On the non-existence of right almost split maps

Šaroch, Jan

doi:10.1007/s00222-016-0712-2

Cited by 13 publications

(19 citation statements)

References 12 publications

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“…To prove that the latter is the case, it is enough to check that L ∈ Add(S). This follows from [16,Lemma 3.4(3)]: indeed, the lemma says that L is the directed union of its direct summands L S = η∈S Im(ρν η ) ∈ Add(S) where S runs through finite subsets of T . It follows that L is the directed union of the ℵ 1 -directed system of submodules L U = η∈U Im(ρν η ) ∈ Add(S) where U runs through countable subsets of (the uncountable set) T .…”

Section: The Main Theoremmentioning

confidence: 96%

“…If we do not insist on proving that M has a perfect decomposition, we can verify the Enochs' conjecture in ZFC for Add(S), where S is any class of countably generated modules, using results from [16]. Fix an Add(S)-precover f : B → C; it exists since S is skeletally small.…”

Section: The Main Theoremmentioning

confidence: 99%

“…Let θ be a cardinal number with cf(θ) = µ and such that C consists of < θpresented modules. Following the proof of [16,Theorem 4.2(1)] from the second paragraph on, we obtain a tree module L such that f splits if and only if Hom R (L, f ) is surjective. To prove that the latter is the case, it is enough to check that L ∈ Add(S).…”

Section: The Main Theoremmentioning

confidence: 99%

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Enochs' conjecture for small precovering classes of modules

Šaroch¹

2021

Preprint

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Enochs' conjecture asserts that each covering class of modules (over any fixed ring) has to be closed under direct limits. Although various special cases of the conjecture have been verified, the conjecture remains open in its full generality. In this short paper, we prove the validity of the conjecture for small precovering classes, i.e. the classes of the form Add(M ) where M is any module, under a mild additional set-theoretic assumption which ensures that there are enough non-reflecting stationary sets. We even show that M has a perfect decomposition if Add(M ) is a covering class. Finally, the additional set-theoretic assumption is shown to be redundant if there exists an n < ω such that M decomposes into a direct sum of ℵn-presented modules.

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Section: The Main Theoremmentioning

confidence: 96%

Section: The Main Theoremmentioning

confidence: 99%

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Enochs' conjecture for small precovering classes of modules

Šaroch¹

2021

Preprint

Self Cite

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“…In [5], Auslander asked whether given a ring R and a module C, there exists a right (left) almost split morphism in Mod-R ending (beginning) in C. The main result of [30] answers this question for right almost split morphisms: Theorem 2.3. Let R be a ring and C ∈ Mod-R. Then there exists a right almost split map f ∈ Hom R (B, C) in Mod-R, iff C is finitely presented and the endomorphism ring of C is local.…”

Section: Tree Modules and Almost Split Sequencesmentioning

confidence: 99%

Tree modules and limits of the approximation theory

Trlifaj¹

2018

Surveys in Representation Theory Of Algebras

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In this expository paper, we present a construction of tree modules and combine it with (infinite dimensional) tilting theory and relative Mittag-Leffler conditions in order to explore limits of the approximation theory of modules. We also present a recent generalization of this construction due tǒ Saroch which applies to factorization properties of maps, and yields a solution of an old problem by Auslander concerning existence of almost split sequences.

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“…Although it was originally introduced in a functorial way, it was mainly applied to studying categories of finitely generated modules over Artin algebras. Throughout thirty years, mathematicians, including Henning Krause [8] [11], Claus Michael Ringel [13] [14], Xiao-Wu Chen and Jue Le [5] [6], JanŠaroch [15] and many others, have been further developing Auslander's idea of morphisms determined by objects. Krause successfully develops this theory in triangulated categories [8].…”

Section: Introductionmentioning

confidence: 99%

Functors and Morphisms Determined by Subcategories

Zhu

2019

Appl Categor Struct

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We study the existence and uniqueness of minimal right determiners in various categories. Particularly in a Hom-finite hereditary abelian category with enough projectives, we prove that the Auslander-Reiten-Smalø-Ringel formula of the minimal right determiner still holds. As an application, we give a formula of minimal right determiners in the category of finitely presented representations of strongly locally finite quivers.2010 Mathematics Subject Classification. 18A05,16G20, 16G70. Key words and phrases. morphisms determined by objects, Krull-Schmidt categories, almost split sequences, strongly locally finite quivers. Lemma 2.2.5. Let S be the poset of all the right determiners of f . If S is closed under arbitary intersections, then f has a unique minimal right determiner. Proof. Because S is closed under intersections, D∈S D will be the minimum element of S.Notice that the converse of Lemma 2.2.5 is not true, see Example 7.0.8.

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On the non-existence of right almost split maps

Cited by 13 publications

References 12 publications

Enochs' conjecture for small precovering classes of modules

Enochs' conjecture for small precovering classes of modules

Tree modules and limits of the approximation theory

Functors and Morphisms Determined by Subcategories

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