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6.2. Class groups and units 169The set of fractional ideals of O is stable under addition, multiplication, quotient andintersection, but the set of proper ideals is not. Nonetheless, we can define an action of invertibleideals on proper ideals by multiplication. For a, b ∈ Prop(O), a ∗ b is defined asa ∗ b = a −1 b .The four basic operations on fractional ideals: addition, multiplication, quotient and intersection,are compatible with localization at a maximal ideal. Moreover, an ideal is invertible if andonly if it is locally invertible, or equivalently locally principal, at each maximal ideal.Proposition 6.2.2 ([216, Satz I.12.4], [251, Theorem 2.7], [63, Corollary 2.1.7]). Let K be anumber field, O K its ring of integers and O an order. Let a be a fractional ideal of O. Then a isinvertible if and only if a p is a principal fractional ideal of O p for every maximal ideal p of O.Definition 6.2.3 (Singular ideal). Let K be a number field, O K its ring of integers and O anorder. A maximal ideal p of O is said to be singular if it is non invertible. Otherwise it is said tobe regular.Equivalently, a maximal ideal is singular if the local ring O p is not a discrete valuationring [251, Theorem 2.17].Definition 6.2.4 (Conductor [216, I.12], [25, 9.1]). Let K be a number field, O K its ring ofintegers and O an order. The conductor of O, denoted by f(O), is the largest ideal of O K containedin O.The conductor can also be described asf(O) = {x ∈ K | xO K ⊂ O} = (O : O K ) .The maximal ideals dividing f are exactly the singular maximal ideals of O [216, Satz I.12.10],[251, Exercise 4.25]. Moreover, if p is regular, then pO K is a maximal ideal. Finally, everyfractional ideal co-prime to the conductor can be uniquely written as a product of maximal idealsco-prime to the conductor.If m = [O K : O] is the index of O in O K , then mO K ⊂ f(O). Furthermore, the rationalprimes p ∈ Q dividing m are exactly those above which O is singular. Contrary to the case ofquadratic fields, orders are not classified anymore by their indices in O K .Finally, for an order O, we define as in Subsection 5.2.2 the Picard or class group Pic(O) ofclasses of invertible ideals, the proper class semigroup Prop(O) of classes of proper ideals, andthe class semigroup of classes of fractional ideals modulo principal ideals. As in the quadraticcase, the set of fractional ideals co-prime to the conductor modulo principal fractional ideals isisomorphic to the Picard group [217].We also have the following fundamental exact sequence relating the Picard group of an orderwith that of the maximal order.Proposition 6.2.5 ([216, Satz I.12.9], [25, Proposition 9.9]). Let K be a number field, O K itsring of integers and O an order of conductor f. The following canonical exact sequence is exact:1 → O ∗ → Õ∗ → ⊕ p|fÕ ∗ p/O ∗ p → Pic(O) → Pic(Õ) → 1 ,where p ranges over the prime ideals of O dividing the conductor f.