[Paper Review] The third homology of SL_2 of local rings

Kevin Hutchinson|arXiv (Cornell University)|Sep 19, 2013

Algebraic Geometry and Number Theory11 references3 citations

TL;DR

This paper computes the third homology of $\mathrm{SL}_2$ over local rings with $\mathbb{Z}[1/2]$-coefficients using a refined Bloch group construction, generalizing Suslin's theorem from fields to local domains. It establishes a localization exact sequence for discrete valuation rings, linking $\mathrm{H}_3(\mathrm{SL}_2(\mathcal{O}_v), \mathbb{Z}[1/2])$ to $\mathrm{H}_3(\mathrm{SL}_2(F), \mathbb{Z}[1/2])$ and a refined scissors congruence group $\mathcal{RP}_1(k)$, enabling explicit calculations for rings like $\mathbb{Z}_p$ when $p \geq 11$. The key contribution is a structural description of $\mathrm{H}_3(\mathrm{SL}_2(A), \mathbb{Z}[1/2])$ in terms of $K^\mathrm{ind}_3(A)[1/2]$ and $\mathcal{RP}_1(k)$, extending classical results to non-field rings.

ABSTRACT

We describe the third homology of SL_2 of local rings over Z[1/2] in terms of a refined Bloch group. We use this to derive a localization sequence for the third homology of SL_2 of certain discrete valuation rings, and to make explicit computations for p-adic integers and other local rings.

Motivation & Objective

• To generalize the Bloch-Wigner exact sequence for $\mathrm{H}_3(\mathrm{SL}_2(F), \mathbb{Z}[1/2])$ from fields to local integral domains.
• To describe $\mathrm{H}_3(\mathrm{SL}_2(A), \mathbb{Z}[1/2])$ for local domains $A$ using a refined Bloch group $\mathcal{RB}(A)$.
• To establish a localization exact sequence relating $\mathrm{H}_3(\mathrm{SL}_2(\mathcal{O}_v), \mathbb{Z}[1/2])$ to $\mathrm{H}_3(\mathrm{SL}_2(F), \mathbb{Z}[1/2])$ and $\mathcal{RP}_1(k)$ for discrete valuation rings.
• To compute $\mathrm{H}_3(\mathrm{SL}_2(\mathbb{Z}_p), \mathbb{Z}[1/2])$ explicitly for $p \geq 11$, showing it is isomorphic to $K^\mathrm{ind}_3(\mathbb{Q}_p)[1/2]$.

Proposed method

• The paper uses the refined scissors congruence group $\mathcal{RP}_1(k)$, defined as a submodule of $\mathcal{RP}(k)$, to model the cokernel of the localization map.
• It constructs a spectral sequence argument to relate $\mathrm{H}_3(\mathrm{SL}_2(A), \mathbb{Z}[1/2])$ to the refined Bloch group $\mathcal{RB}(A)$, showing the edge homomorphism lands in $\mathcal{RB}(A)[1/2]$.
• The method relies on the action of $A^\times / (A^\times)^2$ on homology, with $\mathcal{RB}(A)$ being an $\mathbb{R}_A$-module defined via an explicit presentation.
• It applies results from Mirzaii and Mokari on $K^\mathrm{ind}_3$ for local rings to deduce isomorphisms after inverting 2.
• The proof uses the refined cross ratio map to identify $H_3((L_\bullet)_{\mathrm{SL}_2(A)}) \cong \mathcal{RP}_1(A)$, and shows that a class of order 3 maps to a generator of $\mathcal{RP}_1(A)$.
• It establishes that $\left\langle\!\left\langle x \right\rangle\!\right\rangle D_A = \psi_1(x) - \psi_2(x)$ for units $x \in A^\times$, which implies $\left\langle\!\left\langle x \right\rangle\!\right\rangle D_A = 0$ if $x = \pm \Phi(a)u^2$.

Experimental results

Research questions

• RQ1 How can the third homology of $\mathrm{SL}_2$ over local rings be described in terms of algebraic $K$-theory and scissors congruence groups?
• RQ2 Does the Bloch-Wigner exact sequence for fields extend to local integral domains with finite residue fields?
• RQ3 Can a localization exact sequence be constructed for $\mathrm{H}_3(\mathrm{SL}_2)$ of discrete valuation rings, relating it to the field of fractions and residue field?
• RQ4 What is the structure of $\mathrm{H}_3(\mathrm{SL}_2(\mathbb{Z}_p), \mathbb{Z}[1/2])$ for primes $p \geq 11$?
• RQ5 How does the action of $A^\times / (A^\times)^2$ on $\mathrm{H}_3(\mathrm{SL}_2(A), \mathbb{Z}[1/2])$ relate to the refined Bloch group $\mathcal{RB}(A)$?

Key findings

• The third homology $\mathrm{H}_3(\mathrm{SL}_2(A), \mathbb{Z}[1/2])$ of a local domain $A$ with residue field of at least 10 elements fits into a short exact sequence: $0 \to \mathrm{tor}(\mu_A, \mu_A)[1/2] \to \mathrm{H}_3(\mathrm{SL}_2(A), \mathbb{Z}[1/2]) \to \mathcal{RB}(A)[1/2] \to 0$.
• For any local domain $A$ with sufficiently large residue field, the natural map $\mathrm{H}_3(\mathrm{SL}_2(A), \mathbb{Z}[1/2])_{A^\times} \to K^\mathrm{ind}_3(A)[1/2]$ is an isomorphism.
• For a discrete valuation ring $\mathcal{O}_v$ with residue field $k$ satisfying $U_1 = U_1^2$ and a technical 3-torsion condition, there is a localization exact sequence: $0 \to \mathrm{H}_3(\mathrm{SL}_2(\mathcal{O}_v), \mathbb{Z}[1/2]) \to \mathrm{H}_3(\mathrm{SL}_2(F), \mathbb{Z}[1/2]) \to \mathcal{RP}_1(k)[1/2] \to 0$.
• The refined scissors congruence group $\mathcal{RP}_1(k)$ is explicitly calculable and contains $\mathcal{RB}(k)$ as a submodule, with examples provided in [5].
• For $p \geq 11$, $\mathrm{H}_3(\mathrm{SL}_2(\mathbb{Z}_p), \mathbb{Z}[1/2]) \cong K^\mathrm{ind}_3(\mathbb{Q}_p)[1/2]$, showing the homology is isomorphic to the indecomposable $K_3$ of the $p$-adic field.
• The class $C = C(x,y) \in \mathcal{RP}_1(A)$ of order 3 is independent of $x,y$ and satisfies $3C = 0$, and it lies in $\mathcal{RB}(A)$, confirming the image of the edge homomorphism lies in $\mathcal{RB}(A)[1/2]$.