Modules over Discrete Valuation Domains

Крылов, П. А.; Tuganbaev, A. A.

doi:10.1515/9783110205787

Cited by 24 publications

(12 citation statements)

References 255 publications

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“…In particular, R has a unique maximal ideal, namely, RV ; moreover, the quotient ring R/RV is isomorphic to the division ring of fractions of End U (G a ). Thus, R is a discrete valuation domain (not necessarily commutative), as considered in [KT07]. By [DG70, V.3.6.7], a morphism of unipotent group schemes f : G → H is an isogeny if and only if the associated morphism S −1 M(f ) : S −1 M(H) → S −1 M(G) is an isomorphism.…”

Section: 2])mentioning

confidence: 99%

Commutative algebraic groups up to isogeny. II

Brion¹

2018

Representations of Algebras

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This paper develops a representation-theoretic approach to the isogeny category C of commutative group schemes of finite type over a field k, studied in [Br16]. We construct a ring R such that C is equivalent to the category R-mod of all left R-modules of finite length. We also construct an abelian category of R-modules, R-mod, which is hereditary, has enough projectives, and contains R-mod as a Serre subcategory; this yields a more conceptual proof of the main result of [Br16], asserting that C is hereditary. We show that R-mod is equivalent to the isogeny category of commutative quasi-compact k-group schemes.

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Section: 2])mentioning

confidence: 99%

Commutative algebraic groups up to isogeny. II

Brion¹

2018

Representations of Algebras

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“…The ring End S A is a complete topological ring in the finite topology (see [40,Theorem 16.1]). The notion of the finite topology can be expanded to homomorphism groups.…”

Section: Property 15mentioning

confidence: 99%

“…The module with discrete endomorphism ring is self-small [38]. We define the category Walk of Walker (see [40]). Abelian groups are objects of the category Walk, and Hom(A, B)/ Hom A, T(B) is the set of morphisms Hom W (A, B) from A in B.…”

Section: Idempotent Functors and Localizations In The Category Of Abementioning

confidence: 99%

Idempotent functors and localizations in categories of modules and Abelian groups

Крылов

Tuganbaev²

2012

J Math Sci

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“…When is Hom S (B, A) or Hom S (A, B) a finitely generated free (or projective) End S B-module? The book [12] contains a program of studying the group Hom(A, B) as an End B-module or an End A-module for Abelian groups A and B (Problems [11][12][13]; also see the end of the paper [15]. In [11], there is a characterization of groups A and B for which these modules are injective.…”

Section: Remarkmentioning

confidence: 99%

Colocalizations and dualities for modules and Abelian groups

Крылов

Tuganbaev²

2012

J Math Sci

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We introduce standard colocalizations of modules. In the category of torsion-free Abelian groups, a duality is considered.This paper is close to [14], where we consider idempotent coaugmentation functors in categories of modules and localizations of modules. By dualizing these notions, we obtain idempotent augmentation functors and colocalizations; they are also called cellular covers in literature. Similarly to [14], it is possible and desirable to study general properties of idempotent augmentation functors and colocalizations; we can also study the problem of preserving by them ring or module structures, and so on. Here we only consider some standard colocalizations (see Secs. 1, 2). These standard colocalizations lead to the study of some duality in the category of torsion-free Abelian groups (see Sec. 2). In Sec. 3, we calculate types of the groups A ⊗ B and Hom(B, A), where A and B are torsion-free groups of finite rank. These questions are also inspired by results of Sec. 2.All rings are assumed to be associative and with nonzero identity element; modules are assumed to be unitary and right, unless otherwise stated. Standard Colocalizations of ModulesLet us have two rings S and R and a ring homomorphism e : S → R. Over any R-module A, we can define the structure of an S-module by the relation as = ae(s) for all a ∈ A and s ∈ S. This S-module is called the pullback module (with respect to e). Similarly, left R-modules are pullback left S-modules. In particular, R is an S-S-bimodule; in addition, e is a bimodule homomorphism. The factor module R/e(S) is denoted by R 0 . All R-modules are assumed to be pullback S-modules.We present several definitions from [14]. An R-module A is called a T(e)-module or a T-module with respect to the homomorphism e if the mapping A ⊗ S R → A ⊗ R R, a ⊗ S r → a ⊗ R r, is an isomorphism. For the fixed homomorphism e, we also write a "T-module" instead of a "T(e)-module." An R-module A is called an E(e)-module or an E-module with respect to the homomorphism e or simply an E-module (for the fixed e) if Hom S (R, A) = Hom R (R, A).If R is a T(e)-module as a right R-module, then the ring R is called a T(e)-ring, or a T-ring with respect to e, or a T-ring. If R is an E(e)-module as a right R-module, then the ring R is called an E(e)-ring, or an E-ring with respect to e, or simply an E-ring. In fact, we defined right T-rings and right E-rings.Left T-rings and left E-rings are similarly defined. The property to be a T-ring is left-right symmetrical.Let A be an R-module. The sum W (A) of all submodules C in A such that C is a T(e)-module is called the T(e)-radical or the T-radical with respect to the homomorphism e. The correspondence between any R-module A and the T(e)-radical of A is an idempotent radical in the category of R-modules. W (A) is the largest R-submodule that is a T(e)-module. Now we begin to consider S-modules. Let A and B be two modules. A homomorphism f : B → A is called a colocalization of the module A if for every homomorphism α : B → A, there exists a unique homomor...

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Modules over Discrete Valuation Domains

Cited by 24 publications

References 255 publications

Commutative algebraic groups up to isogeny. II

Commutative algebraic groups up to isogeny. II

Idempotent functors and localizations in categories of modules and Abelian groups

Colocalizations and dualities for modules and Abelian groups

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