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Homology of \(\operatorname{SL}_2\) over function fields. I: Parabolic subcomplexes. (English) Zbl 1400.20044
The author provides interesting, useful calculations of a part of the group homology, with \(\mathbb{Z}[1/2]\) coefficients, of \(\mathrm{SL}_2(k[C])\) where \(C\) is an affine curve over the algebraically closed field \(k\). Let \(\bar{C}\) be a smooth projective curve over \(k\), let \(P_1,\dots, P_s\) be closed points of \(\bar{C}\) and let \(C\) be the affine curve \(\bar{C}\setminus\{ P_1,\dots, P_s\}\). Let \(\Gamma = \mathrm{SL}_2(k[C])\) and let \(\mathfrak{X}_C\) be the associated building. \(\mathfrak{X}_C\) is a product of trees \(\mathfrak{T}_1\times \cdots \times \mathfrak{T}_s\) where \(\mathfrak{T}_i\) is the tree of the valuation \(v_i\) associated to \(P_i\). The vertices of the orbit space \(\Gamma\backslash\mathfrak{X}_C\) are naturally indexed by certain equivalence classes of rank \(2\) vector bundles \(\mathcal{E}\) on \(\bar{C}\) whose restriction to \(C\) is trivial. The author introduces the parabolic subcomplex \(\mathfrak{P}_C\) of \(\mathfrak{X}_C\) consisting of those cells whose stabilizer in \(\Gamma\) contains a non-central non-unipotent element. The orbits of the vertices of \(\mathfrak{P}_C\) are then precisely the equivalence classes of rank \(2\) bundles which decompose as a sum of line bundles. The quotient of \(\mathfrak{X}_C\) modulo the subcomplex \(\mathfrak{P}_C\) is denoted \(\mathfrak{U}_C\), the ‘unknown quotient’. There is thus a long exact sequence in equivariant homology \(\dots \to H_\bullet^{\Gamma}(\mathfrak{P}_C)\to H_\bullet(\Gamma)\to H_\bullet^{\Gamma}(\mathfrak{U}_C)\to \cdots\) (the middle term arising from the contractibility of \(\mathfrak{X}_C\)). The main result of the paper is then a computation of the equivariant homology groups \(H_\bullet^{\Gamma}(\mathfrak{P}_C, \mathbb{Z}[1/2])\) of the parabolic complex, obtained by carefully analysing the structure of the orbit space \(\Gamma\backslash \mathfrak{P}_C\) and the terms of the related isotropy spectral sequence: Let \(\iota\) denote the involution \([L]\mapsto [L^{-1}]\) of the Picard group \(\mathrm{Pic}(C)\). Then \(H_\bullet^{\Gamma}(\mathfrak{P}_C, \mathbb{Z}[1/2])\cong \bigoplus_{[L]\in \mathrm{Pic}(C)/\iota }H_\bullet^{\Gamma}(\mathfrak{P}_C(L),\mathbb{Z}[1/2])\). Here \(\mathfrak{P}_C(L)\) is the connected component of \(\mathfrak{P}_C\) corresponding to \([L]\) and the groups \(H_\bullet^{\Gamma}(\mathfrak{P}_C(L), \mathbb{Z}[1/2])\) can be described in an accessible way: If \(L|_C\not\cong L^{-1}|_C\) then \(H_\bullet^{\Gamma}(\mathfrak{P}_C(L), \mathbb{Z}[1/2])\cong H_\bullet(k[C]^\times, \mathbb{Z}[1/2])\), while if \(L|_C\cong L^{-1}|_C\) there is a long exact sequence of the form \(\dots \to H_\bullet(SN,\mathbb{Z}[1/2])\to H_\bullet^{\Gamma}(\mathfrak{P}_C(L), \mathbb{Z}[1/2])\to \mathcal{RP}^1_\bullet(k)\otimes_\mathbb{Z}\mathbb{Z}[1/2][k[C]^\times/(k[C]^\times)^2]\to \cdots\) where \(SN\) denotes the group of monomial matrices in \(\Gamma\) and the groups \(\mathcal{RP}^1_\bullet(k)\) are ‘refined scissors congruence groups’ defined by the author. These latter groups can be understood quite explicitly in low dimensions.
The author treats also the case \(\Gamma=\mathrm{GL}_2(k[C])\). Furthermore, he shows how to extend the results in some special cases – eg \(C=\mathbb{P}^1(k)\setminus\{ 0,\infty\}\) – to arbitrary infinite fields \(k\). The author also discusses the relationship of his computations, upon taking limits, to the predictions of the Friedlander-Milnor conjecture for \(\mathrm{SL}_2(k(C))\).

MSC:
20G10 Cohomology theory for linear algebraic groups
20G15 Linear algebraic groups over arbitrary fields
20E08 Groups acting on trees
14L30 Group actions on varieties or schemes (quotients)
55N91 Equivariant homology and cohomology in algebraic topology
57T10 Homology and cohomology of Lie groups
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