Homomorphisms

HomsToC(A): AlgEtQ -> SeqEnum[Map]

Precision: RngIntElt                    Default: Precision(GetDefaultRealField())

Returns the sequence of homomorphisms from the algebra \(A\) to a complex field \({{\C}}\). The precision of \({{\C}}\) is given by the optional parameter Precision. The default value is 30.

Example: Complex Embeddings of an Etale Algebra (ex-58e95c)

> _<x> := PolynomialRing(Integers());
> A := EtaleAlgebra(x^2+2);
> homs := HomsToC(A : Precision := 30);
> #homs;
2

Run in calculator (opens in a new tab)Copy input

Hom(A, B, img): AlgEtQ, AlgEtQ, SeqEnum[AlgEtQElt] -> Map

CheckMultiplicative: BoolElt                    Default: false
CheckUnital        : BoolElt                    Default: false
ComputeInverse     : BoolElt                    Default: true

Given étale algebras \(A\) and \(B\) and a sequence img of elements of \(B\) with length equal to the absolute dimension of \(A\), returns the \({{\Q}}\)-algebra homomorphism sending the AbsoluteBasis(A) to img. If ComputeInverse is true and the map is invertible, preimages are defined. If CheckMultiplicative (resp. CheckUnital) is true, multiplicativity (resp. unitality) is checked.

DiagonalEmbedding(K, V): AlgEtQ, AlgEtQ -> Map

NaturalAction(K, V): AlgEtQ, AlgEtQ -> Map

Let \(K=K_1\times\cdots\times K_n\) be a product of distinct number fields and \(V=K_1^{s_1}\times\cdots\times K_n^{s_n}\). Returns the natural component-wise diagonal embedding \(K\to V\).

Example: Homomorphism into a Product Algebra (ex-aab1bc)

> _<x> := PolynomialRing(Integers());
> A := EtaleAlgebra((x^2+2)*(x^2+3));
> B := EtaleAlgebra(Components(A));
> // Build a hom by mapping AbsoluteBasis(A) into B component-wise
> img := [ B!Components(AbsoluteBasis(A)[i]) : i in [1..AbsoluteDimension(A)] ];
> m := Hom(A,B,img : CheckMultiplicative := false, CheckUnital := false,
> ComputeInverse := true);
> // Diagonal embedding on a suitable power algebra
> V, embs, projs := DirectProduct([A,A]);
> d := DiagonalEmbedding(A,V);
> d(One(A));
<1, 1, 1, 1>

Run in calculator (opens in a new tab)Copy input