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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h1102801.png" /> be a finite extension of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h1102802.png" />-adic rationals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h1102803.png" /> endowed with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h1102804.png" />-adic valuation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h1102805.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h1102806.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h1102807.png" /> be its ring of integers (cf. [[Extension of a field|Extension of a field]]; [[Norm on a field|Norm on a field]]; [[P-adic number|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h1102808.png" />-adic number]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h1102809.png" /> be the group ring of a [[Finite group|finite group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028010.png" /> (cf. also [[Group algebra|Group algebra]]; [[Cross product|Cross product]]), with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028011.png" />. An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028013.png" />-Hopf order in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028014.png" /> is a rank-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028015.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028016.png" />-Hopf algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028017.png" /> (cf. [[Hopf algebra|Hopf algebra]]) satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028018.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028019.png" />-Hopf algebras.
+
 
 +
Let $K$ be a finite extension of the $p$-adic rationals $\Q_p$ endowed with the $p$-adic valuation $\nu$ with $\nu(p)=1$ and let $R$ be its ring of integers (cf. [[Extension of a field|Extension of a field]]; [[Norm on a field|Norm on a field]]; [[P-adic number|$p$-adic number]]). Let $KG$ be the group ring of a [[Finite group|finite group]] $G$ (cf. also [[Group algebra|Group algebra]]; [[Cross product|Cross product]]), with $|G|=q$. An $R$-Hopf order in $KG$ is a rank-$q$ $R$-Hopf algebra $H$ (cf. [[Hopf algebra|Hopf algebra]]) satisfying $H\otimes_R K \cong KG$ as $K$-Hopf algebras.
  
There is a method [[#References|[a2]]] for constructing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028020.png" />-Hopf orders in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028021.png" /> using so-called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028022.png" />-adic order-bounded group valuations on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028023.png" />. Given a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028024.png" />-adic order-bounded group valuation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028025.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028026.png" /> be an element in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028027.png" /> of value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028028.png" />. Then the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028029.png" />-Hopf order in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028030.png" /> determined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028031.png" /> (called a Larson order) is of the form
+
There is a method [[#References|[a2]]] for constructing $R$-Hopf orders in $KG$ using so-called $p$-adic order-bounded group valuations on $G$. Given a $p$-adic order-bounded group valuation $\xi$, let $x_{\xi(g)}$ be an element in $R$ of value $\xi(g)$. Then the $R$-Hopf order in $KG$ determined by $\xi$ (called a Larson order) is of the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028032.png" /></td> </tr></table>
+
$$
 +
A(\xi) = R \left[ \left\{ \frac{g-1}{x_{\xi(g)}} : g \in G,\ g \ne 1 \right\}\right].
 +
$$
  
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028033.png" /> Abelian (cf. [[Abelian group|Abelian group]]), the classification of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028034.png" />-Hopf orders in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028035.png" /> is reduced to the case where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028036.png" /> is a [[P-group|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028037.png" />-group]]. Specifically, one takes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028039.png" /> cyclic of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028040.png" />, and assumes that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028041.png" /> contains a primitive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028042.png" />th root of unity, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028043.png" />. In this case, a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028044.png" />-adic order-bounded group valuation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028045.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028046.png" /> is determined by its values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028047.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028048.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028049.png" />, and the Larson order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028050.png" /> is denoted by
+
For $G$ Abelian (cf. [[Abelian group|Abelian group]]), the classification of $R$-Hopf orders in $KG$ is reduced to the case where $G$ is a [[P-group|$p$-group]]. Specifically, one takes $G = C_{p^n}$, $C_{p^n}$ cyclic of order $p^n$, and assumes that $K$ contains a primitive $p^n$th root of unity, denoted by $\zeta_n$. In this case, a $p$-adic order-bounded group valuation $\xi$ on $C_{p^n}$ is determined by its values $\xi(g^{p^i}) = s_i$ for $i=0,\ldots,n-1$, $\langle g \rangle = C_{p^n}$, and the Larson order $A(\xi)$ is denoted by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028051.png" /></td> </tr></table>
+
$$
 +
H(s_{n-1}, \ldots, s_0) = R \left[ \frac{g^{p^{n-1}}-1}{x_{s_{n-1}}}, \ldots, \frac{g-1}{x_{s_0}} \right]
 +
$$
  
It is known [[#References|[a3]]] that every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028052.png" />-Hopf order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028053.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028054.png" /> can be written as a Tate–Oort algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028055.png" />, which in turn can be expressed as the Larson order
+
It is known [[#References|[a3]]] that every $R$-Hopf order $H$ in $KC_p$ can be written as a Tate–Oort algebra $H_b = R[x]/\langle x^p-bx\rangle$, which in turn can be expressed as the Larson order
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028056.png" /></td> </tr></table>
+
$$
 +
H(s) = R \left[ \frac{g-1}{x_s} \right].
 +
$$
  
Thus, every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028057.png" />-Hopf order in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028058.png" /> is Larson. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028059.png" /> this is not the case, though every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028060.png" />-Hopf order does contain a maximal Larson order [[#References|[a2]]].
+
Thus, every $R$-Hopf order in $KC_p$ is Larson. For $n>1$ this is not the case, though every $R$-Hopf order does contain a maximal Larson order [[#References|[a2]]].
  
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028061.png" /> there exists a large class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028062.png" />-Hopf orders in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028063.png" /> (called Greither orders), of the form
+
For $n=2$ there exists a large class of $R$-Hopf orders in $KC_{p^2}$ (called Greither orders), of the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028064.png" /></td> </tr></table>
+
$$
 +
H_v(s,r) = R \left[ \frac{g^p-1}{x_s}, \frac{g-a_v}{x_r} \right],
 +
$$
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028065.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028066.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028067.png" /> are values from a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028068.png" />-adic order-bounded group valuation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028069.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028070.png" /> is an element in the Larson order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028071.png" /> (see [[#References|[a1]]]). The parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028072.png" /> is an element in the units group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028073.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028074.png" /> is the ramification index of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028075.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028076.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028077.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028078.png" />, then the Greither order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028079.png" /> is the Larson order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028080.png" />; moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028081.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028082.png" />.
+
$\langle g \rangle = C_{p^2}$, where $s$ and $r$ are values from a $p$-adic order-bounded group valuation on $C_{p^2}$ and $a_v$ is an element in the Larson order $H(s)$ (see [[#References|[a1]]]). The parameter $v$ is an element in the units group $U_{s'e + \langle re/p \rangle} \cap U_{\langle s'e /p \rangle + re}$, where $e$ is the ramification index of $p$ in $R$, and $s' = 1/(p-1)-s$. If $v=1$, then the Greither order $H_1(s,r)$ is the Larson order $H(s,r)$; moreover, $H_v(s,r) \cong H_w(s,r)$ if and only if $v/w \in U_{s'e+re}$.
  
Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028083.png" />, the linear dual <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028084.png" /> of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028085.png" />-Hopf order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028086.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028087.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028088.png" />-Hopf order in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028089.png" />. One has
+
Since $\zeta_n \in K$, the linear dual $H^*$ of the $R$-Hopf order $H$ in $KC_{p^n}$ is an $R$-Hopf order in $KC_{p^n}$. One has
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028090.png" /></td> </tr></table>
+
$$
 +
H(s)^* \cong R \left[ \frac{g-1}{x_{s'}} \right]
 +
$$
  
 
and
 
and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028091.png" /></td> </tr></table>
+
$$
 +
H_v(s,r)^* \cong R \left[ \frac{g^p-1}{x_{r'}} , \frac{g-a_{v'}}{x_{s'}} \right],
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028092.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028093.png" /> (see [[#References|[a5]]]). It is known [[#References|[a4]]] that an arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028094.png" />-Hopf order in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028095.png" /> is either a Greither order or the linear dual of a Greither order. Thus, every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028096.png" />-Hopf order in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028097.png" /> can be written in the form
+
where $v'=1+\zeta_2 - v$, $r'=1/(p-1)-r$ (see [[#References|[a5]]]). It is known [[#References|[a4]]] that an arbitrary $R$-Hopf order in $KC_{p^2}$ is either a Greither order or the linear dual of a Greither order. Thus, every $R$-Hopf order in $KC_{p^2}$ can be written in the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028098.png" /></td> </tr></table>
+
$$
 +
A_w(b,a) = R \left[ \frac{g^p-1}{x_b}, \frac{g-a_w}{x_a} \right]
 +
$$
  
for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h11028099.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h110280100.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h110280101.png" />.
+
for some $a,b,w$.
  
The construction of Greither orders can be generalized to give a complete classification of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h110280102.png" />-Hopf orders in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h110280103.png" />, as well as a class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h110280104.png" />-Hopf orders in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h110280105.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h110280106.png" />, which are not Larson (see [[#References|[a5]]]). However, the complete classification of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h110280107.png" />-Hopf orders in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h110280108.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h110280109.png" />, remains an open problem.
+
The construction of Greither orders can be generalized to give a complete classification of $R$-Hopf orders in $KC_{p^3}$, as well as a class of $R$-Hopf orders in $KC_{p^n}$, $n>2$, which are not Larson (see [[#References|[a5]]]). However, the complete classification of $R$-Hopf orders in $KC_{p^n}$, $n>3$, remains an open problem.
  
 
See also [[Hopf orders, applications of|Hopf orders, applications of]].
 
See also [[Hopf orders, applications of|Hopf orders, applications of]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C. Greither,  "Extensions of finite group schemes, and Hopf Galois theory over a complete discrete valuation ring"  ''Math. Z.'' , '''210'''  (1992)  pp. 37–67</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R.G. Larson,  "Hopf algebra orders determined by group valuations"  ''J. Algebra'' , '''38'''  (1976)  pp. 414–452</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J. Tate,  F. Oort,  "Group schemes of prime order"  ''Ann. Sci. Ecole Norm. Super. (4)'' , '''3'''  (1970)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R.G. Underwood,  "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h110280110.png" />-Hopf algebra orders in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h110280111.png" />"  ''J. Algebra'' , '''169'''  (1994},)  pp. 418–440</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  R.G. Underwood,  "The valuative condition and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h110280112.png" />-Hopf algebra orders in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110280/h110280113.png" />"  ''Amer. J. Math. (4)'' , '''118'''  (1996)  pp. 701–743</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  C. Greither,  "Extensions of finite group schemes, and Hopf Galois theory over a complete discrete valuation ring"  ''Math. Z.'' , '''210'''  (1992)  pp. 37–67 {{ZBL|0737.11038}}</TD></TR>
 +
<TR><TD valign="top">[a2]</TD> <TD valign="top">  R.G. Larson,  "Hopf algebra orders determined by group valuations"  ''J. Algebra'' , '''38'''  (1976)  pp. 414–452 {{ZBL|0407.20007}}</TD></TR>
 +
<TR><TD valign="top">[a3]</TD> <TD valign="top">  J. Tate,  F. Oort,  "Group schemes of prime order"  ''Ann. Sci. Ecole Norm. Super. (4)'' , '''3'''  (1970)</TD></TR>
 +
<TR><TD valign="top">[a4]</TD> <TD valign="top">  R.G. Underwood,  "$R$-Hopf algebra orders in $KC_{p^2}$"  ''J. Algebra'' , '''169'''  (1994},)  pp. 418–440 {{ZBL|0820.16036}}</TD></TR>
 +
<TR><TD valign="top">[a5]</TD> <TD valign="top">  R.G. Underwood,  "The valuative condition and $R$-Hopf algebra orders in $KC_{p^3}$"  ''Amer. J. Math. (4)'' , '''118'''  (1996)  pp. 701–743 {{ZBL|0857.16039}}</TD></TR>
 +
 
 +
<TR><TD valign="top">[b1]</TD> <TD valign="top">  R.G. Underwood,  "An Introduction to Hopf Algebras" Springer (2011) {{ISBN|978-0-387-72765-3}}  {{ZBL|1234.16022}}</TD></TR>
 +
</table>
 +
 
 +
{{TEX|done}}

Latest revision as of 04:20, 15 February 2024

Let $K$ be a finite extension of the $p$-adic rationals $\Q_p$ endowed with the $p$-adic valuation $\nu$ with $\nu(p)=1$ and let $R$ be its ring of integers (cf. Extension of a field; Norm on a field; $p$-adic number). Let $KG$ be the group ring of a finite group $G$ (cf. also Group algebra; Cross product), with $|G|=q$. An $R$-Hopf order in $KG$ is a rank-$q$ $R$-Hopf algebra $H$ (cf. Hopf algebra) satisfying $H\otimes_R K \cong KG$ as $K$-Hopf algebras.

There is a method [a2] for constructing $R$-Hopf orders in $KG$ using so-called $p$-adic order-bounded group valuations on $G$. Given a $p$-adic order-bounded group valuation $\xi$, let $x_{\xi(g)}$ be an element in $R$ of value $\xi(g)$. Then the $R$-Hopf order in $KG$ determined by $\xi$ (called a Larson order) is of the form

$$ A(\xi) = R \left[ \left\{ \frac{g-1}{x_{\xi(g)}} : g \in G,\ g \ne 1 \right\}\right]. $$

For $G$ Abelian (cf. Abelian group), the classification of $R$-Hopf orders in $KG$ is reduced to the case where $G$ is a $p$-group. Specifically, one takes $G = C_{p^n}$, $C_{p^n}$ cyclic of order $p^n$, and assumes that $K$ contains a primitive $p^n$th root of unity, denoted by $\zeta_n$. In this case, a $p$-adic order-bounded group valuation $\xi$ on $C_{p^n}$ is determined by its values $\xi(g^{p^i}) = s_i$ for $i=0,\ldots,n-1$, $\langle g \rangle = C_{p^n}$, and the Larson order $A(\xi)$ is denoted by

$$ H(s_{n-1}, \ldots, s_0) = R \left[ \frac{g^{p^{n-1}}-1}{x_{s_{n-1}}}, \ldots, \frac{g-1}{x_{s_0}} \right] $$

It is known [a3] that every $R$-Hopf order $H$ in $KC_p$ can be written as a Tate–Oort algebra $H_b = R[x]/\langle x^p-bx\rangle$, which in turn can be expressed as the Larson order

$$ H(s) = R \left[ \frac{g-1}{x_s} \right]. $$

Thus, every $R$-Hopf order in $KC_p$ is Larson. For $n>1$ this is not the case, though every $R$-Hopf order does contain a maximal Larson order [a2].

For $n=2$ there exists a large class of $R$-Hopf orders in $KC_{p^2}$ (called Greither orders), of the form

$$ H_v(s,r) = R \left[ \frac{g^p-1}{x_s}, \frac{g-a_v}{x_r} \right], $$

$\langle g \rangle = C_{p^2}$, where $s$ and $r$ are values from a $p$-adic order-bounded group valuation on $C_{p^2}$ and $a_v$ is an element in the Larson order $H(s)$ (see [a1]). The parameter $v$ is an element in the units group $U_{s'e + \langle re/p \rangle} \cap U_{\langle s'e /p \rangle + re}$, where $e$ is the ramification index of $p$ in $R$, and $s' = 1/(p-1)-s$. If $v=1$, then the Greither order $H_1(s,r)$ is the Larson order $H(s,r)$; moreover, $H_v(s,r) \cong H_w(s,r)$ if and only if $v/w \in U_{s'e+re}$.

Since $\zeta_n \in K$, the linear dual $H^*$ of the $R$-Hopf order $H$ in $KC_{p^n}$ is an $R$-Hopf order in $KC_{p^n}$. One has

$$ H(s)^* \cong R \left[ \frac{g-1}{x_{s'}} \right] $$

and

$$ H_v(s,r)^* \cong R \left[ \frac{g^p-1}{x_{r'}} , \frac{g-a_{v'}}{x_{s'}} \right], $$

where $v'=1+\zeta_2 - v$, $r'=1/(p-1)-r$ (see [a5]). It is known [a4] that an arbitrary $R$-Hopf order in $KC_{p^2}$ is either a Greither order or the linear dual of a Greither order. Thus, every $R$-Hopf order in $KC_{p^2}$ can be written in the form

$$ A_w(b,a) = R \left[ \frac{g^p-1}{x_b}, \frac{g-a_w}{x_a} \right] $$

for some $a,b,w$.

The construction of Greither orders can be generalized to give a complete classification of $R$-Hopf orders in $KC_{p^3}$, as well as a class of $R$-Hopf orders in $KC_{p^n}$, $n>2$, which are not Larson (see [a5]). However, the complete classification of $R$-Hopf orders in $KC_{p^n}$, $n>3$, remains an open problem.

See also Hopf orders, applications of.

References

[a1] C. Greither, "Extensions of finite group schemes, and Hopf Galois theory over a complete discrete valuation ring" Math. Z. , 210 (1992) pp. 37–67 Zbl 0737.11038
[a2] R.G. Larson, "Hopf algebra orders determined by group valuations" J. Algebra , 38 (1976) pp. 414–452 Zbl 0407.20007
[a3] J. Tate, F. Oort, "Group schemes of prime order" Ann. Sci. Ecole Norm. Super. (4) , 3 (1970)
[a4] R.G. Underwood, "$R$-Hopf algebra orders in $KC_{p^2}$" J. Algebra , 169 (1994},) pp. 418–440 Zbl 0820.16036
[a5] R.G. Underwood, "The valuative condition and $R$-Hopf algebra orders in $KC_{p^3}$" Amer. J. Math. (4) , 118 (1996) pp. 701–743 Zbl 0857.16039
[b1] R.G. Underwood, "An Introduction to Hopf Algebras" Springer (2011) ISBN 978-0-387-72765-3 Zbl 1234.16022
How to Cite This Entry:
Hopf order. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hopf_order&oldid=16468
This article was adapted from an original article by R.G. Underwood (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article