Skip to main content
We gratefully acknowledge support from the Simons Foundation, member institutions, and all contributors. Donate
arXiv logo
Cornell University Logo

Mathematics > Number Theory

arXiv:1501.01307 (math)
[Submitted on 6 Jan 2015 (v1), last revised 3 Feb 2019 (this version, v3)]

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

Authors:Thomas Church, Benson Farb, Andrew Putman
View PDF
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.
Comments: 36 pages; final version; to appear in Amer. J. Math
Subjects: Number Theory (math.NT); Algebraic Topology (math.AT); Group Theory (math.GR); Geometric Topology (math.GT)
Cite as: arXiv:1501.01307 [math.NT]
  (or arXiv:1501.01307v3 [math.NT] for this version)
  https://doi.org/10.48550/arXiv.1501.01307
Journal reference: Amer. J. Math. 141 (2019), no. 5, 1375-1419
Related DOI: https://doi.org/10.1353/ajm.2019.0036

Submission history

From: Andrew Putman [view email]
[v1] Tue, 6 Jan 2015 21:00:07 UTC (43 KB)
[v2] Wed, 17 Aug 2016 15:39:27 UTC (47 KB)
[v3] Sun, 3 Feb 2019 04:39:43 UTC (49 KB)
Full-text links:

Access Paper:

  • View PDF
  • TeX Source
  • Other Formats
view license
Current browse context:
math.NT
< prev   |   next >
new | recent | 2015-01
Change to browse by:
math
math.AT
math.GR
math.GT

References & Citations

export BibTeX citation

Bookmark

BibSonomy logo Reddit logo

Bibliographic and Citation Tools

Bibliographic Explorer (What is the Explorer?)
Connected Papers (What is Connected Papers?)
Litmaps (What is Litmaps?)
scite Smart Citations (What are Smart Citations?)

Code, Data and Media Associated with this Article

alphaXiv (What is alphaXiv?)
CatalyzeX Code Finder for Papers (What is CatalyzeX?)
DagsHub (What is DagsHub?)
Gotit.pub (What is GotitPub?)
Hugging Face (What is Huggingface?)
Papers with Code (What is Papers with Code?)
ScienceCast (What is ScienceCast?)

Demos

Replicate (What is Replicate?)
Hugging Face Spaces (What is Spaces?)
TXYZ.AI (What is TXYZ.AI?)

Recommenders and Search Tools

Influence Flower (What are Influence Flowers?)
CORE Recommender (What is CORE?)

arXivLabs: experimental projects with community collaborators

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.

Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)