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Κατεύθυνση Θεωρητικά Μαθηματικά

Library of the School of Science

Library of the School of Science

2023-09-19

2023

Mitos Achilleas

Ιωάννης Εμμανουήλ, Καθηγητής, Τμήμα Μαθηματικών, ΕΚΠΑ

Μιχαήλ Συκιώτης, Αναπληρωτής Καθηγητής, Τμήμα Μαθηματικών, ΕΚΠΑ

Ιωάννης Ντόκας, Επίκουρος Καθηγητής, Τμήμα Μαθηματικών, ΕΚΠΑ

Μιχαήλ Συκιώτης, Αναπληρωτής Καθηγητής, Τμήμα Μαθηματικών, ΕΚΠΑ

Ιωάννης Ντόκας, Επίκουρος Καθηγητής, Τμήμα Μαθηματικών, ΕΚΠΑ

A short thesis on Shelah's Singular Compactness Theorem for modules

English

A short thesis on Shelah's Singular Compactness Theorem for modules

This thesis revolves around Shelah’s Singular Compactness Theorem, first presented in a paper of his in 1975. The theorem states that if a group is λ-free, meaning that each subgroup of cardinality <λ is free, then the group is λ+-free (where of course λ+ is the successor cardinal of λ). In particular, when the group has cardinality λ, then this theorem implies that the group itself is free.

The theorem saw many re-prints both in its statement (often generalizing the notions of group and free) and in its proof. We focus on a recent such attempt by authors Saroch and Stovicek (2018), whose article states the theorem as follows: if M is a κ-presented module of a ring with enough idempotents, C is a filter-closed class of modules and there is an infinite cardinal v such that for all successor cardinals ν < λ < κ the module M is (C,λ)-projective, then for any module N in the class C we have Ext1(M,N) = 0. In particular, when C is the class of all left R-modules then M is projective.

The thesis is divided into three chapters. The first two give the preliminary knowledge on Homological Algebra and Set Theory that is required for one to study the third chapter. This last chapter starts off with a collection of past results and the history of the theorem, and continues to analyze the tools developed in the aforementioned article by Saroch and Stovicek with the aim of presenting this impressive result in a clear and easy-to-follow manner.

The theorem saw many re-prints both in its statement (often generalizing the notions of group and free) and in its proof. We focus on a recent such attempt by authors Saroch and Stovicek (2018), whose article states the theorem as follows: if M is a κ-presented module of a ring with enough idempotents, C is a filter-closed class of modules and there is an infinite cardinal v such that for all successor cardinals ν < λ < κ the module M is (C,λ)-projective, then for any module N in the class C we have Ext1(M,N) = 0. In particular, when C is the class of all left R-modules then M is projective.

The thesis is divided into three chapters. The first two give the preliminary knowledge on Homological Algebra and Set Theory that is required for one to study the third chapter. This last chapter starts off with a collection of past results and the history of the theorem, and continues to analyze the tools developed in the aforementioned article by Saroch and Stovicek with the aim of presenting this impressive result in a clear and easy-to-follow manner.

Science

singular, compactness, Shelah, modules, projective

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0

Yes

18

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