[Paper Review] Homology of SL2 over function fields I: parabolic subcomplexes

Matthias Wendt|arXiv (Cornell University)|Apr 23, 2014

Algebraic Geometry and Number Theory7 references3 citations

TL;DR

This paper computes the equivariant homology of the parabolic subcomplex in the building for $\mathrm{SL}_2(k[C])$, where $C$ is a smooth affine curve over an algebraically closed field $k$. Using $\mathbb{Z}[1/2]$-coefficients, it provides explicit formulas in terms of the unit group $k[C]^\times$ and refined scissors congruence groups $\mathcal{RP}^1_\bullet(k)$, generalizing Suslin's result for $s=1$ and extending the understanding of homology in function field settings.

ABSTRACT

The present paper studies the homology of the groups $SL_2(k[C])$ and $GL_2(k[C])$ where $C=\overline{C}\setminus\{P_1,\dots,P_s\}$ is a smooth affine curve over an algebraically closed field $k$. It is well-known that these groups act on a product of trees and the quotients can be described in terms of certain equivalence classes of vector bundles on the complete curve. There is a natural subcomplex of cells with non-unipotent isotropy group. The paper provides explicit formulas for the equivariant homology of this "parabolic subcomplex". These formulas also describe the homology of $SL_2(k[C])$ above degree s, with finite coefficients away from the characteristic of $k$, generalizing a result of Suslin for the case s=1.

Motivation & Objective

• To understand the homology of $\mathrm{SL}_2(k[C])$ for affine curves $C$ over algebraically closed fields, where standard analytic methods fail.
• To analyze the structure of the parabolic subcomplex $\mathfrak{P}_C$ in the building $\mathfrak{X}_C$ associated with $\mathrm{SL}_2(k[C])$.
• To compute the equivariant homology of $\mathfrak{P}_C$ explicitly, especially above degree $s$, generalizing Suslin's result for $s=1$.
• To relate the homology to refined scissors congruence groups $\mathcal{RP}^1_\bullet(k)$ and the unit group $k[C]^\times$ via spectral sequence techniques.

Proposed method

• The group $\mathrm{SL}_2(k[C])$ acts on a product of trees, and the quotient $\mathrm{SL}_2(k[C])\backslash\mathfrak{X}_C$ is analyzed via the isotropy spectral sequence.
• The parabolic subcomplex $\mathfrak{P}_C$ is defined as the subcomplex of cells with non-unipotent stabilizer, and its connected components are indexed by $\mathcal{K}(C) = \operatorname{Pic}(C)/\iota$, where $\iota$ is the inversion map.
• Equivariant homology is computed using $\mathbb{Z}[1/2]$-coefficients to simplify the spectral sequence, which degenerates in key cases.
• For components where $\mathcal{L}|_C \not\cong \mathcal{L}|_C^{-1}$, the homology is isomorphic to $\mathrm{H}_\bullet(\mathbb{Z}[1/2], k[C]^\times)$.
• For components with $\mathcal{L}|_C \cong \mathcal{L}|_C^{-1}$, a long exact sequence involving $\tilde{\mathcal{SN}}$ (monomial matrices) and $\mathcal{RP}^1_\bullet(k)$ is derived.
• The results are extended to $\mathrm{PGL}_2$ via similar spectral sequence analysis and comparison with the $\mathrm{SL}_2$ case.

Experimental results

Research questions

• RQ1 How can the equivariant homology of the parabolic subcomplex $\mathfrak{P}_C$ for $\mathrm{SL}_2(k[C])$ be computed explicitly over function fields?
• RQ2 What is the role of the Picard group modulo inversion in indexing the connected components of $\mathfrak{P}_C$?
• RQ3 How do refined scissors congruence groups $\mathcal{RP}^1_\bullet(k)$ arise in the homology computation of $\mathfrak{P}_C$?
• RQ4 What is the structure of the homology when $\mathcal{L}|_C \cong \mathcal{L}|_C^{-1}$, and how does it differ from the non-self-dual case?
• RQ5 Can the spectral sequence for the action on the building be simplified using $\mathbb{Z}[1/2]$-coefficients to yield explicit formulas?

Key findings

• The equivariant homology of $\mathfrak{P}_C$ decomposes as a direct sum over $\mathcal{K}(C) = \operatorname{Pic}(C)/\iota$, with each component indexed by a class $[\mathcal{L}]$.
• For $[\mathcal{L}] \in \mathcal{K}(C)$ with $\mathcal{L}|_C \not\cong \mathcal{L}|_C^{-1}$, the homology of $\mathfrak{P}_C(\mathcal{L})$ is isomorphic to $\mathrm{H}_\bullet(k[C]^\times, \mathbb{Z}[1/2])$.
• For $[\mathcal{L}] \in \mathcal{K}(C)$ with $\mathcal{L}|_C \cong \mathcal{L}|_C^{-1}$, there is a long exact sequence involving $\mathcal{RP}^1_\bullet(k)$ and $\mathrm{H}_\bullet(\tilde{\mathcal{SN}}, \mathbb{Z}[1/2])$, where $\tilde{\mathcal{SN}}$ is the group of monomial matrices in $\mathrm{SL}_2(k[C])$.
• The spectral sequence for the complex $C^\mathrm{alt}_\bullet(F)$ degenerates in a controlled way, with differentials isomorphic to the maps $\mathcal{RP}^1_q(F) \to \mathrm{H}_{q-1}(\mathrm{N}(F), \mathbb{Z}[1/2])$.
• The groups $\mathcal{RP}^1_q(F)$ are identified as kernels of differentials $d^1$ and $d^{q-1}$ in the spectral sequence, and are trivial for $q \leq 1$.
• The low-degree part of the long exact sequence recovers the Bloch-Wigner sequence, and $\mathcal{RP}^1_2(F) \cong I(F)$, the fundamental ideal in the Witt ring of $F$.