On Reading Chow Groups with Coefficients by Rost

1. Cycle premodules

Let \(B/k\) be a scheme. The word scheme here means a localization of a separated scheme of finite type over a field. We mean by a field over \(B\) a field \(F\) together with a morphism \(Spec(F)\rightarrow B\) s.t. F is finitely generated (of finite transcedental degree) over \(k\). By a valuation over \(B\) we mean a discrete valuation \(\nu\) of rank 1 together with a morphism \(Spec(\mathcal{O}_\nu)\rightarrow B\) s.t. \(\nu\) is of geometric type over \(k\). The latter means that \(\mathcal{O}_\nu\) is the localization of an integral domain of finite type over \(k\) in a regular point of codimension \(1\). Alternatively, valuations of geometric type may be characterized by: \(k\subset\mathcal{O}_\nu\), the quotient field \(F\) and the residue class field \(\kappa(\nu)\) are finitely generated over \(k\) and \(tr.deg(F|k)=tr.deg(\kappa(\nu)|k)+1\).

In the following the letters \(\phi\), \(\psi\) stand for homomorphisms of fields over \(B\) and all maps between various \(M(F)\), \(M(E)\) are understood as homomorphisms of graded abelian groups.

1.1

(Definition) Let \(\mathcal{F}(B)\) be the class of fields over \(B\). A cycle premodule \(M\) consists of a function \(M:\mathcal{F}(B)\rightarrow\mathcal{A}\) to the calss of abelian groups together with a \(\mathbb{Z}/2\) or a \(\mathbb{Z}\) grading and with the following data D1-D4 and rules R1a-R3e.

• (D1) For each \(\phi:F\rightarrow E\) there is \(\phi_*:M(F)\rightarrow M(E)\) of degree \(0\).
• (D2) For each finite \(\phi:F\rightarrow E\) there is \(\phi^*:M(E)\rightarrow M(F)\) of degree \(0\).
• (D3) For each \(F\) the group \(M(F)\) is equipped with a left \(K_*F\)-module structure denoted by \(x\cdot\rho\) for \(x\in K_*F\) and \(\rho\in M(F)\). The product respects the grading.
• (D4) For a valuation \(\nu\) on \(F\) there is \(\partial_\nu:M(F)\rightarrow M(\kappa(\nu))\) of degree \(-1\).

For a prime \(\pi\) of \(\nu\) on \(F\) we put

\[ s_\nu^\pi:M(F)\rightarrow M(\kappa(\nu)), \] \[ s_\nu^\pi(\rho)=\partial_\nu(\{-\pi\}\cdot\rho). \]

Note that this map is of degree \(0\).

• (R1a) \(\phi_*\) is functorial.
• (R1b) \(\phi^*\) is functorial.
• (R1c) In the cocartesian diagram

for \(p\in Spec(R)\) let \(\phi_p:L\rightarrow R/p\) and \(\psi_p:E\rightarrow R/p\) be the natural maps. Moreover let \(l_p\) be the length of the localized ring \(R_p\). Then

\[ \psi_*\phi^*=\sum_p l_0\cdot (\phi_p^*\circ\psi_{p*}). \]

• (R2) For \(\phi:F\rightarrow E\), \(x\in K_*F\), \(y\in K_*E\), \(\rho\in M(F)\), \(\mu\in M(E)\) one has (with \(\phi\) finite if required)

• (R2a) \(\phi_*(x\cdot\rho)=\phi_*(x)\cdot\phi_*(\rho)\).

• (R2b) \(\phi^*(\phi_*(x)\cdot\mu)=x\cdot\phi^*(\mu)\).

• (R2c) \(\phi^*(y\cdot\phi_*(\rho))=\phi^*(y)\cdot\rho\).

• (R3a) Let \(\phi:E\rightarrow F\) and let \(\nu\) bve a valuation on \(F\) which restricts to a nontrivial valuation \(\omega\) on \(E\) with ramification index \(e\). Let \(\bar{\phi}:\kappa(\omega)\rightarrow\kappa(\nu)\) be the induced map. Then

\[ \partial_\nu\circ\phi_*=e\cdot\bar{\phi}_*\circ\partial_\omega. \]

• (R3b) Let \(\phi:F\rightarrow E\) be finite and let \(\nu\) be a valuation on \(F\). For the extensions \(\omega\) of \(\nu\) to \(E\) let \(\phi_\omega:\kappa(\nu)\rightarrow\kappa(\omega)\) be the induced maps. Then

\[ \partial_\nu\circ\phi^*=\sum_\omega\phi_\omega^*\circ\partial_\omega. \]

Wait, all the extensions? I cannot get a clear picture on this

• (R3c) Let \(\phi:E\rightarrow F\) and let \(\nu\) be a valuation on \(F\) which is trivial on \(E\). Then

\[ \partial_\nu\circ\phi_*=0. \]

Trivial means every unit is valued \(0\). For example local fields of char \(p\).

• (R3d) Let \(\phi\), \(\nu\) be as in (R3c), let \(\bar{\phi}:E\rightarrow\kappa(\nu)\) be the induced map and \(\pi\) be a prime of \(\nu\). Then

\[ s_\nu^\pi\circ\phi_*=\bar{\phi}_*. \]

This is the common definition of residue map in Milnor’s K ring.

• (R3e) For valuation \(\nu\) on \(F\), a \(\nu\)-unit \(u\) and \(\rho\in M(F)\) one has

\[ \partial_\nu(\{u\}\cdot\rho)=-\{\bar{u}\}\cdot\partial_\nu(\rho). \]

These maps \(\phi_*\) and \(\phi^*\) are called the restriction and corestriction homomorphisms, respectively. We use the notations \(\phi_*=r_{E|F}\), \(\phi^*=c_{E|F}\) uif there is no ambiguity.

Note that (R2c) with \(y=1\in K_0E\) gives (follows from basic property of Milnor’s K ring)

• (R2d) For finite \(\phi:F\rightarrow E\) one has

\[ \phi^*\circ\phi_*=(deg\phi)\cdot id. \]

Moreover (R1c) implies

• (R2e) For finite totally inseparable \(\phi:F\rightarrow E\) one has

\[ \phi_*\circ\phi^*=(deg\phi)\cdot id. \]

We consider \(M(F)\) also as a right \(K_*F\)-module via

\[ \rho\cdot x=(-1)^{nm}x\cdot\rho \]

for \(x\in K_nF\) and \(\rho\in M_m(F)\).