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Integrality in the Steinberg module and ...
Church, Thomas...

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Integrality in the Steinberg module and the top-dimensional cohomology of SL_n(O_K) by Church, Thomas ( Author )
Lemonjar Open Access
N.A
06-01-2015
We prove a new structural result for the spherical Tits building attached to SL_n(K) for many number fields K, and more generally for the fraction fields of many Dedekind domains O: the Steinberg module St_n(K) is generated by integral apartments if and only if the ideal class group cl(O) is trivial. We deduce this integrality by proving that the complex of partial bases of O^n is Cohen-Macaulay. We apply this to prove new vanishing and nonvanishing results for H^{vcd}(SL_n(O_K); Q), where O_K is the ring of integers in a number field and vcd is the virtual cohomological dimension of SL_n(O_K). The (non)vanishing depends on the (non)triviality of the class group of O_K. We also obtain a vanishing theorem for the cohomology H^{vcd}(SL_n(O_K); V) with twisted coefficients V.
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Article
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36.88 KB
English
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MYR 0.01
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https://arxiv.org/abs/1501.01307
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