General construction of residue map

Question

In a literature, I found the following object: let $k$ is a field of characteristic zero and $U=\mathbf{P}^1_k\setminus\{0,1,\infty\}$, then there are residue maps

$\mathrm{Res}_i:H^1(U,\mathbb{Q}_p(1))\to\mathbb{Q}_p$

for $i=0,1$. I could not imagine what these map should be. So, I would like to ask what these maps are and how to construct them, and whether there is a way of explicit computation.

Answer 1

Composing with pullback to the base change over the algebraic closure of $k$, the real content is with $k$ an algebraically closed base field (of char. 0), $U$ a non-empty proper open subset of a smooth proper connected curve $X$ over $k$, and $x \in X(k) - U(k)$ a choice of missing point: given such data one seeks to make a "residue map" ${\rm{res}}_x: {\rm{H}}^1(U,\mu_n) \rightarrow \mathbf{Z}/(n)$ compatible with multiplicative change in $n > 0$ (then passing to the inverse limit with $n$ varying through powers of a prime $p$ and inverting $p$ gives a "residue map" ${\rm{res}}_x: {\rm{H}}^1(U, \mathbf{Q}_p(1)) \rightarrow \mathbf{Q}_p$, and then sum over all such $x$).

If $K$ is the fraction field of the henselization (or completion) of the local ring $O_{X,x}$ then by composing with pullback along ${\rm{Spec}}(K) \rightarrow U$ we can forget about $(U,x)$ and focus on a strictly henselian discrete valuation ring $R$ (e.g., a complete dvr with separably closed residue field) with fraction field $K$: we seek to construct a natural map ${\rm{H}}^1(K, \mu_n) \rightarrow \mathbf{Z}/(n)$ for $n>0$ that is a unit in $R$ (all $n$ when the residue characteristic is 0) such that:

(i) it is compatible with multiplicative change in $n$,

(ii) when we begin with a punctured smooth curve $U$ over $k=\mathbf{C}$, the resulting map ${\rm{H}}^1(U, \mu_n) \rightarrow \mathbf{Z}/(n)$ is identified via Artin's comparison isomorphism with the mod-$n$ reduction of the map ${\rm{H}}^1(U(\mathbf{C}), \mathbf{Z}(1)) \rightarrow \mathbf{Z}$ induced map the "analytic residue map" ${\rm{H}}^1(U(\mathbf{C}), \mathbf{C}) \rightarrow \mathbf{C}$ given by $(1/2\pi i)\int_{\gamma_x,i}$ (using integration along an $i$-oriented loop $\gamma_x$ around $x$).

Of course, (ii) justifies the name for the construction.

To make the construction fulfilling (i), we just use Kummer theory: the $n$-torsion Kummer sequence $1 \to \mu_n \rightarrow \mathbf{G}_{\rm{m}} \stackrel{t^n}{\rightarrow} \mathbf{G}_{\rm{m}} \rightarrow 1$ identifies ${\rm{H}}^1(K, \mu_n)$ with $K^{\times}/(K^{\times})^n$, and since $R^{\times}$ is $n$-divisible (as $R$ is strictly henselian with residue characteristic not divisible by $n$) the short exact sequence $$1 \rightarrow R^{\times} \rightarrow K^{\times} \stackrel{{\rm{ord}}}{\to} \mathbf{Z} \rightarrow 1$$ defined by the normalized ord-function on $K^{\times}$ visibly identifies $K^{\times}/(K^{\times})^n$ with $\mathbf{Z}/(n)$. (sending the class of $c \in K^{\times}$ to ${\rm{ord}}(c) \bmod n$).

That is the construction, and its compatibility with multiplicative change in $n$ is an elementary exercise in unraveling the definitions, especially formulating the sense in which the formation of the $n$th-power Kummer sequence is compatible with multiplicative change in $n$.

When we begin over $\mathbf{C}$, it identifies the construction with one for the fraction field of the discrete valuation ring $\mathbf{C}\{z\}$ of convergent power series near 0 which has nothing at all to do with the particular curve but does make contact with meromorphic constructions. To prove the compatibility property in (ii), we can shrink $U$ so it misses enough points that there exists a finite map $f:U\rightarrow \mathbf{A}^1- \{0\}$ with a simple zero at $x$. Hence, the pullback of the $\mu_n$-torsor over $\mathbf{A}^1 - \{0\}$ corresponding to the $n$th root of the standard coordinate pulls back to a class carried to $1 \in \mathbf{Z}/(n)$ via the above construction.

The compatibility with the classical residue map at $x$ (through the procedure described above) is thereby reduced to showing that the class in ${\rm{H}}^1(\mathbf{C}^{\times}, \mu_n)$ corresponding to the $\mu_n$-torsor $t^n:\mathbf{C}^{\times} \rightarrow \mathbf{C}^{\times}$ is the reduction of a class in ${\rm{H}}^1(\mathbf{C}^{\times}, \mathbf{Z}(1)) \subset {\rm{H}}^1(\mathbf{C}^{\times}, \mathbf{C})$ with "analytic residue" equal to 1. But that explicit $\mu_n$-torsor is the pushout along $\mathbf{Z}(1) \twoheadrightarrow \mu_n$ of the $\mathbf{Z}(1)$-torsor ${\rm{exp}}: \mathbf{C} \rightarrow \mathbf{C}^{\times}$. In the language of monodromy along loops, this latter covering space carries the $i$-oriented loop $\gamma$ around 0 through $1 = e^0$ to $2\pi i \in \mathbf{Z}(1)$. Hence, applying $(1/2\pi i)\int_{\gamma,i}$ carries this class to 1, so we get the desired consistency.