Asymptotic properties of infinite directed unions of local quadratic transforms

Directed Unions of Local Quadratic Transforms of Regular Local Rings and Pullbacks

“…These observations raise the question of the ideal-theoretic structure of a quadratic Shannon extension of a regular local ring R with dim R > 2, a question that was taken up in [21] and [22]. In this section we recall some of the results from [21] and [22] with special emphasis on non-archimedean quadratic Shannon extensions, a class of Shannon extensions that we classify in Sections 3 and 4.…”

“…The boundary valuation ring is given by a valuation from the nonzero elements of the quotient field of R to a totally ordered abelian group of rank at most 2 [22, Theorem 6.4 and Corollary 8.6]. In [22] the following two mappings on the quotient field of R are introduced as invariants of a quadratic Shannon extension. The first, e, takes values in Z ∪ {∞}, while the second, w, takes values in R ∪ {−∞, +∞}.…”

“…The first, e, takes values in Z ∪ {∞}, while the second, w, takes values in R ∪ {−∞, +∞}. Both e and w are used in [22] to decompose the boundary valuation v of the quadratic Shannon extension into a function that takes its values in R ⊕ R with the lexicographic ordering. The function e is defined in terms of the transform (aR n ) R n+i of a principal ideal aR n in R n+i for i > n; see [25] for the general definition of the (weak) transform of an ideal and [21] for more on the properties of the transform in our setting.…”

“…Moving beyond dimension two, examples due to David Shannon [30, Examples 4.7 and 4.17] show that the union S = i R i of an iterated sequence of local quadratic transforms of a regular local ring of Krull dimension > 2 need not be a valuation ring. The recent articles [21,22] address the structure of such rings and how this structure encodes asymptotic properties of the sequence {R i } ∞ i=0 . We call S a quadratic Shannon extension of R. In general, a quadratic Shannon extension need not be a valuation ring nor a Noetherian ring, although it is always an intersection of two such rings (see Theorem 2.2).…”

“…A quadratic Shannon extension S is nonarchimedean if there is an element x in the maximal ideal of S such that i>0 x i S = 0. The class of non-archimedean quadratic Shannon extensions is analyzed in detail in [21] and [22]. We carry this analysis further in the present article by using techniques from multiplicative ideal theory to classify a non-archimedean quadratic Shannon extension as the pullback of a valuation ring of rational rank one along a homomorphism from a regular local ring onto its residue field.…”