22,556 research outputs found
Finite depth and Jacobson-Bourbaki correspondence
We introduce a notion of depth three tower of three rings C < B < A with
depth two ring extension A | B recovered when B = C. If A = \End B_C and B | C
is a Frobenius extension, this captures the notion of depth three for a
Frobenius extension in arXiv:math/0107064 and arXiv:math/0108067, such that if
B | C is depth three, then A | C is depth two (a phenomenon of finite depth
subfactors, see arXiv:math/0006057). We provide a similar definition of finite
depth Frobenius extension with embedding theorem utilizing a depth three
subtower of the Jones tower. If A, B and C correspond to a tower of subgroups G
> H > K via the group algebra over a fixed base ring, the depth three condition
is the condition that subgroup K has normal closure K^G contained in H. For a
depth three tower of rings, there is a pre-Galois theory for the ring \End
{}_BA_C and coring (A \o_B A)^C involving Morita context bimodules and left
coideal subrings. This is applied in two sections to a specialization of a
Jacobson-Bourbaki correspondence theorem for augmented rings to depth two
extensions with depth three intermediate division rings.Comment: 26 pp., depth three towers with new section on finite depth, and
correction
Explicit integral Galois module structure of weakly ramified extensions of local fields
Let L/K be a finite Galois extension of complete local fields with finite
residue fields and let G=Gal(L/K). Let G_1 and G_2 be the first and second
ramification groups. Thus L/K is tamely ramified when G_1 is trivial and we say
that L/K is weakly ramified when G_2 is trivial. Let O_L be the valuation ring
of L and let P_L be its maximal ideal. We show that if L/K is weakly ramified
and n is congruent to 1 mod |G_1| then P_L^n is free over the group ring
O_K[G], and we construct an explicit generating element. Under the additional
assumption that L/K is wildly ramified, we then show that every free generator
of P_L over O_K[G] is also a free generator of O_L over its associated order in
the group algebra K[G]. Along the way, we prove a `splitting lemma' for local
fields, which may be of independent interest.Comment: 13 pages, numerous changes since v3 including those recommended by
referee's report. To appear in Proceedings of the AM
Construction of self-dual normal bases and their complexity
Recent work of Pickett has given a construction of self-dual normal bases for
extensions of finite fields, whenever they exist. In this article we present
these results in an explicit and constructive manner and apply them, through
computer search, to identify the lowest complexity of self-dual normal bases
for extensions of low degree. Comparisons to similar searches amongst normal
bases show that the lowest complexity is often achieved from a self-dual normal
basis
The determination of integral closures and geometric applications
We express explicitly the integral closures of some ring extensions; this is
done for all Bring-Jerrard extensions of any degree as well as for all general
extensions of degree < 6; so far such an explicit expression is known only for
degree < 4 extensions. As a geometric application we present explicitly the
structure sheaf of every Bring-Jerrard covering space in terms of coefficients
of the equation defining the covering; in particular, we show that a degree-3
morphism f : Y --> X is quasi-etale if and only if the first Chern class of the
sheaf f_*(O_Y) is trivial (details in Theorem 5.3). We also try to get a
geometric Galoisness criterion for an arbitrary degree-n finite morphism; this
is successfully done when n = 3 and less satifactorily done when n = 5.Comment: Advances in Mathematics, to appear (no changes, just add this info
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