cover coincide in some interesting cases.     

On the comparison theorem for étale cohomology of non-Archimedean analytic spaces

Let ϕ:Y →X be a morphism of finite type between schemes of locally finite type over a non-Archimedean fieldk, and letF be an étale constructible sheaf onY. In [Ber2] we proved that if the torsion orders ofF are prime to the characteristic of the residue field ofk then the canonical homomorphisms (R ^Q ϱ*F)^an →R ^q ϱ _* ^an F ^an are isomorphisms. In this paper we extend the above result to the class of sheavesF with torsion orders prime to the characteristic ofk.     

The quotient field as a torsion-free covering module

R will denote a commutative integral domain with quotient fieldQ. A torsion-free cover of a moduleM is a torsion-free moduleF and anR-epimorphism σ:F→M such that given any torsion-free moduleG and λ∈Hom _R (G, M) there exists μ∈Hom _R (G,F) such that σμ=λ. It is known that ifM is a maximal ideal ofR, R→R/M is a torsion-free cover if and only ifR is a maximal valuation ring. LetE denote the injective hull ofR/M thenR→R/M extends to a homomorphismQ→E. We give necessary and sufficient conditions forQ→E to be a torsion-free cover.     

A WEAKER FORM OF BAER'S SPLITTING PROBLEM OVER VALUATION DOMAINS

Abstract

Eklof-Fuchs [3] have shown that over an arbitrary valuation domain R, the modules B which satisfy Ext 1/R (B,T) = 0 for all torsion R-modules T are precisely the free R-modules. Here we modify the problem and describe all R-modules B for which Ext 1/R (B, T) vanishes for all bounded and for all divisible torsion R-modules T. It is well known that if R is a descrete rank one valuation domain then all torsion—free R-modules B have this property.     

A GENERALIZATION OF AUSLANDER–BUCHSBAUM AND BASS FORMULAS

Abstract

Given a contravariant functor F : 𝒞 → 𝒮ets for some category 𝒞, we say that F (𝒞) (or F) is generated by a pair (X, x) where X is an object of 𝒞 and x ∈ F(X) if for any object Y of 𝒞 and any y ∈ F(Y), there is a morphism f : Y → X such that F(f)(x) = y. Furthermore, when Y = X and y = x, any f : X → X such that F(f)(x) = x is an automorphism of X, we say that F is minimally generated by (X, x). This paper shows that if the ring R is left noetherian, then there exists a minimal generator for the functor ?xt (?, M) : ? → 𝒮ets, where M is a left R-module and ? is the class (considered as full subcategory of left R-modules) of injective left R-modules.     

Complexes of Gorenstein Flat Modules and Gorenstein Cotorsion Modules

A complex (C, δ) is called strongly Gorenstein flat if C is exact and Ker δ ⁿ is Gorenstein flat in R-Mod for all n ∈ ?. Let 𝒮𝒢 stand for the class of strongly Gorenstein flat complexes. We show that a complex C of left R-modules over a right coherent ring R is in the right orthogonal class of 𝒮𝒢 if and only if C ⁿ is Gorenstein cotorsion in R-Mod for all n ∈ ? and Hom^.(G, C) is exact for any strongly Gorenstein flat complex G. Furthermore, a bounded below complex C over a right coherent ring R is in the right orthogonal class of 𝒮𝒢 if and only if C ⁿ is Gorenstein cotorsion in R-Mod for all n ∈ ?. Finally, strongly Gorenstein flat covers and 𝒮𝒢^⊥-envelopes of complexes are considered. For a right coherent ring R, we show that every bounded below complex has a 𝒮𝒢^⊥-envelope.     

Gorenstein flatness and injectivity over Gorenstein rings

Let R be a Gorenstein ring.We prove that if I is an ideal of R such that R/I is a semi-simple ring,then the Gorenstein flat dimension of R/I as a right R-module and the Gorenstein injective dimension of R/I as a left R-module are identical.In addition,we prove that if R→S is a homomorphism of rings and SE is an injective cogenerator for the category of left S-modules,then the Gorenstein flat dimension of S as a right R-module and the Gorenstein injective dimension of E as a left R-module are identical.We also give some applications of these results.