The Auslander–Gruson–Jensen recollement

For any ring R, the Auslander–Gruson–Jensen functor D A...
Ausführliche Beschreibung

For any ring R, the Auslander–Gruson–Jensen functor D A : fp ( R - Mod , Ab ) → ( mod - R , Ab ) op is the exact functor which sends a representable functor ( X , − ) to the tensor functor − ⊗ X . We show that this functor admits a fully faithful right adjoint D R and a fully faithful left adjoint D L . That is, we show that D A is part of a recollement of abelian categories. In particular, this shows that D A is a localisation and a colocalisation which gives an equivalence of categories fp ( R - Mod , Ab ) { F : D A F = 0 } ≃ ( mod - R , Ab ) op . We show that { F : D A F = 0 } is the Serre subcategory of fp ( R - Mod , Ab ) consisting of finitely presented functors which arise from a pure-exact sequence. As an application of our main result, we show that the 0-th right pure-derived functor of a finitely presented functor R - Mod → Ab is also finitely presented. Ausführliche Beschreibung

Gespeichert in:

Autor*in:	Dean, Samuel [verfasserIn] Russell, Jeremy [verfasserIn]

Format:	E-Artikel
Sprache:	Englisch

Erschienen:	2018

Schlagwörter:	Recollements Finitely presented functors

Übergeordnetes Werk:	Enthalten in: Journal of algebra - San Diego, Calif. : Elsevier, 1964, 511, Seite 440-454
Übergeordnetes Werk:	volume:511 ; pages:440-454

DOI / URN:	10.1016/j.jalgebra.2018.05.032

Katalog-ID:	ELV000166618

Internformat


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520			\|a For any ring R, the Auslander–Gruson–Jensen functor D A : fp ( R - Mod , Ab ) → ( mod - R , Ab ) op is the exact functor which sends a representable functor ( X , − ) to the tensor functor − ⊗ X . We show that this functor admits a fully faithful right adjoint D R and a fully faithful left adjoint D L . That is, we show that D A is part of a recollement of abelian categories. In particular, this shows that D A is a localisation and a colocalisation which gives an equivalence of categories fp ( R - Mod , Ab ) { F : D A F = 0 } ≃ ( mod - R , Ab ) op . We show that { F : D A F = 0 } is the Serre subcategory of fp ( R - Mod , Ab ) consisting of finitely presented functors which arise from a pure-exact sequence. As an application of our main result, we show that the 0-th right pure-derived functor of a finitely presented functor R - Mod → Ab is also finitely presented.
650		4	\|a Recollements
650		4	\|a Finitely presented functors
700	1		\|a Russell, Jeremy \|e verfasserin \|4 aut
773	0	8	\|i Enthalten in \|t Journal of algebra \|d San Diego, Calif. : Elsevier, 1964 \|g 511, Seite 440-454 \|h Online-Ressource \|w (DE-627)266890423 \|w (DE-600)1468947-9 \|w (DE-576)10337311X \|x 1090-266X \|7 nnns
773	1	8	\|g volume:511 \|g pages:440-454
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912			\|a GBV_ILN_2065
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912			\|a GBV_ILN_4333
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912			\|a GBV_ILN_4335
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912			\|a GBV_ILN_4367
912			\|a GBV_ILN_4393
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936	b	k	\|a 31.20 \|j Algebra: Allgemeines
951			\|a AR
952			\|d 511 \|h 440-454

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matchkey_str	article:1090266X:2018----::hasadrrsnesn
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bklnumber	31.20
publishDate	2018
allfields	10.1016/j.jalgebra.2018.05.032 doi (DE-627)ELV000166618 (ELSEVIER)S0021-8693(18)30372-7 DE-627 ger DE-627 rda eng 510 DE-600 31.20 bkl Dean, Samuel verfasserin (orcid)0000-0002-9583-1258 aut The Auslander–Gruson–Jensen recollement 2018 nicht spezifiziert zzz rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier For any ring R, the Auslander–Gruson–Jensen functor D A : fp ( R - Mod , Ab ) → ( mod - R , Ab ) op is the exact functor which sends a representable functor ( X , − ) to the tensor functor − ⊗ X . We show that this functor admits a fully faithful right adjoint D R and a fully faithful left adjoint D L . That is, we show that D A is part of a recollement of abelian categories. In particular, this shows that D A is a localisation and a colocalisation which gives an equivalence of categories fp ( R - Mod , Ab ) { F : D A F = 0 } ≃ ( mod - R , Ab ) op . We show that { F : D A F = 0 } is the Serre subcategory of fp ( R - Mod , Ab ) consisting of finitely presented functors which arise from a pure-exact sequence. As an application of our main result, we show that the 0-th right pure-derived functor of a finitely presented functor R - Mod → Ab is also finitely presented. Recollements Finitely presented functors Russell, Jeremy verfasserin aut Enthalten in Journal of algebra San Diego, Calif. : Elsevier, 1964 511, Seite 440-454 Online-Ressource (DE-627)266890423 (DE-600)1468947-9 (DE-576)10337311X 1090-266X nnns volume:511 pages:440-454 GBV_USEFLAG_U SYSFLAG_U GBV_ELV SSG-OPC-MAT 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_105 GBV_ILN_110 GBV_ILN_150 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_702 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2027 GBV_ILN_2034 GBV_ILN_2038 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2056 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2190 GBV_ILN_2336 GBV_ILN_2507 GBV_ILN_2522 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 AR 511 440-454
spelling	10.1016/j.jalgebra.2018.05.032 doi (DE-627)ELV000166618 (ELSEVIER)S0021-8693(18)30372-7 DE-627 ger DE-627 rda eng 510 DE-600 31.20 bkl Dean, Samuel verfasserin (orcid)0000-0002-9583-1258 aut The Auslander–Gruson–Jensen recollement 2018 nicht spezifiziert zzz rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier For any ring R, the Auslander–Gruson–Jensen functor D A : fp ( R - Mod , Ab ) → ( mod - R , Ab ) op is the exact functor which sends a representable functor ( X , − ) to the tensor functor − ⊗ X . We show that this functor admits a fully faithful right adjoint D R and a fully faithful left adjoint D L . That is, we show that D A is part of a recollement of abelian categories. In particular, this shows that D A is a localisation and a colocalisation which gives an equivalence of categories fp ( R - Mod , Ab ) { F : D A F = 0 } ≃ ( mod - R , Ab ) op . We show that { F : D A F = 0 } is the Serre subcategory of fp ( R - Mod , Ab ) consisting of finitely presented functors which arise from a pure-exact sequence. As an application of our main result, we show that the 0-th right pure-derived functor of a finitely presented functor R - Mod → Ab is also finitely presented. Recollements Finitely presented functors Russell, Jeremy verfasserin aut Enthalten in Journal of algebra San Diego, Calif. : Elsevier, 1964 511, Seite 440-454 Online-Ressource (DE-627)266890423 (DE-600)1468947-9 (DE-576)10337311X 1090-266X nnns volume:511 pages:440-454 GBV_USEFLAG_U SYSFLAG_U GBV_ELV SSG-OPC-MAT 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_105 GBV_ILN_110 GBV_ILN_150 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_702 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2027 GBV_ILN_2034 GBV_ILN_2038 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2056 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2190 GBV_ILN_2336 GBV_ILN_2507 GBV_ILN_2522 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 AR 511 440-454
allfields_unstemmed	10.1016/j.jalgebra.2018.05.032 doi (DE-627)ELV000166618 (ELSEVIER)S0021-8693(18)30372-7 DE-627 ger DE-627 rda eng 510 DE-600 31.20 bkl Dean, Samuel verfasserin (orcid)0000-0002-9583-1258 aut The Auslander–Gruson–Jensen recollement 2018 nicht spezifiziert zzz rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier For any ring R, the Auslander–Gruson–Jensen functor D A : fp ( R - Mod , Ab ) → ( mod - R , Ab ) op is the exact functor which sends a representable functor ( X , − ) to the tensor functor − ⊗ X . We show that this functor admits a fully faithful right adjoint D R and a fully faithful left adjoint D L . That is, we show that D A is part of a recollement of abelian categories. In particular, this shows that D A is a localisation and a colocalisation which gives an equivalence of categories fp ( R - Mod , Ab ) { F : D A F = 0 } ≃ ( mod - R , Ab ) op . We show that { F : D A F = 0 } is the Serre subcategory of fp ( R - Mod , Ab ) consisting of finitely presented functors which arise from a pure-exact sequence. As an application of our main result, we show that the 0-th right pure-derived functor of a finitely presented functor R - Mod → Ab is also finitely presented. Recollements Finitely presented functors Russell, Jeremy verfasserin aut Enthalten in Journal of algebra San Diego, Calif. : Elsevier, 1964 511, Seite 440-454 Online-Ressource (DE-627)266890423 (DE-600)1468947-9 (DE-576)10337311X 1090-266X nnns volume:511 pages:440-454 GBV_USEFLAG_U SYSFLAG_U GBV_ELV SSG-OPC-MAT 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_105 GBV_ILN_110 GBV_ILN_150 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_702 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2027 GBV_ILN_2034 GBV_ILN_2038 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2056 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2190 GBV_ILN_2336 GBV_ILN_2507 GBV_ILN_2522 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 AR 511 440-454
allfieldsGer	10.1016/j.jalgebra.2018.05.032 doi (DE-627)ELV000166618 (ELSEVIER)S0021-8693(18)30372-7 DE-627 ger DE-627 rda eng 510 DE-600 31.20 bkl Dean, Samuel verfasserin (orcid)0000-0002-9583-1258 aut The Auslander–Gruson–Jensen recollement 2018 nicht spezifiziert zzz rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier For any ring R, the Auslander–Gruson–Jensen functor D A : fp ( R - Mod , Ab ) → ( mod - R , Ab ) op is the exact functor which sends a representable functor ( X , − ) to the tensor functor − ⊗ X . We show that this functor admits a fully faithful right adjoint D R and a fully faithful left adjoint D L . That is, we show that D A is part of a recollement of abelian categories. In particular, this shows that D A is a localisation and a colocalisation which gives an equivalence of categories fp ( R - Mod , Ab ) { F : D A F = 0 } ≃ ( mod - R , Ab ) op . We show that { F : D A F = 0 } is the Serre subcategory of fp ( R - Mod , Ab ) consisting of finitely presented functors which arise from a pure-exact sequence. As an application of our main result, we show that the 0-th right pure-derived functor of a finitely presented functor R - Mod → Ab is also finitely presented. Recollements Finitely presented functors Russell, Jeremy verfasserin aut Enthalten in Journal of algebra San Diego, Calif. : Elsevier, 1964 511, Seite 440-454 Online-Ressource (DE-627)266890423 (DE-600)1468947-9 (DE-576)10337311X 1090-266X nnns volume:511 pages:440-454 GBV_USEFLAG_U SYSFLAG_U GBV_ELV SSG-OPC-MAT 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_105 GBV_ILN_110 GBV_ILN_150 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_702 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2027 GBV_ILN_2034 GBV_ILN_2038 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2056 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2190 GBV_ILN_2336 GBV_ILN_2507 GBV_ILN_2522 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 AR 511 440-454
allfieldsSound	10.1016/j.jalgebra.2018.05.032 doi (DE-627)ELV000166618 (ELSEVIER)S0021-8693(18)30372-7 DE-627 ger DE-627 rda eng 510 DE-600 31.20 bkl Dean, Samuel verfasserin (orcid)0000-0002-9583-1258 aut The Auslander–Gruson–Jensen recollement 2018 nicht spezifiziert zzz rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier For any ring R, the Auslander–Gruson–Jensen functor D A : fp ( R - Mod , Ab ) → ( mod - R , Ab ) op is the exact functor which sends a representable functor ( X , − ) to the tensor functor − ⊗ X . We show that this functor admits a fully faithful right adjoint D R and a fully faithful left adjoint D L . That is, we show that D A is part of a recollement of abelian categories. In particular, this shows that D A is a localisation and a colocalisation which gives an equivalence of categories fp ( R - Mod , Ab ) { F : D A F = 0 } ≃ ( mod - R , Ab ) op . We show that { F : D A F = 0 } is the Serre subcategory of fp ( R - Mod , Ab ) consisting of finitely presented functors which arise from a pure-exact sequence. As an application of our main result, we show that the 0-th right pure-derived functor of a finitely presented functor R - Mod → Ab is also finitely presented. Recollements Finitely presented functors Russell, Jeremy verfasserin aut Enthalten in Journal of algebra San Diego, Calif. : Elsevier, 1964 511, Seite 440-454 Online-Ressource (DE-627)266890423 (DE-600)1468947-9 (DE-576)10337311X 1090-266X nnns volume:511 pages:440-454 GBV_USEFLAG_U SYSFLAG_U GBV_ELV SSG-OPC-MAT 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_105 GBV_ILN_110 GBV_ILN_150 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_702 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2027 GBV_ILN_2034 GBV_ILN_2038 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2056 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2190 GBV_ILN_2336 GBV_ILN_2507 GBV_ILN_2522 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 AR 511 440-454
language	English
source	Enthalten in Journal of algebra 511, Seite 440-454 volume:511 pages:440-454
sourceStr	Enthalten in Journal of algebra 511, Seite 440-454 volume:511 pages:440-454
format_phy_str_mv	Article
bklname	Algebra: Allgemeines
institution	findex.gbv.de
topic_facet	Recollements Finitely presented functors
dewey-raw	510
isfreeaccess_bool	false
container_title	Journal of algebra
authorswithroles_txt_mv	Dean, Samuel @@aut@@ Russell, Jeremy @@aut@@
publishDateDaySort_date	2018-01-01T00:00:00Z
hierarchy_top_id	266890423
dewey-sort	3510
id	ELV000166618
language_de	englisch
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author	Dean, Samuel
spellingShingle	Dean, Samuel ddc 510 bkl 31.20 misc Recollements misc Finitely presented functors The Auslander–Gruson–Jensen recollement
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topic_title	510 DE-600 31.20 bkl The Auslander–Gruson–Jensen recollement Recollements Finitely presented functors
topic	ddc 510 bkl 31.20 misc Recollements misc Finitely presented functors
topic_unstemmed	ddc 510 bkl 31.20 misc Recollements misc Finitely presented functors
topic_browse	ddc 510 bkl 31.20 misc Recollements misc Finitely presented functors
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title	The Auslander–Gruson–Jensen recollement
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title_full	The Auslander–Gruson–Jensen recollement
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title_sort	the auslander–gruson–jensen recollement
title_auth	The Auslander–Gruson–Jensen recollement
abstract	For any ring R, the Auslander–Gruson–Jensen functor D A : fp ( R - Mod , Ab ) → ( mod - R , Ab ) op is the exact functor which sends a representable functor ( X , − ) to the tensor functor − ⊗ X . We show that this functor admits a fully faithful right adjoint D R and a fully faithful left adjoint D L . That is, we show that D A is part of a recollement of abelian categories. In particular, this shows that D A is a localisation and a colocalisation which gives an equivalence of categories fp ( R - Mod , Ab ) { F : D A F = 0 } ≃ ( mod - R , Ab ) op . We show that { F : D A F = 0 } is the Serre subcategory of fp ( R - Mod , Ab ) consisting of finitely presented functors which arise from a pure-exact sequence. As an application of our main result, we show that the 0-th right pure-derived functor of a finitely presented functor R - Mod → Ab is also finitely presented.
abstractGer	For any ring R, the Auslander–Gruson–Jensen functor D A : fp ( R - Mod , Ab ) → ( mod - R , Ab ) op is the exact functor which sends a representable functor ( X , − ) to the tensor functor − ⊗ X . We show that this functor admits a fully faithful right adjoint D R and a fully faithful left adjoint D L . That is, we show that D A is part of a recollement of abelian categories. In particular, this shows that D A is a localisation and a colocalisation which gives an equivalence of categories fp ( R - Mod , Ab ) { F : D A F = 0 } ≃ ( mod - R , Ab ) op . We show that { F : D A F = 0 } is the Serre subcategory of fp ( R - Mod , Ab ) consisting of finitely presented functors which arise from a pure-exact sequence. As an application of our main result, we show that the 0-th right pure-derived functor of a finitely presented functor R - Mod → Ab is also finitely presented.
abstract_unstemmed	For any ring R, the Auslander–Gruson–Jensen functor D A : fp ( R - Mod , Ab ) → ( mod - R , Ab ) op is the exact functor which sends a representable functor ( X , − ) to the tensor functor − ⊗ X . We show that this functor admits a fully faithful right adjoint D R and a fully faithful left adjoint D L . That is, we show that D A is part of a recollement of abelian categories. In particular, this shows that D A is a localisation and a colocalisation which gives an equivalence of categories fp ( R - Mod , Ab ) { F : D A F = 0 } ≃ ( mod - R , Ab ) op . We show that { F : D A F = 0 } is the Serre subcategory of fp ( R - Mod , Ab ) consisting of finitely presented functors which arise from a pure-exact sequence. As an application of our main result, we show that the 0-th right pure-derived functor of a finitely presented functor R - Mod → Ab is also finitely presented.
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title_short	The Auslander–Gruson–Jensen recollement
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up_date	2024-07-06T17:04:19.060Z
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score	7.398814

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The Auslander–Gruson–Jensen recollement

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