Ideal Class Monoid

Let \(R\) be an order in an étale algebra \(A\) over \({{\Bbb Q}}\). Ideal multiplication induces the structure of commutative monoid on the set of ideal classes of \(R\), which we then call ideal class monoid of \(R\). We denote it by \({\cal I}(R)\). The unit element of \({\cal I}(R)\) is the class of any principal fractional \(R\)-ideal.

We have a partitioning \({\cal I}(R) = \bigsqcup_S {\cal I}_S(R)\) where the disjoint union is taken over the overorders \(S\) of \(R\) and \({\cal I}_S(R)\) is the subset of \({\cal I}(R)\) consisting of ideal classes with multiplicator ring \(S\). The computation is then performed by first computing \({\cal W}(R)\) and then observing that for each overorder \(S\) of \(R\), the Picard group \({\rm Pic}(S)\) acts freely on \({\cal I}_S(R)\) with quotient space \({\cal W}_S(R)\).

ICM_bar(S): AlgEtQOrd -> SeqEnum

GRH: BoolElt                    Default: false

Returns the ideal classes of fractional \(S\)-ideals having Multiplicator Ring equal to \(S\). This is the same as the orbit of the action of PicardGroup(S) on WKICM_bar(S).

ICM(S): AlgEtQOrd -> SeqEnum

GRH: BoolElt                    Default: false

Returns the ideal class monoid of the order \(S\), that is, a set of representatives for the isomorphism classes of the fractional \(S\)-ideals.

We also provide an abstract representation of the ideal class monoid. The abstract monoid has type AlgEtQICM and classes have type AlgEtQICMElt. An ideal class can be created via ! starting from an overorder or a fractional ideal, and a (deterministic) representative can be recovered using RepresentativeMap. Each class is internally a pair consisting of a weak equivalence class and an element of the abstract representation of \({\rm Pic}(S)\).

The second method to compute the ideal class monoid of an order \(R\), returns an abstract representation of \({\cal I}(R)\) with type AlgEtQICM together with a map to a set of representatives. Each class has type AlgEtQICMElt, and it is internally represented as a pair consisting of a weak equivalence class (of type AlgEtQWECMElt) and an element of the representation of \({\rm Pic}(S)\) (as an abstract abelian group), where \(S\) is the corresponding multiplicator ring. This representation is more efficient than the previous one, since it does not need to compute in advance and store a representative for each ideal class. Ideal classes can be created using the coercion operator ! starting from an overorder \(S\) of \(R\) or a fractional \(S\)-ideal. Ideal classes can be multiplied using the operator *.

IdealClassMonoidAbstract(R): AlgEtQOrd -> AlgEtQICM, Map

Given an order \(R\), returns the abstract ideal class monoid \({\rm icm}\) together with a map (with preimages) sending each class to a representative.

icm ! x: AlgEtQICM, Any -> AlgEtQICMElt

Coerce \(x\) into the abstract ideal class monoid when possible, returning an ideal class.

x in icm: AlgEtQICMElt, AlgEtQICM -> BoolElt

Parent(x): AlgEtQICMElt -> AlgEtQICM

WEClass(x): AlgEtQICMElt -> AlgEtQWECMElt

PicClass(x): AlgEtQICMElt -> GrpAbElt, Map

Ideal(x): AlgEtQICMElt -> AlgEtQIdl

MultiplicatorRing(x): AlgEtQICMElt -> AlgEtQOrd

x1 eq x2: AlgEtQICMElt, AlgEtQICMElt -> BoolElt

x * y: AlgEtQICMElt, AlgEtQICMElt -> AlgEtQICMElt

x ^ n: AlgEtQICMElt, RngIntElt -> AlgEtQICMElt

IsOne(x): AlgEtQICMElt -> BoolElt

IsInvertibleInMultiplicatorRing(x): AlgEtQICMElt -> BoolElt

Order(icm): AlgEtQICM -> AlgEtQOrd

RepresentativeMap(icm): AlgEtQICM -> Map

icm1 eq icm2: AlgEtQICM, AlgEtQICM -> BoolElt

# icm: AlgEtQICM -> RngInt

Classes(icm): AlgEtQICM -> SeqEnum[AlgEtQICMElt]

Representatives(icm): AlgEtQICM -> SeqEnum[AlgEtQIdl]

One(icm): AlgEtQICM -> AlgEtQICMElt

Random(icm): AlgEtQICM -> AlgEtQICMElt

(More standard intrinsics.)