Some new approaches to integer-valued polynomial ...

Commutative Algebra and Applications, 1–15

© de Gruyter 2009

Some new approaches to integer-valued polynomial rings Jesse Elliott Abstract. We present some new results on and approaches to integer-valued polynomial rings. One of our results is that, for any PvMD D, the domain Int(D) of integer-valued polynomials on D is locally free as a D-module if Int(Dp ) = Int(D)p for every prime ideal p of D. This fact allows us in particular to strengthen the main results of J. Algebra 318 (2007), 68–92, to prove, for example, that the multivariable integer-valued polynomial ring Int(Dn ) decomposes as the n-th tensor power of Int(D) over D for any such PvMD D. We also present a survey of some new techniques for studying integer-valued polynomial rings—such as universal properties, tensor product decompositions, pullback constructions, and Bhargava rings—that may prove useful. Keywords. Integer-valued polynomials, PvMD, Krull domain, domain extensions. AMS classification. 13F20, 13F05, 13G05, 13B02.

1 Introduction Because they possess a rich theory and provide an excellent source of examples and counterexamples, integer-valued polynomial rings have attained some prominence in the theory of non-Noetherian commutative rings. As with ordinary polynomial rings, much of commutative algebra has some bearing on the subject. Today there remain many open problems concerning, for example, their module structure, ideal structure, prime spectrum, and Krull dimension. Their definition is simple: for any integral domain D with quotient field K , any set X , and any subset E of K X , the ring of integer-valued polynomials on E is the subring Int(E, D) = {f (X ) ∈ K [X ] : f (E ) ⊂ D} of the polynomial ring K [X ]. As is standard in the literature, we write Int(DX ) = Int(DX , D) and Int(D) = Int(D, D). We also write Int(Dn ) = Int(DX ) if X is a set of cardinality n. Historically, many of the results currently known about integer-valued polynomial rings were first proved for number rings and later generalized to larger classes of domains. Research on integer-valued polynomial rings began with some challenging questions surrounding their module structure, and in particular with the search for module bases for them. In more recent years it was discovered that integer-valued polynomial rings also have intricate ideal structures and prime spectra, and questions surrounding their ring-theoretic properties were formulated. From this research several natural settings in which to study integer-valued polynomial rings have emerged.

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In particular, they have been studied in connection with Dedekind domains, almost Dedekind and Prüfer domains, Noetherian domains, Krull domains, Pv MD’s, and Mori domains, among many other classes of domains. It is a general thesis of this article that insight into integer-valued polynomial rings may be gained not only from a moduletheoretic or ring-theoretic point of view, but also from a category-theoretic viewpoint. For example, at the most basic level they can be characterized via universal properties, and these universal properties motivate questions concerning integer-valued polynomial rings that have not been considered before. We begin in Section 2 by examining a condition first studied in [11], and later studied in [8] and given the name polynomial regularity. This condition generalizes another important condition, namely, the condition Int(S −1 D) = S −1 Int(D), appearing throughout the literature. In Section 3 we examine int primes, as defined in [5], and their relation to t-maximal ideals. We prove that the int primes of an arbitrary H-domain (as defined in [12]) are precisely its prime conductor ideals having finite residue field. We also show that, for any Krull domain D, or more generally for any Pv MD D such that Int(Dp ) = Int(D)p for every prime ideal p of D, the domain Int(D) of integer-valued polynomials on D is locally free as a D-module. This fact allows us in particular to strengthen several of the main results in [8], including [8, Theorem 1.3]. It also represents some progress toward a classification of those domains D such that Int(D) is flat as a D-module. Moreover, it allows us to prove in Section 4 that the multivariable integer-valued polynomial ring Int(Dn ) decomposes as the n-th tensor power of Int(D) over D for any such Pv MD D. The tensor product decomposition offers a new approach to studying integer-valued polynomial rings and is discussed further in Section 4. In Section 5 we motivate some universal characterizations of integer-valued polynomial rings. One of the main ideas is this. In contrast to characterizing the polynomially dense subsets of a given domain, we consider the problem of characterizing those domains that contain a given domain as a polynomially dense subset. We say that an extension A of a domain D is polynomially complete if D is a polynomially dense subset of A, that is, if every polynomial with coefficients in the quotient field of A mapping D into A actually maps all of A into A. It turns out that, for any set X , the domain Int(DX ) is the free polynomially complete extension of D generated by the set X , provided only that D is not a finite field [8, Proposition 2.4]. In Section 6 we present some connections to A + XB [X ] and other pullback constructions and to Bhargava rings. Finally, in the last section, we define some important D-module lattices contained in Int(D) and exhibit their relation to the factorial ideals n!D E , as defined in [6, Definition 1.2]. The main results in these last two sections are at this stage somewhat philosophical in nature, although experts in ring-theoretic pullback constructions or module lattices who are also interested in the study of integer-valued polynomial rings may find these approaches useful.

Some new approaches to integer-valued polynomial rings

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2 The condition Int(D, A) = AInt(D) The condition Int(S −1 D) = S −1 Int(D) for a multiplicative subset S of D is wellknown to be extremely important to the study of integer-valued polynomial rings. If D is Noetherian or even Mori then this condition is known to hold for all multiplicative subsets S of D [4, Proposition 2.1], but even for almost Dedekind domains the condition is rather subtle, as seen for example by [17, Theorem 2.4]. The condition Int(S −1 D) = S −1 Int(D) can be subsumed under another condition that appears in the literature. As in [8, Section 3], we say that an extension A of a domain D, by which we mean a domain containing D together with its Dalgebra structure, is polynomially regular if Int(D, A) = AInt(D), where AInt(D) denotes the A-module (or, equivalently, the A-algebra) generated by Int(D). Since Int(D, S −1 D) = Int(S −1 D) by [3, Corollary I.2.6], it follows that Int(S −1 D) = S −1 Int(D) if and only if S −1 D is a polynomially regular extension of D. Thus the condition Int(S −1 D) = S −1 Int(D) can be subsumed under the polynomial regularity condition. The condition of polynomial regularity appears in the literature as early as 1993 in an important paper by Gerboud [11], although the condition is not given a name there. There, in another guise, appears the following result, whose proof is trivial. Proposition 2.1. For any polynomially regular extension A of a domain D, the following conditions are equivalent. (1) D is a polynomially dense subset of A, that is, Int(D, A) = Int(A). (2) Int(A) is equal to the A-module generated by Int(D). (3) Int(A) ⊃ Int(D). Note that, for an extension A of a domain D that is not necessarily polynomially regular, conditions (1) and (2) each imply condition (3), but (1) need not imply (2) and (2) need not imply (1). For example, if D is a domain containing a multiplicative subset S for which Int(S −1 D) 6= S −1 Int(D), then A = S −1 D is an extension of D satisfying (1) but not (2). Conversely, the extension A = Z[T /2] of D = Z[T ] satisfies condition (2) but not condition (1), by [8, Example 7.3]. The following is another result from [11]. Theorem 2.2. Let D be a Dedekind domain. Then any extension of D is polynomially regular, and for any extension A of D, the domain D is a polynomially dense subset of A if and only if the extension A of D has trivial residue field extensions, and is unramified, at every nonzero prime ideal of D with finite residue field. The latter of the two equivalent conditions of the theorem is to be understood as follows: for every nonzero prime ideal p of D with finite residue field and every prime ideal P of A lying over p one has A/P = D/p and pAP = PAP . The result above was recently extended to flat extensions of Krull domains [8, Theorem 1.3]. In particular, it was shown that every flat extension of a Krull domain is polynomially regular. In the next section we will prove a stronger version of that theorem, Theorem 3.8.

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3 Int primes and the condition Int(D) ⊂ S −1 D[X] Another condition important for studying integer-valued polynomial rings is the condition Int(D) ⊂ S −1 D[X ], or equivalently S −1 Int(D) = S −1 D[X ], for multiplicative subsets S of D. Recall that a conductor ideal of D is an ideal of the form (aD :D bD) for some a, b ∈ D with a 6= 0. Define dn (X ) ∈ Z[X ] = Z[X0 , X1 , . . . , Xn ] by Y dn (X ) = (Xi − Xj ). 0≤j