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# MultiprojectiveVariety ** Ring -- change the coefficient ring of a multi-projective variety

## Synopsis

• Operator: **
• Usage:
X ** K
• Inputs:
• X, , defined over a coefficient ring F
• K, a ring, the new coefficient ring (which must be a field)
• Outputs:
• , a multi-projective variety defined over K, obtained by coercing the coefficients of the multi-forms defining X into K

## Description

It is necessary that all multi-forms in the old coefficient ring F can be automatically coerced into the new coefficient ring K.

 i1 : use ring PP_QQ^{2,3}; i2 : X = projectiveVariety ideal(x1_2^2-x1_1*x1_3,x1_1*x1_2-x1_0*x1_3,x1_1^2-x1_0*x1_2,x0_1^2-x0_0*x0_2); o2 : ProjectiveVariety, surface in PP^2 x PP^3 i3 : ideal X 2 2 2 o3 = ideal (x1 - x1 x1 , x1 x1 - x1 x1 , x1 - x1 x1 , x0 - x0 x0 ) 2 1 3 1 2 0 3 1 0 2 1 0 2 o3 : Ideal of QQ[x0 ..x0 , x1 ..x1 ] 0 2 0 3 i4 : K = ZZ/65521; i5 : X' = X ** K; o5 : ProjectiveVariety, surface in PP^2 x PP^3 i6 : ideal X' 2 2 2 o6 = ideal (x1 - x1 x1 , x1 x1 - x1 x1 , x1 - x1 x1 , x0 - x0 x0 ) 2 1 3 1 2 0 3 1 0 2 1 0 2 o6 : Ideal of K[x0 ..x0 , x1 ..x1 ] 0 2 0 3