---
- CiteKey: "Gon95"
- Type: journalArticle
- Title: "Geometry of Configurations, Polylogarithms, and Motivic Cohomology," 
- Author: "Goncharov, A. B.;"  
- Year: 1995 
- DOI: 10.1006/aima.1995.1045
- Collections: "Syllabus; Talbot 2022,"
---

# Geometry of Configurations, Polylogarithms, and Motivic Cohomology

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- **URL**: <https://www.sciencedirect.com/science/article/pii/S0001870885710456>
- **URI**: <http://zotero.org/users/1049732/items/LC6IIRYL>
- **Local File**: [Goncharov - 1995 - Geometry of Configurations, Polylogarithms, and Mo.pdf](file:///home/zack/Zotero/storage/KH68V9ME/Goncharov%20-%201995%20-%20Geometry%20of%20Configurations,%20Polylogarithms,%20and%20Mo.pdf)
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## Abstract

Let A be a discrete valuation ring with field of fractions F and residue field k such that |k|≠2,3,4,5,7,8,9,16,27,32,64<math><mo stretchy="false" is="true">|</mo><mi is="true">k</mi><mo stretchy="false" is="true">|</mo><mo is="true">≠</mo><mn is="true">2</mn><mo is="true">,</mo><mn is="true">3</mn><mo is="true">,</mo><mn is="true">4</mn><mo is="true">,</mo><mn is="true">5</mn><mo is="true">,</mo><mn is="true">7</mn><mo is="true">,</mo><mn is="true">8</mn><mo is="true">,</mo><mn is="true">9</mn><mo is="true">,</mo><mn is="true">16</mn><mo is="true">,</mo><mn is="true">27</mn><mo is="true">,</mo><mn is="true">32</mn><mo is="true">,</mo><mn is="true">64</mn></math>. We prove that there is a natural exact sequence where RP1(k)<math><msub is="true"><mrow is="true"><mi mathvariant="script" is="true">RP</mi></mrow><mrow is="true"><mn is="true">1</mn></mrow></msub><mo stretchy="false" is="true">(</mo><mi is="true">k</mi><mo stretchy="false" is="true">)</mo></math> is the refined scissors congruence group of k. Let Γ0(mA)<math><msub is="true"><mrow is="true"><mi mathvariant="normal" is="true">Γ</mi></mrow><mrow is="true"><mn is="true">0</mn></mrow></msub><mo stretchy="false" is="true">(</mo><msub is="true"><mrow is="true"><mi mathvariant="fraktur" is="true">m</mi></mrow><mrow is="true"><mi is="true">A</mi></mrow></msub><mo stretchy="false" is="true">)</mo></math> denote the congruence subgroup consisting of matrices in SL2(A)<math><msub is="true"><mrow is="true"><mi mathvariant="normal" is="true">SL</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></msub><mo stretchy="false" is="true">(</mo><mi is="true">A</mi><mo stretchy="false" is="true">)</mo></math> whose lower off-diagonal entry lies in the maximal ideal mA<math><msub is="true"><mrow is="true"><mi mathvariant="fraktur" is="true">m</mi></mrow><mrow is="true"><mi is="true">A</mi></mrow></msub></math>. We also prove that there is an exact sequence where I2(k)<math><msup is="true"><mrow is="true"><mi is="true">I</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></msup><mo stretchy="false" is="true">(</mo><mi is="true">k</mi><mo stretchy="false" is="true">)</mo></math> is the second power of the fundamental ideal of the Grothendieck-Witt ring GW(k)<math><mrow is="true"><mi mathvariant="normal" is="true">GW</mi></mrow><mo stretchy="false" is="true">(</mo><mi is="true">k</mi><mo stretchy="false" is="true">)</mo></math> and P‾(k)<math><mover accent="true" is="true"><mrow is="true"><mi mathvariant="script" is="true">P</mi></mrow><mo is="true">‾</mo></mover><mo stretchy="false" is="true">(</mo><mi is="true">k</mi><mo stretchy="false" is="true">)</mo></math> is a certain quotient of the scissors congruence group (in the sense of Dupont-Sah) P(k)<math><mi mathvariant="script" is="true">P</mi><mo stretchy="false" is="true">(</mo><mi is="true">k</mi><mo stretchy="false" is="true">)</mo></math> of k.
For an infinite field F, we study the kernel of the map and the cokernel of We give conjectural estimates of these kernels and cokernels and prove our conjectures for n≤4<math><mi is="true">n</mi><mo is="true">≤</mo><mn is="true">4</mn></math>.

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