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1. CJM 2000 (vol 52 pp. 1269)
Well Ramified Extensions of Complete Discrete Valuation Fields with Applications to the Kato Conductor |
Well Ramified Extensions of Complete Discrete Valuation Fields with Applications to the Kato Conductor We study extensions $L/K$ of complete discrete valuation fields $K$
with residue field $\oK$ of characteristic $p > 0$, which we do not
assume to be perfect. Our work concerns ramification theory for such
extensions, in particular we show that all classical properties which
are true under the hypothesis {\it ``the residue field extension
$\oL/\oK$ is separable''} are still valid under the more general
hypothesis that the valuation ring extension is monogenic. We also
show that conversely, if classical ramification properties hold true
for an extension $L/K$, then the extension of valuation rings is
monogenic. These are the ``{\it well ramified}'' extensions. We show
that there are only three possible types of well ramified extensions
and we give examples. In the last part of the paper we consider, for
the three types, Kato's generalization of the conductor, which we show
how to bound in certain cases.
Categories:11S, 11S15, 11S20 |
2. CJM 1998 (vol 50 pp. 1007)
Galois module structure of ambiguous ideals in biquadratic extensions Let $N/K$ be a biquadratic extension of algebraic number fields, and
$G=\Gal (N/K)$. Under a weak restriction on the ramification filtration
associated with each prime of $K$ above $2$, we explicitly describe the
$\bZ[G]$-module structure of each ambiguous ideal of $N$. We find under
this restriction that in the representation of each ambiguous ideal as a
$\bZ[G]$-module, the exponent (or multiplicity) of each indecomposable
module is determined by the invariants of ramification, alone.
For a given group, $G$, define ${\cal S}_G$ to be the set of
indecomposable $\bZ[G]$-modules, ${\cal M}$, such that there
is an extension, $N/K$, for which $G\cong\Gal (N/K)$, and ${\cal M}$
is a $\bZ[G]$-module summand of an ambiguous ideal of $N$. Can
${\cal S}_G$ ever be infinite? In this paper we answer this
question of Chinburg in the affirmative.
Keywords:Galois module structure, wild ramification Categories:11R33, 11S15, 20C32 |
3. CJM 1997 (vol 49 pp. 722)
Galois module structure of the integers in wildly ramified $C_p\times C_p$ extensions Let $L/K$ be a finite Galois extension of local fields which are finite
extensions of $\bQ_p$, the field of $p$-adic numbers. Let $\Gal (L/K)=G$,
and $\euO_L$ and $\bZ_p$ be the rings of integers in $L$ and $\bQ_p$,
respectively. And let $\euP_L$ denote the maximal ideal of $\euO_L$. We
determine, explicitly in terms of specific indecomposable $\bZ_p[G]$-modules,
the $\bZ_p[G]$-module structure of $\euO_L$ and $\euP_L$, for $L$, a
composite of two arithmetically disjoint, ramified cyclic extensions of
$K$, one of which is only weakly ramified in the sense of Erez \cite{erez}.
Keywords:Galois module structure---integral representation. Categories:11S15, 20C32 |