  
  [1X1 [33X[0;0YModule Presentations[133X[101X
  
  
  [1X1.1 [33X[0;0YFunctors[133X[101X
  
  [1X1.1-1 FunctorStandardModuleLeft[101X
  
  [33X[1;0Y[29X[2XFunctorStandardModuleLeft[102X( [3XR[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya functor[133X
  
  [33X[0;0YThe  argument is a homalg ring [23XR[123X. The output is a functor which takes a left
  presentation as input and computes its standard presentation.[133X
  
  [1X1.1-2 FunctorStandardModuleRight[101X
  
  [33X[1;0Y[29X[2XFunctorStandardModuleRight[102X( [3XR[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya functor[133X
  
  [33X[0;0YThe argument is a homalg ring [23XR[123X. The output is a functor which takes a right
  presentation as input and computes its standard presentation.[133X
  
  [1X1.1-3 FunctorGetRidOfZeroGeneratorsLeft[101X
  
  [33X[1;0Y[29X[2XFunctorGetRidOfZeroGeneratorsLeft[102X( [3XR[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya functor[133X
  
  [33X[0;0YThe  argument is a homalg ring [23XR[123X. The output is a functor which takes a left
  presentation as input and gets rid of the zero generators.[133X
  
  [1X1.1-4 FunctorGetRidOfZeroGeneratorsRight[101X
  
  [33X[1;0Y[29X[2XFunctorGetRidOfZeroGeneratorsRight[102X( [3XR[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya functor[133X
  
  [33X[0;0YThe argument is a homalg ring [23XR[123X. The output is a functor which takes a right
  presentation as input and gets rid of the zero generators.[133X
  
  [1X1.1-5 FunctorLessGeneratorsLeft[101X
  
  [33X[1;0Y[29X[2XFunctorLessGeneratorsLeft[102X( [3XR[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya functor[133X
  
  [33X[0;0YThe  argument  is  a homalg ring [23XR[123X. The output is functor which takes a left
  presentation as input and computes a presentation having less generators.[133X
  
  [1X1.1-6 FunctorLessGeneratorsRight[101X
  
  [33X[1;0Y[29X[2XFunctorLessGeneratorsRight[102X( [3XR[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya functor[133X
  
  [33X[0;0YThe  argument  is a homalg ring [23XR[123X. The output is functor which takes a right
  presentation as input and computes a presentation having less generators.[133X
  
  [1X1.1-7 FunctorDualLeft[101X
  
  [33X[1;0Y[29X[2XFunctorDualLeft[102X( [3XR[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya functor[133X
  
  [33X[0;0YThe  argument is a homalg ring [23XR[123X that has an involution function. The output
  is  functor  which  takes  a  left  presentation [3XM[103X as input and computes its
  Hom(M, R) as a left presentation.[133X
  
  [1X1.1-8 FunctorDualRight[101X
  
  [33X[1;0Y[29X[2XFunctorDualRight[102X( [3XR[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya functor[133X
  
  [33X[0;0YThe  argument is a homalg ring [23XR[123X that has an involution function. The output
  is  functor  which  takes  a  right presentation [3XM[103X as input and computes its
  Hom(M, R) as a right presentation.[133X
  
  [1X1.1-9 FunctorDoubleDualLeft[101X
  
  [33X[1;0Y[29X[2XFunctorDoubleDualLeft[102X( [3XR[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya functor[133X
  
  [33X[0;0YThe  argument is a homalg ring [23XR[123X that has an involution function. The output
  is  functor which takes a left presentation [3XM[103X as input and computes its [3XHom(
  Hom(M, R), R )[103X as a left presentation.[133X
  
  [1X1.1-10 FunctorDoubleDualRight[101X
  
  [33X[1;0Y[29X[2XFunctorDoubleDualRight[102X( [3XR[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya functor[133X
  
  [33X[0;0YThe  argument is a homalg ring [23XR[123X that has an involution function. The output
  is functor which takes a right presentation [3XM[103X as input and computes its [3XHom(
  Hom(M, R), R )[103X as a right presentation.[133X
  
  
  [1X1.2 [33X[0;0YGAP Categories[133X[101X
  
  [1X1.2-1 IsLeftOrRightPresentationMorphism[101X
  
  [33X[1;0Y[29X[2XIsLeftOrRightPresentationMorphism[102X( [3Xobject[103X ) [32X filter[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X[133X
  
  [33X[0;0YThe   GAP   category   of  morphisms  in  the  category  of  left  or  right
  presentations.[133X
  
  [1X1.2-2 IsLeftPresentationMorphism[101X
  
  [33X[1;0Y[29X[2XIsLeftPresentationMorphism[102X( [3Xobject[103X ) [32X filter[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X[133X
  
  [33X[0;0YThe GAP category of morphisms in the category of left presentations.[133X
  
  [1X1.2-3 IsRightPresentationMorphism[101X
  
  [33X[1;0Y[29X[2XIsRightPresentationMorphism[102X( [3Xobject[103X ) [32X filter[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X[133X
  
  [33X[0;0YThe GAP category of morphisms in the category of right presentations.[133X
  
  [1X1.2-4 IsLeftOrRightPresentation[101X
  
  [33X[1;0Y[29X[2XIsLeftOrRightPresentation[102X( [3Xobject[103X ) [32X filter[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X[133X
  
  [33X[0;0YThe  GAP  category of objects in the category of left presentations or right
  presentations.[133X
  
  [1X1.2-5 IsLeftPresentation[101X
  
  [33X[1;0Y[29X[2XIsLeftPresentation[102X( [3Xobject[103X ) [32X filter[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X[133X
  
  [33X[0;0YThe GAP category of objects in the category of left presentations.[133X
  
  [1X1.2-6 IsRightPresentation[101X
  
  [33X[1;0Y[29X[2XIsRightPresentation[102X( [3Xobject[103X ) [32X filter[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X[133X
  
  [33X[0;0YThe GAP category of objects in the category of right presentations.[133X
  
  
  [1X1.3 [33X[0;0YConstructors[133X[101X
  
  [1X1.3-1 PresentationMorphism[101X
  
  [33X[1;0Y[29X[2XPresentationMorphism[102X( [3XA[103X, [3XM[103X, [3XB[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(A,B)[123X[133X
  
  [33X[0;0YThe  arguments  are  an object [23XA[123X, a homalg matrix [23XM[123X, and another object [23XB[123X. [23XA[123X
  and  [23XB[123X shall either both be objects in the category of left presentations or
  both  be  objects  in  the  category of right presentations. The output is a
  morphism  [23XA \rightarrow B[123X in the the category of left or right presentations
  whose underlying matrix is given by [23XM[123X.[133X
  
  [1X1.3-2 AsMorphismBetweenFreeLeftPresentations[101X
  
  [33X[1;0Y[29X[2XAsMorphismBetweenFreeLeftPresentations[102X( [3Xm[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(F^r,F^c)[123X[133X
  
  [33X[0;0YThe  argument is a homalg matrix [23Xm[123X. The output is a morphism [23XF^r \rightarrow
  F^c[123X  in  the  the  category of left presentations whose underlying matrix is
  given  by [23Xm[123X, where [23XF^r[123X and [23XF^c[123X are free left presentations of ranks given by
  the number of rows and columns of [23Xm[123X.[133X
  
  [1X1.3-3 AsMorphismBetweenFreeRightPresentations[101X
  
  [33X[1;0Y[29X[2XAsMorphismBetweenFreeRightPresentations[102X( [3Xm[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(F^c,F^r)[123X[133X
  
  [33X[0;0YThe  argument is a homalg matrix [23Xm[123X. The output is a morphism [23XF^c \rightarrow
  F^r[123X  in  the  the category of right presentations whose underlying matrix is
  given by [23Xm[123X, where [23XF^r[123X and [23XF^c[123X are free right presentations of ranks given by
  the number of rows and columns of [23Xm[123X.[133X
  
  [1X1.3-4 AsLeftPresentation[101X
  
  [33X[1;0Y[29X[2XAsLeftPresentation[102X( [3XM[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Yan object[133X
  
  [33X[0;0YThe  argument is a homalg matrix [23XM[123X over a ring [23XR[123X. The output is an object in
  the  category  of  left  presentations  over  [23XR[123X.  This  object  has [23XM[123X as its
  underlying matrix.[133X
  
  [1X1.3-5 AsRightPresentation[101X
  
  [33X[1;0Y[29X[2XAsRightPresentation[102X( [3XM[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Yan object[133X
  
  [33X[0;0YThe  argument is a homalg matrix [23XM[123X over a ring [23XR[123X. The output is an object in
  the  category  of  right  presentations  over  [23XR[123X.  This  object has [23XM[123X as its
  underlying matrix.[133X
  
  [1X1.3-6 AsLeftOrRightPresentation[101X
  
  [33X[1;0Y[29X[2XAsLeftOrRightPresentation[102X( [3XM[103X, [3Xl[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Yan object[133X
  
  [33X[0;0YThe  arguments  are  a  homalg  matrix  [23XM[123X and a boolean [23Xl[123X. If [23Xl[123X is [10Xtrue[110X, the
  output  is  an  object in the category of left presentations. If [23Xl[123X is [10Xfalse[110X,
  the  output  is  an  object  in the category of right presentations. In both
  cases, the underlying matrix of the result is [23XM[123X.[133X
  
  [1X1.3-7 FreeLeftPresentation[101X
  
  [33X[1;0Y[29X[2XFreeLeftPresentation[102X( [3Xr[103X, [3XR[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Yan object[133X
  
  [33X[0;0YThe  arguments  are a non-negative integer [23Xr[123X and a homalg ring [23XR[123X. The output
  is an object in the category of left presentations over [23XR[123X. It is represented
  by the [23X0 \times r[123X matrix and thus it is free of rank [23Xr[123X.[133X
  
  [1X1.3-8 FreeRightPresentation[101X
  
  [33X[1;0Y[29X[2XFreeRightPresentation[102X( [3Xr[103X, [3XR[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Yan object[133X
  
  [33X[0;0YThe  arguments  are a non-negative integer [23Xr[123X and a homalg ring [23XR[123X. The output
  is  an  object  in  the  category  of  right  presentations  over  [23XR[123X.  It is
  represented by the [23Xr \times 0[123X matrix and thus it is free of rank [23Xr[123X.[133X
  
  [1X1.3-9 UnderlyingMatrix[101X
  
  [33X[1;0Y[29X[2XUnderlyingMatrix[102X( [3XA[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya homalg matrix[133X
  
  [33X[0;0YThe  argument  is an object [23XA[123X in the category of left or right presentations
  over a homalg ring [23XR[123X. The output is the underlying matrix which presents [23XA[123X.[133X
  
  [1X1.3-10 UnderlyingHomalgRing[101X
  
  [33X[1;0Y[29X[2XUnderlyingHomalgRing[102X( [3XA[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya homalg ring[133X
  
  [33X[0;0YThe  argument  is an object [23XA[123X in the category of left or right presentations
  over a homalg ring [23XR[123X. The output is [23XR[123X.[133X
  
  [1X1.3-11 Annihilator[101X
  
  [33X[1;0Y[29X[2XAnnihilator[102X( [3XA[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(I, F)[123X[133X
  
  [33X[0;0YThe  argument is an object [23XA[123X in the category of left or right presentations.
  The output is the embedding of the annihilator [23XI[123X of [23XA[123X into the free module [23XF[123X
  of  rank [23X1[123X. In particular, the annihilator itself is seen as a left or right
  presentation.[133X
  
  [1X1.3-12 LeftPresentationsAsFreydCategoryOfCategoryOfRows[101X
  
  [33X[1;0Y[29X[2XLeftPresentationsAsFreydCategoryOfCategoryOfRows[102X( [3XR[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya category[133X
  
  [33X[0;0YThe  argument  is  a  homalg  ring  [23XR[123X.  The  output  is the category of left
  presentations  over  [23XR[123X,  constructed  internally as the [10XFreydCategory[110X of the
  [10XCategoryOfRows[110X  of [3XR[103X. Only available if the package [10XFreydCategoriesForCAP[110X is
  available.[133X
  
  [1X1.3-13 RightPresentationsAsFreydCategoryOfCategoryOfColumns[101X
  
  [33X[1;0Y[29X[2XRightPresentationsAsFreydCategoryOfCategoryOfColumns[102X( [3XR[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya category[133X
  
  [33X[0;0YThe  argument  is  a  homalg  ring  [23XR[123X.  The  output is the category of right
  presentations  over  [23XR[123X,  constructed  internally as the [10XFreydCategory[110X of the
  [10XCategoryOfColumns[110X  of [3XR[103X. Only available if the package [10XFreydCategoriesForCAP[110X
  is available.[133X
  
  [1X1.3-14 LeftPresentations[101X
  
  [33X[1;0Y[29X[2XLeftPresentations[102X( [3XR[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya category[133X
  
  [33X[0;0YThe  argument  is  a  homalg  ring  [23XR[123X.  The  output  is the category of left
  presentations over [23XR[123X.[133X
  
  [1X1.3-15 RightPresentations[101X
  
  [33X[1;0Y[29X[2XRightPresentations[102X( [3XR[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya category[133X
  
  [33X[0;0YThe  argument  is  a  homalg  ring  [23XR[123X.  The  output is the category of right
  presentations over [23XR[123X.[133X
  
  
  [1X1.4 [33X[0;0YAttributes[133X[101X
  
  [1X1.4-1 UnderlyingHomalgRing[101X
  
  [33X[1;0Y[29X[2XUnderlyingHomalgRing[102X( [3XR[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya homalg ring[133X
  
  [33X[0;0YThe  argument  is  a  morphism  [23X\alpha[123X  in  the  category  of  left or right
  presentations over a homalg ring [23XR[123X. The output is [23XR[123X.[133X
  
  [1X1.4-2 UnderlyingMatrix[101X
  
  [33X[1;0Y[29X[2XUnderlyingMatrix[102X( [3Xalpha[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya homalg matrix[133X
  
  [33X[0;0YThe  argument  is  a  morphism  [23X\alpha[123X  in  the  category  of  left or right
  presentations. The output is its underlying homalg matrix.[133X
  
  
  [1X1.5 [33X[0;0YNon-Categorical Operations[133X[101X
  
  [1X1.5-1 StandardGeneratorMorphism[101X
  
  [33X[1;0Y[29X[2XStandardGeneratorMorphism[102X( [3XA[103X, [3Xi[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(F, A)[123X[133X
  
  [33X[0;0YThe  argument  is an object [23XA[123X in the category of left or right presentations
  over  a  homalg ring [23XR[123X with underlying matrix [23XM[123X and an integer [23Xi[123X. The output
  is  a morphism [23XF \rightarrow A[123X given by the [23Xi[123X-th row or column of [23XM[123X, where [23XF[123X
  is a free left or right presentation of rank [23X1[123X.[133X
  
  [1X1.5-2 CoverByFreeModule[101X
  
  [33X[1;0Y[29X[2XCoverByFreeModule[102X( [3XA[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(F,A)[123X[133X
  
  [33X[0;0YThe  argument is an object [23XA[123X in the category of left or right presentations.
  The  output is a morphism from a free module [23XF[123X to [23XA[123X, which maps the standard
  generators of the free module to the generators of [23XA[123X.[133X
  
  
  [1X1.6 [33X[0;0YNatural Transformations[133X[101X
  
  [1X1.6-1 NaturalIsomorphismFromIdentityToStandardModuleLeft[101X
  
  [33X[1;0Y[29X[2XNaturalIsomorphismFromIdentityToStandardModuleLeft[102X( [3XR[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya      natural      transformation     [23X\mathrm{Id}     \rightarrow
            \mathrm{StandardModuleLeft}[123X[133X
  
  [33X[0;0YThe  argument is a homalg ring [23XR[123X. The output is the natural isomorphism from
  the identity functor to the left standard module functor.[133X
  
  [1X1.6-2 NaturalIsomorphismFromIdentityToStandardModuleRight[101X
  
  [33X[1;0Y[29X[2XNaturalIsomorphismFromIdentityToStandardModuleRight[102X( [3XR[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya      natural      transformation     [23X\mathrm{Id}     \rightarrow
            \mathrm{StandardModuleRight}[123X[133X
  
  [33X[0;0YThe  argument is a homalg ring [23XR[123X. The output is the natural isomorphism from
  the identity functor to the right standard module functor.[133X
  
  [1X1.6-3 NaturalIsomorphismFromIdentityToGetRidOfZeroGeneratorsLeft[101X
  
  [33X[1;0Y[29X[2XNaturalIsomorphismFromIdentityToGetRidOfZeroGeneratorsLeft[102X( [3XR[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya      natural      transformation     [23X\mathrm{Id}     \rightarrow
            \mathrm{GetRidOfZeroGeneratorsLeft}[123X[133X
  
  [33X[0;0YThe  argument is a homalg ring [23XR[123X. The output is the natural isomorphism from
  the identity functor to the functor that gets rid of zero generators of left
  modules.[133X
  
  [1X1.6-4 NaturalIsomorphismFromIdentityToGetRidOfZeroGeneratorsRight[101X
  
  [33X[1;0Y[29X[2XNaturalIsomorphismFromIdentityToGetRidOfZeroGeneratorsRight[102X( [3XR[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya      natural      transformation     [23X\mathrm{Id}     \rightarrow
            \mathrm{GetRidOfZeroGeneratorsRight}[123X[133X
  
  [33X[0;0YThe  argument is a homalg ring [23XR[123X. The output is the natural isomorphism from
  the  identity  functor  to  the  functor that gets rid of zero generators of
  right modules.[133X
  
  [1X1.6-5 NaturalIsomorphismFromIdentityToLessGeneratorsLeft[101X
  
  [33X[1;0Y[29X[2XNaturalIsomorphismFromIdentityToLessGeneratorsLeft[102X( [3XR[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya      natural      transformation     [23X\mathrm{Id}     \rightarrow
            \mathrm{LessGeneratorsLeft}[123X[133X
  
  [33X[0;0YThe argument is a homalg ring [23XR[123X. The output is the natural morphism from the
  identity functor to the left less generators functor.[133X
  
  [1X1.6-6 NaturalIsomorphismFromIdentityToLessGeneratorsRight[101X
  
  [33X[1;0Y[29X[2XNaturalIsomorphismFromIdentityToLessGeneratorsRight[102X( [3XR[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya      natural      transformation     [23X\mathrm{Id}     \rightarrow
            \mathrm{LessGeneratorsRight}[123X[133X
  
  [33X[0;0YThe argument is a homalg ring [23XR[123X. The output is the natural morphism from the
  identity functor to the right less generator functor.[133X
  
  [1X1.6-7 NaturalTransformationFromIdentityToDoubleDualLeft[101X
  
  [33X[1;0Y[29X[2XNaturalTransformationFromIdentityToDoubleDualLeft[102X( [3XR[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya      natural      transformation     [23X\mathrm{Id}     \rightarrow
            \mathrm{FunctorDoubleDualLeft}[123X[133X
  
  [33X[0;0YThe argument is a homalg ring [23XR[123X. The output is the natural morphism from the
  identity functor to the double dual functor in left Presentations category.[133X
  
  [1X1.6-8 NaturalTransformationFromIdentityToDoubleDualRight[101X
  
  [33X[1;0Y[29X[2XNaturalTransformationFromIdentityToDoubleDualRight[102X( [3XR[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya      natural      transformation     [23X\mathrm{Id}     \rightarrow
            \mathrm{FunctorDoubleDualRight}[123X[133X
  
  [33X[0;0YThe argument is a homalg ring [23XR[123X. The output is the natural morphism from the
  identity functor to the double dual functor in right Presentations category.[133X
  
