Hopf orders, applications of
Let be a finite extension of the
-adic rationals
endowed with the
-adic valuation
with
and let
be its ring of integers (cf. also Extension of a field; Norm on a field;
-adic number). Let
be the ramification index of
in
and assume that
contains a primitive
rd root of unity. If
is an
-Hopf order in
(cf. Hopf order), then the group scheme
can be resolved, i.e., involved in a short exact sequence of group schemes
(taken in the flat topology). Here,
and
are represented by the
-Hopf algebras
and
, respectively (
an indeterminate). One uses this short exact sequence in the long exact sequence in cohomology to construct
, which is identified with the group of
-Galois extensions of
(cf. also Galois extension). One has
, where the class
corresponds to the isomorphism class
of the
-Galois extension
![]() |
with (see [a2]). Moreover, if
,
, is the dual of the Larson order
(cf. Hopf order), then
can be involved in the short exact sequence of group schemes
, where
and
are represented by the
-Hopf algebras
and
for appropriate polynomials
, respectively (see [a3]). Work has been completed to obtain a resolution of
when
is an arbitrary
-Hopf order in
(see [a4]). This yields (via the long exact sequence in cohomology) a characterization of all
-Galois extensions. In general, if
is an
-Hopf order in
, then
is an
-Galois extension if and only if
is an
-Galois algebra (see [a1]).
Examples of -Galois extensions where
is a Tate–Oort–Larson order in
can be recovered using the corresponding classification theorem. For example, if
is a Kummer extension of prime degree, then
is an
-Galois algebra if and only if the ramification number
of
satisfies
(see [a1]). Therefore,
is an
-Galois extension, thus
![]() |
for some parameter . Knowledge of Tate–Oort Galois extensions can also be used to characterize the ring of integers of certain degree-
extensions. For example, if
,
, is a Greither order with
and
, then there exists a finite extension
so that
is an
-Galois algebra. In this case,
is of the form
![]() |
where is an element in
and
is an
-Galois extension (see [a2]).
References
[a1] | L. Childs, "Taming wild extensions with Hopf algebras" Trans. Amer. Math. Soc. , 304 (1987) Zbl 0632.12013 |
[a2] | 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 |
[a3] | T. Sekiguchi, N. Suwa, "Théories de Kummer–Artin–Schreier–Witt" C.R. Acad. Sci. Ser. I , 319 (1994) pp. 1–21 Zbl 0845.14023 |
[a4] | R.G. Underwood, "The group of Galois extensions in $KC_{p^2}$" Trans. Amer. Math. Soc. , 349 (1997) pp. 1503–1514 Zbl 0957.16010 |
Hopf orders, applications of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hopf_orders,_applications_of&oldid=42722