4
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ Have you heard about the Gysin exact sequence? $\endgroup$ – abx Aug 17 '17 at 16:20
  • $\begingroup$ @abx I have never. Is it related with my maps? $\endgroup$ – User0829 Aug 17 '17 at 16:36
3
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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