Reduction of exact structures

Brüstle, Thomas; Hassoun, Souheila; Langford, Denis; Roy, Sunny

doi:10.48550/arxiv.1809.01282

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Characterization Of (Jhp)1

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2019

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“…Remark 4.5. The weakly length-like function constructed in the proof of (2) ⇒ (1) is the same as the length given in [BHLR,Section 6].…”

Section: Characterization Of (Jhp)mentioning

confidence: 99%

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The Jordan-Hölder property and Grothendieck monoids of exact categories

Enomoto

2019

Preprint

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We investigate the Jordan-Hölder property (JHP) in exact categories. First we introduce a new invariant of exact categories, the Grothendieck monoids, and show that (JHP) holds if and only if the Grothendieck monoid is free. Moreover, we give a criterion for this which only uses the Grothendieck group and the number of simple objects. Next we apply these results to the representation theory of artin algebras. For a large class of exact categories including functorially finite torsion(-free) classes, (JHP) holds precisely when the number of projectives is equal to that of simples. We study torsion-free classes in type A quiver in detail using the combinatorics of symmetric groups. In particular, we show that simples correspond to Bruhat inversions of a c-sortable element, and give the combinatorial criterion for (JHP).

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“…Remark 4.5. The weakly length-like function constructed in the proof of (2) ⇒ (1) is the same as the length given in [BHLR,Section 6].…”

Section: Characterization Of (Jhp)mentioning

confidence: 99%

“…Example A (c.f. [BHLR,Example 6.9]). Let k be a field and define E as the category of finitedimensional k-vector spaces whose dimensions are not equal to 1.…”

Section: Introductionmentioning

confidence: 99%

The Jordan-Hölder property and Grothendieck monoids of exact categories

Enomoto

2019

Preprint

View full text Add to dashboard Cite

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Reduction of exact structures

Cited by 1 publication

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The Jordan-Hölder property and Grothendieck monoids of exact categories

The Jordan-Hölder property and Grothendieck monoids of exact categories

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