MORAVA K-THEORIES AND LOCALISATION 15
2.3. Operations in E theory. We now turn to the study of operations in E-
theory and iT-theory. There is a well-known connection between this and the study
of the Morava stabiliser group, but no really adequate account of this. For this and
a variety of technical reasons we have chosen to use a more traditional approach.
Write 2* = E*E for the (non-commutative) ring of operations in JS-cohomology.
Note that K*E = Kom(K*E, K*) (by Proposition 2.16 or by the general theory of
modules over afield spectrum) and K*E =
K*BP®BP*E*
because E* is Landweber
exact. Because In is an invariant ideal, it is easy to check that K*E is the same as
the ring E* = E(n)* studied in [Rav86, Chapter VI]. We thus have
K*E = X*=K4tk\k 0]/(tf - v£-Hk).
Here |£&| =
2(pk
1), so K*E and E* = K*E are in even degrees. It follows from
Proposition 2.5 that E* = E*E is pro-free, and that E* = E*//
n
E*. Moreover,
because In is an invariant ideal in BP* one can check that I
n
S* = E*In, and thus
that E* is a quotient ring of E*.
Our main result is as follows.
Theorem 2.24. The ring E* is left Noetherian in the graded sense.
This is proved after Proposition 2.28, using Hopf algebras. Another possibility
would be to make the connection with the Morava stabiliser group and Lazard's
work on profinite group theory. In some respects this would be more conceptual,
but it introduces additional technicalities that we have preferred to avoid.
As the theory of non-commutative Noetherian rings is less familiar than the
commutative version, we start with some elementary remarks. Let R be a possibly
non-commutative ring. Unless otherwise specified, we shall take "ideal" to mean
"left ideal" and "module" to mean "left module". As in the commutative case, one
checks easily that the following are equivalent:
(a) Every ideal J R is finitely generated.
(b) Every ascending chain JQ J\ ... of ideals is eventually constant.
(c) Any submodule of a finitely generated module over R is finitely generated.
(d) Every ascending chain of submodules of a finitely generated module over R
is eventually constant.
If so, we say that R is (left) Noetherian. If R is a graded ring and all ideals and
modules are required to be homogeneous, then the corresponding conditions are
again equivalent; if they hold, we say that R is Noetherian in the graded sense.
Lemma 2.25. Let R—*Sbea map of rings such that S is finitely generated and
free both as a left R-module and a right R-module. Then S is left Noetherian if and
only if R is left Noetherian. Similarly for the graded case.
Proof. Suppose that R is Noetherian. Then any ascending chain of ideals in S
is a chain of iJ-submodules of a finitely generated i?-module, and thus eventually
constant.
Conversely, suppose that S is Noetherian, and that S = (B^ aiR. Then the
map
Jt-SJ = @aiJ~($J
i i
embeds the lattice of ideals in R into the Noetherian lattice of ideals in S, so R is
Noetherian.
Previous Page Next Page