数理解析研究所講究録 1200 巻 2001 年 1-12

1

ON THE UNIFICATION OF KUMMER AND ARTIN-SCHREIER-WITT
THEORIESTSUTOMU SEKIGUCHI*)

1. MOTIVATION

Our aim of this report is to give an explanation of the final version of our theory which
unifies the Kummer theory and Artin-Schreier-Witt theory. The details of this report canbe seen in the Bordeaux preprint [15].
First, we review the Kummer theory.
Let be an integer with $n\geq 2$ , and $K$ be afield of charcteristic with
and$K\supset\mu_{n}=\{\zeta|\zeta^{n}=1\}$ .
$n$

$q$

$q\Lambda n$Theorem 1.1 (Kummer Theory).
$L/K:n$ -cyclic Galois extension$\Leftrightarrow\exists a\in K^{*}\mathrm{s}.\mathrm{t}$. $L=K(\sqrt[n]{a})$$arrow$$L=K\otimes_{K[x,x^{-1}]}K[X, X^{-1}]\uparrow$$K[X, X^{-1}]\uparrow$$X^{n}[perp]\uparrow$$\Leftrightarrow\exists a\in K^{*}\mathrm{s}.\mathrm{t}$.
$K$$\Leftrightarrow\exists f$:$\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}Karrow \mathrm{G}_{m,K}\mathrm{s}.\mathrm{t}$.

$arrow$$X$$X$$arrow\dashv$$a$$\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}L$$K[X, X^{-1}]$$arrow$$\mathrm{G}_{m,K}$$\downarrow\square \downarrow\theta_{n}$,$\mathrm{G}_{m,K}$$\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}K$$\vec{f}$where$\theta_{n}$:$\mathrm{G}_{m,K}arrow \mathrm{G}_{m.K}$;$x\mapsto x^{n}$.

Namely, the Kummer theory implies that the following exact sequence (s0-called theKummer exact sequence)of sheaves on the fppf (or \’etale) site on
is essential$K$
in the world of cyclic coverings of :$\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}K$$arrow\theta_{n}$$1arrow\mu_{n,K}arrow$$\mathrm{G}_{m,K}$$t$$\mathrm{G}_{m,K}$$\mapsto$$arrow 1$$t^{n}$

In fact, from the exact sequence, for any $K$-scheme $X$ we can deduce the exact sequence:$\mathrm{G}_{m,K}(X)arrow \mathrm{G}_{m,K}(X)arrow \mathrm{H}^{1}(X, \mu_{n,K})\theta_{n}\partialarrow \mathrm{H}^{1}(X, \mathrm{G}_{m,K})arrow \mathrm{H}^{1}(X, \mathrm{G}_{m,K})$.

Partially supported by Grant-in-Aid for Scientific Research #11640045 and TAO unit “Security
System Project for Electronic Society$*)$

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TSUTOMU SEKIGUCHI

Here$\mathrm{H}^{1}(X, \mu_{n,K})=\mathrm{t}\mathrm{h}\mathrm{e}$$\mathrm{H}^{1}(X, \mathrm{G}_{m,K})=0$

set of isomorphism classes of unramified$\mu_{n}$coverings of $X$

for suitable $X$ ’s by Hilbert Theorem 90

Next we review the Artin-Schreier-Witt theory.
Let be afield of positive characteristic .
: the group scheme of Witt vectors of length
$k$

$p$

$W_{n,k}$$n$$\wp:W_{n,k}arrow W_{n,k;}x\mapsto x^{(p)}-x$

Theorem 1.2 (Artin-Schreier-Witt Theory).
$K/k:p^{n}$-cyclic Galois dension. $K=k(\wp^{-1}(a))$$\Leftrightarrow\exists a\in W_{n}(k)\mathrm{s}.\mathrm{t}$$arrow$$K=k\otimes_{k[\mathrm{X}]}k[\mathrm{X}]\uparrow$$\Leftrightarrow\exists a=$.

$(a_{0}, a_{1}, \ldots, a_{n-1})\in W_{n}(k)\mathrm{s}.\mathrm{t}$$k$$a_{i}$$arrow$$\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}K$$\Leftrightarrow\exists f$:$k[\mathrm{X}]\uparrow\wp^{*}$$\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}k$$arrow W_{n,k}\mathrm{s}.\mathrm{t}$.

$arrow$$k[\mathrm{X}]$$arrow\dashv X_{\dot{l}}$$W_{n,k}$$\downarrow\square \downarrow \mathrm{p}$,$W_{n,k}$$\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}k$$\vec{f}$Here$\mathrm{X}=$$(X_{0}, X_{1}, \ldots, X_{n-1})$.

Namely, the Artin-Schreier-Witt theory implies that the following exact sequence (s0called the Artin-Schreier-Witt exact sequence) of sheaves on the fppf (or \’etale) siteon theis essential:$\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}k$$0arrow \mathbb{Z}/p^{n}\mathbb{Z}arrow$$W_{n,k}$$x$

In fact, from the exact sequence, for any$k$$arrow\wp$$\mapsto$$W_{n,k}$$arrow 0$.

$x^{(p)}-x$

scheme $X$ we can deduce the exact sequence:$W_{n,k}(X)arrow W_{n,k}\wp(X)arrow \mathrm{H}^{1}(X, \mathbb{Z}/p^{n})\partialarrow \mathrm{H}^{1}(X, Wn,k)arrow \mathrm{H}^{1}(X, Wn,k)$.

Here$\mathrm{H}^{1}(X, \mathbb{Z}/p^{n})=\mathrm{t}\mathrm{h}\mathrm{e}$$\mathrm{H}^{1}(X, W_{n,k})=0$

set of isomorphism classes of unramified$\mathbb{Z}/p^{n}\mathbb{Z}$coverings of $X$

for affine schemes $X$

Therefore, the Kummer theory implies that in the world of unramified -cyclic coverings in characteristic 0, the Kummer exact sequence is the Buddha, and any suchcoverings is deduced from the sequence. On the other hand, the Artin-Schreier-Witttheory imples that in the world of unramified -cyclic coverings in characteristic , the$p^{n}$$p^{n}$$p$

3
ON THE UNFICATION OF KUMMER AND ARTIN-SCHREIER-WITT THEORIESArtin-Schreier-Witt exact sequence is the Buddha, and any such covering is deduced from the sequence. But our religion asserts that every Buddha should be deduced
from the unique eseential Buddha. Hence, behind the two Buddhas, there should exist amore essential Buddha unifying them.
$(\mathrm{M}\mathrm{a}\mathrm{h}\overline{\mathrm{a}}\mathrm{v}\mathrm{a}\mathrm{i}\mathrm{r}\mathrm{o}\mathrm{c}\mathrm{a}\mathrm{n}\mathrm{a}\mathrm{h}.)$So we arrive at the following problems:$\bullet$$\bullet$$\bullet$Search for the Buddha unifying the Kummer and ASW sequences.
Construct the deformations of the group schemes of Witt vectors of finite length
to tori.
Such deformations should keep the filtarations of the group schemes of Witt vectori.

2. 1DIMENSIONAL CASE
Let
be aDVR with {the plane curve over :$(A, \mathrm{m})$$.\{.\mathrm{A}$$=K$ and $A/\mathrm{m}=k$ , and$\lambda\in \mathrm{m}\backslash \{0\}$. Now we look at$A$$\mathrm{C}$:$\mathrm{Y}^{2}Z-\lambda X\mathrm{Y}Z-X^{3}=0\subset \mathrm{P}^{2}$,

whose generic fibre is anodal curve and the special fibre is acuspidal curve. Therefore thePicard scheme of the curve gives adeformation of an additive group scheme to atorus:$\mathrm{P}\mathrm{i}\mathrm{c}^{0}(\mathrm{C}/A)\cong \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}A[X, 1/(1+\lambda X)]$,

with group law $x\cdot y=\lambda xy+x+y$ . Hereafter we denote this group scheme by$\mathcal{G}^{(\lambda)}:=\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}A[X, 1/(1+\lambda X)]$

The important fact is that any deformations of’s. In fact, we have the following.

$\mathrm{G}_{a}$to$\mathcal{G}^{(\lambda)}$

:

.

over

$\mathrm{G}_{m}$$A$

are only the type of$\mathcal{G}^{(\lambda)}$Theorem 2.1 ([17, Th. 2.5]). Let be a flat group scheme overwith gereric fibre
and special fibre. Then there exists a non-zero element Aof , uniquely up to unit
factors, such that$\mathcal{G}$$\mathrm{G}_{m}$$\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}$$\mathrm{G}_{a}$$\mathrm{m}$$\mathcal{G}\cong \mathcal{G}^{(\lambda)}$.

3. HIGHER DIMENSIONAL CASE
If we obtain adeformation
filtaration$0arrow$$\mathrm{W}\mathrm{n}-\mathrm{i}$$\mathbb{Z}/p$of$arrow$$\mathrm{G}_{a,k}$to$\mathbb{Z}/p^{n}$$\mathrm{G}_{m,K}^{n-1}$$arrow$$arrow$$W_{n,k}$, then since the Witt vectors has the$\mathbb{Z}/p^{n-1}$$arrow 0$$\cap$$\cap$$\cap$$0arrow$$\mathrm{W}\mathrm{n}-\mathrm{i}$$arrow$$W_{n-1,k}$$arrow 0$,

we can expect the next one fits into an extension.

$0arrow \mathcal{G}^{(\lambda)}arrow \mathcal{W}_{n+1}arrow \mathcal{W}_{n}arrow 0\in \mathrm{E}\mathrm{x}\mathrm{t}^{1}(\mathcal{W}_{n}, \mathcal{G}^{(\lambda)})$4
TSUTOMU SEKIGUCHIDefinition 3.1. Let (A, m) be a $DVR_{f}$ andthe densions$\lambda_{1}$,,$\lambda_{2}$$\lambda_{n}\in \mathrm{m}\backslash \{0\}$\ldots ,If.

$\mathcal{W}_{n}$is given by$0arrow \mathcal{G}^{(\lambda_{2})}arrow \mathcal{W}_{2}arrow \mathcal{G}^{(\lambda_{1})}arrow 0\in \mathrm{E}\mathrm{x}\mathrm{t}^{1}(\mathcal{G}^{(\lambda_{1})}, \mathcal{G}^{(\lambda_{2})})$$0arrow \mathcal{G}^{(\lambda_{3})}arrow \mathcal{W}_{3}arrow \mathcal{W}_{2}arrow 0\in \mathrm{E}\mathrm{x}\mathrm{t}^{1}(\mathcal{W}_{2}, \mathcal{G}^{(\lambda_{3})})$$0arrow \mathcal{G}^{\langle\lambda_{n})}arrow \mathcal{W}_{n}arrow \mathcal{W}_{n-1}arrow 0\in \mathrm{E}\mathrm{x}\mathrm{t}^{1}(\mathcal{W}_{n-1}, \mathcal{G}^{(\lambda_{n})})$

we call it a group scheme of type$(\lambda_{1}, \lambda_{2},$$\lambda_{n})$\ldots ,,.

) for agroup scheme,(Wz,of type
To compute the groupthe following exact sequence of sheaves on each the small Zariski, fppf or etale site on
is essential:$\mathrm{E}\mathrm{x}\mathrm{t}^{1}$$\mathcal{G}^{(\lambda_{\ell+1})}$$(\lambda_{1}, \lambda_{2}, \ldots, \lambda_{\ell})$$\mathcal{W}_{\ell}$$\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}A$$arrow\rho^{(\lambda)}$$arrow\alpha^{(\lambda)}$$0arrow$$\mathcal{G}^{(\lambda)}$$x$$\mathrm{G}_{m,A}$$\iota_{*}\mathrm{G}_{m,A_{\lambda}}$$arrow 0$.

$1+\lambda x$$\mapsto$$t$$\mapsto$

mod A

$t$

is the canonical inclusion.
where :
By an explicit computation of cocycles, we have
Proposition 3.1.
$\iota$$\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(A/\lambda)arrow \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}$$A$.

$\mathrm{E}\mathrm{x}\mathrm{t}^{1}(\mathcal{G}^{(\lambda)}, \mathrm{G}_{m,A})=0$

Therefore inductively we have$\mathrm{E}\mathrm{x}\mathrm{t}^{1}(\mathcal{W}_{\ell}, \mathrm{G}_{m,A})=0$

for any group scheme of type$(\lambda_{1}, \lambda_{2}, \ldots, \lambda_{\ell})$,.

Hence, by using the above exact sequence we obtain the following.

Theorem 3.2. Let
we have$\mathcal{W}_{n}$be a group scheme of type$(\lambda_{1}, \lambda_{2}, \ldots, \lambda_{n})$, and A$\mathrm{E}\mathrm{x}\mathrm{t}^{1}(\mathcal{E}, \mathcal{G}^{(\lambda)})\cong \mathrm{H}\mathrm{o}\mathrm{m}(\mathcal{E}, \iota_{*}\mathrm{G}_{m,A/\lambda})/(\rho^{(\lambda)})_{*}(\mathrm{H}\mathrm{o}\mathrm{m}(\mathcal{E}, \mathrm{G}_{m,A}))$$\in \mathrm{m}\backslash \{0\}$. Then.

From this theorem, we can deduce the following.
Theorem 3.3. Let
homomorphism$\mathcal{W}_{n}$be a group scheme of type$(\lambda_{1}, \lambda_{2}, \ldots, \lambda_{n})$. Then there eists$D_{\ell:}\mathcal{W}_{\ell}arrow\iota_{*}\mathrm{G}_{m,A/\lambda_{\ell+1}}$

for each $\ell(2\leq\ell\leq n-1)$ , and each$\mathcal{W}_{\ell}\cong \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}A[X_{0},$$\ldots$$\frac{1}{D_{1}(X_{0})+\lambda_{2}X_{1}}$$\mathcal{W}_{\ell}$,is given by$X_{\ell-1}$, .. . ’,$\frac{1}{1+\lambda_{1}X_{0}}$,$\frac{1}{D_{\ell-1}(X_{0},\ldots,X_{\ell-2})+\lambda_{\ell}X_{\ell-1}}$

].

$a$

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ON THE UNFICATION OF KUMMER AND ARTIN-SCHREIER-WITT THEORIES

Moreover, the group law of$\alpha^{(\ell)}$:is the one which makes the morphism$\mathcal{W}_{\ell}$$(\mathrm{G}_{m,A})^{\ell}$$arrow$$\mathcal{W}_{\ell}$(1 %$(X_{0}, \ldots, X_{\ell-1})$$\mapsto$$\lambda_{1}X_{0}$

... ’

,

$D_{1}(X_{0})+\lambda_{2}X_{1}$,$D_{\ell-1}(X_{0}, \ldots, X_{\ell-2})+\lambda_{\ell}X_{\ell-1})$

a group-schematic homomor phism.
Definition 3.2. Suppose that$A$

dominates$\mathbb{Z}_{(p)}[\mu_{p^{n}}]$, and put$\lambda=\lambda(1)$. We call a group

KASW groupfor each satisfying$(\lambda)^{n}=(^{\frac{n}{\lambda,\lambda,\ldots,\lambda}})$,over of type,,scheme:scheme over , if there xists an inclusion
commtative diagram$\mathcal{W}_{1}=\mathcal{G}^{(\lambda)}$$A$$\mathcal{W}_{n}$$\mathcal{W}_{2}$$\ldots$$A$$i_{\ell}$$0arrow 0arrow$$(\mathbb{Z}/p)_{A}\mathcal{G}^{(\lambda)}\downarrow i_{1}$$(\mathbb{Z}/p^{\ell})_{A}w_{\ell}^{1^{i_{\ell}}}$$arrowarrow$$a$$\mathbb{Z}/p^{\ell}arrow \mathcal{W}_{\ell}$$(\mathbb{Z}/p^{\ell-1})_{A}w_{\ell-1}^{1^{\dot{l}\ell-1}}$$arrowarrow r_{\ell}$$\ell$$a$$arrow 0arrow 0$.
Once we obtain aKASW group scheme, then it embodies the unified Kummer-ArtinSchreier-Witt theory.
be a KASW group scheme over A. Let $B$ andTheorem 3.4 (KASW theory). Let
are local flat -algebras such that is an unramified -cyclic covering over B. Then
, and the covering
there eists an A morphism :is given by the fibre product$\mathcal{W}_{n}$

$C$

$A$

$C$

$f$

$p^{n}$$\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}Carrow \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}B$$\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{B}arrow \mathcal{W}_{n}/(\mathbb{Z}/p^{n})$$\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}B\mathrm{S}\mathrm{p}\mathrm{e}_{1}\mathrm{c}C\coprod_{f,arrow}^{arrow},$$\mathcal{W}_{n}/(\mathbb{Z}/p^{n}\mathcal{W}_{n}\downarrow.

)$.
By these argument, our work is concentrated upon the calculation of.
namely of$\mathrm{E}\mathrm{x}\mathrm{t}^{1}(\mathcal{W}_{n}, \mathcal{G}^{(\lambda)})$$\mathrm{H}\mathrm{o}\mathrm{m}(\mathcal{W}_{n}, \mathrm{G}_{m,A/\lambda})$4. DETERMINATION OF$\mathrm{E}\mathrm{x}\mathrm{t}^{1}(\mathcal{W}_{n}, \mathcal{G}^{(\lambda)})$We provide some notations.
$(A, \mathrm{m})$$\Phi_{n}(\mathrm{T})$:DVR dominating$\mathbb{Z}_{(p)}$,$\lambda\in \mathrm{m}\backslash \{0\}$$=T_{0}^{p^{n}}+pT_{1}^{p^{n-1}}+\cdots+p^{n}T_{n}$$\tilde{a}:=(a, 0,0, \ldots)\in \mathrm{W}\{\mathrm{A}$) for: Witt polynomial$A\in A$for $b=(b_{0}, b_{1}, \ldots)$: $W_{A}arrow W_{A};[p]b:=$: $W_{A}arrow W_{A}$ :Verschiebung endomorphism$F:W_{A}arrow W_{A}$ :the generalized Frobenius endomorphism$[p]$$(0, b_{0}^{p}, b_{1}^{p}, \ldots)$$V$$F^{(\lambda)}:=F-(\lambda^{p-1})^{\sim}$

For $a\in W(A)$ , we define$T_{a}$: $W(A)arrow W(A)$ by$\Phi_{n}(T_{a}x)=a_{0}^{p^{n}}\Phi_{n}(x)$ $+pa_{1}^{p^{n-1}}\Phi_{n-1}(x)+\cdots+p^{n}a_{n}\Phi_{0}(x)$$(n\geq 0)$,

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TSUTOMU SEKIGUCHI

for $x\in W(A)$ . Then we have
If is aring (not necessarily a$T_{a}= \sum_{k\geq 0}V^{k}\cdot\tilde{a}_{k}$$A$$\mathbb{Z}_{(p)}$.

-algebra),$\overline{W}_{n}(A)=\{(a_{0}, a_{1}, \ldots, a_{n-1})\in W_{n}(A)$;$a$:is nilpotent for all

$i\}$

and

;

$\overline{W}(A)=\{$ $(a_{0}, a_{1}, a_{2}, \ldots)\in W_{n}(A)$$a$:is nilpotent for all and$i$$a:=0$ for all but afinite number of

$i\}$

.

Moreover we need to deform the Artin-Hasse exponential series$E_{p}(X):= \exp(X+\frac{X^{p}}{p}+\frac{X^{p^{2}}}{p^{2}}+\cdots)$$=e^{X}e^{\frac{\mathrm{x}^{\mathrm{p}}}{\mathrm{p}}}e^{\mathrm{x}^{\mathrm{p}^{2}}}\mathrm{p}$$\overline{2}...\in \mathbb{Z}_{(p)}[[X]]$.

The well-known formulacan be seen thatis adeformation of. Prom this point of view, we obtain the deformations of Artin-Hasse
exponential series:$\lim_{\lambdaarrow 0}(1+\lambda x)^{\alpha/\lambda}=e^{\alpha x}$$(1+\lambda x)^{\alpha/\lambda}$$e^{\alpha x}$$E_{p}(U, \Lambda;X):=(1+\Lambda X)^{\mathrm{x}}\prod_{k=1}^{\infty}(U1+\Lambda^{p^{k}}X^{p^{k}})^{\frac{1}{\mathrm{p}^{k}}((_{\mathrm{X}}^{U})^{\mathrm{p}^{k}}-(\frac{U}{\Lambda})^{\mathrm{p}^{k-1}})}$$\in \mathbb{Z}_{(p)}[U, \Lambda][[X]]$.

Moreover for aWitt vector $a\in W(A)$ , we define aformal power series as follows:$E_{p}(a, \lambda;X):=\prod_{k=0}^{\infty}E_{p}(a_{k}, \lambda^{p^{k}}; X^{p^{k}})$$=(1+ \lambda X)^{9_{\lambda}\mathit{1}}\prod_{k=1}^{\infty}(1+\lambda^{p^{k}}X^{p^{k}})^{\frac{1}{\mathrm{p}^{k}\lambda \mathrm{P}^{k}}\Phi_{k-1}(F^{(\lambda)}a)}$The boundary of this power series $Ep(a, \lambda;X)$ is given by the following.
,$( \partial E_{p}(a, \lambda;\cdot))(X, \mathrm{Y})=,\frac{E_{p}(a,\lambda,X)E_{p}(a,\lambda\cdot \mathrm{Y})}{E_{p}(a,\lambda\cdot X+\mathrm{Y}+\lambda X\mathrm{Y})}.$$= \prod_{k=1}^{\infty}(\frac{(1+\lambda^{p^{k}}X^{p^{k}})(1+\lambda^{p^{k}}\mathrm{Y}^{p^{k}})}{1+\lambda^{p^{k}}(X+\mathrm{Y}+\lambda X\mathrm{Y})^{p^{k}}})^{\frac{1}{\mathrm{p}^{k_{\lambda}\mathrm{p}^{k}}}\Phi_{k-1}(F^{(\lambda)}a)}$Now replacingwith aWitt vector $b=$
equation, we define acocyle as follows.
$F^{(\lambda)}a$$(b_{0}, b_{1}, \ldots)$in the right hand side of this$F_{p}(b, \lambda;X, \mathrm{Y}):=\prod_{k=1}^{\infty}(\frac{(1+\lambda^{p^{k}}X^{p^{k}})(1+\lambda^{p^{k}}\mathrm{Y}^{p^{k}})}{1+\lambda^{p^{k}}(X+\mathrm{Y}+\lambda X\mathrm{Y})^{p^{k}}})^{\frac{1}{\mathrm{p}^{k_{\lambda}\mathrm{p}^{k}}}\Phi_{k-1}(b)}$$\in \mathbb{Z}_{(p)}[b, \lambda][[X, \mathrm{Y}]]$.

Using these deformed Artin-Hasse exponential series, we can obtain the following.

7
ON THE UNFICATION OF KUMMER AND ARTIN-SCHREIER-WITT THEORIESTheorem 4.1 (Explicit Formula in 1Dimensional Case).
$\xi_{0}^{1}$:$\mathrm{K}\mathrm{e}\mathrm{r}(\overline{W}(A/\lambda_{2})arrow\overline{W}(A/\lambda_{2}))F^{(\lambda_{1})}$$a$$\xi_{1}^{0}$:$arrow\sim$$\mathrm{H}\mathrm{o}\mathrm{m}(\mathcal{G}^{(\lambda_{1})}, \iota_{*}\mathrm{G}_{m,A/\lambda_{2}})$$E_{p}(a, \lambda_{1;}X)$$\mapsto$$\mathrm{C}\mathrm{o}\mathrm{k}\mathrm{e}\mathrm{r}(\overline{W}(A/\lambda_{2})arrow\overline{W}(A/\lambda_{2}))F^{(\lambda_{1})}$$b$$arrow\sim$,$\mathrm{H}_{0}^{2}(\mathcal{G}^{(\lambda_{1})}, \iota_{*}\mathrm{G}_{m,A/\lambda_{2}})$$\mapsto$.

$F_{p}(b, \lambda_{1;}X, \mathrm{Y})$Therefore$\xi_{0}^{1}$:$\frac{\mathrm{K}\mathrm{e}\mathrm{r}(\overline{W}(A/\lambda_{2})arrow\overline{W}(A/\lambda_{2}))F^{(\lambda_{1})}}{<\overline{\lambda_{1}}>}arrow\sim\frac{\mathrm{H}\mathrm{o}\mathrm{m}(\mathcal{G}^{(\lambda_{1})},\iota_{*}\mathrm{G}_{m,A/\lambda_{2}})}{\mathrm{H}\mathrm{o}\mathrm{m}(\mathcal{G}^{(\lambda_{1})},\mathrm{G}_{m,A})}\simarrow \mathrm{E}\mathrm{x}\mathrm{t}^{1}(\mathcal{G}^{(\lambda_{1})}, \mathcal{G}^{(\lambda_{2})})$

In higher dimentional case, we need more notations. For avector
define$[p]E_{p}(\mathrm{U}, \Lambda;X):=E_{p}([p]\mathrm{U}, \Lambda;X)$$\mathrm{U}=$$(U_{0}, U_{1}, \ldots)$

.
, we

,

$[p]F_{p}(\mathrm{U}, \Lambda;X, \mathrm{Y}):=F_{p}([p]\mathrm{U}, \Lambda;X, \mathrm{Y})$.

$H(X, \mathrm{Y}):=\frac{1}{\Lambda_{2}}\{F_{p}(\mathrm{U}, \Lambda_{1}; X, \mathrm{Y})-1\}$,

Moreover$G_{p}( \mathrm{A}, \Lambda_{2;}E):=\prod_{\ell\geq 1}(\frac{1+(E-1)^{p^{\ell}}}{[p]^{\ell}E})^{\frac{1}{\mathrm{p}^{\ell}\mathrm{A}_{2}^{\mathrm{p}^{\ell}}}\Phi_{\ell-1}(\mathrm{A})}$,$G_{p}( \mathrm{A}, \Lambda_{2;}F):=\prod_{\ell\geq 1}(\frac{1+(F-1)^{p^{\ell}}}{[p]^{\ell}F})^{\frac{1}{\mathrm{p}^{\ell}\mathrm{A}_{2}^{\mathrm{p}^{\ell}}}\Phi_{\ell-1}(\mathrm{A})}$$\in \mathbb{Z}_{(p)}[\mathrm{A}, \frac{\mathrm{u}}{\Lambda_{2}}, \Lambda_{1}, \Lambda_{2}][[X, \mathrm{Y}]]$For aseries of variables$j\leq i)$ , we denote$\Lambda_{1}$,$\Lambda_{2}$,$\ldots$, and aseries of vectors$\mathrm{A}^{i}=(\mathrm{A}_{\ell}^{i})_{1\leq\ell\leq i}:=$We define vectors$\mathrm{B}_{j}^{i}(1\leq j<i)$$(\begin{array}{l}\mathrm{A}_{1}^{i}\mathrm{A}_{2}^{i}\vdots\mathrm{A}_{i}^{i}\end{array})$and$\mathrm{A}_{j}^{i}=$$(A_{j0}^{i}, A_{j1}^{i}, \ldots)$$(\Lambda_{\ell})_{1\leq\ell\leq i}:=(\begin{array}{l}\Lambda_{1}\Lambda_{2}\vdots\Lambda\end{array})$.

inductively by,$\mathrm{B}_{1}^{2}:=\frac{1}{\Lambda_{2}}F^{(\Lambda_{1})}\mathrm{A}_{1}^{1}$

and for $k\geq 2$ ,$\mathrm{B}_{j}^{k+1}:=\frac{1}{\Lambda_{k+1}}(F^{(\Lambda_{\mathrm{j}})}\mathrm{A}_{j}^{k}-\Sigma_{\ell=j+1\mathrm{B}_{\mathrm{j}}^{\ell}}^{k}T\mathrm{A}_{\ell}^{k})$$\{$$\mathrm{B}_{k}^{k+1}:=\frac{1}{\Lambda_{k+1}}F^{(\Lambda_{k})}\mathrm{A}_{k}^{k}$.

$1\leq j\leq k-1$$(1\leq i;1\leq$

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TSUTOMU SEKIGUCHI

Using these symboles, we define triangle matrices Un’s by.

.

$-.\cdot..T_{\mathrm{B}_{2}^{3}}-T_{\mathrm{B}_{1}^{3}}...\cdot$$U^{n}:=[_{0}^{0}F^{(\Lambda_{1})}00$$F^{(\Lambda_{2})}-T_{\mathrm{B}_{1}^{2}}000$$-T_{\mathrm{B}_{n-1}^{n}}F^{(\Lambda_{n})}-T_{\mathrm{B}_{2}^{n}]}-T_{\mathrm{B}_{1}^{n}}.\cdot.$$.\cdot 0^{\cdot}..\cdot$

We define inductively aseries of formal power series Dk{X0$D_{0}=1$$\geq 1$$\ldots$,$X_{k-1}$)’ by$\mathrm{s}$,$D_{1}(X_{0})=E_{p}(\mathrm{A}_{1}^{1}, \Lambda_{1;}X_{0})$and for&,,,$D_{k+1}(X_{0}, X_{1}, \ldots,\mathrm{x}_{k})=E_{p}(\mathrm{A}^{k+1}, (\Lambda_{\ell})_{1\leq\ell\leq k+1;X_{0},X_{1}},$$\ldots$,$X_{k})$.

$:= \prod_{\dot{|}=1}^{k+1}E_{p}(\mathrm{A}^{k+1}., \Lambda:;\frac{X_{\dot{|}-1}}{D_{\dot{|}-1}(X_{0},\ldots,X_{\dot{l}}-1)})$

Hereafter, we put$\mathrm{X}=$$(X_{0}, X_{1}, \ldots)$,$\mathrm{Y}$$=(\mathrm{Y}_{0}, \mathrm{Y}_{1}, \ldots)$and$\Sigma:=\mathrm{X}\dotplus \mathrm{Y}\in W$. We define$F^{(k)}:= \partial(D_{k}(\mathrm{X}))=\frac{D_{k}(\mathrm{X})D_{k}(\mathrm{Y})}{D_{k}(\Sigma)}$$H_{k}(\mathrm{X},\mathrm{Y})$$:= \frac{1}{\Lambda_{k+1}}(F^{(k)}-1)$$F_{p}(\mathrm{V}_{1},\Lambda_{1};\mathrm{X},\mathrm{Y})$$:=F_{p}(\mathrm{V}_{1}, \Lambda_{1}; X_{0}, \mathrm{Y}_{0})$$F_{p}((\mathrm{V}_{\dot{1}})_{1\leq:\leq n}, (\mathrm{A}.)_{1\leq:\leq n};\mathrm{X},\mathrm{Y})$$= \prod_{i=1}^{n}F_{p}(\mathrm{V}_{i}, \Lambda:;\dot{.}’\dot{.}\frac{\mathrm{Y}_{\dot{l}}-1}{D_{-1}(\mathrm{Y})}\mathrm{i})D_{-1}(\mathrm{X})X_{-1}$$\cross\prod_{i=2}^{n}F_{p}(\mathrm{V}_{i}, \Lambda:;H_{\dot{|}-1}, \frac{X_{-1}}{D_{i-1}(\mathrm{X})}\dot{.}\dotplus\frac{\mathrm{Y}_{i-1}}{D_{i-1}(\mathrm{Y})})$$\cross.\cdot\prod_{=2}^{n}G_{p}(\mathrm{V}:, \Lambda\dot{.};F^{(:-1)})^{-1}$.

Then the important thing is the following result.
Theorem 4.2. For each $n\geq 1$ , we have$F^{(n)}= \frac{D_{n}(\mathrm{X})D_{n}(\mathrm{Y})}{D_{n}(\Sigma)}=F_{p}(U^{n}\mathrm{A}^{n}, (\Lambda:)_{1\leq:\leq n};\mathrm{X}$,$\mathrm{Y})$.

By using this theorem, we can obtain the explicit determination of. infact, let, and , , ,be aDVR dominating
be non-zero elements of .
We choose Witt vectors$\mathrm{E}\mathrm{x}\mathrm{t}^{1}(\mathcal{W}_{n}, \mathcal{G}^{(\lambda)})$$(A, \mathrm{m})$$\mathbb{Z}(p)$.
$\vec{a}$$=(\overline{a}_{j}^{})_{1\leq j\leq:}\in \mathrm{K}\mathrm{e}\mathrm{r}(U^{:}$:$\lambda$$\lambda_{1}$$\lambda_{2}$$\ldots$$\overline{W}(A/\lambda_{+1}\dot{.})^{:}arrow\overline{W}(A/\lambda_{i+1})^{:})$$\mathrm{m}$9ON THE UNIFICATION OF KUMMER AND ARTIN-SCHREIER-WITT THEORIES

inductively by the following recursive conditions:$U^{1}=F^{(\lambda_{1})}$,($\overline{a}^{1}=\overline{a}_{1}^{1}\in \mathrm{K}\mathrm{e}\mathrm{r}$$b_{1}^{2}= \frac{1}{\lambda_{2}}a_{1}^{1}$,$U^{1}$:$\overline{W}(A/\lambda_{2})arrow\overline{W}(A/\lambda_{2})$$U^{2}=(_{0}^{F^{(\lambda_{1})}}$$F^{()}-T_{\oint_{\lambda_{2}}1)}$),,

and for $k\geq 2$ , we choose$\overline{a}^{k}=(\overline{a}_{i}^{k})_{1\leq i\leq k}\in \mathrm{K}\mathrm{e}\mathrm{r}(U^{k}$:$\overline{W}(A/\lambda_{k+1})^{k}arrow\overline{W}(A/\lambda_{k+1})^{k})$,and we define$1\leq j\leq k-1$$b_{j}^{k+1}:= \frac{1}{\lambda_{k+1}}(F^{(\lambda_{j})}a_{j}^{k}-\sum_{\ell=j+1}^{k}T\mu_{j}a_{\ell)}^{k}$$\{$$b_{k}^{k+1}:= \frac{1}{\lambda_{k+1}}F^{(\lambda_{k})}a_{k}^{k}$,$U^{k+1}:=\{\begin{array}{llllll}F^{(\lambda_{1})} -T\not\in -T_{R} .\cdot -T_{b_{1}^{k+1}}0 F^{(\lambda_{2})} -T\not\in -T_{\mathrm{b}^{k+1}}0 0\ddots \vdots 0 0 \ddots\ddots -T_{b_{k}^{k+1}}0 0 \cdots .0 F^{(\lambda_{k+1}})\end{array}\}$$\overline{a}^{k+1}=(\hat{a_{\dot{l}}}^{+1})_{1\leq i\leq k+1}\in \mathrm{K}\mathrm{e}\mathrm{r}(U^{k+1}$:,$\overline{W}(A/\lambda_{k+2})^{k+1}arrow\overline{W}(A/\lambda_{k+2})^{k+1})$.

We define formal power series $D_{k}(\mathrm{X})=D_{k}(X\circ, \ldots,X_{k-1})(k\geq 1)$ by$D_{0}=1$,$D_{1}(X_{0})--E_{p}(a_{1}^{1}, \lambda_{1;}X_{0})$,and for $k\geq 1$ ,$D_{k+1}(X_{0}, X_{1}, \ldots,X_{k})=E_{p}(a^{k+1}, (\lambda_{\ell})_{1\leq\ell\leq k+1;}X_{0},$ $X_{1}$,$\ldots$,$X_{k})$$:= \dot{.}\prod_{=1}^{k+1}E_{p}(a_{i}^{k+1}, \lambda:;\dot{.}\frac{X_{\dot{|}-1}}{D_{-1}(X_{0},\ldots,X_{\dot{\iota}-1})})$.

We put$\mathcal{W}_{n}:=\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}A[X_{0}, \ldots, X_{n-1}, \frac{1}{1+\lambda_{1}X_{0}}, \frac{1}{D_{1}(X_{0})+\lambda_{2}X_{1}}, \ldots, \frac{1}{D_{n-1}(\mathrm{X})+\lambda_{n}X_{n-1}}]$.

Theorem 4.3 (Explicit Formula in General Case). Let $B=A/\lambda$ . Then we have$\xi_{0}^{n}:\mathrm{K}\mathrm{e}\mathrm{r}(\overline{W}(B)^{n}arrow\overline{W}(U^{n}B)^{n})\simarrow \mathrm{H}\mathrm{o}\mathrm{m}(\mathcal{W}_{n,B}, \mathrm{G}_{m,B})$;$\overline{v}^{n}=(\overline{v}_{i}^{n})_{1\leq i\leq n}\mapsto E_{p}(\overline{v}^{n}, (\lambda_{i})_{1\leq\dot{|}\leq n};X_{0},$ $X_{1}$$\xi_{1}^{n}$:$\mathrm{C}\mathrm{o}\mathrm{k}\mathrm{e}\mathrm{r}(\overline{W}(B)^{n}arrow\overline{W}(U^{n}B)^{n})arrow \mathrm{H}_{0}^{2}\sim(\mathcal{W}_{n,B}, \mathrm{G}_{m,B})$$\overline{w}^{n}=(\overline{w}_{i}^{n})_{1\leq i\leq n}\mapsto F_{p}(\overline{w}^{n}, (\lambda:)_{1\leq i\leq n};\mathrm{X},$$\mathrm{Y})$,.

$\ldots$, $X_{n-1})$

10
TSUTOMU SEKIGUCHI

Theorem 4.4.
$\overline{\xi}_{0}^{n}$$w/iere$ $<c^{0}$$=$,$c^{1}$,$\ldots$

:

,

$.$

$\frac{\mathrm{K}\mathrm{e}\mathrm{r}(U^{n}\cdot\overline{W}(B)^{n}arrow\overline{W}(B/\lambda)^{n})}{<c^{0},c^{1},\ldots,c^{n-1}>}\simarrow \mathrm{E}\mathrm{x}\mathrm{t}^{1}(\mathcal{W}_{n,B}, \mathcal{G}_{B}^{(\lambda)})$, $c^{n-1}>is$ the subgroup generated by the vectors $c^{0}=$$d=(a^{\ell},\overline{\lambda}_{\ell+1},0, \ldots, 0),\ldots$$(a^{1},\overline{\lambda}_{2},0, \ldots, 0),\ldots$’’$c^{n-1}=(a^{n-1},\tilde{\lambda}_{n})$$(\tilde{\lambda}_{1},0, \ldots, 0),c^{1}$.

5. REDUCTIONS OF EXTENSIONSThe special fibres of the group schemes of type$(\lambda_{1}, \lambda_{2},$$\lambda_{n})$\ldots ,can be decided as follows.

Theorem 5.1. Let$\mathcal{W}_{n}=\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}A[X_{0}, X_{1}, \ldots, X_{n-1}, \frac{1}{1+\lambda_{1}X_{0}}, \frac{1}{D_{1}(X_{0})+\lambda_{2}X_{1}}, \ldots, \frac{1}{D_{n-1}(\mathrm{X})+\lambda_{n}X_{n-1}}]$be the group scheme of type$(\lambda_{1}, \lambda_{2}, \ldots, \lambda_{n-1})$

defined by$D_{1}(X_{0})=E_{p}(a_{1}^{1}, \lambda_{1)}.X_{0})$and for $1\leq k\leq n-2_{f}$$D_{k+1}(\mathrm{X})=E_{p}(a^{k+1}, (\lambda_{\ell})_{1\leq\ell\leq k+1};X_{0},$$X_{1}$,$\ldots$,$X_{k}$),

and$\overline{a}^{k}\in \mathrm{K}\mathrm{e}\mathrm{r}(U^{k} :\overline{W}(A/\lambda_{k+1})^{k}|arrow\overline{W}(A/\lambda_{k})^{k})$.

Here$(i=2, \ldots, n)$$b^{i}={}^{t}(b_{1}^{i}, b_{2}^{i}, \ldots, b_{i-1}^{i})=\frac{1}{\lambda}\dot{.}U^{i-1}a^{i-1}$

If$b_{\ell}^{k}\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d} \mathrm{m})$we havefor $3\leq k\leq n$ , $1\leq\ell\leq k-2$ , and.

$b_{k-1}^{k}\equiv(1,0, , \ldots)$$\mathcal{W}_{n,k}=\mathcal{W}_{n}\otimes_{A}k=W_{n,k}$$(\mathrm{m}\mathrm{o}\mathrm{d} m)$, then.

6. CONDITIONS FOR KRSW GROUP SCHEMES
Let$\mathcal{W}_{n}=\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}A[X_{0},$$\ldots$,$X_{n-1}$,$\frac{1}{1+\lambda_{(1)}X_{0}’}\frac{1}{D_{1}(X_{0})+\lambda_{(1)}X_{1}}$be aKASW group scheme over aDVR
are given by,... ’$(A,\mathrm{m})$$\frac{1}{D_{n-1}(X_{0},\ldots,X_{n-2})+\lambda_{(1)}X_{n-1}}$, and Abe an element of$D_{:}(\mathrm{X})=E_{p}(a^{i}, (\lambda_{(1)})^{n};\mathrm{X})$with$\overline{a}^{\dot{l}}\in \mathrm{K}\mathrm{e}\mathrm{r}(U^{:} :.

\overline{W}(A/\lambda(1))arrow\overline{W}(A/\lambda(1)))$

We look at the exact sequence.

$0arrow(\mathbb{Z}/p^{n})_{A}-\dot{.}3\mathcal{W}_{n}arrow \mathcal{W}_{n}/(\mathbb{Z}/p^{n})_{A}\psi_{n}arrow 0$$\mathrm{m}\backslash \{0\}$]. Here$D_{:}$’s

11
ON THE UNFICATION OF KUMMER AND ARTIN-SCHREIER-WITT THEORIESThen we have$i_{n}^{*}$:$\mathrm{E}\mathrm{x}\mathrm{t}_{A}^{1}(\mathcal{W}_{n}, \mathcal{G}^{(\lambda)})$$arrow \mathrm{E}\mathrm{x}\mathrm{t}_{A}^{1}((\mathbb{Z}/p^{n})_{A}, \mathcal{G}^{(\lambda)})$

$||l$

$||l$

$\frac{\mathrm{K}\mathrm{e}\mathrm{r}(\hat{W}(A/\lambda)^{n}\hat{W}(A/\lambda))U^{n}}{<\mathrm{c}^{0},\mathrm{c}^{1},\ldots,\mathrm{c}^{n-1}>}$$a^{n}$$arrow$$(1+\lambda A)/(1+\lambda A)^{p^{n}}$$\mapsto$$\Pi_{r>0}(E_{p}(a_{1,r}^{n}, \lambda_{(1)}^{p^{r}}; 1)^{p^{n}}\Pi_{i=2}^{n}E_{p}(a_{i,r}^{n}, \lambda_{(1)}^{p^{r}}; (\dot{.}\frac{\mathrm{C}-1}{D_{-}(i_{n}(1))}\mathrm{i})^{p^{r}})^{p^{n}})$Under these notations, we have the following.
Theorem 6.1. Let
by the isomorphism$\mathcal{W}_{n+1}\in \mathrm{E}\mathrm{x}\mathrm{t}^{1}(\mathcal{W}_{n}, \mathcal{G}^{(\lambda)})$be the extension corresponding to a vector$a^{n}=(a_{i}^{n})_{1\leq i\leq n}$$. \frac{\mathrm{K}\mathrm{e}\mathrm{r}(U^{n}\cdot\overline{W}(A/\lambda)^{n}arrow\overline{W}(A/\lambda)^{n})}{(c^{1},c^{2},\ldots,c^{n-1})}\simarrow \mathrm{E}\mathrm{x}\mathrm{t}^{1}(\mathcal{W}_{n}, \mathcal{G}^{(\lambda)})$Then these exisis an inclusion$(\mathbb{Z}/p^{n+1})_{A}\subset \mathcal{W}_{n+1}$.

fitting into a commutative diagram$0arrow(\mathbb{Z}/p)_{A}arrow(\mathbb{Z}/p^{n+1})_{A}arrow(\mathbb{Z}/p^{n})_{A}arrow 0$$i_{1}\mathcal{G}^{(\lambda)}\downarrow$$i_{n}\mathcal{W}_{n}\downarrow$$i_{n+1}\mathcal{W}_{n+1}\downarrow$$0arrow$$arrow 0$ ,$arrow$$arrow$

if and only if$E_{p}(a^{n}, (\lambda, \ldots, \lambda);i_{n}(1))^{p^{n}}=\zeta_{1}$.

Using these results, we construct explicitly the KASW group schemes.
Theorem 6.2 (Main Theorem). For each positive integer , we construct explicitly
standard KASW group scheme
over
.
$n$

$\mathcal{W}_{n}$$a$$\mathbb{Z}_{(p)}[\mu_{p^{n}}]$, the quotient
Finally, we remark that for aKASW group scheme, and which is given explicitly.
is agroup scheme of type$\mathcal{W}_{n}$$\mathcal{V}_{n}:=\mathcal{W}_{n}/(\mathbb{Z}/p^{n})$$(\lambda_{(1)}^{p})^{n}$

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550-568(1980)(Tsutomu Sekiguchi) DEpARTMENT OF MATHEMATICS, FACULTY OF SCIENCE AND ENGENEERING,

Chuo UNIVERSITY 1-13-27 KASUGA BUNKYO-KU Tokyo 112-8551, JApAN-mail address: sekiguti@math.chu0-u.ac.jp$E$