Abstract
We aim at analyzing a Prüfer extension A ⊂ R in terms of the set S(R∕A) of nontrivial PM-valuations v on R over A (i.e. with A ⊂ A v ), called the restricted PM-spectrum of R over A, in order to understand the lattice of overrings of A in R. We engage S(R∕A) as partially ordered set (v ≤ w iff A v ⊂ A w ), although it would be more comprehensive to view S(R∕A) as a topological space, namely as subspace of the valuation spectrum Spv(R), equipped with one of its well established spectral topologies, whose specialization relation restricts to the partial ordering above. We exhibit several types of Prüfer extensions, where the poset viewpoint has seizable success.
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Notes
- 1.
This means Proposition 5.9 in Chap. 1. If we cite this result in its own section we just write Proposition 9. If we cite this result in another chapter we write Proposition 1.5.9.
- 2.
Formally this definition makes sense for an arbitrary ring extension, but the set Z(B∕A) will be useful only in the case that A is ws in B.
- 3.
There are various topologies on this set which give spectral spaces relevant for applications. Here Spv(R) means the same as in [HK].
- 4.
Recall [Vol. I, Definition 3 in II §7].
- 5.
Recall the notations Q(A), M(A), P(A) from [Vol. I, Chap. I §4 & §5].
- 6.
The letter P stands for “Prüfer”, CR stands for “completely reducible”.
- 7.
Recall that P(A) denotes the Prüfer hull of A, cf. [Vol. I, Definition 3 in I §5].
- 8.
Recall that ω(R∕A) denotes the set of minimal elements in the poset S(R∕A).
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Knebusch, M., Kaiser, T. (2014). Overrings and PM-Spectra. In: Manis Valuations and Prüfer Extensions II. Lecture Notes in Mathematics, vol 2103. Springer, Cham. https://doi.org/10.1007/978-3-319-03212-2_1
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DOI: https://doi.org/10.1007/978-3-319-03212-2_1
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