5.2 Hom_R(M,N)

Let M be given as R^m/Im(A₀), N as R^p/Im(B₀), where $R=K[x_1,\ldots , x_n]/(h_1,\ldots , h_p)\,$ together with free resolutions

$$\ldots \longrightarrow F_k \stackrel{A_{k-1}}{\longrightarrow... ...}{\longrightarrow} F_0=R^m \longrightarrow M \longrightarrow 0$$

and

$$\ldots \longrightarrow G_k \stackrel{B_{k-1}}{\longrightarrow... ...{\longrightarrow} G_0=R^p \longrightarrow N \longrightarrow 0 .$$

We get the following commutative diagram with exact columns and rows:

$$\begin{array}{ccccccccc} & & 0 & \\ & & \downarrow & \\ & &... ...& \longleftarrow & F_1^{\ast } \otimes G_1 \\ \par\end{array}$$

.

Algorithm 5..2  

$$Hom_R(M,N)= Im\,(modulo\,(A_0^T \otimes id_{G_0},\,id_{F_1^{\ast}} \otimes B_0))\,/\:Im\,(id_{F_0^{\ast }}\otimes B_0)$$

is a free module modulo the image of the matrix

$$modulo\,(modulo\,(A_0^T \otimes id_{G_0},\,id_{F_1^{\ast}} \otimes B_0)\,,\:id_{F_0^{\ast }}\otimes B_0)$$