Tensor Products and Tor

TensorProduct(M, N): ModMPol, ModMPol -> ModMPol, Map

Given \(R-\)modules \(M\) and \(N\), return the tensor product \(M\otimes_R N\) as an ambient module \(T\), together with the associated map \(f: M\times N \rightarrow T\). If \(M\) and \(N\) are graded, then \(T\) is graded also.

TensorProduct(C, N): ModCpx, ModMPol -> ModMPol

Given a complex \(C\) of \(R\)-modules and an \(R\)-module \(N\), return \(C\otimes_R N\). This is a new complex whose \(i\)-th term is \(C_i \otimes_R N\) (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 \(- \otimes_R N\) (see [Eisenbud, 1995, p.64]).

Tor(i, M, N): RngIntElt, ModMPol, ModMPol -> ModMPol

Given an integer \(i\ge 0\) and \(R\)-modules \(M\) and \(N\), return \({\rm Tor}^i(M, N)\). This is the homology at the \(i\)-th term of the complex \(C\otimes_R N\) where \(C\) is a free resolution of \(M\).

Example: Hom (ex-ff1e8f)

We construct a tensor product and some \({\rm Tor}\) modules for the same modules from the previous example.

> 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]>;
> T, f := TensorProduct(M, N);
> T;
Graded Module R^6/<relations>
Relations (Groebner basis):
[x^2, y*z, 0, 0, 0, 0],
[0, 0, 0, 0, x^2, y*z],
[0, 0, 0, 0, 0, x*y*z - y*z^2],
[x*y - y*z, 0, 0, 0, 0, 0],
[0, x*y - y*z, 0, 0, 0, 0],
[y*z, 0, 0, -y*z, x*y, 0],
[y, 0, x, 0, y, 0],
[0, y, 0, x, 0, y],
[0, y^2 - y*z, 0, 0, 0, 0],
[0, 0, 0, y^2 - y*z, 0, 0],
[0, 0, 0, 0, 0, y^2 - y*z],
[y*z^2, 0, 0, -y*z^2, 0, -y*z^2],
[0, y*z^2, 0, y*z^2, 0, y*z^2]

Run in calculatorCopy input

Note that \(f\) maps the cartesian product of \(M\) and \(N\) into \(T\).

> f(<M.1, N.1>);
[1, 0, 0, 0, 0, 0]
> [f(<m, n>): n in Basis(N), m in Basis(M)];
[
    [1, 0, 0, 0, 0, 0],
    [0, 1, 0, 0, 0, 0],
    [0, 0, 1, 0, 0, 0],
    [0, 0, 0, 1, 0, 0],
    [0, 0, 0, 0, 1, 0],
    [0, 0, 0, 0, 0, 1]
]

Run in calculatorCopy input

Finally we construct associated Tor modules.

> Tor(0, M, N);
Graded Module R^6/<relations>
Relations:
[y, 0, x, 0, y, 0],
[0, y, 0, x, 0, y],
[0, 0, 0, 0, x*y - y*z, 0],
[0, 0, 0, 0, 0, x*y - y*z],
[y*z, x^2, 0, 0, 0, 0],
[x*y*z - y*z^2, 0, 0, 0, 0, 0],
[y^2 - y*z, 0, 0, 0, 0, 0],
[0, 0, y*z, x^2, 0, 0],
[0, 0, x*y*z - y*z^2, 0, 0, 0],
[0, 0, y^2 - y*z, 0, 0, 0],
[0, 0, 0, 0, y*z, x^2],
[0, 0, 0, 0, y^2 - y*z, 0],
[0, 0, 0, 0, x*y*z - y*z^2, 0]
> Tor(1, M, N);
Graded Module R^2/<relations> with grading [3, 3]
Relations:
[y - z,     0],
[    z,    -y],
[  z^2,  -x*y],
[    0,     0]
> Tor(2, M, N);
Free Reduced Module R^0

Run in calculatorCopy input