Ideal approximation theory

Fu, Xianhui; Asensio, Pedro A. Guil; Herzog, Ivo; Torrecillas, Blas

doi:10.1016/j.aim.2013.05.020

Cited by 38 publications

(66 citation statements)

References 22 publications

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“…Phantom morphisms, which have their origin in homotopy theory [14], were introduced by Gnacadja [9] in the category of modules over a finite group ring, and considered by Herzog for a general module category in [12]. In [8] phantom morphisms with respect to the exact substructure F have been defined, and also the dual notion of phantom morphisms, the cophantom morphisms have been introduced.…”

Section: Has a Solution Over M If And Only If The System Of Linear Eqmentioning

confidence: 99%

Ziegler partial morphisms in additive exact categories

et al. 2020

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Section: Has a Solution Over M If And Only If The System Of Linear Eqmentioning

confidence: 99%

Ziegler partial morphisms in additive exact categories

et al. 2020

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“…[6,Section 5], [5, Theorem 2.1]). We refer to [10] for a recent study of this kind of special preenveloping situation which involves homomorphisms instead of objects. Even the orthogonality used in this paper does not cover the (co)silting case (cf.…”

Section: Silting Classesmentioning

confidence: 99%

Torsion classes generated by silting modules

Breaz

Žemlička

2018

Arkiv För Matematik

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We study the classes of modules which are generated by a silting module. In the case of either hereditary or perfect rings it is proved that these are exactly the torsion T such that the regular module has a special T -preenvelope. In particular, every torsion enveloping class in Mod-R are of the form Gen(T ) for a minimal silting module T . For the dual case we obtain for general rings that the covering torsion-free classes of modules are exactly the classes of the form Cogen(T ), where T is a cosilting module.

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“…For example, if G = Z/2×Z/2 is the Klein 4-group, and the characteristic of k is 2, then J 3 = 0 in k [G], so that Theorem 28 implies that Φ 2 = 0 in the stable category k[G]-Mod, a result established by Benson and Gnacadja [7, §4.6] when k is countable. On the other hand, it is a consequence of the Pure Semisimple Conjecture for QF rings [29,Cor 5.3] that a QF ring is phantomless if and only if it is of finite representation type [22,Prop 41]. Because the group algebra k[Z/2 × Z/2] is not of finite representation type [6,Thm 4.4.4], Φ = 0 in the stable category.…”

Section: Introductionmentioning

confidence: 99%

Powers of the phantom ideal

Fu¹,

Herzog²

2016

Proc. London Math. Soc.

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Abstract. It is proved that if G is a finite group, then the order of G is a proper upper bound for the phantom number of G. This answers a question of Benson and Gnacadja. More specifically, if k is a field whose characteristic divides the order of G, and Φ is the ideal of phantom morphisms in the stable category k[G]-Mod of modules over the group algebra k[G], then Φ n−1 = 0, where n is the nilpotency index of the Jacobson radical J of k [G]. If R is a semiprimary ring, with J n = 0, and Φ denotes the phantom ideal in the module category R-Mod, then Φ n is the ideal of morphisms that factor through a projective module. If R is a right coherent ring and every cotorsion left R-module has a coresolution of length n by pure injective modules, then Φ n+1 is the ideal of morphisms that factor through a flat module.These results are obtained by introducing the mono-epi (ME) exact structure on the morphisms of an exact category (A; E ). This exact structure (Arr(A); ME) is used to develop further the ideal approximation theory of (A; E ), by proving new versions of Salce's Lemma, the Ghost Lemma of Christensen, and Wakamatsu's Lemma. Salce's Lemma states that if (A; E ) has enough injective morphisms and projective morphisms, then the map I → I ⊥ on ideals is a bijective correspondence between the class of special precovering ideals of (A; E ) and that of its special preenveloping ideals. The exact category (Arr(A); ME) of morphisms allows us to introduce the notion of an extension i ⋆ j of morphisms in an exact category (A; E ), and the notion of an extension I ⋄ J of ideals of A. The Ghost Lemma, instrumental in proving the consequences above, asserts that the class of special precovering (resp., special preenveloping) ideals is closed under products and extensions and that the bijective correspondence of Salce's Lemma satisfies (IJ )Wakamatsu's Lemma is the statement that if a covering ideal I is closed under extensions I ⋄I = I, then I is a special precovering ideal that possesses a syzygy ideal Ω(I) ⊆ I ⊥ generated by objects.

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Ideal approximation theory

Cited by 38 publications

References 22 publications

Ziegler partial morphisms in additive exact categories

Ziegler partial morphisms in additive exact categories

Torsion classes generated by silting modules

Powers of the phantom ideal

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