Quelle Morphisms.gi Sprache: unbekannt

# SPDX-License-Identifier: GPL-2.0-or-later
# homalg: A homological algebra meta-package for computable Abelian categories
#
# Implementations
#

##  Implementations of homalg procedures for morphisms of abelian categories.

####################################
#
# methods for operations:
#
####################################

##
InstallMethod( Resolution, ### defines: Resolution (ResolutionOfSeq for a single map)
        "for homalg static morphisms",
        [ IsInt, IsStaticMorphismOfFinitelyGeneratedObjectsRep ],

  function( _q, phi  )
    local q, S, T, d_S, d_T, index_pair, j, d_S_j, d_T_j, phi_j, cm;

    q := _q;

    S := Source( phi );
    T := Range( phi );

    d_S := Resolution( q, S );
    d_T := Resolution( q, T );

    if q < 0 then
        q := Maximum( List( [ S, T ], LengthOfResolution ) );
        d_S := Resolution( q, S );
        d_T := Resolution( q, T );
    fi;

    index_pair := PairOfPositionsOfTheDefaultPresentations( phi );

    if not IsBound( phi!.resolutions ) then
        phi!.resolutions := rec( );
    fi;

    if IsBound( phi!.resolutions.(String( index_pair )) ) then
        cm := phi!.resolutions.(String( index_pair ));
        j := HighestDegree( cm );
        phi_j := HighestDegreeMorphism( cm );
    else
        j := 0;

        d_S_j := CoveringEpi( S );
        d_T_j := CoveringEpi( T );

        phi_j := CompleteImageSquare( d_S_j, phi, d_T_j );

        cm := HomalgChainMorphism( phi_j, d_S, d_T );

        phi!.resolutions.(String( index_pair )) := cm;
    fi;

    #=====# begin of the core procedure #=====#

    while j < q do

        j := j + 1;

        d_S_j := CertainMorphism( d_S, j );
        d_T_j := CertainMorphism( d_T, j );

        phi_j := CompleteImageSquare( d_S_j, phi_j, d_T_j );

        Add( cm, phi_j );

    od;

    SetIsMorphism( cm, true );

    return cm;

end );

InstallMethod( Resolution,
        "for homalg static morphisms",
        [ IsStaticMorphismOfFinitelyGeneratedObjectsRep ],

  function( phi )

    return Resolution( -1, phi );

end );

## [HS. Prop. II.9.6]
InstallMethod( CokernelSequence,
        "for homalg static morphisms",
        [ IsStaticMorphismOfFinitelyGeneratedObjectsRep ],

  function( phi )
    local nu, epsilon, C;

    nu := ImageObjectEmb( phi );

    epsilon := CokernelEpi( phi );

    C := HomalgComplex( epsilon );

    Add( C, nu );

    return C;

end );

## [HS. Prop. II.9.6]
InstallMethod( CokernelCosequence,
        "for homalg static morphisms",
        [ IsStaticMorphismOfFinitelyGeneratedObjectsRep ],

  function( phi )
    local nu, epsilon, C;

    nu := ImageObjectEmb( phi );

    epsilon := CokernelEpi( phi );

    C := HomalgCocomplex( nu );

    Add( C, epsilon );

    return C;

end );

## [HS. Prop. II.9.6]
InstallMethod( KernelSequence,
        "for homalg static morphisms",
        [ IsStaticMorphismOfFinitelyGeneratedObjectsRep ],

  function( phi )
    local mu, eta, C;

    mu := KernelEmb( phi );

    eta := ImageObjectEpi( phi );

    C := HomalgComplex( eta );

    Add( C, mu );

    return C;

end );

## [HS. Prop. II.9.6]
InstallMethod( KernelCosequence,
        "for homalg static morphisms",
        [ IsStaticMorphismOfFinitelyGeneratedObjectsRep ],

  function( phi )
    local mu, eta, C;

    mu := KernelEmb( phi );

    eta := ImageObjectEpi( phi );

    C := HomalgCocomplex( mu );

    Add( C, eta );

    return C;

end );

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