TITEL: Localizations over the Steenrod Algebra. The lost Chapter
AUTHOR: Mara D. Neusel
EMAIL: mdn@sunrise.uni-math.gwdg.de
mneusel@cfgauss.uni-math.gwdg.de
neusel@math.umn.edu
maramara@steenrod.mast.queensu.ca
AMS CODE: 55S10 Steenrod Algebra, 13BXX Ring Extensions and Related Topics,
55XX Algebraic Topology, 13XX Commutative Rings and Algebras
KEY WORDS: Steenrod Algebra, Unstable Algebras over the Steenrod Algebras,
Unstable Part, Localizations, Noetherianess, Integral Closure,
Dickson Algebra
ABSTRACT:
Let H be an unstable algebra over the Steenrod algebra, and let
S\subset \H be a multiplicatively closed subset. The localization
at S, i.e. S^{-1}H, inherits an action of the Steenrod algebra
from H, which is, however, in general no longer unstable. In this note
we consider the following three statements.
(1) H is Noetherian,
(2) the integral closure, \overline{H_{S^{-1}H}},
of H in the localization with respect to S
is Noetherian,
(3) \overline{H_{S^{-1}H}}= Un(S^{-1}H).
where Un(-) denotes the unstable part.
If the set S contains only (nonzero) non zero divisors and the algebras
are reduced then
(1) is equivalent to (2).
If S contains zero divisors, then only (1) \Rightarrow (2)
remains true, to show the converse is false we construct a counter
example.
The implication (2) \Rightarrow (3) is always true, while
its converse (3) \Rightarrow (2) needs a weird bunch of technical
assumptions to remain true. However, none of them can be removed:
we illustrate this also with examples.
Finally, as a technical tool, we characterize
Delta-finite algebras.