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The A-decomposability of the Singer construction
Let RsM denote the Singer construction on an unstable module M over the Steenrod algebra A at the prime two; RsM is canonically a subobject of Ps⊗M, where Ps=F2[x1,…,xs] with generators of degree one and F2 is the field with two elements. Passage to A-indecomposables gives the natural transformation...
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| Published in: | Journal of algebra 2019-01, Vol.517, p.186-206 |
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| Main Authors: | , |
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| Language: | English |
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| cites | cdi_FETCH-LOGICAL-c346t-1d51e45639153a241ba2f086fca5a546fe2d58f994630b2e08c14b70854f96733 |
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| container_title | Journal of algebra |
| container_volume | 517 |
| creator | Hưng, Nguyễn H.V. Powell, Geoffrey |
| description | Let RsM denote the Singer construction on an unstable module M over the Steenrod algebra A at the prime two; RsM is canonically a subobject of Ps⊗M, where Ps=F2[x1,…,xs] with generators of degree one and F2 is the field with two elements. Passage to A-indecomposables gives the natural transformation RsM→F2⊗A(Ps⊗M), which identifies with the dual of the composition of the Singer transfer and the Lannes–Zarati homomorphism.
The main result of the paper proves the weak generalized algebraic spherical class conjecture, which was proposed by the first author. Namely, this morphism is trivial on elements of positive degree when s>2. The condition s>2 is necessary, as exhibited by the spherical classes of Hopf invariant one and those of Kervaire invariant one. |
| doi_str_mv | 10.1016/j.jalgebra.2018.09.030 |
| format | article |
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The main result of the paper proves the weak generalized algebraic spherical class conjecture, which was proposed by the first author. Namely, this morphism is trivial on elements of positive degree when s>2. The condition s>2 is necessary, as exhibited by the spherical classes of Hopf invariant one and those of Kervaire invariant one.</description><identifier>ISSN: 0021-8693</identifier><identifier>EISSN: 1090-266X</identifier><identifier>DOI: 10.1016/j.jalgebra.2018.09.030</identifier><language>eng</language><publisher>Elsevier Inc</publisher><subject>Algebraic Topology ; Destabilization ; Indecomposables ; Mathematics ; Singer functor ; Steenrod algebra ; Unstable module</subject><ispartof>Journal of algebra, 2019-01, Vol.517, p.186-206</ispartof><rights>2018 Elsevier Inc.</rights><rights>Distributed under a Creative Commons Attribution 4.0 International License</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c346t-1d51e45639153a241ba2f086fca5a546fe2d58f994630b2e08c14b70854f96733</citedby><cites>FETCH-LOGICAL-c346t-1d51e45639153a241ba2f086fca5a546fe2d58f994630b2e08c14b70854f96733</cites><orcidid>0000-0001-6879-3180 ; 0000-0003-2564-1202</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>230,314,780,784,885,27924,27925</link.rule.ids><backlink>$$Uhttps://hal.science/hal-02324725$$DView record in HAL$$Hfree_for_read</backlink></links><search><creatorcontrib>Hưng, Nguyễn H.V.</creatorcontrib><creatorcontrib>Powell, Geoffrey</creatorcontrib><title>The A-decomposability of the Singer construction</title><title>Journal of algebra</title><description>Let RsM denote the Singer construction on an unstable module M over the Steenrod algebra A at the prime two; RsM is canonically a subobject of Ps⊗M, where Ps=F2[x1,…,xs] with generators of degree one and F2 is the field with two elements. Passage to A-indecomposables gives the natural transformation RsM→F2⊗A(Ps⊗M), which identifies with the dual of the composition of the Singer transfer and the Lannes–Zarati homomorphism.
The main result of the paper proves the weak generalized algebraic spherical class conjecture, which was proposed by the first author. Namely, this morphism is trivial on elements of positive degree when s>2. The condition s>2 is necessary, as exhibited by the spherical classes of Hopf invariant one and those of Kervaire invariant one.</description><subject>Algebraic Topology</subject><subject>Destabilization</subject><subject>Indecomposables</subject><subject>Mathematics</subject><subject>Singer functor</subject><subject>Steenrod algebra</subject><subject>Unstable module</subject><issn>0021-8693</issn><issn>1090-266X</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2019</creationdate><recordtype>article</recordtype><recordid>eNqFkE1LAzEQhoMoWD_-guzVw66TzyY3S1ErFDxYwVvIZpM2y3ZTkrXQf--WqldPAzPv88I8CN1hqDBg8dBWrenWrk6mIoBlBaoCCmdogkFBSYT4PEcTAIJLKRS9RFc5twAYcyYnCFYbV8zKxtm43cVs6tCF4VBEXwzj4T30a5cKG_s8pC87hNjfoAtvuuxuf-Y1-nh-Ws0X5fLt5XU-W5aWMjGUuOHYMS6owpwawnBtiAcpvDXccCa8Iw2XXikmKNTEgbSY1VOQnHklppReo_tT78Z0epfC1qSDjiboxWypjzsglLAp4Xs8ZsUpa1PMOTn_B2DQR0e61b-O9NGRBqVHRyP4eALd-Mk-uKSzDa63rgnJ2UE3MfxX8Q32-3D7</recordid><startdate>20190101</startdate><enddate>20190101</enddate><creator>Hưng, Nguyễn H.V.</creator><creator>Powell, Geoffrey</creator><general>Elsevier Inc</general><general>Elsevier</general><scope>AAYXX</scope><scope>CITATION</scope><scope>1XC</scope><orcidid>https://orcid.org/0000-0001-6879-3180</orcidid><orcidid>https://orcid.org/0000-0003-2564-1202</orcidid></search><sort><creationdate>20190101</creationdate><title>The A-decomposability of the Singer construction</title><author>Hưng, Nguyễn H.V. ; Powell, Geoffrey</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c346t-1d51e45639153a241ba2f086fca5a546fe2d58f994630b2e08c14b70854f96733</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2019</creationdate><topic>Algebraic Topology</topic><topic>Destabilization</topic><topic>Indecomposables</topic><topic>Mathematics</topic><topic>Singer functor</topic><topic>Steenrod algebra</topic><topic>Unstable module</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Hưng, Nguyễn H.V.</creatorcontrib><creatorcontrib>Powell, Geoffrey</creatorcontrib><collection>CrossRef</collection><collection>Hyper Article en Ligne (HAL)</collection><jtitle>Journal of algebra</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Hưng, Nguyễn H.V.</au><au>Powell, Geoffrey</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>The A-decomposability of the Singer construction</atitle><jtitle>Journal of algebra</jtitle><date>2019-01-01</date><risdate>2019</risdate><volume>517</volume><spage>186</spage><epage>206</epage><pages>186-206</pages><issn>0021-8693</issn><eissn>1090-266X</eissn><abstract>Let RsM denote the Singer construction on an unstable module M over the Steenrod algebra A at the prime two; RsM is canonically a subobject of Ps⊗M, where Ps=F2[x1,…,xs] with generators of degree one and F2 is the field with two elements. Passage to A-indecomposables gives the natural transformation RsM→F2⊗A(Ps⊗M), which identifies with the dual of the composition of the Singer transfer and the Lannes–Zarati homomorphism.
The main result of the paper proves the weak generalized algebraic spherical class conjecture, which was proposed by the first author. Namely, this morphism is trivial on elements of positive degree when s>2. The condition s>2 is necessary, as exhibited by the spherical classes of Hopf invariant one and those of Kervaire invariant one.</abstract><pub>Elsevier Inc</pub><doi>10.1016/j.jalgebra.2018.09.030</doi><tpages>21</tpages><orcidid>https://orcid.org/0000-0001-6879-3180</orcidid><orcidid>https://orcid.org/0000-0003-2564-1202</orcidid></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0021-8693 |
| ispartof | Journal of algebra, 2019-01, Vol.517, p.186-206 |
| issn | 0021-8693 1090-266X |
| language | eng |
| recordid | cdi_hal_primary_oai_HAL_hal_02324725v1 |
| source | ScienceDirect Journals |
| subjects | Algebraic Topology Destabilization Indecomposables Mathematics Singer functor Steenrod algebra Unstable module |
| title | The A-decomposability of the Singer construction |
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