Localizations of integer-valued polynomials and of their Picard group

We prove a necessary and sufficient criterion for the ring of integer-valued polynomials to behave well under localization. Then, we study how the Picard group of $\mathrm{Int}(D)$ and the quotient group $\mathcal{P}(D):=\mathrm{Pic}(\mathrm{Int}(D))/\mathrm{Pic}(D)$ behave in relation to Jaffard, weak Jaffard and pre-Jaffard families; in particular, we show that $\mathcal{P}(D)\simeq\bigoplus\mathcal{P}(T)$ when $T$ ranges in a Jaffard family of $D$, and study when similar isomorphisms hold when $T$ ranges in a pre-Jaffard family. In particular, we show that the previous isomorphism holds when $D$ is an almost Dedekind domain such that the ring integer-valued polynomials behave well under localization and such that the maximal space of $D$ is scattered with respect to the inverse topology.


Introduction
Let D be an integral domain with quotient field K.A polynomial f (X) ∈ K[X] is integer-valued over D if f (d) ∈ D for every d ∈ D; the set of all integer-valued polynomials is a ring, denoted by Int(D).The ring of integer-valued polynomials presents several properties that makes it a very interesting subject of study: for example, it is a simple example of a construction that does not involve limits, infinite families of indeterminates, or intersections of complicated families of rings, and that rather consistently produces rings that are non-Noetherian, even starting from a Noetherian ring.Furthermore, this construction can be tailored to several topics (for example, considering polynomials that are integer-valued only on a subset) in order to obtain examples of phenomenon that are difficult to obtain with other constructions.We refer the reader to the book [2] for background and results about integer-valued polynomials.
One particular problem of the theory of integer-valued polynomials is its relationship with localization: given a domain D and a multiplicatively closed set, under what hypothesis the equality S −1 Int(D) = Int(S −1 D) holds?Several special cases have been proved (see e.g.[2, Section 1.2], [3, Proposition 2.1], [8]); we give in Section 3 a necessary and sufficient criterion for this to happen, involving the conductor (D : f (D)), and show how the known criterion descend from ours.We also deal not only with localizations but, more generally, with flat overrings of the base domain D.
We then concentrate on generalizing globalization properties for the Picard group Pic(Int(D)) of Int(D).Unless Int(D) is trivial (i.e., unless Int(D) = D[X]), the Picard group of Int(D) is usually much larger than the Picard goup of D, and can be calculated only in very special circumstances (for example, for discrete valuation domains and for some kinds of one-dimensional Noetherian local domains [2,Chapter 6]).To obtain a description of Pic(Int(D)) in more cases, the main tool is globalization: for example, when D is a one-dimensional Noetherian domain, there is always an exact sequence Pic(Int(D M )) −→ 0, which allows at least to understand the main features of Pic(Int(D)).In this context, our first result (given in two different forms in Theorems 4.4 and 4.7) gives a generalization of the previous exact sequence, proving that a similar result holds if, instead of the family {D M | M ∈ Max(D)}, one takes a Jaffard family of D, a particular kind of family of flat overrings with strong independence properties (see Section 2.2 for a precise definition).The result becomes more striking using the int-polynomial Picard group P(D), defined as the quotient between Pic(Int(D)) and the image of the canonical inclusion of Pic(D): in this terminology, the theorem guarantees that P(D) and the direct sum {P(T ) | T ∈ Θ} are isomorphic for any Jaffard family Θ.
In Sections 6 and 7, we further generalize this result by considering weak Jaffard families and pre-Jaffard families, that are obtained by relaxing the conditions defining a Jaffard family.In the former case, we obtain in Theorem 6.4 an exact sequence 0 −→ T ∈Θ T =T∞ P(T ) −→ P(D) −→ P(D, T ∞ ) −→ 0 (see below for the definition of T ∞ and P(D, T ∞ )).For pre-Jaffard families, we use the result on weak Jaffard families to set up a transfinite inductive reasoning (which uses the derived sequence of the pre-Jaffard family, see Section 2.2) that allows to prove, under some additional hypothesis, the existence of an exact sequence 0 −→ T ∈Θ\Tα P(T ) −→ P(D) −→ P(D, T α ) −→ 0 (see below for the definition of T α ).In particular, when also the pre-Jaffard family is sharp, one obtain an isomorphism P(D) ≃ {P(T ) | T ∈ Θ}, just like in the case of Jaffard families.In particular, such an isomorphism holds when D is an almost Dedekind domain such that Int(D) behaves well under localization and such that the maximal space of D is scattered in the inverse topology (Corollary 7.6).

Preliminaries
Throughout the paper, D is an integral domain with quotient field K.
An overring of D is a ring contained between D and K; we denote by Over(D) the set of all overrings of D. A flat overring is an overring that is flat as a D-module; in particular, every localization and every quotient ring of D is a flat overring.If T is a flat overring of D, then for every prime ideal P of T we have T P = D P ∩D ; in particular, every flat overring is an intersection of localizations of D, and every (prime) ideal of T is the extension of a (prime) ideal of D [1,18].
Let I be a D-submodule of K and A ⊆ K.The conductor of A in I is (I : A) := {x ∈ K | xA ⊆ I}; moreover, (I : A) = (I : AD), where we denote by AD the D-submodule generated by A. The conductor is always a D-submodule of K, and can be (0).If T is a flat overring of D and J is a finitely generated D-module, then (I : J)T = (IT : JT ) [15,Theorem 7.4].
A fractional ideal of D is a D-submodule I of K such that (D : I) = (0), i.e. such that xI ⊆ D for some nonzero x ∈ K.A fractional ideal I is invertible if there is a fractional ideal J such that IJ = D; equivalently, I is invertible if it is finitely generated and locally principal (i.e., ID M is principal for every M ∈ Max(D)).The set of invertible ideal is an abelian group, denoted by Inv(D), having as a subgroup the set Princ(D) of principal fractional ideals of D; the quotient Inv(D)/Princ(D) is called the Picard group of D, and is denoted by Pic(D).
2.1.Topologies.Let D be an integral domain.The spectrum Spec(D) of D can be endowed, in addition the usual Zariski topology, with another topology, called the inverse topology.The inverse topology is defined as the topology having, as a subbasis of open sets, the closed sets of the Zariski topology.Under the inverse topology, the spectrum is still a compact T 1 space.
The set Over(D) of the overrings of D can be endowed with a natural topology, called the Zariski topology, whose subbasic open sets are the ones in the form as x 1 , . . ., x n varies in K.The Zariski topology on Over(D) is intimately connected with the Zariski topology on the spectrum Spec(D) of D: for example, the localization map P → D P is a topological embedding when Spec(D) and Over(D) are endowed with the respective Zariski topologies [7,Lemma 2.4].Moreover, the Zariski topology has several good properties: for example, it is a spectral space, in the sense that there is a ring R (not explicitly constructed) such that Spec(R) ≃ Over(D) [9,Proposition 3.5].
The inverse topology on Over(D) is the topology such that the B(x 1 , . . ., x n ) are a subbasis of closed sets.This topology is closely connected with the properties of representations of D as intersection of overrings (see e.g.[16]).Properties of the inverse topology, in the context of spectral spaces, can be found in [6].
We shall use many times the following result [10, Corollary 5]: if Θ ⊆ Over(D) is compact, with respect to the Zariski topology, and if I is a flat D-submodule of K, then 2.2.Jaffard and pre-Jaffard families.D be an integral domain with quotient field K.We say that a subset Θ ⊆ Over(D) is a pre-Jaffard family of D if the following conditions hold [21]: In particular, if Θ is a pre-Jaffard family and P is a nonzero prime ideal of D, then there is exactly one T ∈ Θ such that P T = T .
A family Θ of overrings of D is locally finite if every nonzero x ∈ D is a nonunit in only finitely many elements of Θ; if Θ = {D M | M ∈ Max(D)} is locally finite, we say that D itself is locally finite.Any locally finite family of overrings is compact, with respect to the Zariski topology (see e.g. the proof of [10,Corollary 8]).
A Jaffard family is a pre-Jaffard family that is locally finite.Jaffard families enjoy several good factorization properties that make them a non-local generalization of h-local domains and thus of Dedekind domains; see for example [11,Section 6.3], [19,Section 4] or [20].
We say that an overring T of D is a Jaffard overring if T belongs to a Jaffard family of D. Given a Jaffard family Θ of D, we can construct a well-ordered decreasing chain {N α (D)} of subsets of Θ and a corresponding ascending chain {T α } of overrings of D in the following way.Given an ordinal number α, we set: [21, Section 6] • if α = 0, N 0 (Θ) := Θ and T 0 := D; T .
• if α = γ + 1 is a limit ordinal, then N α (D) is the set of all elements of N γ (D) that are not Jaffard overrings of T γ ; Note that, in [21], the set N α (D) was denoted simply by Θ α .Each N α (D) is a pre-Jaffard family of T α (in particular, it is compact with respect to the Zariski topology) [21, Proposition 6.1], and it is a closed subset of Θ, with respect to the inverse topology.We call {T α } the derived sequence of Θ.
If N 1 (Θ) is a single element T ∞ , we say that Θ is a weak Jaffard family pointed at T ∞ .Weak Jaffard families are usually the stepping stones in inductive arguments used to generalize properties of Jaffard families to pre-Jaffard families.

2.3.
Homology.We shall frequently use a basic results of homological algebra, the snake lemma: is a commutative diagram of abelian groups (or, more generally, of modules over a ring R) with exact rows, then the sequence

When integer-valued polynomials localize
In this section, we find a necessary and sufficient criterion for Int(D) to localize at a flat overring, i.e., for when the equality Int(D)T = Int(T ) holds.Before doing so, we introduce a notion that generalizes Jaffard families and Jaffard overrings.Definition 3.1.Let D be an integral domain and Θ ⊆ Over(D).We say that Θ is a t-Jaffard family of D if: We say that an overring T of D is a t-Jaffard overring if it belongs to a t-Jaffard family of D.
Note that, in particular, every Jaffard family is a t-Jaffard family, and thus every Jaffard overring is a t-Jaffard overring.The converse does not hold: for example, if D is a Krull domain, then the family of localizations at its prime ideals of height 1 is a t-Jaffard family, but it is not a Jaffard family unless D has dimension 1.
The following proposition can be seen as a variant of [19, Lemma Joining the two characterizations, we have our criterion.Proof.It is enough to apply the two conditions of Proposition 3.4.
As a consequence, we get back several known results about the possibility of localizing the ring Int(D).Recall that a Mori domain is a domain whose divisorial ideals satisfy the ascending chain condition.We end this section with a result which will be useful alter.

The Picard group
When Int(D) is nontrival, a direct calculation of its Picard group can only be done under very special circumstances, for example when D is a discrete valuation ring or an analytically irreducible one-dimensional domain [2, Proposition VIII.2.8 and Corollary VIII.3.10].To reach more cases, the main tool is globalization: for example, if D is onedimensional Noetherian domain, then there is a short exact sequence [2, Theorem VIII.1.9] In this section, we begin to extend the use of this kind of exact sequence by considering the case of Jaffard families.
for some arbitrary family Θ of flat overrings.Proof.Since T is flat, it is the colimit of a directed set {M i } of free D-modules; since each of these is contained in the quotient field of D, there are x i such that  We now prove the first theorem of this section.While very similar to the localization result for Dedekind domains, Theorem 4.4 includes in its statement the group Pic(D, Θ), which may not be easy to calculate.In the next theorem, we trade its presence with the one of the Picard groups Pic(T ); we first show how they are related.T ∈Θ Pic(Int(T )) 0 The first row is exact by Proposition 4.5, while the second one from Theorem 4.4; on the other hand, the leftmost vertical map is the identity and the other two vertical maps are injective.By the snake lemma, there is an exact sequence 0 −→ coker ι D −→ coker ι Θ −→ 0. By definition, coker ι D is just P(D), while coker ι Θ is the direct sum P(T ), and thus we have the isomorphism.The sequence (which is exact by definition) 0 −→ Pic(D) −→ Pic(Int(D)) −→ P(D) −→ 0 then becomes the one in the statement by substituting P(D) with the direct sum.
A domain is h-local if every nonzero ideal is contained in only finitely many maximal ideals and every nonzero prime ideal is contained in only one maximal ideal.The previous theorems immediately give the following.Proof.We first note that, if D is one-dimensional and locally finite, then D is h-local; likewise, if D is one-dimensional and Noetherian, then it is locally finite.Hence it is enough to prove the claim for D since each subset of Max(D) is compact.Hence, Θ is independent and thus a Jaffard family.By Theorem 4.4 there is an exact sequence Let P be a maximal ideal of T .Then, T P is a valuation domain of dimension strictly greater than 1, and thus by [2, Proposition I. is trivial.Hence, the sequence in the statement is exact.
To show that it is split, it is enough to note that Pic(Int(D M )) is always a free group (if D M is not discrete since in that case Pic(Int(D M )) = Pic(D M [X]) = (0), if D M is discrete by [5,Proposition 7.7]).The isomorphism follows.

The surjectivity of the extension map
A consequence of Theorem 4.4 (or rather, of its proof) is that when T is a Jaffard overring then the extension map Pic(Int(D)) −→ Pic(Int(D)T ) = Pic(Int(T )) is surjective.This property is in general not true, not even for an extension map Pic(D) −→ Pic(T ) where D ⊆ T is a flat extension: D may be a local ring (so Pic(D) is trivial), while the Picard group of a flat overring may not be trivial.Moreover, even if the surjectivity hold, it need not to pass to integer-valued polynomials: we will give in Example 6.5 below an example where Pic(D) −→ Pic(T ) is surjective, while Pic(Int(D)) −→ Pic(Int(T )) is not.In this section, we collect some sufficient conditions for this surjectivity to hold, which will be useful later, as well as a direct application to the calculation of Pic(Int(D)) for one-dimensional domains.Proof.We first note that, for every finite subset A ⊆ T , there is an S ∈ L containing A: indeed, each a ∈ A is contained in some S a ∈ L, and since L is a sublattice of Over(D), there is an S ∈ L containing all S a and thus all of A.
Let I := (x 1 , . . ., x n ) be an invertible ideal of T , and let J := (y 1 , . . ., y m ) be its inverse.Then, x i y j ∈ T for every i, j, and there are r ij ∈ T such that 1 = i,j r ij x i y j .Therefore, there is an S ∈ L that contains all x i , all x i y j and all r ij .
. ., t n , then h ∈ Int(D)S.Hence, we can apply Lemma 5.1 to L 1 .
We shall apply this criteria in Propositions 6.1 and 7.2 below; we conclude this section by showing that for one-dimensional Prüfer domains we can exclude some maximal ideals with infinite residue field while controlling the change in the Picard group.By construction, a ∈ D Q for every Q ∈ Θ 2 , and thus a ∈ S 2 .Moreover, if P is a maximal ideal of D such that P T = T , then a ∈ D P , and thus P ∈ Θ 2 ; hence S 2 ⊆ T .It follows that S 2 ∈ L, and thus a belongs to the union of the elements of L. Since a was arbitrary, T is equal to the union, and we can apply Lemma 5.2.Proposition 5.4.Let D be a one-dimensional Prüfer domain, let Then, there is an exact sequence In particular, P(D) ≃ P(D, T ).
Proof.Let X be the closure of X in Max(D), with respect to the inverse topology.Then, X is a closed set of Spec(D), with respect to the inverse topology, and thus it is compact in the Zariski topology; hence, also , and thus local and integrally closed.Hence, the direct product in the previous sequence vanishes.Moreover, ID N is principal for every invertible ideal I of D; hence, Pic(D, Θ \ X) = Pic(D), and Pic(D, Θ) = Pic(D, T ).Thus, the exact sequence becomes To conclude, we note that the rightmost map of the sequence is the extension map, which is surjective by Lemma 5.3.Hence, the sequence of the statement is exact.
To prove the isomorphism, we apply the same method of Theorem 4. The rows are exact (by definition and by the first part of the proof), while the vertical maps are injective (and the leftmost one is the identity).By the snake lemma, the cokernels of the other two vertical maps are isomorphic; since they are, respectively, P(D) and P(D, T ), the claim is proved.

Weak Jaffard families
We now start to study how to extend Theorem 4.4 towards weak Jaffard and pre-Jaffard families.In these cases, we have two problems: first, the equality Int(D)T = Int(T ) may not hold (see Example 6.5 below); second, the cokernel of the map Pic(D) −→ Pic(Int(D)) cannot reduce to the direct sum, and in general it may be difficult to actually determine it inside the direct product of the various Pic(Int(T )) or Pic(Int(D)T ).The first problem cannot be resolved with our methods, and, for the most part, we will have to use the equality Int(D)T = Int(T ) as an additional hypothesis; to solve the second problem, on the other hand, our strategy will be to write the cokernel as the middle element of some other exact sequences, using this knowledge to write exact sequences involving the int-polynomial Picard groups.
We study in this section the case of weak Jaffard families, which will then be used as an inductive step in the next section (where we will deal with pre-Jaffard families).We first show that the direct sum belongs to the kernel, and to do so we need to show that it is actually inside ∆.If To conclude, we need to show that the map ∆ −→ Int(D)T ∞ is surjective.However, this map factorizes the extension map Int(D) −→ Int(D)T ∞ , which is surjective by Proposition 6.1, and thus it is surjective itself.The claim is proved.
We now transform this result using int-polynomial Picard groups; the following lemma has the same role of Proposition 4.5.Lemma 6.3.Let Θ be a weak Jaffard family pointed at T ∞ .Then, there is an exact sequence • W n extends to W n+1 and Z n+1,n+1 in L n+1 ; • for i = 1, . . ., n, Z i,n has a unique extension to L n+1 , namely Z i,n+1 ; • V ⊂ W n is an immediate extension; • for each i, the extension V ⊂ Z i,n is trivial on value groups, while the extension of residue fields has degree at least n.
Let L := n L n .Then, the integral closure V of V in L is a onedimensional Prüfer domain whose localization at the maximal ideals are the extensions of V to L, namely W ∞ := n W n and, for each In particular, V is equal to the intersection of all W i,∞ , and Θ := {W ∞ , Z i,∞ | i ∈ N} is a weak Jaffard family of V .
The residue field of each Z i,∞ is infinite, and thus Int(Z i,∞ ) is trivial; therefore, also Int(V ) is trivial, and thus Int(V )W ∞ = W ∞ [X].However, W ∞ is a DVR with finite residue field, and thus Int(W ∞ ) is not trivial [2, Proposition I. 3.16]; it follows that Int(V )W ∞ = Int(W ∞ ).
7. Pre-Jaffard families Proposition 6.2 is, in some ways, the best result that is possible to obtain without adding more hypothesis.However, if Int(D)T ∞ = Int(T ∞ ) (something that need not to happen, see Example 6.5), then one may in principle repeat the process by taking a weak Jaffard family Θ ′ of T ∞ and apply the same result; hopefully, this can lead to a finer description of ∆ and thus of Pic(Int(D)) and P(D).The purpose of this section is to systematize this idea by using the notions of pre-Jaffard family and of its derived sequence (see Section 2.2); we use throughout the section the notation introduced therein.Lemma 7.1.Let Θ be a pre-Jaffard family of D, and let γ be a limit ordinal.Then, γ<α T γ = T α .
Proof.Let R be the union of T γ , for γ < α.Then, R is the union of a chain of flat overrings, and thus it is itself flat; moreover, R ⊆ T α since T γ ⊆ T α when γ < α.If R = T α , then (since T α is flat too) there should be a nonzero prime ideal P of D such that P R = R and P T α = T α .Since Θ is a Jaffard family, there is a unique T ∈ Θ such that P T = T ; by construction, T / ∈ N α (Θ), and since α is a limit ordinal there is a β < α such that T / ∈ N β (Θ).In this case, we have P T β = T β , and thus P R = R since T β ⊆ R.This is a contradiction, and thus R = T α , as claimed.

Proposition 3 . 2 .
Let T be a t-Jaffard overring of D. Then, for every fractional ideal I of D, we have (D : I)T = (T : IT ).Proof.Let Θ be a t-Jaffard family of D containing T , and letA := {S | S ∈ Θ \ {T }}.Since Θ \ {T } is locally finite, it is compact, and thus, by [10, Corollary 5] I)T = (T ∩ A : I)T = ((T : I) ∩ (A : I))T = (T : I)T ∩ (A : I)T.We have (T : I)T = (T : IT )T = (T : IT ); on the other hand, (A : I) is an A-ideal, and thus (A : I)T = (A : I)AT = K.Hence, (D : I)T = (T : IT ), as claimed.We now go back to studying integer-valued polynomials.The following is a slight generalization of [2, Theorem I.2.1].Proposition 3.3.Let T be a flat overring of D and f 2.1] (since D P is a localization of D), and f (t) ∈ D P since T ⊆ D P (by the flatness of T ).This is a contradiction, and thus f (t) ∈ f (D)T and f (D)T = f (T )T .Proposition 3.4.Let T be a flat overring of D, and let and thus each f (X)d i ∈ Int(D).Hence f (X) ∈ Int(D)T .Conversely, suppose f ∈ Int(D)T .If T = D P for some prime ideal P , then Int(D)D P = Int(D) P and thus there is an s ∈ D \ P such that sf ∈ Int(D); hence s ∈ (D : D f (D)) and (D : D f (D))D P = D P .For the general case, if f ∈ Int(D)T then f ∈ Int(D)D P for all prime ideals P of D such that P T = T (as T is flat, T ⊆ D P for all such P ), and thus (D : D f (D))D P = D P ; the claim now follows from the fact that all maximal ideals of T are extensions of prime ideals of D.(

Theorem 3 . 5 .
Let T be a flat overring of D. Then, Int(D)T = Int(T ) if and only if (D : D f (D))T = (T : T f (D)T ) for every f ∈ K[X].

Proposition 3 . 6 .
Let T be a flat overring of D. Suppose that one of the following conditions hold: (a) T is a Jaffard overring of D; (b) T is a t-Jaffard overring of D; (c) D is one-dimensional and locally finite; (d) [2, Theorem I.2.3]D is Noetherian; (e) [3, Proposition 2.1] D is Mori.Then, we have Int(D)T = Int(T ).Proof.If T is a Jaffard or t-Jaffard overring, then (D : I)T = (T : IT ) for all ideals I, and thus in particular for I = f (D)D.If D is one-dimensional and locally finite, then every flat overring is a Jaffard overring and we are in the previous case.If D is Noetherian, then f (D)D is finitely generated, and thus we can bring the flat overring inside the conductor.If D is Mori, then every ideal is strictly v-finite, and thus there is a finitely generated ideal J ⊆ f (D)D such that (D : f (D)) = (D : J).Hence, (D : f (D))T = (D : J)T = (T : JT ) ⊇ (T : f (D)T ) ⊇ (D : f (D))T and thus (D : f (D))T = (T : f (D)T ).

Proposition 3 . 7 .
Let D be an integral domain, T a flat overring and Λ be a complete family of flat overrings of T .If Int(D)S = Int(S) for every S ∈ Λ, then Int(D)T = Int(T ).Proof.Let f ∈ Int(T ): then, f (T ) ⊆ T ⊆ S for every S ∈ Λ, i.e., f ∈ Int(T, S) = Int(S) = Int(D)S.Hence, f ∈ S∈Λ Int(D)S = S∈Λ Int(D)T S = Int(D)T, since Λ is complete over T .The claim is proved.

Definition 4 . 1 .
Let T be a flat overring of D. Then extension map of Picard groups is the group homomorphism ψ D,T : Pic(D) −→ Pic(T ), [I] −→ [IT ].If Θ is a family of flat overrings of D, the Picard group of D relative to Θ is Pic(D, Θ) := {[I] ∈ Pic(D) | [IT ] = [T ] for all T ∈ Θ}.If Θ = {T } we write Pic(D, T ) := Pic(D, {T }).Since ψ D,T is a group homomorphism, Pic(D, Θ) is always a subgroup of Pic(D) (indeed, it is the intersection of the kernels of the ψ D,T , as T ranges in Θ).When every element of Θ is local, Pic(D, Θ) = Pic(D).The starting point of the globalization results of [2, Chapter VIII] is an extension map from Int(D) to the direct product of Int(D) M = Int(D)D M , as M ranges among the maximal ideals of D [2, Proposition VIII.1.6].Likewise, our study begins by examining the extension map (1)

Lemma 4 . 2 .
Let T be a flat overring of D. Then, Int(D)T is a flat overring of Int(D).
and thus it is flat over Int(D).

Theorem 4 . 4 .
Let Θ be a Jaffard family of D. Then, there is an exact sequence 0 −→ Pic(D, Θ) −→ Pic(Int(D)) −→ T ∈Θ Pic(Int(T )) −→ 0. Proof.We first note that, by Proposition 3.6(a), we have Int(D)T = Int(T ), and thus Pic(Int(D)T ) = Pic(Int(T )).Let ∆ be the image of the extension map π Θ : Pic(Int(D)) −→ {Pic(Int(T )) | T ∈ Θ}.We claim that its image is just the direct sum.Indeed, if [I] ∈ Pic(Int(D)) then by [2, Remark VIII.1.5]we can suppose that I is an integral unitary ideal of Int(D); in particular, I contains a nonzero constant a.Since Θ is a Jaffard family, it is locally finite, and thus aT = T for all but finitely many T ∈ Θ; hence, IInt(T ) = Int(T ) for all but finitely many T , and thus [IInt(T )] is almost always equal to [Int(T )].It follows that ∆ is contained in the direct sum.To prove the converse, it is enough to show that, for any T ∈ Θ and any [J] ∈ Pic(Int(T )), there is a [I] ∈ Pic(Int(D)) such that [IInt(T )] = [J] and [IInt(S)] = [Int(S)] for every S ∈ Θ, S = T .Again by [2, Remark VIII.1.5]we can suppose that J is an integral unitary ideal of Int(T ); moreover, we can suppose that J = (f 1 , . . ., f n )Int(T ) for some f 1 , . . ., f n ∈ Int(D).Let L := J ∩D = J ∩T ∩D = I ∩D: then, LS = S for every S ∈ Θ, S = T .Then, I ′ := (f 1 , . . ., f n )Int(D) + LInt(D) is contained in I and finitely generated, but I ′ Int(T ) = J and I ′ Int(S) = Int(S), as well as I ′ K[X] = K[X].It follows that I ′ = I, and thus I is finitely generated.We show that I is locally principal.Let M be a maximal ideal of Int(D): if M ∩ D = (0) then Int(D) M is a localization of K[X], and thus IInt(D) M is principal.If M ∩ D = (0), then Int(D) M contains Int(D)S for some S ∈ Θ, and thus IInt(D) M = IInt(S)Int(D) M .If S = T , then IInt(D) M = IInt(S)Int(D) M = Int(D) M is principal.If S = T , then IInt(T ) = J is invertible, and thus IInt(D) M is principal since Int(D) M is a localization of Int(T ).Thus I is locally principal and thus invertible; therefore the direct sum is in the image of ψ Θ .The claim is proved.

Corollary 4 . 8 .
Let D be an integral domain such that one of the following conditions holds.(a) D is h-local; (b) D is one-dimensional and locally finite; (c) D is a one-dimensional Noetherian domain.Then, there is an exact sequence 0 −→ Pic(D) −→ Pic(Int(D)) −→ M ∈Max(D) Pic(Int(D M )) −→ 0. In particular, P(D) ≃ M ∈Max(D) P(D M ) ≃ M ∈Max(D) Pic(Int(D M )).
} is a Jaffard family, and thus the claim follows either from Theorem 4.4 (since Pic(D, Θ) = Pic(D)) or by Theorem 4.7 (since P(D M ) = Pic(Int(D M )) as D M is local).

Lemma 5 . 1 .
Let D be an integral domain, T a flat overring of D, and let L be a sublattice of Over(D) such that {S | S ∈ L} = T .If the extension map Pic(D) −→ Pic(S) is surjective for every S ∈ L, then the extension map Pic(D) −→ Pic(T ) is surjective.

Lemma 5 . 2 .
Let D be an integral domain, T a flat overring of D, and let L be a sublattice of Over(D) such that {S | S ∈ Λ} = T .If the extension map Pic(Int(D)) −→ Pic(Int(D)S) is surjective for every S ∈ L, then the extension map Pic(Int(D)) −→ Pic(Int(D)T ) is surjective.

Lemma 5 . 3 .
Let D be a one-dimensional Prüfer domain, and let T be a flat overring of D. Then, the extension map Pic(Int(D)) −→ Pic(Int(D)T ) is surjective.Proof.Let L be the family of all Jaffard overrings of D contained in T .Then, L is a sublattice of Over(D), since the product of two Jaffard overrings is a Jaffard overring, and the extension map Pic(Int(D)) −→ Pic(Int(D)S) = Pic(Int(S)) is surjective for all such S. Take a ∈ T , and let I := (D : D a) = a −1 D ∩ D. Since D is a Prüfer domain, I is finitely generated; therefore, both the closed set V (I) and the open set D(I) ∩ Max(D) of Max(D) are compact in the Zariski topology of Max(D).Let Θ 1 := {P ∈ Max(D) | P ∈ V (I)} and Θ 2 := {Q ∈ Max(D) | Q ∈ D(I) ∩ Max(D)}, and let S i := {D P | P ∈ Θ 1 }.Applying [21, Propostion 4.8] to Θ := {D M | M ∈ Max(D)}, we obtain that {S 1 , S 2 } is a pre-Jaffard family of D; being finite, it is a Jaffard family, and thus S 1 and S 2 are Jaffard overrings.

Proof.
The extension map Pic(D) −→ Pic(T ∞ ) is surjective, with kernel Pic(D, T ∞ ).In particular, the kernel contains Pic(D, Θ), and thus the extension map induces a surjective map Pic(D) Pic(D, Θ) −→ Pic(T ∞ ) with kernel Pic(D, T ∞ ) Pic(D, Θ) .We claim that this group is isomorphic to T ∈Θ\{T∞} Pic(T ).Indeed, consider the extension map φ : Pic(D, T ∞ ) −→ T ∈Θ\{T∞} Pic(T ).Note that φ is well-defined since, if IT ∞ = T ∞ , then IS = S only for finitely many S ∈ Θ [21, Proposition 5.3(a)].Moreover, φ is surjective: indeed, let [I] ∈ Pic(T ), with I ⊆ T , and set J := I ∩ D.Then, J is an invertbile ideal of D such that JS = S for all S ∈ Θ, S = T , and in particular JT ∞ = T ∞ .Therefore, φ([J]) is the element of the direct sum whose only nonzero coefficient is the one corresponding to T , which is equal to [I].Thus, φ is surjective.The kernel of φ is given by all [I] ∈ Pic(D, T ∞ ) that become principal in Pic(T ) for each T ∈ Θ; that is, by definition, ker φ = Pic(D, Θ).Thus Pic(D, T ∞ ) Pic(D, Θ) ≃ T ∈Θ\{T∞} Pic(T ).The exactness of the sequence of the statement follows.Theorem 6.4.Let Θ be a weak Jaffard family of D pointed at T ∞ .Then, there is an exact sequence 0 −→ T ∈Θ\{T∞} P(T ) −→ P(D) −→ P(D, T ∞ ) −→ 0. the cokernel of π Θ .The first row is defined (and is exact) by Lemma 6.3, while the second row is exact by Proposition 6.2.All vertical maps are injective: the side ones since Pic(A) −→ Pic(Int(D)A) is always injective, while the middle one because the kernel of the natural map Pic(D) −→ ∆ is exactly Pic(D, Θ).By the snake lemma, the sequence of cokernels 0 −→ T ∈Θ\{T∞} P(T ) −→ G −→ P(D, T ∞ ) −→ 0 is exact.Moreover, the quotient G between ∆ and Pic(thus, we obtain exactly the exact sequence of the statement.Example 6.5.Let p be a prime number, and let V := Z (p) .Applying repeatedly [17, Chapter 6, Theorem 4], we can construct a chain of extensions