Integrality in the Steinberg module and the top-dimensional cohomology of SL_n(O_K)

Abstract

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.

36 pages; final version; to appear in Amer. J. Math

References in corpus (1)

• The complex of partial bases for F_n and finite generation of the Torelli subgroup of Aut(F_n)

Cited by in corpus (17)

• Stability in the high-dimensional cohomology of congruence subgroups
• Non-integrality of some Steinberg modules
• On the codimension-two cohomology of ${\mathrm{SL}}_n(\mathbb{Z})$
• The dualizing module and top-dimensional cohomology group of ${\text{GL}}_n(\mathcal{O})$
• On the generalized Bykovskii presentation of Steinberg modules
• The Steinberg representation is irreducible
• The free factor complex and the dualizing module for the automorphism group of a free group
• On the top-dimensional cohomology of arithmetic Chevalley groups
• The common basis complex and the partial decomposition poset
• Cohen--Macaulay Complexes, Duality Groups, and the dualizing module of $\mathrm{Out}(F_N)$
• Posets arising from decompositions of objects in a monoidal category
• Steinberg homology, modular forms, and real quadratic fields
• Apartment classes of integral symplectic groups
• Cohomology of congruence subgroups of SL_3(Z), Steinberg modules, and real quadratic fields
• Top-degree rational cohomology in the symplectic group of a number ring
• Cutting and Pasting in the Torelli subgroup of Out($F_n$)
• Connectivity of partial basis complexes of freely decomposable groups