Equivalences between categories of modules and categories of comodules

We show the close connection between apparently different Galois theories for comodules introduced recently in [J. Gomez-Torrecillas and J. Vercruysse, Comatrix corings and Galois Comodules over firm rings, Algebr. Represent. Theory, 10 （2007）, 271 306] and [Wisbauer, On Galois comodules, Comm. Algebra 34 （2006）, 2683-2711]. Furthermore we study equivalences between categories of comodules over a coring and modules over a firm ring. We show that these equivalences are related to Galois theory for comodules.     

On Partial Tilting Complexes

Let R be a commutative noetherian ring and A a noetherian R-algebra. Let P ^? ∈ 𝒦^b(𝒫 _A ) with Hom_𝒦(Mod-A)(P ^?, P ^?[i]) = 0 for i > 0. We will provide a sufficient condition for P ^? to be a direct summand of a silting complex. Also, in case Hom_𝒦(Mod-A)(P ^?, P ^?[i]) = 0 for i ≠ 0, we will provide a sufficient condition for P ^? to be a direct summand of a tilting complex.     

Classification des Objets Galoisiens de U q (𝔤) à Homotopie PrèS

Pour toute algèbre enveloppante quantique U_q(𝔤) de Drinfeld–Jimbo et toute famille λ = (λ_ij)1≤i ∈ k^? d'éléments inversibles du corps de base, nous construisons explicitement par générateurs et relations un objet galoisien A_λ de U_q(𝔤) et nous montrons que tout objet galoisien de U_q(𝔤) est homotope à un unique objet de la forme A_λ.

For any Drinfeld–Jimbo quantum enveloping algebra U_q(𝔤) and for any family λ = (λ_ij)_{1≤i ∈ k^? of invertible elements of the base field, we explicitly construct a Galois object Aλ of Uq(𝔤) by generators and relations and we prove that any Galois object of Uq(𝔤) is homotopic to a unique object of type Aλ.}     

Vertex Poisson Algebras Associated with Courant Algebroids

We study relationships between vertex Poisson algebras and Courant algebroids. For any ?-graded vertex Poisson algebra A = ? _n∈? A _(n), we show that A ₍₁₎ is a Courant A ₍₀₎-algebroid. On the other hand, for any Courant 𝒜-algebroid ?, we construct an ?-graded vertex Poisson algebra A = ? _n∈? A _(n) such that A ₍₀₎ is 𝒜 and the Courant 𝒜-algebroid A ₍₁₎ is isomorphic to ? as a Courant 𝒜-algebroid.     

On Derived Equivalences for Selfinjective Algebras

We show that if A is a representation-finite selfinjective Artin algebra, then every P ^? ? K ^b(𝒫 _A ) with Hom _K(Mod?A)(P ^?,P ^?[i]) = 0 for i ≠ 0 and add(P ^?) = add(νP ^?) is a direct summand of a tilting complex, and that if A, B are derived equivalent representation-finite selfinjective Artin algebras, then there exists a sequence of selfinjective Artin algebras A = B ₀, B ₁,…, B _m  = B such that, for any 0 ≤ i < m, B _i+1 is the endomorphism algebra of a tilting complex for B _i of length ≤ 1.     

On Galois Algebras Satisfying the Fundamental Theorem

Let B be a Galois algebra over a commutative ring R with Galois group G such that B ^H is a separable subalgebra of B for each subgroup H of G. Then it is shown that B satisfies the fundamental theorem if and only if B is one of the following three types: (1) B is an indecomposable commutative Galois algebra, (2) B = Re ⊕ R(1 ? e) where e and 1 ? e are minimal central idempotents in B, and (3) B is an indecomposable Galois algebra such that for each separable subalgebra A, V _B (A) = ?∑ _g∈G(A) J _g , and the centers of A and B ^G(A) are the same where V _B (A) is the commutator subring of A in B, J _g  = {b ∈ B | bx = g(x)b for each x ∈ B} for a g ∈ G, and G(A) = {g ∈ G | g(a) = a for all a ∈ A}.     

Über Die Assoziierten Primideale des Bidualen

Let (R, 𝔪) be a commutative, noetherian, local ring, E the injective hull of the residue field R/𝔪, and M ^○○ = Hom _R (Hom _R (M, E), E) the bidual of an R-module M. We investigate the elements of Ass(M ^○○) as well as those of Coatt(M) = {𝔭 ∈ Spec(R)|𝔭 = Ann _R (Ann _M (𝔭))} and provide criteria for equality in one of the two inclusions Ass(M) ? Ass(M ^○○) ? Coatt(M). If R is a Nagata ring and M a minimax module, i.e., an extension of a finitely generated R-module by an artinian R-module, we show that Ass(M ^○○) = Ass(M) ∪ {𝔭 ∈ Coatt(M)| R/𝔭 is incomplete}.     

Invertibility of Matrices Over Subrings

Given rings R ? S, consider the division closure 𝒟(R, S) and the rational closure ?(R, S) of R in S. If S is commutative, then 𝒟(R, S) = ?(R, S) = RT ^?1, where T = {t ∈ R | t ^?1 ∈ S}. We show that this is also true if we assume only that R is commutative.     

S-Noetherian Rings of the Forms 𝒜[X] and 𝒜[[X]]

Let 𝒜 = (A _n ) _n≥0 be an ascending chain of commutative rings with identity, S ? A ₀ a multiplicative set of A ₀, and let 𝒜[X] (respectively, 𝒜[[X]]) be the ring of polynomials (respectively, power series) with coefficient of degree i in A _i for each i ∈ ?. In this paper, we give necessary and sufficient conditions for the rings 𝒜[X] and 𝒜[[X]] to be S ? Noetherian.     

On the Constructive Inverse Problem in Differential Galois Theory#

We give sufficient conditions for a differential equation to have a given semisimple group as its Galois group. For any group G with G ⁰ = G ₁ · ··· · G _r , where each G _i is a simple group of type A_?, C_?, D_?, E₆, or E₇, we construct a differential equation over C(x) having Galois group G.     

Rings Over which the Krull Dimension and the Noetherian Dimension of All Modules Coincide

We denote by 𝒜(R) the class of all Artinian R-modules and by 𝒩(R) the class of all Noetherian R-modules. It is shown that 𝒜(R) ? 𝒩(R) (𝒩(R) ? 𝒜(R)) if and only if 𝒜(R/P) ? 𝒩(R/P) (𝒩(R/P) ? 𝒜(R/P)), for all centrally prime ideals P (i.e., ab ∈ P, a or b in the center of R, then a ∈ P or b ∈ P). Equivalently, if and only if 𝒜(R/P) ? 𝒩(R/P) (𝒩(R/P) ? 𝒜(R/P)) for all normal prime ideals P of R (i.e., ab ∈ P, a, b normalize R, then a ∈ P or b ∈ P). We observe that finitely embedded modules and Artinian modules coincide over Noetherian duo rings. Consequently, 𝒜(R) ? 𝒩(R) implies that 𝒩(R) = 𝒜(R), where R is a duo ring. For a ring R, we prove that 𝒩(R) = 𝒜(R) if and only if the coincidence in the title occurs. Finally, if Q is the quotient field of a discrete valuation domain R, it is shown that Q is the only R-module which is both α-atomic and β-critical for some ordinals α,β ≥ 1 and in fact α = β = 1.     

Correspondences of Coclosed Submodules

We establish an order-preserving bijective correspondence between the sets of coclosed elements of some bounded lattices related by suitable Galois connections. As an application, we deduce that if M is a finitely generated quasi-projective left R-module with S = End _R (M) and N is an M-generated left R-module, then there exists an order-preserving bijective correspondence between the sets of coclosed left R-submodules of N and coclosed left S-submodules of Hom _R (M, N).     

Pure ideals quotient categories and infinite group-graded rings

Abstract

Adapting the idea of twisted tensor products to the category of conic algebras (CA), i.e., finitely generated graded algebras, we define a family of objects hom ^?[?, 𝒜] there, one for each pair 𝒜, ? ∈ CA, with analogous properties to its internal coHom objects hom [?, 𝒜], but representing spaces of transformations whose coordinate rings and the ones of their respective domains do not commute among themselves. They give rise to a CA ^op -based category different from that defined by the function (𝒜, ?) ?  hom [?, 𝒜]. The mentioned non commutativity is controlled by a collection of twisting maps τ_{𝒜, ?}. We show, under certain circumstances, that (bi)algebras end ^?[𝒜] ?  hom ^?[𝒜, 𝒜] are counital 2-cocycle twistings of the corresponding coEnd objects end [𝒜]. This fact generalizes the twist equivalence (at a semigroup level) between, for instance, the quantum groups G L _q (n) and their multiparametric versions.     

When Essential Extensions of Finitely Generated Modules are Finitely Generated

For certain classes 𝒞 of R-modules, including singular modules or modules with locally Krull dimensions, it is investigated when every module in 𝒞 with a finitely generated essential submodule is finitely generated. In case 𝒞 = Mod-R, this means E(M)/M is Noetherian for any finitely generated module M_R. Rings R with latter property are studied and shown that they form a class 𝒬 properly between the class of pure semisimple rings and the class of certain max rings. Duo rings in 𝒬 are precisely Artinian rings. If R is a quasi continuous ring in 𝒬 then R ? A ⊕ T where A is a semisimple Artinian ring and T ∈ 𝒬 with Z(T_T) ≤_ess T_T.     

Regularity and Substructures of Hom

ABSTRACT

We define “regular” for maps in a Hom group. This notion specializes to the well-known notions of (Von Neumann) regular in rings and modules. A map f ∈ Hom _R (A,M) is regular if and only if Ker(f) ?^⊕ A and Im(f) ?^⊕ M. There exists a unique maximal regular End(M)-End(A)-submodule in Hom _R (A,M). We study regularity in Hom _R (A ₁ ⊕ A ₂, M ₁ ⊕ M ₂). The existence of a regular function Hom _R (A,M) implies the existence of projective summands of Hom _R (A,M) _{End R (A)} and of _{End R ( M )} Hom _R (A,M). We consider regularity in endomorphism rings, and generalize a theorem of Ware-Zelmanowitz. We examine connections between the maximum regular bimodule and other substructures of Hom, mention two generalizations of regularity, and raise some questions.     

Localization of Monoid Objects and Hochschild Homology

Let A be a (not necessarily commutative) monoid object in an abelian symmetric monoidal category (C, ?,1) satisfying certain conditions. In this paper, we continue our study of the localization M _S of any A-module M with respect to a subset S ? Hom _A?Bimod (A, A) that is closed under composition. In particular, we prove the following theorem: if P is an A-bimodule such that P is symmetric as a bimodule over the center Z(A) of A, we have isomorphisms HH _*(A, P) _S  ? HH _*(A, P _S ) ? HH _*(A _S , P _S ) of Hochschild homology groups.     

Purity and Self-Small Groups

Let A be an abelian group. A group B is A-solvable if the natural map Hom(A, B) ?  _E(A) A → B is an isomorphism. We study pure subgroups of A-solvable groups for a self-small group A of finite torsion-free rank. Particular attention is given to the case that A is in , the class of self-small mixed groups G with G/tG? ? ⁿ for some n < ω. We obtain a new characterization of the elements of , and demonstrate that  differs in various ways from the class  ? of torsion-free abelian groups of finite rank despite the fact that the quasi-category ?  is dual to a full subcategory of ?  ?.     

Completely 𝒲-Resolved Complexes

Let R be a ring and 𝒲 a self-orthogonal class of left R-modules which is closed under finite direct sums and direct summands. A complex C of left R-modules is called a 𝒲-complex if it is exact with each cycle Z ⁿ (C) ∈ 𝒲. The class of such complexes is denoted by 𝒞_𝒲. A complex C is called completely 𝒲-resolved if there exists an exact sequence of complexes D _· = … → D _?1 → D ₀ → D ₁ → … with each term D _i in 𝒞_𝒲 such that C = ker(D ₀ → D ₁) and D _· is both Hom(𝒞_𝒲, ?) and Hom(?, 𝒞_𝒲) exact. In this article, we show that C = … → C ^?1 → C ⁰ → C ¹ → … is a completely 𝒲-resolved complex if and only if C ⁿ is a completely 𝒲-resolved module for all n ∈ ?. Some known results are obtained as corollaries.     

Sequentially Cohen-Macaulay Modules Under Base Change

ABSTRACT

Assume that ?:(R, ± 𝔪) → (S, ± 𝔫) is a local flat homomorphism between commutative Noetherian local rings R and S. Let M be a finitely generated R-module. We investigate the ascent and descent of sequentially Cohen-Macaulay properties between the R-module M and the S-module M ? _R  S.