Homological symbols and the Quillen conjecture. (English) Zbl 1207.20041
Let \(R=\mathbb{Z}[\tfrac1\ell,\xi_\ell]\) for an odd regular prime \(\ell\), where \(\xi_\ell\) is a primitive \(\ell\)-th root of unity. Consider in the bi-graded algebra \(A=\bigoplus_{i,j=0}^\infty H_i(\text{GL}_j(R),\mathbb{F}_\ell)\) the subspace \(V=H_*(\text{GL}_1(R),\mathbb{F}_\ell)\).
The author conjectures that the kernel of the map from the tensor algebra on \(V\) to \(A\) is generated by certain relations coming from \(H_*(\text{GL}_1(R)^{\times 2},\mathbb{F}_\ell)\). Now take \(\ell=5\). Using the computer algebra system GAP the author confirms the conjecture for this \(\ell\) and computes both \(H_2(\text{GL}_2(\mathbb{Z}[\tfrac15,\xi_5]),\mathbb{F}_5)\) and \(H_2(\text{SL}_2(\mathbb{Z}[\frac15,\xi_5]),\mathbb{F}_5)\).
The author conjectures that the kernel of the map from the tensor algebra on \(V\) to \(A\) is generated by certain relations coming from \(H_*(\text{GL}_1(R)^{\times 2},\mathbb{F}_\ell)\). Now take \(\ell=5\). Using the computer algebra system GAP the author confirms the conjecture for this \(\ell\) and computes both \(H_2(\text{GL}_2(\mathbb{Z}[\tfrac15,\xi_5]),\mathbb{F}_5)\) and \(H_2(\text{SL}_2(\mathbb{Z}[\frac15,\xi_5]),\mathbb{F}_5)\).
Reviewer: Wilberd van der Kallen (Utrecht)
MSC:
| 20G10 | Cohomology theory for linear algebraic groups |
| 19C20 | Symbols, presentations and stability of \(K_2\) |
| 20G30 | Linear algebraic groups over global fields and their integers |
Software:
GAPReferences:
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