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Infinitesimal dilogarithm on curves over truncated polynomial rings

Sinan Unver

Abstract

Let $C$ be a smooth and projective curve over the truncated polynomial ring $k_m:=k[t]/(t^m), $ where $k$ is a field of characteristic 0. Using a candidate for the motivic cohomology group ${\rm H}^{3}_{\pazocal{M}}(C,\mathbb{Q}(3))$ based on the Bloch complex of weight 3, we construct regulators to $k$ for every $m<r<2m.$ Specializing this construction, we obtain an invariant $ρ_{m,r}(f \wedge g \wedge h)$ of rational functions $f,$ $g$ and $h$ on $C.$ The current work is a twofold generalization of our work on the infinitesimal Chow dilogarithm: we sheafify the previous construction and therefore do not restrict ourselves to triples of rational functions and we construct the regulator for any $m<r<2m,$ rather than only for $m=2.$ We also define regulators of cycles, which we expect to give a complete set of invariants for the infinitesimal part of ${\rm CH}^{2}(k_{m},3). $ This generalizes Park's work, where the additive Chow cycles, namely the case of cycles close to 0, is handled for $r=m+1.$ In this paper, we generalize the reciprocity theorem to pairs of cycles which are the same modulo $(t^m)$ and for any $m<r<2m.$ We expect the theory of the paper to give regulators on categories of motives over rings with nilpotents.

Infinitesimal dilogarithm on curves over truncated polynomial rings

Abstract

Let be a smooth and projective curve over the truncated polynomial ring where is a field of characteristic 0. Using a candidate for the motivic cohomology group based on the Bloch complex of weight 3, we construct regulators to for every Specializing this construction, we obtain an invariant of rational functions and on The current work is a twofold generalization of our work on the infinitesimal Chow dilogarithm: we sheafify the previous construction and therefore do not restrict ourselves to triples of rational functions and we construct the regulator for any rather than only for We also define regulators of cycles, which we expect to give a complete set of invariants for the infinitesimal part of This generalizes Park's work, where the additive Chow cycles, namely the case of cycles close to 0, is handled for In this paper, we generalize the reciprocity theorem to pairs of cycles which are the same modulo and for any We expect the theory of the paper to give regulators on categories of motives over rings with nilpotents.

Paper Structure

This paper contains 17 sections, 37 theorems, 179 equations.

Table of Contents

  1. Introduction
  2. Additive dilogarithm of higher modulus
  3. Infinitesimal Milnor $K$-theory of local rings
  4. Construction of maps from $B_{2}(R_m) \otimes R_{m} ^{\times}$ to $\Omega^{1}_{R}$
  5. Preliminaries on the construction
  6. The regulator maps from ${\rm H}^{2}(R_{m},\mathbb{Q}(3))$ to $\Omega^{1} _{R}$
  7. Construction of the maps from $\Lambda ^3 (R_{2m-1},(t^m))^{\times}$ to $\Omega^{1} _{R/k}$
  8. Description of the construction
  9. Proof that $\Omega_{m,r}$ is well-defined
  10. Behaviour of $\omega_{m,r}$ with respect to automorphisms of $R_{2m-1}$ which are identity modulo $(t^m)$
  11. Behaviour of $res(\omega_{m,r,\sigma})$ with respect to automorphisms of $R_{m}$
  12. Variant of the residue map for different liftings.
  13. The residue of $\omega_{m,r}$ on good liftings.
  14. Construction of $\rho$ and a regulator on curves
  15. Regulators on curves.
  16. ...and 2 more sections

Key Result

Proposition 2.0.1

With the notation above, for $\alpha \in \mathbb{Q}[R_{\infty} ^{\flat}]$ and $2 \leq m<r<2m,$ we have and this function descends through the canonical projections to define a map from $B_{2}(R_{m})$ to $R,$ denoted by the same notation.

Theorems & Definitions (79)

  • Proposition 2.0.1
  • proof
  • Definition 2.0.2
  • Theorem 2.0.3
  • Proposition 3.0.1
  • proof
  • Lemma 3.0.2
  • proof
  • Lemma 3.0.3
  • proof
  • ...and 69 more