  
  [1X8 [33X[0;0YBicomplexes[133X[101X
  
  [33X[0;0YEach  bicomplex  in  [5Xhomalg[105X  has  an  underlying  complex  of complexes. The
  bicomplex  structure  is  simply  the addition of the known sign trick which
  induces  the obvious equivalence between the category of bicomplexes and the
  category  of  complexes  with  complexes  as  objects and chain morphisms as
  morphisms.  The  majority  of  filtered  complexes  in  algebra and geometry
  (unlike  topology)  arise  as  the total complex of a bicomplex. Hence, most
  spectral sequences in algebra are spectral sequences of bicomplexes. Indeed,
  bicomplexes  in [5Xhomalg[105X are mainly used as an input for the spectral sequence
  machinery.[133X
  
  
  [1X8.1 [33X[0;0YBicomplexes: Category and Representations[133X[101X
  
  [1X8.1-1 IsHomalgBicomplex[101X
  
  [33X[1;0Y[29X[2XIsHomalgBicomplex[102X( [3XBC[103X ) [32X Category[133X
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0YThe [5XGAP[105X category of [5Xhomalg[105X bi(co)complexes.[133X
  
  [33X[0;0Y(It is a subcategory of the [5XGAP[105X category [10XIsHomalgObject[110X.)[133X
  
  [1X8.1-2 IsBicomplexOfFinitelyPresentedObjectsRep[101X
  
  [33X[1;0Y[29X[2XIsBicomplexOfFinitelyPresentedObjectsRep[102X( [3XBC[103X ) [32X Representation[133X
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0YThe  [5XGAP[105X representation of bicomplexes (homological bicomplexes) of finitley
  generated [5Xhomalg[105X objects.[133X
  
  [33X[0;0Y(It is a representation of the [5XGAP[105X category [2XIsHomalgBicomplex[102X ([14X8.1-1[114X), which
  is      a      subrepresentation      of      the     [5XGAP[105X     representation
  [10XIsFinitelyPresentedObjectRep[110X.)[133X
  
  [1X8.1-3 IsBicocomplexOfFinitelyPresentedObjectsRep[101X
  
  [33X[1;0Y[29X[2XIsBicocomplexOfFinitelyPresentedObjectsRep[102X( [3XBC[103X ) [32X Representation[133X
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0YThe  [5XGAP[105X  representation  of  bicocomplexes  (cohomological  bicomplexes) of
  finitley generated [5Xhomalg[105X objects.[133X
  
  [33X[0;0Y(It is a representation of the [5XGAP[105X category [2XIsHomalgBicomplex[102X ([14X8.1-1[114X), which
  is      a      subrepresentation      of      the     [5XGAP[105X     representation
  [10XIsFinitelyPresentedObjectRep[110X.)[133X
  
  
  [1X8.2 [33X[0;0YBicomplexes: Constructors[133X[101X
  
  [1X8.2-1 HomalgBicomplex[101X
  
  [33X[1;0Y[29X[2XHomalgBicomplex[102X( [3XC[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X bicomplex[133X
  
  [33X[0;0YThis  constructor creates a bicomplex (homological bicomplex) given a [5Xhomalg[105X
  complex  of  (co)complexes  [3XC[103X  (-->  [2XHomalgComplex[102X ([14X6.2-1[114X)), resp. creates a
  bicocomplex   (cohomological   bicomplex)   given   a  [5Xhomalg[105X  cocomplex  of
  (co)complexes  [3XC[103X (--> [2XHomalgCocomplex[102X ([14X6.2-2[114X)). Using the usual sign-trick a
  complex of complexes gives rise to a bicomplex and vice versa.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XZZ := HomalgRingOfIntegers( );[127X[104X
    [4X[28XZ[128X[104X
    [4X[25Xgap>[125X [27XM := HomalgMatrix( "[ 2, 3, 4,   5, 6, 7 ]", 2, 3, ZZ );[127X[104X
    [4X[28X<A 2 x 3 matrix over an internal ring>[128X[104X
    [4X[25Xgap>[125X [27XM := LeftPresentation( M );[127X[104X
    [4X[28X<A non-torsion left module presented by 2 relations for 3 generators>[128X[104X
    [4X[25Xgap>[125X [27Xd := Resolution( M );[127X[104X
    [4X[28X<A non-zero right acyclic complex containing a single morphism of left modules\[128X[104X
    [4X[28X at degrees [ 0 .. 1 ]>[128X[104X
    [4X[25Xgap>[125X [27Xdd := Hom( d );[127X[104X
    [4X[28X<A non-zero acyclic cocomplex containing a single morphism of right modules at\[128X[104X
    [4X[28X degrees [ 0 .. 1 ]>[128X[104X
    [4X[25Xgap>[125X [27XC := Resolution( dd );[127X[104X
    [4X[28X<An acyclic cocomplex containing a single morphism of right complexes at degre\[128X[104X
    [4X[28Xes [ 0 .. 1 ]>[128X[104X
    [4X[25Xgap>[125X [27XCC := Hom( C );[127X[104X
    [4X[28X<A non-zero acyclic complex containing a single morphism of left cocomplexes a\[128X[104X
    [4X[28Xt degrees [ 0 .. 1 ]>[128X[104X
    [4X[25Xgap>[125X [27XBC := HomalgBicomplex( CC );[127X[104X
    [4X[28X<A non-zero bicomplex containing left modules at bidegrees [ 0 .. 1 ]x[128X[104X
    [4X[28X[ -1 .. 0 ]>[128X[104X
    [4X[25Xgap>[125X [27XDisplay( BC );[127X[104X
    [4X[28X * *[128X[104X
    [4X[28X * *[128X[104X
    [4X[25Xgap>[125X [27XUU := UnderlyingComplex( BC );[127X[104X
    [4X[28X<A non-zero acyclic complex containing a single morphism of left cocomplexes a\[128X[104X
    [4X[28Xt degrees [ 0 .. 1 ]>[128X[104X
    [4X[25Xgap>[125X [27XIsIdenticalObj( UU, CC );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XtBC := TransposedBicomplex( BC );[127X[104X
    [4X[28X<A non-zero bicomplex containing left modules at bidegrees [ -1 .. 0 ]x[128X[104X
    [4X[28X[ 0 .. 1 ]>[128X[104X
    [4X[25Xgap>[125X [27XDisplay( tBC );[127X[104X
    [4X[28X * *[128X[104X
    [4X[28X * *[128X[104X
  [4X[32X[104X
  
  
  [1X8.3 [33X[0;0YBicomplexes: Properties[133X[101X
  
  [1X8.3-1 IsBisequence[101X
  
  [33X[1;0Y[29X[2XIsBisequence[102X( [3XBC[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0YCheck if all maps in [3XBC[103X are well-defined.[133X
  
  [1X8.3-2 IsBicomplex[101X
  
  [33X[1;0Y[29X[2XIsBicomplex[102X( [3XBC[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0YCheck if [3XBC[103X is bicomplex.[133X
  
  [1X8.3-3 IsTransposedWRTTheAssociatedComplex[101X
  
  [33X[1;0Y[29X[2XIsTransposedWRTTheAssociatedComplex[102X( [3XBC[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0YCheck  if  [3XBC[103X  is  transposed  with  respect  to  the  associated complex of
  complexes.[133X
  [33X[0;0Y(no method installed).[133X
  
  
  [1X8.4 [33X[0;0YBicomplexes: Attributes[133X[101X
  
  [1X8.4-1 TotalComplex[101X
  
  [33X[1;0Y[29X[2XTotalComplex[102X( [3XBC[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X (co)complex[133X
  
  [33X[0;0YThe associated total complex.[133X
  
  [1X8.4-2 SpectralSequence[101X
  
  [33X[1;0Y[29X[2XSpectralSequence[102X( [3XBC[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X (co)homological spectral sequence[133X
  
  [33X[0;0YThe associated spectral sequence.[133X
  
  
  [1X8.5 [33X[0;0YBicomplexes: Operations and Functions[133X[101X
  
  [1X8.5-1 UnderlyingComplex[101X
  
  [33X[1;0Y[29X[2XUnderlyingComplex[102X( [3XBC[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X complex[133X
  
  [33X[0;0YThe  (co)complex  of  (co)complexes underlying the (co)homological bicomplex
  [3XBC[103X.[133X
  
  [1X8.5-2 ByASmallerPresentation[101X
  
  [33X[1;0Y[29X[2XByASmallerPresentation[102X( [3XB[103X ) [32X method[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X bicomplex[133X
  
  [33X[0;0YSee [2XByASmallerPresentation[102X ([14X6.5-2[114X) on complexes.[133X
  
  [4X[32X  Code  [32X[104X
    [4XInstallMethod( ByASmallerPresentation,[104X
    [4X        "for homalg bicomplexes",[104X
    [4X        [ IsHomalgBicomplex ],[104X
    [4X        [104X
    [4X  function( B )[104X
    [4X    [104X
    [4X    ByASmallerPresentation( UnderlyingComplex( B ) );[104X
    [4X    [104X
    [4X    IsZero( B );[104X
    [4X    [104X
    [4X    return B;[104X
    [4X    [104X
    [4Xend );[104X
  [4X[32X[104X
  
  [33X[0;0YThis method performs side effects on its argument [3XB[103X and returns it.[133X
  
