## Orders and signatures of higher level on a commutative ring

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##### Date

1994##### Author

Walter, Leslie J.

##### Type

Thesis##### Degree Level

Doctoral##### Abstract

One obtains orders of higher level in a commutative ring A by pulling back the higher
level orders in the residue fields of its prime ideals. Since inclusion relationships can
hold amongst the higher level orders in a field (unlike the level 1 situation), there
may exist orders in the ring A which are not contained in a unique order maximal
with respect to inclusion. However, if the specializations of an order P are defined
to be those orders Çª âŠ‡ P such that Çª P âŠ† Çª âˆ© -Çª, every higher level order in A is contained in a unique maximal specialization. The real spectrum of A relative to a higher level preorder T is defined to be the set SperTA of all orders in A containing T. As with the ordinary real spectrum of Coste and Roy, SperTA is given a compact topology in which the closed points are precisely the orders in A maximal with respect to specialization. For 2-primary level, we show that an
abstract higher level version of the Hormander-Å ojasiewicz Inequality holds and
use it to characterize the basic sets in SperTA.
A higher level signature on a commutative ring A is a pull-back Ïƒ of a higher
level signature on the residue field of some prime ideal Ã¾ with Ïƒ(Ã¾) = 0. If T
is a higher level preorder in A and Ïƒ(T) = {0, 1} then Ïƒ is called a T-signature.
Specializations of T-signatures are defined just as for orders and every T-signature
is shown to have a unique maximal specialization. Each T-signature a determines
a unique order in A containing T which is maximal with respect to specialization iff
Ïƒ is. Generalizing a result of M. Marshall, we show for a higher level preorder T in
a commutative ring satisfying a certain simple axiom, the space XT of all maximal
T-signatures can be embedded in the character group of a suitable abelian group
GT of finite even exponent and under this embedding, the pair (XT, GT) is a space
of signatures in the sense of Mulcahy and Marshall.