PUBLICATIONS DU DÉPARTEMENT DE MATHÉMATIQUES DE LYON ALAIN BOUVIER Survey on Locally Factorial Krull Domains Publications du Département de Mathématiques de Lyon, 1980, tome 17, fascicule 1 , p. 1-33 <http://www.numdam.org/item?id=PDML_1980__17_1_1_0> © Université de Lyon, 1980, tous droits réservés. L’accès aux archives de la série « Publications du Département de mathématiques de Lyon » im- plique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pé- nale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Publications du Département de Mathématiques Lyon 1980 t. 17-1 SURVEY ON LOCALLY FACTORIAL KRULL DOMAINS Alain Bouvier In the following diagram, we consider some classes of rings which have been introduced to study integrally closed domains: Discrète valuation ring •^^ Noetherians S (DVR) '•^N. non ' \. χ . _ noeterian^^ -, . / Dedekind valuation ^ . ^/ ι \ Factorial u + ^ (UFD) * Regular ι bezout y ^-^ST ><Γ*" \ ^^^^ / locally ^ \ ηηΤΛ , . \( factorial \ GCD-domain s ,( -,Λ , . \ ^ νKrull domains Prîif er - \ ν ^ Krull \ \ f locally " ^N domains \ | factorial j ι -x \ ν domains \ ' i/^ completely j locally % integrally î GCD-domains / closed Integrally closed 1 Survey on locally factorial Krull domains Assuming furthermore that the domains are noetherian the above diagram collapses and becomes the diagram below· Valuation = DVR l Principal = Bezout UFD = GCD Prufer = Dedekind Regular locally GCD = locally factorial = locally factorial Krull l Krull = integrally closed In this note we shall prove some interesting properties of locally factorial Krull domains; namely: 1) They are the Krull domains such that every divisor- ial idéal Is invertible; in comparison, Dedekind domain have the property that non-zero ideals are invertible and UFD1s that every divisorial is principal. 2) For a locally factorial Krull domain A , the Picard group Pic(A) is equal to the class group c£(A) ; c A thus the quotient group ^ ^|pic(A) indicates how far the Krull domain A is from being a locally factorial domain. 2 Survey on locally factorial Krull domains 3) In order to prove a ring A is factorial it is convenient to prove that A is locally factorial and satisfies few more conditions (for instance A is noetherian on Pic(A) = 0 ). Several of thèse results, like in [GR], are proved for noetherian domains; what is yet true without this hypothesis? Before the study of locally factorial Krull domains (§3) we indicate a few properties of domains satisfying local conditions: locally GCD-domains and locally UFD. We end this lecture by posing some questions related to this topic. I want to thank D.D. Anderson for the stimulating letters he sent to me, P. Ribenboim and A, Geramita for their help while I was preparing this lecture. 3 Survey on locally factorial Krull domains § 1. Preliminaries To make this paper easy to read, we summarize in this first paragraph the terminology and notations used in the following ones. More détails can be found in [Β], [BG], or [F]. (1-1) Let A be a domain. We write Κ = Frac(A) its field of quotients, Spec(A) its prime spectrum, Max(A) its maximal spectrum and X^^CA) the set of height one prime ideals of A A non null (fractional) idéal is called divisorial or a v-ideal if it is the intersection of any family of principal ideals. Let D(A) be the set of such ideals and I(A) the monoid of non null ideals in A . We write Div(A) for the set of divisors of A ; that is, the quotient of I(A) by the Artin congruence defined by I Ξ J if and only if A : I = A : J . Let div : HA) -+ biv(A) be the canonical surjection. We write I ^ = A:I and since I = (I 1)""1 is a v-ideal, we call it the ν-idéal associated to I . If 1' is a prime, we sometimes write p^n^ instead of (Pn) . We say I is a v-ideal η of finite tvpe if I = (Σ a.A) for some a. ε Κ . Let D (A) be the set of v-ideals of finite type, P(A) the set of principal ideals and Cart(A) the Cartier group of invertible ideals of Aj one has the folJowing inclusion: 4 Survey on locally factorial Krull domains P(A) <=—> Cart(A)^—* Dt(A)^-—>D(A)<=—->I(A) A domain A is a Dedekind domain if and only if Cart(A) = I(A) or if and only if D(A) = I(A) Principal UFD Dedekind ι : 1 ι 1 p(A)c >Cart(A)* > D(A) c > I(A) We will complète this diagram later (§3)· We say a ring A [resp. on idéal I of A] satisfies locally a property Ρ if each Ap [resp. each IAp] satisfies the property Ρ for every Ρ ε Spec(A). For instance, a ring A is locally a UFU if every Ap for } ε Spec(A) is an UFD. An idéal 1 is invertible if and only if I is finitely generated and locally principal. (1-2) Let ^i^iei a iamuy of subrings of a field Κ . We say this family satisfies (FC) ("finiteness condition") if every non real élément χ ε Κ is a unit in every A^ but finitely many of them. A domain A is a Krull domain if there exists a family ^ΐ^£ει °^ discrète valuation rings in Frac(A) , satisfying the finiteness condition and such that A = f\ i 5 Survey on locally fictorial Krull domains Recall that if A is a Krull domain, then A . /Ο Ap , the Ap are discret valuation rings, the ΡεΧνι'(Α) V V family (Απ) satisfies (FC) and if (V.).T is P (i) 1 ιεΙ another family of discrète valuation rings satisfying the same conditions then for every Ρ ε X^\A) , their exists i ε I such that An = V. The AN are called the Ρ ι Ρ essential valuation rings of A . If A is a Krull domain, then Div A is a free abelian group with {div P} n. as a basis; the ΡεΧνΐ;(Α) subgroup of principal divisors is denoted by Prin(A) , the quotient group Div(A)|ppin(A) = et(A) is called the class group of A and the canonical image of Cart(A) in c£(A) the Picard group of A denoted by Pic(A) · In a Dedekind domain, cl(A) = Pic(A). Let Y be a subset of X^"^(A) ; the ring Αγ = ^""^ Ap is a Krull ring called a subintersection χ ρεγ Y of A ; for instance one can prove every ring of quotients is a subintersection. (1-3) Let A be a Krull ring, Αγ a subintersection, Y the set complément of Y in X(1)(A) and G (Y) the group generated by the canonical image of Y in c£(A)· then there exists a canonical map cl(A) c£(Ay) and CLABORN [C ] proved that the following séquence of groups i s exact : 6 Survey on locally factorial Krull domains LCLAB] 0 + G (Y) + cl(A) + c£(Ay) + 0 In particular, if S "''A is a ring of quotients of A^ then c£(A) + cl(S ^A) is surjective and its kernel is generated by the images of the height one primes which meet S (1-4) Spécial notations and terminology AR overring of a domain A is a domain Β such that A C Β dFrac(A). Following Gilmer, we write A(x) the quotient ring S A[x] where S is the set of polynomials whose coefficients generate the unit idéal. If S is a multiplicatively closed set of ideals of A , then A0 will dénote the S-transform of the generalized quotient ring of A with respect to S (see [ARB]). A domain A is cohérent if the intersection of two ideals finitely generated is an idéal finitely generated. Ail the rings are domains; for instance the valuation rings or the Krull rings are actually domains. 7 Survey on locally factorial Krull domains § 2 · Local Properties ( 2-1) GCD-domains A GCD-domain is a domain A satisfying the following équivalent conditions: i) aA Λ bA is principal for any a,b € Frac(A) ; ii) A: (A: aA + bA) is principal for any a,b 6 Frac(A) ; iii) Every finitely gençrated v-ideal is principal (i.e. Dt(A) = P(A)). For instance, UFD1s, Bezout rings and so valuation rings are GCD-domains. If A is a noetherian or a Krull domain one checks that A is a GCD-domain if and only if A is a UFD. A GCD-domain A is integrally closed and Pic(A) = 0 . But an integrally closed domain A with Pic(A) =0 is not necessarily a GCD-domain. Let f ε A[x] ; the v-ideal of A generated by the coefficients of f is a principal idéal, denoted cCf)^ and called the v-content of f . If c(f) = aA , then f = af# with c(f*) = A . A polynomial f ε A[x] such that c(g) = A is called a v-primitive polynomial. Any Ρ ε Spec(A[x]) such that Ρ f\ A = (0) contains a v-primitive polynomial. Properties of v-contents can be found in [Me] and [T]. For instance, if A is a GCD-domain and f ε KLx] , then there exists a ε c(f)""^ such that 8 Survey on locally factorial Krull domains fK[x] Π ALx] = afAlx] If A is a GCD-domain^the rings A[x], A(x) and any localizations of A are also GCD-domain.