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### 7 Relative affine varieties

#### 7.1 Attributes

##### 7.1-1 GraphOfRingMorphism
 ‣ GraphOfRingMorphism( phi ) ( attribute )

Returns: an object in a Zariski coframe of an affine variety

Compute the graph the morphism of the morphism phi of affine rings.

##### 7.1-2 FunctorClosureOfProjectionBetweenZariskiCoframes
 ‣ FunctorClosureOfProjectionBetweenZariskiCoframes( arg ) ( attribute )

##### 7.1-3 ClosureOfProjection
 ‣ ClosureOfProjection( arg ) ( attribute )

##### 7.1-4 ClosureOfImage
 ‣ ClosureOfImage( phi ) ( attribute )

Returns: a constructible object as a union of formal multiple differences

Compute the image closure of the morphism defined by the morphism phi of affine rings.

##### 7.1-5 FunctorPreimageOfProjectionBetweenZariskiCoframes
 ‣ FunctorPreimageOfProjectionBetweenZariskiCoframes( arg ) ( attribute )

#### 7.2 Operations

##### 7.2-1 PreimageOfProjection
 ‣ PreimageOfProjection( arg1, arg2 ) ( operation )

##### 7.2-2 PreimageOfProjection
 ‣ PreimageOfProjection( arg1, arg2 ) ( operation )

##### 7.2-3 FiberOfProjectionOverBasePoint
 ‣ FiberOfProjectionOverBasePoint( arg1, arg2 ) ( operation )

##### 7.2-4 FiberOfProjectionOverBasePoint
 ‣ FiberOfProjectionOverBasePoint( arg1, arg2 ) ( operation )

##### 7.2-5 TangentSpaceOfFiberAtPoint
 ‣ TangentSpaceOfFiberAtPoint( V, p_base, p_fiber ) ( operation )

Returns: an object in a Zariski coframe of an affine variety

Compute the tangent space of the fiber of V over closed base point p_base at the closed point p_fiber as an affine subspace of the ambient space of V intersecting (p_base, p_fiber).

##### 7.2-6 TangentSpaceOfFiberAtPoint
 ‣ TangentSpaceOfFiberAtPoint( V, p_base, p_fiber ) ( operation )

##### 7.2-7 DimensionsOfFibrationAtClosedPoint
 ‣ DimensionsOfFibrationAtClosedPoint( V, p_base, p_fiber ) ( operation )

Returns: a list

##### 7.2-8 DimensionsOfFibrationAtClosedPoint
 ‣ DimensionsOfFibrationAtClosedPoint( V, p_base, p_fiber ) ( operation )

##### 7.2-9 EmbeddedComplementOfTangentSpaceOfFiberAtPoint
 ‣ EmbeddedComplementOfTangentSpaceOfFiberAtPoint( V, p_base, p_fiber ) ( operation )

Returns: an object in a Zariski coframe

Compute a compolement of the tangent space of the fiber of V over the closed base point p_base at the closed point p_fiber as an affine subspace of the ambient space of V intersecting (p_base, p_fiber).

##### 7.2-10 EmbeddedComplementOfTangentSpaceOfFiberAtPoint
 ‣ EmbeddedComplementOfTangentSpaceOfFiberAtPoint( V, p_base, p_fiber ) ( operation )

##### 7.2-11 ClosedSubsetWithGenericallyZeroDimensionalFibers
 ‣ ClosedSubsetWithGenericallyZeroDimensionalFibers( V, p_base, p_fiber ) ( operation )

Returns: an object in a Zariski coframe of an affine variety

##### 7.2-12 ClosedSubsetOfBaseWithFreeFibersOverComplementOrEmpty
 ‣ ClosedSubsetOfBaseWithFreeFibersOverComplementOrEmpty( arg ) ( attribute )

##### 7.2-13 ClosedSubsetOfBaseWithFreeFibersOverComplement
 ‣ ClosedSubsetOfBaseWithFreeFibersOverComplement( arg ) ( attribute )
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