Last edited by Kem
Sunday, August 2, 2020 | History

4 edition of Covers, envelopes, and cotorsion theories found in the catalog.

# Covers, envelopes, and cotorsion theories

## by Edgar E. Enochs

Written in English

Subjects:
• Torsion theory (Algebra),
• Algebra, Homological.

• Edition Notes

Includes bibliographical references (p. [105]-109) and index.

Classifications The Physical Object Statement Edgar E. Enochs and Luis Oyonarte. Contributions Oyonarte, Luis. LC Classifications QA251.3 .E56 2002 Pagination vi, 113 p. : Number of Pages 113 Open Library OL3564044M ISBN 10 1590334515 LC Control Number 2002033741

We prove general results about completeness of cotorsion theories and existence of covers and envelopes in locally presentable abelian categories, extending the well-established theory for module categories and Grothendieck categories. These results are then applied to the categories of contramodules over topological rings, which provide examples and counterexamples. flat ring epimorphisms of countable type - leonid positselskiMissing: Covers.

flat covers and flat cotorsion modules lifted to F. Hence F -> M is a flat precover as i?-modules since F is flat as an.R-module. If R is a complete local ring, . Cotorsion pairs have been used to study covers and envelopes [13], [12], particularly in the proof of the ﬂat cover conjecture [4]. They have also been used in tilting theory [1] and in the representation theory of Artin algebras [26].

In this paper, we introduce and study the Gorenstein relative homology theory for unbounded complexes of modules over arbitrary associative rings, which is defined using special Gorenstein flat precovers. We compare the Gorenstein relative homology to the Tate/unbounded homology and get some results that improve the known ones. Covers, Envelopes and Cotorsion Theories (with Luis Oyonarte), Nova Science Publishers, pages (). The flat cover conjecture and its solution (with Overtoun Jenda), International Symposium on Ring Theory, , Birkhauser, Berlin.

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### Covers, envelopes, and cotorsion theories by Edgar E. Enochs Download PDF EPUB FB2

ISBN: OCLC Number: Description: vi, pages: illustrations ; 24 cm: Contents: Cotorsion Theories --Covers and Envelopes --Homological Properties of Cotorsion Theories --Homological Dimensions Relative to Cotorsion Theories --Cotorsion Theories and Balance --Existence of Covers and Envelopes in Categories --Cotorsion Theories with Enough Projectives.

Free shipping for non-business customers when ordering books at De Gruyter Online. Please find details to our shipping fees here. RRP: Recommended Retail Price. Print Flyer; Overview; Content; Book Book Series. Previous chapter.

Next chapter. envelopes 7 Covers, Envelopes, and Cotorsion Theories. 30,00 € / $/ £ Get Access to Full Text. Covers, Envelopes and Cotorsion Theories (Hardback) Edgar E. and cotorsion theories book Enochs (author), Luis Oyonarte (author). Free shipping for non-business customers when ordering books at De Gruyter Online. Please find details to our shipping fees here. RRP: Recommended Retail Price. Print Flyer; Overview; Content; Book Book Series. Previous chapter. Next chapter. Chapter 7. Covers, Envelopes, and Cotorsion Theories. 30,00 € /$ / £ Get Access to Full. COVERS, ENVELOPES, AND COTORSION THEORIES. Cotorsion theories are analogs of the classical torsion theories where.

Hom is replaced b y Ext. Similarly, one can deﬁne F-torsion theories. Covers and Envelopes in the Category of Complexes of Modules collects these scattered notes and results into a single, concise volume that provides an account of recent developments in the theory and presents several new and important ideas.

Abstract: We prove general results about completeness of cotorsion theories and existence of covers and envelopes in locally presentable abelian categories, extending the well-established theory for module categories and cotorsion theories book Grothendieck categories. These results are then applied to the categories of contramodules over topological rings, which provide examples and counterexamples.

The importance of each of the numerous envelopes, covers and cotorsion theories depends very much on the ring in case. We will illustrate this throughout the text in the case of domains, in particular the Pru¨fer and the Dedekind ones.

For example, if Ris a Pru¨fer domain then the complete. Since its beginnings inthe theory of covers and envelopes has been an important branch of associative algebra and homological algebra. This book attempts to clarify the relationship between cotorsion theories and the theory of covers and envelopes, as well.

We then prove the existence of flat covers and cotorsion envelopes of complexes, giving some examples. This generalizes the earlier work (J. Algebra (), 86. Having discussed the place of contramodule categories in the general category theory, let us now say a few words about covers, envelopes, and cotorsion theories.

The classical flat cover conjecturehas two approaches to its solution developed in the literature. In fact, the paper, where this conjecture was first proved, contains two proofs.

In this paper, we study the existence of L ⊥ -envelopes, L-envelopes, D ⊥ -envelopes, D-covers, and L-covers where L and D denote the classes of modules of injective and projective dimension less than or equal to a natural number n, prove that over any ring R, special D ⊥ -preenvelopes and special D-precovers always the ring is noetherian, the same holds for L.

The purpose of this paper is to establish relative cohomology theories based on cotorsion pairs in the setting of unbounded complexes of modules over R.

Let (풜, ℬ) be a complete hereditary cotorsion pair in R-Mod. Then (d g 풜 ̃, ℬ ̃) and (풜 ̃, d g ℬ ̃) are complete hereditary cotorsion pairs in. Since the injective envelope and projective cover were defined by Eckmann and Bas in the s, they have had great influence on the development of homological algebra, ring theory and module theory.

In the s, Enochs introduced the flat cover and conjectured that every module has such a cover. We consider the problem of characterizing the commutative domains such that every module admits a divisible envelope or a $\mathcal{P}_1$-cover, Theory 11(1), 43 ( Cotorsion theories induced by tilting and cotilting modules, Proc.

AGRAM', Contemp. Math. This book provides the uniform methods and systematic treatment to study general envelopes and covers with the emphasis on the existence of flat cover. It shows that Enochs' conjecture is true for a large variety of interesting rings, and then presents the applications of the results.

This book focuses the study on envelopes and covers. It emphasises on DG-injective and DG-projective complexes and flat and DG-flat covers.

It covers. topics including Zorn's Lemma for categories, preserving and reflecting covers by functors, and orthogonality in the category of complexes. This book provides the uniform methods and systematic treatment to study general envelopes and covers with the emphasis on the existence of flat cover.

It shows that Enochs' conjecture is true for. This proves the flat cover conjecture for (possibly unbounded) chain complexes and also gives us a new flat'' model category structure on.

In the last section we use the theory of model categories to show that we can define using a flat resolution of and a cotorsion coresolution of. Covers, envelopes, and cotorsion theories in locally. torsion free, and cotorsion. A number of these results can be found in the book [5], but see also [1], [2], [4], [6], [8], [13], and [14].

This paper considers two diﬀerent setups which we now describe, and proves for each that certain classes are covering. Filtered colimits. .Cotorsion pairs have been used to study covers and envelopes [13], [12], particularly in the proof of the ﬂat cover conjecture [4].

They have also been used in tilting theory [1] and in the representation theory of Artin algebras [26]. The most obvious example of a cotorsion pair is when D = A, in which case E is the class of injective objects.Buy a Kindle Kindle eBooks Kindle Unlimited Prime Reading Best Sellers & More Kindle Book Deals Kindle Singles Newsstand Manage content and devices Advanced Search Edgar E.

EnochsMissing: Covers.