Factorization and Primes

Factorization(I): AlgEtQIdl -> Tup

Given an integral \(S\)-ideal \(I\) coprime with the conductor of \(S\) (hence invertible in \(S\)), returns its factorization into a product of primes of \(S\).

PrimesAbove(I): AlgEtQIdl -> SeqEnum[AlgAssEtOrdIdl]

Given an integral \(S\)-ideal \(I\), returns the sequence of maximal ideals \(P\) of \(S\) above \(I\).

SingularPrimes(R): AlgEtQOrd -> SeqEnum[AlgAssEtOrdIdl]

Returns the non-invertible primes of the order \(R\).

PlacesAboveRationalPrime(E, p): AlgEtQ, RngIntElt -> SeqEnum[AlgEtQIdl]

Given an étale algebra and a rational prime, returns the primes of the maximal order of the algebra containing the rational prime.

NonInvertiblePrimes(R): AlgEtQOrd -> SetIndx

Returns the non-invertible primes of the order \(R\).

IsPrime(I): AlgEtQIdl -> BoolElt

IsMaximal(I): AlgEtQIdl -> BoolElt

IsMaximalIdeal(I): AlgEtQIdl -> BoolElt

Given an integral \(S\)-ideal \(I\), returns if the ideal is a prime fractional ideal of \(S\), that is a maximal \(S\) ideal.

Valuation(x, P): AlgEtQElt, AlgEtQIdl -> RngIntElt

Valuation(I, P): AlgEtQIdl, AlgEtQIdl -> RngIntElt

Valuation at the prime \(P\) of an element \(x\) or of a fractional ideal \(I\) of the maximal order.

InertiaDegree(P): AlgEtQIdl -> RngIntElt

RamificationIndex(P): AlgEtQIdl -> RngIntElt

For a prime \(P\) of the maximal order \({\cal O}\), returns its inertia degree and ramification index.

IsBassAtPrime(S, P): AlgEtQOrd, AlgEtQIdl -> BoolElt

Check if the order is Bass at the prime ideal \(P\), that is, if every overorder of \(S\) is Gorenstein at the primes above \(P\).

IsBass(S): AlgEtQOrd -> BoolElt

Check if the order \(S\) is Bass, that is, if every overorder of \(S\) is Gorenstein.

IsGorensteinAtPrime(S, P): AlgEtQOrd, AlgEtQIdl -> BoolElt

Check if the order \(S\) is Gorenstein at the prime ideal \(P\), that is, if every fractional ideal \(I\) with \((I:I)=S\) is locally principal at \(P\).

IsGorenstein(O): AlgEtQOrd -> BoolElt

Checks if the order \(O\) is Gorenstein, that is if the TraceDualIdeal of \(O\) is invertible, or equivalently, if all fractional ideals \(I\) with \((I:I)=O\) are invertible.

Uniformizers(PPs): SeqEnum[AlgEtQIdl] -> SeqEnum

Given a sequence of primes \(P\) of the maximal order, returns a sequence of elements \(t_P\) such that \(t_P\) is a uniformizer at \(P\) and a unit at every other prime in the sequence.