Ideal mutations in triangulated categories and generalized Auslander-Reiten theory

We introduce the notion of ideal mutations in a triangulated category, which generalizes the version of Iyama and Yoshino by replacing approximations by objects of a subcategory with approximations by morphisms of an ideal. As applications, for a Hom -finite Krull-Sc...
Ausführliche Beschreibung

Gespeichert in:

Autor*in:	Zhang, Yaohua [verfasserIn] Zhu, Bin [verfasserIn]

Format:	E-Artikel
Sprache:	Englisch

Erschienen:	2024

Schlagwörter:	Auslander-Reiten theory Ideal mutation Ideal approximation theory Auslander-Reiten quiver Auslander-Reiten sequence

Übergeordnetes Werk:	Enthalten in: Journal of algebra - San Diego, Calif. : Elsevier, 1964, 644, Seite 191-221
Übergeordnetes Werk:	volume:644 ; pages:191-221

DOI / URN:	10.1016/j.jalgebra.2023.12.032

Katalog-ID:	ELV06702713X

Internformat


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520			\|a We introduce the notion of ideal mutations in a triangulated category, which generalizes the version of Iyama and Yoshino by replacing approximations by objects of a subcategory with approximations by morphisms of an ideal. As applications, for a Hom -finite Krull-Schmidt triangulated category T over an algebraically closed field K. (1) We generalize a theorem of Jorgensen to a more general setting; (2) We provide a method to detect whether T has Auslander-Reiten triangles or not by checking the necessary and sufficient conditions on its Jacobson radical J : (i) J is functorially finite, (ii) Gh J = CoGh J , and (iii) Gh J -source maps coincide with Gh J -sink maps; (3) We generalize the classical Auslander-Reiten theory by using ideal mutations.
650		4	\|a Auslander-Reiten theory
650		4	\|a Ideal mutation
650		4	\|a Ideal approximation theory
650		4	\|a Auslander-Reiten quiver
650		4	\|a Auslander-Reiten sequence
700	1		\|a Zhu, Bin \|e verfasserin \|4 aut
773	0	8	\|i Enthalten in \|t Journal of algebra \|d San Diego, Calif. : Elsevier, 1964 \|g 644, Seite 191-221 \|h Online-Ressource \|w (DE-627)266890423 \|w (DE-600)1468947-9 \|w (DE-576)10337311X \|x 1090-266X \|7 nnns
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912			\|a GBV_ILN_224
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912			\|a GBV_ILN_370
912			\|a GBV_ILN_602
912			\|a GBV_ILN_702
912			\|a GBV_ILN_2001
912			\|a GBV_ILN_2003
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912			\|a GBV_ILN_2059
912			\|a GBV_ILN_2061
912			\|a GBV_ILN_2064
912			\|a GBV_ILN_2088
912			\|a GBV_ILN_2106
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912			\|a GBV_ILN_2152
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936	b	k	\|a 31.20 \|j Algebra: Allgemeines \|q VZ
951			\|a AR
952			\|d 644 \|h 191-221

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bklnumber	31.20
publishDate	2024
allfields	10.1016/j.jalgebra.2023.12.032 doi (DE-627)ELV06702713X (ELSEVIER)S0021-8693(24)00018-8 DE-627 ger DE-627 rda eng 510 VZ 31.20 bkl Zhang, Yaohua verfasserin (orcid)0000-0001-5343-3473 aut Ideal mutations in triangulated categories and generalized Auslander-Reiten theory 2024 nicht spezifiziert zzz rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier We introduce the notion of ideal mutations in a triangulated category, which generalizes the version of Iyama and Yoshino by replacing approximations by objects of a subcategory with approximations by morphisms of an ideal. As applications, for a Hom -finite Krull-Schmidt triangulated category T over an algebraically closed field K. (1) We generalize a theorem of Jorgensen to a more general setting; (2) We provide a method to detect whether T has Auslander-Reiten triangles or not by checking the necessary and sufficient conditions on its Jacobson radical J : (i) J is functorially finite, (ii) Gh J = CoGh J , and (iii) Gh J -source maps coincide with Gh J -sink maps; (3) We generalize the classical Auslander-Reiten theory by using ideal mutations. Auslander-Reiten theory Ideal mutation Ideal approximation theory Auslander-Reiten quiver Auslander-Reiten sequence Zhu, Bin verfasserin aut Enthalten in Journal of algebra San Diego, Calif. : Elsevier, 1964 644, Seite 191-221 Online-Ressource (DE-627)266890423 (DE-600)1468947-9 (DE-576)10337311X 1090-266X nnns volume:644 pages:191-221 GBV_USEFLAG_U GBV_ELV SYSFLAG_U SSG-OPC-MAT GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_150 GBV_ILN_151 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_370 GBV_ILN_602 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 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_2034 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2088 GBV_ILN_2106 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2470 GBV_ILN_2507 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 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_4338 GBV_ILN_4367 GBV_ILN_4393 GBV_ILN_4700 31.20 Algebra: Allgemeines VZ AR 644 191-221
spelling	10.1016/j.jalgebra.2023.12.032 doi (DE-627)ELV06702713X (ELSEVIER)S0021-8693(24)00018-8 DE-627 ger DE-627 rda eng 510 VZ 31.20 bkl Zhang, Yaohua verfasserin (orcid)0000-0001-5343-3473 aut Ideal mutations in triangulated categories and generalized Auslander-Reiten theory 2024 nicht spezifiziert zzz rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier We introduce the notion of ideal mutations in a triangulated category, which generalizes the version of Iyama and Yoshino by replacing approximations by objects of a subcategory with approximations by morphisms of an ideal. As applications, for a Hom -finite Krull-Schmidt triangulated category T over an algebraically closed field K. (1) We generalize a theorem of Jorgensen to a more general setting; (2) We provide a method to detect whether T has Auslander-Reiten triangles or not by checking the necessary and sufficient conditions on its Jacobson radical J : (i) J is functorially finite, (ii) Gh J = CoGh J , and (iii) Gh J -source maps coincide with Gh J -sink maps; (3) We generalize the classical Auslander-Reiten theory by using ideal mutations. Auslander-Reiten theory Ideal mutation Ideal approximation theory Auslander-Reiten quiver Auslander-Reiten sequence Zhu, Bin verfasserin aut Enthalten in Journal of algebra San Diego, Calif. : Elsevier, 1964 644, Seite 191-221 Online-Ressource (DE-627)266890423 (DE-600)1468947-9 (DE-576)10337311X 1090-266X nnns volume:644 pages:191-221 GBV_USEFLAG_U GBV_ELV SYSFLAG_U SSG-OPC-MAT GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_150 GBV_ILN_151 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_370 GBV_ILN_602 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 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_2034 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2088 GBV_ILN_2106 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2470 GBV_ILN_2507 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 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_4338 GBV_ILN_4367 GBV_ILN_4393 GBV_ILN_4700 31.20 Algebra: Allgemeines VZ AR 644 191-221
allfields_unstemmed	10.1016/j.jalgebra.2023.12.032 doi (DE-627)ELV06702713X (ELSEVIER)S0021-8693(24)00018-8 DE-627 ger DE-627 rda eng 510 VZ 31.20 bkl Zhang, Yaohua verfasserin (orcid)0000-0001-5343-3473 aut Ideal mutations in triangulated categories and generalized Auslander-Reiten theory 2024 nicht spezifiziert zzz rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier We introduce the notion of ideal mutations in a triangulated category, which generalizes the version of Iyama and Yoshino by replacing approximations by objects of a subcategory with approximations by morphisms of an ideal. As applications, for a Hom -finite Krull-Schmidt triangulated category T over an algebraically closed field K. (1) We generalize a theorem of Jorgensen to a more general setting; (2) We provide a method to detect whether T has Auslander-Reiten triangles or not by checking the necessary and sufficient conditions on its Jacobson radical J : (i) J is functorially finite, (ii) Gh J = CoGh J , and (iii) Gh J -source maps coincide with Gh J -sink maps; (3) We generalize the classical Auslander-Reiten theory by using ideal mutations. Auslander-Reiten theory Ideal mutation Ideal approximation theory Auslander-Reiten quiver Auslander-Reiten sequence Zhu, Bin verfasserin aut Enthalten in Journal of algebra San Diego, Calif. : Elsevier, 1964 644, Seite 191-221 Online-Ressource (DE-627)266890423 (DE-600)1468947-9 (DE-576)10337311X 1090-266X nnns volume:644 pages:191-221 GBV_USEFLAG_U GBV_ELV SYSFLAG_U SSG-OPC-MAT GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_150 GBV_ILN_151 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_370 GBV_ILN_602 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 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_2034 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2088 GBV_ILN_2106 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2470 GBV_ILN_2507 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 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_4338 GBV_ILN_4367 GBV_ILN_4393 GBV_ILN_4700 31.20 Algebra: Allgemeines VZ AR 644 191-221
allfieldsGer	10.1016/j.jalgebra.2023.12.032 doi (DE-627)ELV06702713X (ELSEVIER)S0021-8693(24)00018-8 DE-627 ger DE-627 rda eng 510 VZ 31.20 bkl Zhang, Yaohua verfasserin (orcid)0000-0001-5343-3473 aut Ideal mutations in triangulated categories and generalized Auslander-Reiten theory 2024 nicht spezifiziert zzz rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier We introduce the notion of ideal mutations in a triangulated category, which generalizes the version of Iyama and Yoshino by replacing approximations by objects of a subcategory with approximations by morphisms of an ideal. As applications, for a Hom -finite Krull-Schmidt triangulated category T over an algebraically closed field K. (1) We generalize a theorem of Jorgensen to a more general setting; (2) We provide a method to detect whether T has Auslander-Reiten triangles or not by checking the necessary and sufficient conditions on its Jacobson radical J : (i) J is functorially finite, (ii) Gh J = CoGh J , and (iii) Gh J -source maps coincide with Gh J -sink maps; (3) We generalize the classical Auslander-Reiten theory by using ideal mutations. Auslander-Reiten theory Ideal mutation Ideal approximation theory Auslander-Reiten quiver Auslander-Reiten sequence Zhu, Bin verfasserin aut Enthalten in Journal of algebra San Diego, Calif. : Elsevier, 1964 644, Seite 191-221 Online-Ressource (DE-627)266890423 (DE-600)1468947-9 (DE-576)10337311X 1090-266X nnns volume:644 pages:191-221 GBV_USEFLAG_U GBV_ELV SYSFLAG_U SSG-OPC-MAT GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_150 GBV_ILN_151 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_370 GBV_ILN_602 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 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_2034 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2088 GBV_ILN_2106 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2470 GBV_ILN_2507 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 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_4338 GBV_ILN_4367 GBV_ILN_4393 GBV_ILN_4700 31.20 Algebra: Allgemeines VZ AR 644 191-221
allfieldsSound	10.1016/j.jalgebra.2023.12.032 doi (DE-627)ELV06702713X (ELSEVIER)S0021-8693(24)00018-8 DE-627 ger DE-627 rda eng 510 VZ 31.20 bkl Zhang, Yaohua verfasserin (orcid)0000-0001-5343-3473 aut Ideal mutations in triangulated categories and generalized Auslander-Reiten theory 2024 nicht spezifiziert zzz rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier We introduce the notion of ideal mutations in a triangulated category, which generalizes the version of Iyama and Yoshino by replacing approximations by objects of a subcategory with approximations by morphisms of an ideal. As applications, for a Hom -finite Krull-Schmidt triangulated category T over an algebraically closed field K. (1) We generalize a theorem of Jorgensen to a more general setting; (2) We provide a method to detect whether T has Auslander-Reiten triangles or not by checking the necessary and sufficient conditions on its Jacobson radical J : (i) J is functorially finite, (ii) Gh J = CoGh J , and (iii) Gh J -source maps coincide with Gh J -sink maps; (3) We generalize the classical Auslander-Reiten theory by using ideal mutations. Auslander-Reiten theory Ideal mutation Ideal approximation theory Auslander-Reiten quiver Auslander-Reiten sequence Zhu, Bin verfasserin aut Enthalten in Journal of algebra San Diego, Calif. : Elsevier, 1964 644, Seite 191-221 Online-Ressource (DE-627)266890423 (DE-600)1468947-9 (DE-576)10337311X 1090-266X nnns volume:644 pages:191-221 GBV_USEFLAG_U GBV_ELV SYSFLAG_U SSG-OPC-MAT GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_150 GBV_ILN_151 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_370 GBV_ILN_602 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 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_2034 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2088 GBV_ILN_2106 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2470 GBV_ILN_2507 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 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_4338 GBV_ILN_4367 GBV_ILN_4393 GBV_ILN_4700 31.20 Algebra: Allgemeines VZ AR 644 191-221
language	English
source	Enthalten in Journal of algebra 644, Seite 191-221 volume:644 pages:191-221
sourceStr	Enthalten in Journal of algebra 644, Seite 191-221 volume:644 pages:191-221
format_phy_str_mv	Article
bklname	Algebra: Allgemeines
institution	findex.gbv.de
topic_facet	Auslander-Reiten theory Ideal mutation Ideal approximation theory Auslander-Reiten quiver Auslander-Reiten sequence
dewey-raw	510
isfreeaccess_bool	false
container_title	Journal of algebra
authorswithroles_txt_mv	Zhang, Yaohua @@aut@@ Zhu, Bin @@aut@@
publishDateDaySort_date	2024-01-01T00:00:00Z
hierarchy_top_id	266890423
dewey-sort	3510
id	ELV06702713X
language_de	englisch
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author	Zhang, Yaohua
spellingShingle	Zhang, Yaohua ddc 510 bkl 31.20 misc Auslander-Reiten theory misc Ideal mutation misc Ideal approximation theory misc Auslander-Reiten quiver misc Auslander-Reiten sequence Ideal mutations in triangulated categories and generalized Auslander-Reiten theory
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topic_title	510 VZ 31.20 bkl Ideal mutations in triangulated categories and generalized Auslander-Reiten theory Auslander-Reiten theory Ideal mutation Ideal approximation theory Auslander-Reiten quiver Auslander-Reiten sequence
topic	ddc 510 bkl 31.20 misc Auslander-Reiten theory misc Ideal mutation misc Ideal approximation theory misc Auslander-Reiten quiver misc Auslander-Reiten sequence
topic_unstemmed	ddc 510 bkl 31.20 misc Auslander-Reiten theory misc Ideal mutation misc Ideal approximation theory misc Auslander-Reiten quiver misc Auslander-Reiten sequence
topic_browse	ddc 510 bkl 31.20 misc Auslander-Reiten theory misc Ideal mutation misc Ideal approximation theory misc Auslander-Reiten quiver misc Auslander-Reiten sequence
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title	Ideal mutations in triangulated categories and generalized Auslander-Reiten theory
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title_full	Ideal mutations in triangulated categories and generalized Auslander-Reiten theory
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title_sort	ideal mutations in triangulated categories and generalized auslander-reiten theory
title_auth	Ideal mutations in triangulated categories and generalized Auslander-Reiten theory
abstract	We introduce the notion of ideal mutations in a triangulated category, which generalizes the version of Iyama and Yoshino by replacing approximations by objects of a subcategory with approximations by morphisms of an ideal. As applications, for a Hom -finite Krull-Schmidt triangulated category T over an algebraically closed field K. (1) We generalize a theorem of Jorgensen to a more general setting; (2) We provide a method to detect whether T has Auslander-Reiten triangles or not by checking the necessary and sufficient conditions on its Jacobson radical J : (i) J is functorially finite, (ii) Gh J = CoGh J , and (iii) Gh J -source maps coincide with Gh J -sink maps; (3) We generalize the classical Auslander-Reiten theory by using ideal mutations.
abstractGer	We introduce the notion of ideal mutations in a triangulated category, which generalizes the version of Iyama and Yoshino by replacing approximations by objects of a subcategory with approximations by morphisms of an ideal. As applications, for a Hom -finite Krull-Schmidt triangulated category T over an algebraically closed field K. (1) We generalize a theorem of Jorgensen to a more general setting; (2) We provide a method to detect whether T has Auslander-Reiten triangles or not by checking the necessary and sufficient conditions on its Jacobson radical J : (i) J is functorially finite, (ii) Gh J = CoGh J , and (iii) Gh J -source maps coincide with Gh J -sink maps; (3) We generalize the classical Auslander-Reiten theory by using ideal mutations.
abstract_unstemmed	We introduce the notion of ideal mutations in a triangulated category, which generalizes the version of Iyama and Yoshino by replacing approximations by objects of a subcategory with approximations by morphisms of an ideal. As applications, for a Hom -finite Krull-Schmidt triangulated category T over an algebraically closed field K. (1) We generalize a theorem of Jorgensen to a more general setting; (2) We provide a method to detect whether T has Auslander-Reiten triangles or not by checking the necessary and sufficient conditions on its Jacobson radical J : (i) J is functorially finite, (ii) Gh J = CoGh J , and (iii) Gh J -source maps coincide with Gh J -sink maps; (3) We generalize the classical Auslander-Reiten theory by using ideal mutations.
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title_short	Ideal mutations in triangulated categories and generalized Auslander-Reiten theory
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up_date	2024-07-06T19:49:43.068Z
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Ideal mutations in triangulated categories and generalized Auslander-Reiten theory

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