{"title":"论 SLn(Z) 的同调性","authors":"Avner Ash","doi":"10.1016/j.aim.2024.109868","DOIUrl":null,"url":null,"abstract":"<div><p>Denote the virtual cohomological dimension of <span><math><msub><mrow><mi>SL</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>Z</mi><mo>)</mo></math></span> by <span><math><msub><mrow><mi>ν</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mi>n</mi><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>/</mo><mn>2</mn></math></span>. Let <em>St</em> denote the Steinberg module of <span><math><msub><mrow><mi>SL</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>Q</mi><mo>)</mo></math></span> tensored with <span><math><mi>Q</mi></math></span>. Let <span><math><mi>S</mi><msub><mrow><mi>h</mi></mrow><mrow><mo>•</mo></mrow></msub><mo>→</mo><mi>S</mi><mi>t</mi></math></span> denote the sharbly resolution of the Steinberg module. By Borel-Serre duality, <span><math><msup><mrow><mi>H</mi></mrow><mrow><msub><mrow><mi>ν</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>−</mo><mi>i</mi></mrow></msup><mo>(</mo><msub><mrow><mi>SL</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>Z</mi><mo>)</mo><mo>,</mo><mi>Q</mi><mo>)</mo></math></span> is isomorphic to <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>(</mo><msub><mrow><mi>SL</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>Z</mi><mo>)</mo><mo>,</mo><mi>S</mi><mi>t</mi><mo>)</mo></math></span>. The latter is isomorphic to the sharbly homology <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>(</mo><msub><mrow><mo>(</mo><mi>S</mi><msub><mrow><mi>h</mi></mrow><mrow><mo>•</mo></mrow></msub><mo>)</mo></mrow><mrow><msub><mrow><mi>SL</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>Z</mi><mo>)</mo></mrow></msub><mo>)</mo></math></span>. We produce nonzero classes in <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>(</mo><msub><mrow><mi>SL</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>Z</mi><mo>)</mo><mo>,</mo><mi>S</mi><mi>t</mi><mo>)</mo></math></span>, for certain small <em>i</em>, in terms of sharbly cycles and cosharbly cocycles.</p></div>","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the cohomology of SLn(Z)\",\"authors\":\"Avner Ash\",\"doi\":\"10.1016/j.aim.2024.109868\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Denote the virtual cohomological dimension of <span><math><msub><mrow><mi>SL</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>Z</mi><mo>)</mo></math></span> by <span><math><msub><mrow><mi>ν</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mi>n</mi><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>/</mo><mn>2</mn></math></span>. Let <em>St</em> denote the Steinberg module of <span><math><msub><mrow><mi>SL</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>Q</mi><mo>)</mo></math></span> tensored with <span><math><mi>Q</mi></math></span>. Let <span><math><mi>S</mi><msub><mrow><mi>h</mi></mrow><mrow><mo>•</mo></mrow></msub><mo>→</mo><mi>S</mi><mi>t</mi></math></span> denote the sharbly resolution of the Steinberg module. By Borel-Serre duality, <span><math><msup><mrow><mi>H</mi></mrow><mrow><msub><mrow><mi>ν</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>−</mo><mi>i</mi></mrow></msup><mo>(</mo><msub><mrow><mi>SL</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>Z</mi><mo>)</mo><mo>,</mo><mi>Q</mi><mo>)</mo></math></span> is isomorphic to <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>(</mo><msub><mrow><mi>SL</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>Z</mi><mo>)</mo><mo>,</mo><mi>S</mi><mi>t</mi><mo>)</mo></math></span>. The latter is isomorphic to the sharbly homology <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>(</mo><msub><mrow><mo>(</mo><mi>S</mi><msub><mrow><mi>h</mi></mrow><mrow><mo>•</mo></mrow></msub><mo>)</mo></mrow><mrow><msub><mrow><mi>SL</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>Z</mi><mo>)</mo></mrow></msub><mo>)</mo></math></span>. We produce nonzero classes in <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>(</mo><msub><mrow><mi>SL</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>Z</mi><mo>)</mo><mo>,</mo><mi>S</mi><mi>t</mi><mo>)</mo></math></span>, for certain small <em>i</em>, in terms of sharbly cycles and cosharbly cocycles.</p></div>\",\"PeriodicalId\":1,\"journal\":{\"name\":\"Accounts of Chemical Research\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":16.4000,\"publicationDate\":\"2024-08-02\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Accounts of Chemical Research\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0001870824003839\",\"RegionNum\":1,\"RegionCategory\":\"化学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"CHEMISTRY, MULTIDISCIPLINARY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0001870824003839","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
Denote the virtual cohomological dimension of by . Let St denote the Steinberg module of tensored with . Let denote the sharbly resolution of the Steinberg module. By Borel-Serre duality, is isomorphic to . The latter is isomorphic to the sharbly homology . We produce nonzero classes in , for certain small i, in terms of sharbly cycles and cosharbly cocycles.
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