View source for Hopf orders, applications of

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h1102901.png" /> be a finite extension of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h1102902.png" />-adic rationals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h1102903.png" /> endowed with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h1102904.png" />-adic valuation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h1102905.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h1102906.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h1102907.png" /> be its ring of integers (cf. also [[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/h110290/h1102908.png" />-adic number]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h1102909.png" /> be the ramification index of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029010.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029011.png" /> and assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029012.png" /> contains a primitive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029013.png" />rd root of unity. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029014.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029015.png" />-Hopf order in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029016.png" /> (cf. [[Hopf order|Hopf order]]), then the [[Group scheme|group scheme]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029017.png" /> can be resolved, i.e., involved in a short [[Exact sequence|exact sequence]] of group schemes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029018.png" /> (taken in the flat topology). Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029020.png" /> are represented by the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029021.png" />-Hopf algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029023.png" />, respectively (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029024.png" /> an indeterminate). One uses this short exact sequence in the long exact sequence in [[Cohomology|cohomology]] to construct <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029025.png" />, which is identified with the group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029026.png" />-Galois extensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029027.png" /> (cf. also [[Galois extension|Galois extension]]). One has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029028.png" />, where the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029029.png" /> corresponds to the isomorphism class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029030.png" /> of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029031.png" />-Galois extension

<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/h110290/h11029032.png" /></td> </tr></table>

with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029033.png" /> (see [[#References|[a2]]]). Moreover, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029035.png" />, is the dual of the Larson order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029036.png" /> (cf. [[Hopf order|Hopf order]]), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029037.png" /> can be involved in the short exact sequence of group schemes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029038.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029040.png" /> are represented by the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029041.png" />-Hopf algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029042.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029043.png" /> for appropriate polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029044.png" />, respectively (see [[#References|[a3]]]). Work has been completed to obtain a resolution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029045.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029046.png" /> is an arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029047.png" />-Hopf order in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029048.png" /> (see [[#References|[a4]]]). This yields (via the long exact sequence in cohomology) a characterization of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029049.png" />-Galois extensions. In general, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029050.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029051.png" />-Hopf order in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029052.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029053.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029054.png" />-Galois extension if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029055.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029056.png" />-Galois algebra (see [[#References|[a1]]]).

Examples of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029057.png" />-Galois extensions where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029058.png" /> is a Tate–Oort–Larson order in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029059.png" /> can be recovered using the corresponding classification theorem. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029060.png" /> is a [[Kummer extension|Kummer extension]] of prime degree, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029061.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029062.png" />-Galois algebra if and only if the ramification number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029063.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029064.png" /> satisfies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029065.png" /> (see [[#References|[a1]]]). Therefore, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029066.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029067.png" />-Galois extension, thus

<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/h110290/h11029068.png" /></td> </tr></table>

for some parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029069.png" />. Knowledge of Tate–Oort Galois extensions can also be used to characterize the ring of integers of certain degree-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029070.png" /> extensions. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029071.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029072.png" />, is a [[Greither order]] with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029073.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029074.png" />, then there exists a finite extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029075.png" /> so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029076.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029077.png" />-Galois algebra. In this case, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029078.png" /> 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/h110290/h11029079.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029080.png" /> is an element in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029081.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029082.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029083.png" />-Galois extension (see [[#References|[a2]]]).

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Childs,   "Taming wild extensions with Hopf algebras"  ''Trans. Amer. Math. Soc.'' , '''304'''  (1987)  {{ZBL|0632.12013}}</TD></TR>
<TR><TD valign="top">[a2]</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">[a3]</TD> <TD valign="top">  T. Sekiguchi,   N. Suwa,   "Théories de Kummer–Artin–Schreier–Witt"  ''C.R. Acad. Sci. Ser. I'' , '''319'''  (1994)  pp. 1–21  {{ZBL|0845.14023}}</TD></TR>
<TR><TD valign="top">[a4]</TD> <TD valign="top">  R.G. Underwood,   "The group of Galois extensions in $KC_{p^2}$"  ''Trans. Amer. Math. Soc.'' , '''349'''  (1997)  pp. 1503–1514  {{ZBL|0957.16010}}</TD></TR>
</table>