Torsion Pairs and Simple-Minded Systems in Triangulated Categories

Abstract Let 𝒯 be a Hom-finite triangulated Krull-Schmidt category over a field k. Inspired by a definition of Koenig and Liu (Q. J. Math 63(3), 653-674, 2012), we say that a family 𝒮 ⊆ 𝒯 of pairwise orthogonal bricks is a simple-minded system if its closure under extensions is all of 𝒯. We construc...
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Autor*in:	Dugas, Alex [verfasserIn]

Format:	E-Artikel
Sprache:	Englisch

Erschienen:	2014

Schlagwörter:	Simple-minded system Mutation Torsion pair Derived equivalence Stable equivalence

Übergeordnetes Werk:	Enthalten in: Applied categorical structures - Dordrecht [u.a.] : Springer Science + Business Media B.V, 1993, 23(2014), 3 vom: 11. Feb., Seite 507-526
Übergeordnetes Werk:	volume:23 ; year:2014 ; number:3 ; day:11 ; month:02 ; pages:507-526

Links:	Volltext

DOI / URN:	10.1007/s10485-014-9365-8

Katalog-ID:	SPR010417028

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520			\|a Abstract Let be a Hom-finite triangulated Krull-Schmidt category over a field k. Inspired by a definition of Koenig and Liu (Q. J. Math 63(3), 653-674, 2012), we say that a family ⊆ of pairwise orthogonal bricks is a simple-minded system if its closure under extensions is all of . We construct torsion pairs in associated to any subset of a simple-minded system , and use these to define left and right mutations of relative to . When has a Serre functor ν and and are invariant under ν ∘ [1], we show that these mutations are again simple-minded systems. We are particularly interested in the case where = mod-Λ for a self-injective algebra Λ. In this case, our mutation procedure parallels that introduced by Koenig and Yang for simple-minded collections in Db(mod-Λ) (Koenig and Yang, 2013). It follows that the mutation of the set of simple Λ-modules relative to yields the images of the simple Γ-modules under a stable equivalence mod-Γ → mod-Λ, where Γ is the tilting mutation of Λ relative to .
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912			\|a GBV_ILN_32
912			\|a GBV_ILN_39
912			\|a GBV_ILN_40
912			\|a GBV_ILN_60
912			\|a GBV_ILN_62
912			\|a GBV_ILN_63
912			\|a GBV_ILN_69
912			\|a GBV_ILN_70
912			\|a GBV_ILN_73
912			\|a GBV_ILN_74
912			\|a GBV_ILN_90
912			\|a GBV_ILN_95
912			\|a GBV_ILN_100
912			\|a GBV_ILN_101
912			\|a GBV_ILN_105
912			\|a GBV_ILN_110
912			\|a GBV_ILN_120
912			\|a GBV_ILN_138
912			\|a GBV_ILN_150
912			\|a GBV_ILN_151
912			\|a GBV_ILN_152
912			\|a GBV_ILN_161
912			\|a GBV_ILN_170
912			\|a GBV_ILN_171
912			\|a GBV_ILN_187
912			\|a GBV_ILN_213
912			\|a GBV_ILN_224
912			\|a GBV_ILN_230
912			\|a GBV_ILN_250
912			\|a GBV_ILN_281
912			\|a GBV_ILN_285
912			\|a GBV_ILN_293
912			\|a GBV_ILN_370
912			\|a GBV_ILN_602
912			\|a GBV_ILN_636
912			\|a GBV_ILN_702
912			\|a GBV_ILN_2001
912			\|a GBV_ILN_2003
912			\|a GBV_ILN_2004
912			\|a GBV_ILN_2005
912			\|a GBV_ILN_2006
912			\|a GBV_ILN_2007
912			\|a GBV_ILN_2008
912			\|a GBV_ILN_2009
912			\|a GBV_ILN_2010
912			\|a GBV_ILN_2011
912			\|a GBV_ILN_2014
912			\|a GBV_ILN_2015
912			\|a GBV_ILN_2020
912			\|a GBV_ILN_2021
912			\|a GBV_ILN_2025
912			\|a GBV_ILN_2026
912			\|a GBV_ILN_2027
912			\|a GBV_ILN_2031
912			\|a GBV_ILN_2034
912			\|a GBV_ILN_2037
912			\|a GBV_ILN_2038
912			\|a GBV_ILN_2039
912			\|a GBV_ILN_2044
912			\|a GBV_ILN_2048
912			\|a GBV_ILN_2049
912			\|a GBV_ILN_2050
912			\|a GBV_ILN_2055
912			\|a GBV_ILN_2057
912			\|a GBV_ILN_2059
912			\|a GBV_ILN_2061
912			\|a GBV_ILN_2064
912			\|a GBV_ILN_2065
912			\|a GBV_ILN_2068
912			\|a GBV_ILN_2070
912			\|a GBV_ILN_2086
912			\|a GBV_ILN_2088
912			\|a GBV_ILN_2093
912			\|a GBV_ILN_2106
912			\|a GBV_ILN_2107
912			\|a GBV_ILN_2108
912			\|a GBV_ILN_2110
912			\|a GBV_ILN_2111
912			\|a GBV_ILN_2112
912			\|a GBV_ILN_2113
912			\|a GBV_ILN_2116
912			\|a GBV_ILN_2118
912			\|a GBV_ILN_2119
912			\|a GBV_ILN_2122
912			\|a GBV_ILN_2129
912			\|a GBV_ILN_2143
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912			\|a GBV_ILN_2147
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912			\|a GBV_ILN_2152
912			\|a GBV_ILN_2153
912			\|a GBV_ILN_2188
912			\|a GBV_ILN_2190
912			\|a GBV_ILN_2232
912			\|a GBV_ILN_2336
912			\|a GBV_ILN_2446
912			\|a GBV_ILN_2470
912			\|a GBV_ILN_2472
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912			\|a GBV_ILN_2522
912			\|a GBV_ILN_2548
912			\|a GBV_ILN_4035
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912			\|a GBV_ILN_4112
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912			\|a GBV_ILN_4251
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912			\|a GBV_ILN_4307
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Indexfelder

author_variant	a d ad
matchkey_str	article:15729095:2014----::osopisnsmlmnessesnrag
hierarchy_sort_str	2014
bklnumber	31.27
publishDate	2014
allfields	10.1007/s10485-014-9365-8 doi (DE-627)SPR010417028 (SPR)s10485-014-9365-8-e DE-627 ger DE-627 rakwb eng 510 004 ASE 31.27 bkl Dugas, Alex verfasserin aut Torsion Pairs and Simple-Minded Systems in Triangulated Categories 2014 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract Let 𝒯 be a Hom-finite triangulated Krull-Schmidt category over a field k. Inspired by a definition of Koenig and Liu (Q. J. Math 63(3), 653-674, 2012), we say that a family 𝒮 ⊆ 𝒯 of pairwise orthogonal bricks is a simple-minded system if its closure under extensions is all of 𝒯. We construct torsion pairs in 𝒯 associated to any subset 𝒳 of a simple-minded system 𝒮, and use these to define left and right mutations of 𝒮 relative to 𝒳. When 𝒯 has a Serre functor ν and 𝒮 and 𝒳 are invariant under ν ∘ [1], we show that these mutations are again simple-minded systems. We are particularly interested in the case where 𝒯 = mod-Λ for a self-injective algebra Λ. In this case, our mutation procedure parallels that introduced by Koenig and Yang for simple-minded collections in Db(mod-Λ) (Koenig and Yang, 2013). It follows that the mutation of the set of simple Λ-modules relative to 𝒳 yields the images of the simple Γ-modules under a stable equivalence mod-Γ → mod-Λ, where Γ is the tilting mutation of Λ relative to 𝒳. Simple-minded system (dpeaa)DE-He213 Mutation (dpeaa)DE-He213 Torsion pair (dpeaa)DE-He213 Derived equivalence (dpeaa)DE-He213 Stable equivalence (dpeaa)DE-He213 Enthalten in Applied categorical structures Dordrecht [u.a.] : Springer Science + Business Media B.V, 1993 23(2014), 3 vom: 11. Feb., Seite 507-526 (DE-627)269539077 (DE-600)1475519-1 1572-9095 nnns volume:23 year:2014 number:3 day:11 month:02 pages:507-526 https://dx.doi.org/10.1007/s10485-014-9365-8 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OPC-MAT SSG-OPC-ASE GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 31.27 ASE AR 23 2014 3 11 02 507-526
spelling	10.1007/s10485-014-9365-8 doi (DE-627)SPR010417028 (SPR)s10485-014-9365-8-e DE-627 ger DE-627 rakwb eng 510 004 ASE 31.27 bkl Dugas, Alex verfasserin aut Torsion Pairs and Simple-Minded Systems in Triangulated Categories 2014 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract Let 𝒯 be a Hom-finite triangulated Krull-Schmidt category over a field k. Inspired by a definition of Koenig and Liu (Q. J. Math 63(3), 653-674, 2012), we say that a family 𝒮 ⊆ 𝒯 of pairwise orthogonal bricks is a simple-minded system if its closure under extensions is all of 𝒯. We construct torsion pairs in 𝒯 associated to any subset 𝒳 of a simple-minded system 𝒮, and use these to define left and right mutations of 𝒮 relative to 𝒳. When 𝒯 has a Serre functor ν and 𝒮 and 𝒳 are invariant under ν ∘ [1], we show that these mutations are again simple-minded systems. We are particularly interested in the case where 𝒯 = mod-Λ for a self-injective algebra Λ. In this case, our mutation procedure parallels that introduced by Koenig and Yang for simple-minded collections in Db(mod-Λ) (Koenig and Yang, 2013). It follows that the mutation of the set of simple Λ-modules relative to 𝒳 yields the images of the simple Γ-modules under a stable equivalence mod-Γ → mod-Λ, where Γ is the tilting mutation of Λ relative to 𝒳. Simple-minded system (dpeaa)DE-He213 Mutation (dpeaa)DE-He213 Torsion pair (dpeaa)DE-He213 Derived equivalence (dpeaa)DE-He213 Stable equivalence (dpeaa)DE-He213 Enthalten in Applied categorical structures Dordrecht [u.a.] : Springer Science + Business Media B.V, 1993 23(2014), 3 vom: 11. Feb., Seite 507-526 (DE-627)269539077 (DE-600)1475519-1 1572-9095 nnns volume:23 year:2014 number:3 day:11 month:02 pages:507-526 https://dx.doi.org/10.1007/s10485-014-9365-8 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OPC-MAT SSG-OPC-ASE GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 31.27 ASE AR 23 2014 3 11 02 507-526
allfields_unstemmed	10.1007/s10485-014-9365-8 doi (DE-627)SPR010417028 (SPR)s10485-014-9365-8-e DE-627 ger DE-627 rakwb eng 510 004 ASE 31.27 bkl Dugas, Alex verfasserin aut Torsion Pairs and Simple-Minded Systems in Triangulated Categories 2014 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract Let 𝒯 be a Hom-finite triangulated Krull-Schmidt category over a field k. Inspired by a definition of Koenig and Liu (Q. J. Math 63(3), 653-674, 2012), we say that a family 𝒮 ⊆ 𝒯 of pairwise orthogonal bricks is a simple-minded system if its closure under extensions is all of 𝒯. We construct torsion pairs in 𝒯 associated to any subset 𝒳 of a simple-minded system 𝒮, and use these to define left and right mutations of 𝒮 relative to 𝒳. When 𝒯 has a Serre functor ν and 𝒮 and 𝒳 are invariant under ν ∘ [1], we show that these mutations are again simple-minded systems. We are particularly interested in the case where 𝒯 = mod-Λ for a self-injective algebra Λ. In this case, our mutation procedure parallels that introduced by Koenig and Yang for simple-minded collections in Db(mod-Λ) (Koenig and Yang, 2013). It follows that the mutation of the set of simple Λ-modules relative to 𝒳 yields the images of the simple Γ-modules under a stable equivalence mod-Γ → mod-Λ, where Γ is the tilting mutation of Λ relative to 𝒳. Simple-minded system (dpeaa)DE-He213 Mutation (dpeaa)DE-He213 Torsion pair (dpeaa)DE-He213 Derived equivalence (dpeaa)DE-He213 Stable equivalence (dpeaa)DE-He213 Enthalten in Applied categorical structures Dordrecht [u.a.] : Springer Science + Business Media B.V, 1993 23(2014), 3 vom: 11. Feb., Seite 507-526 (DE-627)269539077 (DE-600)1475519-1 1572-9095 nnns volume:23 year:2014 number:3 day:11 month:02 pages:507-526 https://dx.doi.org/10.1007/s10485-014-9365-8 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OPC-MAT SSG-OPC-ASE GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 31.27 ASE AR 23 2014 3 11 02 507-526
allfieldsGer	10.1007/s10485-014-9365-8 doi (DE-627)SPR010417028 (SPR)s10485-014-9365-8-e DE-627 ger DE-627 rakwb eng 510 004 ASE 31.27 bkl Dugas, Alex verfasserin aut Torsion Pairs and Simple-Minded Systems in Triangulated Categories 2014 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract Let 𝒯 be a Hom-finite triangulated Krull-Schmidt category over a field k. Inspired by a definition of Koenig and Liu (Q. J. Math 63(3), 653-674, 2012), we say that a family 𝒮 ⊆ 𝒯 of pairwise orthogonal bricks is a simple-minded system if its closure under extensions is all of 𝒯. We construct torsion pairs in 𝒯 associated to any subset 𝒳 of a simple-minded system 𝒮, and use these to define left and right mutations of 𝒮 relative to 𝒳. When 𝒯 has a Serre functor ν and 𝒮 and 𝒳 are invariant under ν ∘ [1], we show that these mutations are again simple-minded systems. We are particularly interested in the case where 𝒯 = mod-Λ for a self-injective algebra Λ. In this case, our mutation procedure parallels that introduced by Koenig and Yang for simple-minded collections in Db(mod-Λ) (Koenig and Yang, 2013). It follows that the mutation of the set of simple Λ-modules relative to 𝒳 yields the images of the simple Γ-modules under a stable equivalence mod-Γ → mod-Λ, where Γ is the tilting mutation of Λ relative to 𝒳. Simple-minded system (dpeaa)DE-He213 Mutation (dpeaa)DE-He213 Torsion pair (dpeaa)DE-He213 Derived equivalence (dpeaa)DE-He213 Stable equivalence (dpeaa)DE-He213 Enthalten in Applied categorical structures Dordrecht [u.a.] : Springer Science + Business Media B.V, 1993 23(2014), 3 vom: 11. Feb., Seite 507-526 (DE-627)269539077 (DE-600)1475519-1 1572-9095 nnns volume:23 year:2014 number:3 day:11 month:02 pages:507-526 https://dx.doi.org/10.1007/s10485-014-9365-8 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OPC-MAT SSG-OPC-ASE GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 31.27 ASE AR 23 2014 3 11 02 507-526
allfieldsSound	10.1007/s10485-014-9365-8 doi (DE-627)SPR010417028 (SPR)s10485-014-9365-8-e DE-627 ger DE-627 rakwb eng 510 004 ASE 31.27 bkl Dugas, Alex verfasserin aut Torsion Pairs and Simple-Minded Systems in Triangulated Categories 2014 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract Let 𝒯 be a Hom-finite triangulated Krull-Schmidt category over a field k. Inspired by a definition of Koenig and Liu (Q. J. Math 63(3), 653-674, 2012), we say that a family 𝒮 ⊆ 𝒯 of pairwise orthogonal bricks is a simple-minded system if its closure under extensions is all of 𝒯. We construct torsion pairs in 𝒯 associated to any subset 𝒳 of a simple-minded system 𝒮, and use these to define left and right mutations of 𝒮 relative to 𝒳. When 𝒯 has a Serre functor ν and 𝒮 and 𝒳 are invariant under ν ∘ [1], we show that these mutations are again simple-minded systems. We are particularly interested in the case where 𝒯 = mod-Λ for a self-injective algebra Λ. In this case, our mutation procedure parallels that introduced by Koenig and Yang for simple-minded collections in Db(mod-Λ) (Koenig and Yang, 2013). It follows that the mutation of the set of simple Λ-modules relative to 𝒳 yields the images of the simple Γ-modules under a stable equivalence mod-Γ → mod-Λ, where Γ is the tilting mutation of Λ relative to 𝒳. Simple-minded system (dpeaa)DE-He213 Mutation (dpeaa)DE-He213 Torsion pair (dpeaa)DE-He213 Derived equivalence (dpeaa)DE-He213 Stable equivalence (dpeaa)DE-He213 Enthalten in Applied categorical structures Dordrecht [u.a.] : Springer Science + Business Media B.V, 1993 23(2014), 3 vom: 11. Feb., Seite 507-526 (DE-627)269539077 (DE-600)1475519-1 1572-9095 nnns volume:23 year:2014 number:3 day:11 month:02 pages:507-526 https://dx.doi.org/10.1007/s10485-014-9365-8 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OPC-MAT SSG-OPC-ASE GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 31.27 ASE AR 23 2014 3 11 02 507-526
language	English
source	Enthalten in Applied categorical structures 23(2014), 3 vom: 11. Feb., Seite 507-526 volume:23 year:2014 number:3 day:11 month:02 pages:507-526
sourceStr	Enthalten in Applied categorical structures 23(2014), 3 vom: 11. Feb., Seite 507-526 volume:23 year:2014 number:3 day:11 month:02 pages:507-526
format_phy_str_mv	Article
institution	findex.gbv.de
topic_facet	Simple-minded system Mutation Torsion pair Derived equivalence Stable equivalence
dewey-raw	510
isfreeaccess_bool	false
container_title	Applied categorical structures
authorswithroles_txt_mv	Dugas, Alex @@aut@@
publishDateDaySort_date	2014-02-11T00:00:00Z
hierarchy_top_id	269539077
dewey-sort	3510
id	SPR010417028
language_de	englisch
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author	Dugas, Alex
spellingShingle	Dugas, Alex ddc 510 bkl 31.27 misc Simple-minded system misc Mutation misc Torsion pair misc Derived equivalence misc Stable equivalence Torsion Pairs and Simple-Minded Systems in Triangulated Categories
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topic_title	510 004 ASE 31.27 bkl Torsion Pairs and Simple-Minded Systems in Triangulated Categories Simple-minded system (dpeaa)DE-He213 Mutation (dpeaa)DE-He213 Torsion pair (dpeaa)DE-He213 Derived equivalence (dpeaa)DE-He213 Stable equivalence (dpeaa)DE-He213
topic	ddc 510 bkl 31.27 misc Simple-minded system misc Mutation misc Torsion pair misc Derived equivalence misc Stable equivalence
topic_unstemmed	ddc 510 bkl 31.27 misc Simple-minded system misc Mutation misc Torsion pair misc Derived equivalence misc Stable equivalence
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title	Torsion Pairs and Simple-Minded Systems in Triangulated Categories
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title_full	Torsion Pairs and Simple-Minded Systems in Triangulated Categories
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title_sort	torsion pairs and simple-minded systems in triangulated categories
title_auth	Torsion Pairs and Simple-Minded Systems in Triangulated Categories
abstract	Abstract Let 𝒯 be a Hom-finite triangulated Krull-Schmidt category over a field k. Inspired by a definition of Koenig and Liu (Q. J. Math 63(3), 653-674, 2012), we say that a family 𝒮 ⊆ 𝒯 of pairwise orthogonal bricks is a simple-minded system if its closure under extensions is all of 𝒯. We construct torsion pairs in 𝒯 associated to any subset 𝒳 of a simple-minded system 𝒮, and use these to define left and right mutations of 𝒮 relative to 𝒳. When 𝒯 has a Serre functor ν and 𝒮 and 𝒳 are invariant under ν ∘ [1], we show that these mutations are again simple-minded systems. We are particularly interested in the case where 𝒯 = mod-Λ for a self-injective algebra Λ. In this case, our mutation procedure parallels that introduced by Koenig and Yang for simple-minded collections in Db(mod-Λ) (Koenig and Yang, 2013). It follows that the mutation of the set of simple Λ-modules relative to 𝒳 yields the images of the simple Γ-modules under a stable equivalence mod-Γ → mod-Λ, where Γ is the tilting mutation of Λ relative to 𝒳.
abstractGer	Abstract Let 𝒯 be a Hom-finite triangulated Krull-Schmidt category over a field k. Inspired by a definition of Koenig and Liu (Q. J. Math 63(3), 653-674, 2012), we say that a family 𝒮 ⊆ 𝒯 of pairwise orthogonal bricks is a simple-minded system if its closure under extensions is all of 𝒯. We construct torsion pairs in 𝒯 associated to any subset 𝒳 of a simple-minded system 𝒮, and use these to define left and right mutations of 𝒮 relative to 𝒳. When 𝒯 has a Serre functor ν and 𝒮 and 𝒳 are invariant under ν ∘ [1], we show that these mutations are again simple-minded systems. We are particularly interested in the case where 𝒯 = mod-Λ for a self-injective algebra Λ. In this case, our mutation procedure parallels that introduced by Koenig and Yang for simple-minded collections in Db(mod-Λ) (Koenig and Yang, 2013). It follows that the mutation of the set of simple Λ-modules relative to 𝒳 yields the images of the simple Γ-modules under a stable equivalence mod-Γ → mod-Λ, where Γ is the tilting mutation of Λ relative to 𝒳.
abstract_unstemmed	Abstract Let 𝒯 be a Hom-finite triangulated Krull-Schmidt category over a field k. Inspired by a definition of Koenig and Liu (Q. J. Math 63(3), 653-674, 2012), we say that a family 𝒮 ⊆ 𝒯 of pairwise orthogonal bricks is a simple-minded system if its closure under extensions is all of 𝒯. We construct torsion pairs in 𝒯 associated to any subset 𝒳 of a simple-minded system 𝒮, and use these to define left and right mutations of 𝒮 relative to 𝒳. When 𝒯 has a Serre functor ν and 𝒮 and 𝒳 are invariant under ν ∘ [1], we show that these mutations are again simple-minded systems. We are particularly interested in the case where 𝒯 = mod-Λ for a self-injective algebra Λ. In this case, our mutation procedure parallels that introduced by Koenig and Yang for simple-minded collections in Db(mod-Λ) (Koenig and Yang, 2013). It follows that the mutation of the set of simple Λ-modules relative to 𝒳 yields the images of the simple Γ-modules under a stable equivalence mod-Γ → mod-Λ, where Γ is the tilting mutation of Λ relative to 𝒳.
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title_short	Torsion Pairs and Simple-Minded Systems in Triangulated Categories
url	https://dx.doi.org/10.1007/s10485-014-9365-8
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doi_str	10.1007/s10485-014-9365-8
up_date	2024-07-03T15:59:09.824Z
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fullrecord_marcxml	<?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>01000caa a22002652 4500</leader><controlfield tag="001">SPR010417028</controlfield><controlfield tag="003">DE-627</controlfield><controlfield tag="005">20220110220437.0</controlfield><controlfield tag="007">cr uuu---uuuuu</controlfield><controlfield tag="008">201005s2014 xx \|\|\|\|\|o 00\| \|\|eng c</controlfield><datafield tag="024" ind1="7" ind2=" "><subfield code="a">10.1007/s10485-014-9365-8</subfield><subfield code="2">doi</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-627)SPR010417028</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(SPR)s10485-014-9365-8-e</subfield></datafield><datafield tag="040" ind1=" " ind2=" "><subfield code="a">DE-627</subfield><subfield code="b">ger</subfield><subfield code="c">DE-627</subfield><subfield code="e">rakwb</subfield></datafield><datafield tag="041" ind1=" " ind2=" "><subfield code="a">eng</subfield></datafield><datafield tag="082" ind1="0" ind2="4"><subfield code="a">510</subfield><subfield code="a">004</subfield><subfield code="q">ASE</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">31.27</subfield><subfield code="2">bkl</subfield></datafield><datafield tag="100" ind1="1" ind2=" "><subfield code="a">Dugas, Alex</subfield><subfield code="e">verfasserin</subfield><subfield code="4">aut</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">Torsion Pairs and Simple-Minded Systems in Triangulated Categories</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="c">2014</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="a">Text</subfield><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="a">Computermedien</subfield><subfield code="b">c</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="a">Online-Ressource</subfield><subfield code="b">cr</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="520" ind1=" " ind2=" "><subfield code="a">Abstract Let 𝒯 be a Hom-finite triangulated Krull-Schmidt category over a field k. Inspired by a definition of Koenig and Liu (Q. J. Math 63(3), 653-674, 2012), we say that a family 𝒮 ⊆ 𝒯 of pairwise orthogonal bricks is a simple-minded system if its closure under extensions is all of 𝒯. We construct torsion pairs in 𝒯 associated to any subset 𝒳 of a simple-minded system 𝒮, and use these to define left and right mutations of 𝒮 relative to 𝒳. When 𝒯 has a Serre functor ν and 𝒮 and 𝒳 are invariant under ν ∘ [1], we show that these mutations are again simple-minded systems. We are particularly interested in the case where 𝒯 = mod-Λ for a self-injective algebra Λ. In this case, our mutation procedure parallels that introduced by Koenig and Yang for simple-minded collections in Db(mod-Λ) (Koenig and Yang, 2013). It follows that the mutation of the set of simple Λ-modules relative to 𝒳 yields the images of the simple Γ-modules under a stable equivalence mod-Γ → mod-Λ, where Γ is the tilting mutation of Λ relative to 𝒳.</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Simple-minded system</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Mutation</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Torsion pair</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Derived equivalence</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Stable equivalence</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="773" ind1="0" ind2="8"><subfield code="i">Enthalten in</subfield><subfield code="t">Applied categorical structures</subfield><subfield code="d">Dordrecht [u.a.] : Springer Science + Business Media B.V, 1993</subfield><subfield code="g">23(2014), 3 vom: 11. 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