Synopsis

• Usage:
• Inputs:
• f, , a homomorphism F ** G --> H
• F, , a free module
• G, , a free module
• Outputs:
• , the adjoint homomorphism F --> Hom(G,H)

Description

Recall that ** refers to the tensor product of modules.

 i1 : R = QQ[x_1 .. x_24]; i2 : f = genericMatrix(R,2,4*3) o2 = | x_1 x_3 x_5 x_7 x_9 x_11 x_13 x_15 x_17 x_19 x_21 x_23 | | x_2 x_4 x_6 x_8 x_10 x_12 x_14 x_16 x_18 x_20 x_22 x_24 | 2 12 o2 : Matrix R <--- R i3 : isHomogeneous f o3 = true i4 : g = adjoint(f,R^4,R^3) o4 = | x_1 x_7 x_13 x_19 | | x_2 x_8 x_14 x_20 | | x_3 x_9 x_15 x_21 | | x_4 x_10 x_16 x_22 | | x_5 x_11 x_17 x_23 | | x_6 x_12 x_18 x_24 | 6 4 o4 : Matrix R <--- R

If f is homogeneous, and source f === F ** G (including the grading), then the resulting matrix will be homogeneous.

 i5 : g = adjoint(f,R^4,R^{-1,-1,-1}) o5 = {-1} | x_1 x_7 x_13 x_19 | {-1} | x_2 x_8 x_14 x_20 | {-1} | x_3 x_9 x_15 x_21 | {-1} | x_4 x_10 x_16 x_22 | {-1} | x_5 x_11 x_17 x_23 | {-1} | x_6 x_12 x_18 x_24 | 6 4 o5 : Matrix R <--- R i6 : isHomogeneous g o6 = true