Tensor Products and Tor

TensorProduct(M, N) : ModMPol, ModMPol -> ModMPol, Map
Given R -modules M and N, return the tensor product M tensor R N as an ambient module T, together with the associated map f: M x N -> 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 tensor R N. This is a new complex whose i-th term is Ci tensor R N (where Ci is the i-th term of C); the boundary maps are also derived from those of C in the natural way via the functor - tensor R N (see [Eis95, p.64]).
Tor(i, M, N) : RngIntElt, ModMPol, ModMPol -> ModMPol
Given an integer i≥0 and R-modules M and N, return Tori(M, N). This is the homology at the i-th term of the complex C tensor R N where C is a free resolution of M.

Example PMod_Hom (H116E16)

We construct a tensor product and some 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]
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]
]
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
V2.28, 13 July 2023