\(\text{SL}_n(\mathbb{Z}[t])\) is not \(\text{FP}_{n-1}\). (Q1045213)
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| Language | Label | Description | Also known as |
|---|---|---|---|
| English | \(\text{SL}_n(\mathbb{Z}[t])\) is not \(\text{FP}_{n-1}\). |
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\(\text{SL}_n(\mathbb{Z}[t])\) is not \(\text{FP}_{n-1}\). (English)
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15 December 2009
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In the title, \(n\geq 2\) and the statement means that there is no projective resolution of length \(n\) for \(\mathbb{Z}\) by finitely generated projective modules over the group. Geometry of Euclidean buildings is used in proofs. It was known before that \(\text{SL}_2(\mathbb{Z}[t])\) is not finitely generated and that \(\text{SL}_3(\mathbb{Z}[t])\) is not finitely presented. On the other hand, \(\text{SL}_3(\mathbb{Z}[t])\) is finitely generated (Suslin).
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Euclidean buildings
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projective resolutions
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groups of finite homological type
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infinitely presented groups
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finiteness properties
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