The Hom Module and Ext#
- Hom(M, N): ModMPol, ModMPol -> ModMPol, Map#
Given \(R-\)modules \(M\) and \(N\), return \(H={\rm Hom}_R(M,N)\) as an abstract reduced module and a transfer map \(f: H \rightarrow S\), where \(S\) is the set of all homomorphisms (of type
ModMPolHom) from \(M\) to \(N\). Thus \(H\) is a module representing the set of all homomorphisms from \(M\) to \(N\), while \(f\) maps an element \(h\in H\) to an actual homomorphism from \(M\) to \(N\) (and the inverse image of an element of \(S\) under \(f\) gives a corresponding element of \(H\)). If \(M\) and \(N\) are graded, then \(H\) is graded also, and the degree \(d_f\) of an element \(f\in H\) is the degree of the corresponding homomorphism (so an element in \(M\) of degree \(d\) will be mapped by \(f\) to zero or an element of degree \(d_f + d\) in \(N\)).
- Hom(C, N): ModCpx, ModMPol -> ModMPol#
Given a complex \(C\) of \(R\)-modules and an \(R\)-module \(N\), return \({\rm Hom}_R(C, N)\). This is a new complex whose \(i\)-th term is \({\rm Hom}_R(C_i, N)\) (where \(C_i\) is the \(i\)-th term of \(C\)); the boundary maps are also derived from those of \(C\) in the natural way via the functor \({\rm Hom}_R(-, N)\) (see [Eisenbud, 1995, p.63]). Note that the direction of arrows in this complex is opposite to that of \(C\).
- Ext(i, M, N): RngIntElt, ModMPol, ModMPol -> ModMPol#
Given an integer \(i\ge 0\) and \(R\)-modules \(M\) and \(N\), return \({\rm Ext}^i(M, N)\). This is the homology at the \(i\)-th term of the complex \({\rm Hom}_R(C, N)\) where \(C\) is a free resolution of \(M\).
- Example: Hom (ex-7c5089)#
We construct a Hom module and explicit homomorphisms derived from it.
> R<x,y,z> := PolynomialRing(RationalField(), 3); > M := quo<GradedModule(R, 3) | > [x*y, x*z, y*z], [y, x, y], > [0, x^3 - x^2*z, x^2*y - x*y*z], [y*z, x^2, x*y]>; > N := quo<GradedModule(R, 2) | > [x^2, y^2], [x^2, y*z], [x^2*z, x*y^2]>; > M; Graded Module R^3/<relations> Relations: [ x*y, x*z, y*z], [ y, x, y], [ 0, x^3 - x^2*z, x^2*y - x*y*z], [ y*z, x^2, x*y] > N; Graded Module R^2/<relations> Relations: [ x^2, y^2], [ x^2, y*z], [x^2*z, x*y^2] > H, f := Hom(M, N); > H; Graded Module R^7/<relations> with grading [1, 2, 1, 1, 1, 1, 1] Relations: [x, 0, 0, -z, 0, x, 0], [y, 0, x, 0, y, 0, 0], [y, 0, x, 0, 0, y, 0], [0, 0, 0, 0, 0, 0, y], [-y, 0, -x, 0, -z, 0, z], [x, 0, 0, -y, x, 0, 0], [x*y, y, 0, 0, 0, 0, x*y], [-x*y + x*z, -y + z, 0, 0, 0, 0, x*z - z^2], [x*z, x, 0, 0, 0, 0, 0], [0, y, 0, y^2, -z^2, 0, z^2], [0, y - z, 0, 0, 0, 0, z^2] > h := f(H.1); > h; Module homomorphism (3 by 2) of degree 1 Presentation matrix: [0 z] [x 0] [0 0] > $1 @@ f; [1, 0, 0, 0, 0, 0, 0] > Degree(M.1); 0 > h(M.1); [0, z] > Degree(h(M.1)); 1 > f(Basis(H)); [ Module homomorphism (3 by 2) of degree 1 Presentation matrix: [0 z] [x 0] [0 0], Module homomorphism (3 by 2) of degree 2 Presentation matrix: [ 0 -z^2] [ 0 y*z] [ 0 0], Module homomorphism (3 by 2) of degree 1 Presentation matrix: [ 0 0] [-y 0] [ x 0], Module homomorphism (3 by 2) of degree 1 Presentation matrix: [ 0 0] [ 0 -y] [ 0 x], Module homomorphism (3 by 2) of degree 1 Presentation matrix: [ 0 -z] [ 0 0] [ 0 y], Module homomorphism (3 by 2) of degree 1 Presentation matrix: [ 0 -z] [ 0 0] [ 0 z], Module homomorphism (3 by 2) of degree 1 Presentation matrix: [ 0 y - z] [ 0 0] [ 0 0] ]