Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

I have a couple of questions concerning existence and description of transfers for Bloch groups and scissors congruence groups/pre-Bloch groups.


To fix notation and recall definitions:

From the general algebraic K-theory machinery, we get transfers on $K_3$. In particular, for a finite field extension $E/F$, we get a map $\operatorname{tr}_{E/F}:K_3(E)\to K_3(F)$ such that the composition $K_3(F)\to K_3(E)\stackrel{\operatorname{tr}_{E/F}}{\longrightarrow}K_3(F)$ is multiplication with the degree $[E:F]$.

Now by the work of Bloch, Dupont-Sah, Suslin and others, we have another description of $K_3(F)^{\operatorname{ind}}=K_3(F)/K_3^M(F)$ in terms of an exact sequence $$ 0\to \widetilde{\operatorname{Tor}}(\mu(F),\mu(F))\to K_3(F)^{\operatorname{ind}}\to B(F)\to 0. $$ In the above, the Bloch group $B(F)$ is defined as $$ B(F)=\ker\left(\mathcal{P}(F)\to \Lambda^2(F^\times):[x]\to x\wedge(1-x)\right) $$ and the group $\mathcal{P}(F)$ is the pre-Bloch group or scissors congruence group $$ \mathcal{P}(F)=\left(\bigoplus_{x\in F^\times\setminus\{1\}}\mathbb{Z}[x]\right) /\left([x]-[y]+[y/x]-[(1-x^{-1})/(1-y^{-1})]+[(1-x)/(1-y)]\right), $$ the relation coming from the five-term relation satisfied by the dilogarithm. In particular, elements of the Bloch group $B(F)$ can be written down as linear combinations of symbols $[x], x\in F^\times\setminus\{1\}$ satisfying certain relations.


Now I can formulate my questions on transfers on Bloch groups and scissors congruence groups.

  1. Does the transfer on $K_3$ induce a transfer on the Bloch group? I think that this is not the case in general. The element $[x]+[1-x]\in B(F)$ is independent of $x$, and is typically denoted by $c_F$. The element $c_{\mathbb{R}}$ has exact order $6$ in $B(\mathbb{R})$, and the element $c_{\mathbb{C}}$ is trivial in $B(\mathbb{C})$. This seems to contradict transfers for the Bloch group (it does not contradict transfers for $K_3^{\operatorname{ind}}$ because the torsion moves from $B(F)$ to $\widetilde{\operatorname{Tor}}(\mu(F),\mu(F))$). Are there more torsion elements like this, in particular with other odd orders? Are there further obstructions to the existence of transfers on Bloch groups? If $F$ contains an algebraically closed fields, it follows from work of Suslin and Levine that $B(F)$ is uniquely $\ell$-divisible for $\ell$ different from the characteristic - in particular $c_F=0$. Does the $K$-theory transfer induce a transfer on the Bloch group in this situation? What would be a good reference?

  2. Is there an explicit description of what the transfer map on $K_3$ does on the Bloch group? I would be interested in a description that only uses the definition of the Bloch group via points on $\mathbb{P}^1(F)\setminus\{0,1,\infty\}$ given above.

  3. More generally, are there transfers known on scissors congruence groups/pre-Bloch groups? As written above, these groups are defined in terms of points on $\mathbb{P}^1\setminus\{0,1,\infty\}$ modulo the five-term relation. A very naive approach to the definition of transfers for pre-Bloch groups in an extension $E/F$ would be to sum over $E$-points lying over $F$-points of $\mathbb{P}^1\setminus\{0,1,\infty\}$. Has anyone ever tried to work this out, or are there known obstruction why this cannot provide a transfer? Assuming it works, how would one relate such a naive definition to the definition of transfers for algebraic K-theory? In a related direction, what torsion elements besides those in $B(F)$ are known in the scissors congruence groups over fields which are not algebraically closed (in the algebraically closed case, the scissors congruence groups are uniquely divisible)?

share|improve this question

1 Answer 1

Here is some partial information to the question which I figured out in the meantime. The main question (on transfers for scissors congruence groups) still stands...

Concerning 1: The torsion in $B(F)$ is cyclic, and is related to the roots of unity which are roots of irreducible polynomials of degree $2$ over $F$. For example, $K_3(\mathbb{F}_q)^{\operatorname{ind}}\cong\mathbb{Z}/(q^2-1)$ where $\mathbb{Z}/(q-1)$ comes from the roots of unity in $\mathbb{F}_q$, and $\mathbb{Z}/(q+1)$ comes from the Bloch group. Torsion elements in the Bloch group (such as $[x]+[x^{-1}]$ or $[x]+[1-x]$) seem to be related to automorphisms of $\mathbb{P}^1\setminus\{0,1,\infty\}$. However, I do not know of a general way to write down explicit generators for the torsion in $B(F)$. These torsion elements do obstruct the existence of transfers for $B(F)$ because the torsion element corresponding to $\zeta_n$ is killed in the extension $F(\zeta_n)$.

Concerning 2 and 3: If $E/F$ is a finite Galois extension with group $G$, then $K_3(E)^{\operatorname{ind}}$ is a $G$-module, and by the work of Levine, $K_3(F)^{\operatorname{ind}}=(K_3(E)^{\operatorname{ind}})^G$. I guess the transfers are given by summing over Galois orbits, and that this procedure should also work for the Bloch group.

In the case of a radical extension the relevant facts can be found in papers of Dupont and Sah: assume $F$ is such that $\zeta_n\in F$, let $a\in F$ and consider the Galois extension $F(\sqrt[n]{a})/F$. Then Dupont-Sah prove the following equality in the scissors congruence group: $$ \frac{[a]}{n}=\sum_{0\leq i\leq n-1}[\zeta_n^i a]. $$ In particular, the sum over elements of the Galois orbit in $F(\sqrt[n]{a})$ has a representative over $F$.

Ok, as said before: this is only partial information. I guess, what I want to know is, if the above works generally for arbitrary Galois extensions. How could one generally obtain $F$-representatives of sums over Galois orbits in $E/F$?

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.