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## Synopsis

• Optional inputs:
• DegreeLimit (missing documentation) => ..., default value null,
• MinimalGenerators (missing documentation) => ..., default value true,
• Strategy (missing documentation) => ..., default value null,

## Description

Recall that tensor(Module,Module) and Hom(Module,Module) form an adjoint pair, meaning that there is a natural isomorphism $$\mathrm{Hom}(F\otimes G,H) \cong \mathrm{Hom}(F,\mathrm{Hom}(G,H)).$$

## Synopsis

• Usage:
adjoint(f, F, G)
• Inputs:
• f, , a homomorphism $F \otimes G \to H$
• F, , a free module;
• G, , a free module;
• Outputs:
• , the adjoint homomorphism $F \to \mathrm{Hom}(G,H)$
 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 i7 : f === adjoint'(g, R^{-1,-1,-1}, R^2) o7 = true

## Synopsis

• Usage:
adjoint'(g, G, H)
• Inputs:
• g, , a homomorphism $F \to \mathrm{Hom}(G,H)$ between modules
• G, , a free module;
• H, , a free module;
• Outputs:
• , the adjoint homomorphism $F \otimes G \to H$

If g is homogeneous, and target g === Hom(G,H) (including the grading), then the resulting matrix will be homogeneous.

 i8 : R = QQ[x_1 .. x_12]; i9 : g = genericMatrix(R, 6, 2) o9 = | x_1 x_7 | | x_2 x_8 | | x_3 x_9 | | x_4 x_10 | | x_5 x_11 | | x_6 x_12 | 6 2 o9 : Matrix R <-- R i10 : f = adjoint'(g, R^2, R^3) o10 = | x_1 x_4 x_7 x_10 | | x_2 x_5 x_8 x_11 | | x_3 x_6 x_9 x_12 | 3 4 o10 : Matrix R <-- R i11 : isHomogeneous f o11 = true i12 : g === adjoint(f, R^{-1,-1}, R^2) o12 = true