\contentsline {chapter}{\numberline {1}\leavevmode {\color {Chapter }Introduction}}{19}{chapter.1}
\contentsline {section}{\numberline {1.1}\leavevmode {\color {Chapter }What is the role of the \textsf {homalg} package in the \textsf {homalg} project?}}{19}{section.1.1}
\contentsline {subsection}{\numberline {1.1.1}\leavevmode {\color {Chapter }Philosophy}}{19}{subsection.1.1.1}
\contentsline {subsection}{\numberline {1.1.2}\leavevmode {\color {Chapter }\textsf {homalg} provides ...}}{19}{subsection.1.1.2}
\contentsline {subsection}{\numberline {1.1.3}\leavevmode {\color {Chapter }\textsf {homalg} delegates ...}}{20}{subsection.1.1.3}
\contentsline {subsection}{\numberline {1.1.4}\leavevmode {\color {Chapter }Rings supported in a sufficient way}}{21}{subsection.1.1.4}
\contentsline {subsection}{\numberline {1.1.5}\leavevmode {\color {Chapter }Principal limitation}}{21}{subsection.1.1.5}
\contentsline {subsection}{\numberline {1.1.6}\leavevmode {\color {Chapter }The black box concept}}{22}{subsection.1.1.6}
\contentsline {subsection}{\numberline {1.1.7}\leavevmode {\color {Chapter }\textsf {homalg}'s dictionaries (technical)}}{22}{subsection.1.1.7}
\contentsline {subsection}{\numberline {1.1.8}\leavevmode {\color {Chapter }The advantages of the outsourcing concept}}{22}{subsection.1.1.8}
\contentsline {subsection}{\numberline {1.1.9}\leavevmode {\color {Chapter }Does this mean that \textsf {homalg} has only algorithms for the generic case?}}{23}{subsection.1.1.9}
\contentsline {subsection}{\numberline {1.1.10}\leavevmode {\color {Chapter }The principle of least communication (technical)}}{23}{subsection.1.1.10}
\contentsline {subsection}{\numberline {1.1.11}\leavevmode {\color {Chapter }Building upon the \textsf {homalg} package}}{23}{subsection.1.1.11}
\contentsline {subsection}{\numberline {1.1.12}\leavevmode {\color {Chapter }Frequently asked questions}}{24}{subsection.1.1.12}
\contentsline {section}{\numberline {1.2}\leavevmode {\color {Chapter }This manual}}{24}{section.1.2}
\contentsline {chapter}{\numberline {2}\leavevmode {\color {Chapter }Installation of the \textsf {homalg} Package}}{25}{chapter.2}
\contentsline {chapter}{\numberline {3}\leavevmode {\color {Chapter }Quick Start}}{26}{chapter.3}
\contentsline {section}{\numberline {3.1}\leavevmode {\color {Chapter }Why are all examples in this manual over {\ensuremath {\@mathbb Z}} or ${\ensuremath {\@mathbb Z}}/m{\ensuremath {\@mathbb Z}}$?}}{26}{section.3.1}
\contentsline {section}{\numberline {3.2}\leavevmode {\color {Chapter }\texttt {gap{\textgreater } ExamplesForHomalg();}}}{26}{section.3.2}
\contentsline {section}{\numberline {3.3}\leavevmode {\color {Chapter }A typical example}}{26}{section.3.3}
\contentsline {subsection}{\numberline {3.3.1}\leavevmode {\color {Chapter }HomHom}}{26}{subsection.3.3.1}
\contentsline {chapter}{\numberline {4}\leavevmode {\color {Chapter }Rings}}{32}{chapter.4}
\contentsline {section}{\numberline {4.1}\leavevmode {\color {Chapter }Rings: Category and Representations}}{32}{section.4.1}
\contentsline {subsection}{\numberline {4.1.1}\leavevmode {\color {Chapter }IsHomalgRing}}{32}{subsection.4.1.1}
\contentsline {subsection}{\numberline {4.1.2}\leavevmode {\color {Chapter }IsPreHomalgRing}}{32}{subsection.4.1.2}
\contentsline {subsection}{\numberline {4.1.3}\leavevmode {\color {Chapter }IsHomalgRingElement}}{32}{subsection.4.1.3}
\contentsline {subsection}{\numberline {4.1.4}\leavevmode {\color {Chapter }IsHomalgInternalRingRep}}{32}{subsection.4.1.4}
\contentsline {section}{\numberline {4.2}\leavevmode {\color {Chapter }Rings: Constructors}}{33}{section.4.2}
\contentsline {subsection}{\numberline {4.2.1}\leavevmode {\color {Chapter }HomalgRingOfIntegers (constructor for the integers)}}{33}{subsection.4.2.1}
\contentsline {subsection}{\numberline {4.2.2}\leavevmode {\color {Chapter }HomalgFieldOfRationals (constructor for the field of rationals)}}{33}{subsection.4.2.2}
\contentsline {subsection}{\numberline {4.2.3}\leavevmode {\color {Chapter }\texttt {\char 92\relax }/ (constructor for residue class rings)}}{33}{subsection.4.2.3}
\contentsline {section}{\numberline {4.3}\leavevmode {\color {Chapter }Rings: Properties}}{34}{section.4.3}
\contentsline {subsection}{\numberline {4.3.1}\leavevmode {\color {Chapter }ContainsAField}}{34}{subsection.4.3.1}
\contentsline {subsection}{\numberline {4.3.2}\leavevmode {\color {Chapter }IsRationalsForHomalg}}{34}{subsection.4.3.2}
\contentsline {subsection}{\numberline {4.3.3}\leavevmode {\color {Chapter }IsFieldForHomalg}}{35}{subsection.4.3.3}
\contentsline {subsection}{\numberline {4.3.4}\leavevmode {\color {Chapter }IsDivisionRingForHomalg}}{35}{subsection.4.3.4}
\contentsline {subsection}{\numberline {4.3.5}\leavevmode {\color {Chapter }IsIntegersForHomalg}}{35}{subsection.4.3.5}
\contentsline {subsection}{\numberline {4.3.6}\leavevmode {\color {Chapter }IsResidueClassRingOfTheIntegers}}{35}{subsection.4.3.6}
\contentsline {subsection}{\numberline {4.3.7}\leavevmode {\color {Chapter }IsBezoutRing}}{35}{subsection.4.3.7}
\contentsline {subsection}{\numberline {4.3.8}\leavevmode {\color {Chapter }IsIntegrallyClosedDomain}}{35}{subsection.4.3.8}
\contentsline {subsection}{\numberline {4.3.9}\leavevmode {\color {Chapter }IsUniqueFactorizationDomain}}{35}{subsection.4.3.9}
\contentsline {subsection}{\numberline {4.3.10}\leavevmode {\color {Chapter }IsKaplanskyHermite}}{35}{subsection.4.3.10}
\contentsline {subsection}{\numberline {4.3.11}\leavevmode {\color {Chapter }IsDedekindDomain}}{36}{subsection.4.3.11}
\contentsline {subsection}{\numberline {4.3.12}\leavevmode {\color {Chapter }IsDiscreteValuationRing}}{36}{subsection.4.3.12}
\contentsline {subsection}{\numberline {4.3.13}\leavevmode {\color {Chapter }IsFreePolynomialRing}}{36}{subsection.4.3.13}
\contentsline {subsection}{\numberline {4.3.14}\leavevmode {\color {Chapter }IsWeylRing}}{36}{subsection.4.3.14}
\contentsline {subsection}{\numberline {4.3.15}\leavevmode {\color {Chapter }IsGlobalDimensionFinite}}{36}{subsection.4.3.15}
\contentsline {subsection}{\numberline {4.3.16}\leavevmode {\color {Chapter }IsLeftGlobalDimensionFinite}}{36}{subsection.4.3.16}
\contentsline {subsection}{\numberline {4.3.17}\leavevmode {\color {Chapter }IsRightGlobalDimensionFinite}}{36}{subsection.4.3.17}
\contentsline {subsection}{\numberline {4.3.18}\leavevmode {\color {Chapter }HasInvariantBasisProperty}}{36}{subsection.4.3.18}
\contentsline {subsection}{\numberline {4.3.19}\leavevmode {\color {Chapter }HasLeftInvariantBasisProperty}}{37}{subsection.4.3.19}
\contentsline {subsection}{\numberline {4.3.20}\leavevmode {\color {Chapter }HasRightInvariantBasisProperty}}{37}{subsection.4.3.20}
\contentsline {subsection}{\numberline {4.3.21}\leavevmode {\color {Chapter }IsLocalRing}}{37}{subsection.4.3.21}
\contentsline {subsection}{\numberline {4.3.22}\leavevmode {\color {Chapter }IsSemiLocalRing}}{37}{subsection.4.3.22}
\contentsline {subsection}{\numberline {4.3.23}\leavevmode {\color {Chapter }IsIntegralDomain}}{37}{subsection.4.3.23}
\contentsline {subsection}{\numberline {4.3.24}\leavevmode {\color {Chapter }IsHereditary}}{37}{subsection.4.3.24}
\contentsline {subsection}{\numberline {4.3.25}\leavevmode {\color {Chapter }IsLeftHereditary}}{37}{subsection.4.3.25}
\contentsline {subsection}{\numberline {4.3.26}\leavevmode {\color {Chapter }IsRightHereditary}}{37}{subsection.4.3.26}
\contentsline {subsection}{\numberline {4.3.27}\leavevmode {\color {Chapter }IsHermite}}{38}{subsection.4.3.27}
\contentsline {subsection}{\numberline {4.3.28}\leavevmode {\color {Chapter }IsLeftHermite}}{38}{subsection.4.3.28}
\contentsline {subsection}{\numberline {4.3.29}\leavevmode {\color {Chapter }IsRightHermite}}{38}{subsection.4.3.29}
\contentsline {subsection}{\numberline {4.3.30}\leavevmode {\color {Chapter }IsNoetherian}}{38}{subsection.4.3.30}
\contentsline {subsection}{\numberline {4.3.31}\leavevmode {\color {Chapter }IsLeftNoetherian}}{38}{subsection.4.3.31}
\contentsline {subsection}{\numberline {4.3.32}\leavevmode {\color {Chapter }IsRightNoetherian}}{38}{subsection.4.3.32}
\contentsline {subsection}{\numberline {4.3.33}\leavevmode {\color {Chapter }IsArtinian (for rings)}}{38}{subsection.4.3.33}
\contentsline {subsection}{\numberline {4.3.34}\leavevmode {\color {Chapter }IsLeftArtinian}}{38}{subsection.4.3.34}
\contentsline {subsection}{\numberline {4.3.35}\leavevmode {\color {Chapter }IsRightArtinian}}{39}{subsection.4.3.35}
\contentsline {subsection}{\numberline {4.3.36}\leavevmode {\color {Chapter }IsOreDomain}}{39}{subsection.4.3.36}
\contentsline {subsection}{\numberline {4.3.37}\leavevmode {\color {Chapter }IsLeftOreDomain}}{39}{subsection.4.3.37}
\contentsline {subsection}{\numberline {4.3.38}\leavevmode {\color {Chapter }IsRightOreDomain}}{39}{subsection.4.3.38}
\contentsline {subsection}{\numberline {4.3.39}\leavevmode {\color {Chapter }IsPrincipalIdealRing}}{39}{subsection.4.3.39}
\contentsline {subsection}{\numberline {4.3.40}\leavevmode {\color {Chapter }IsLeftPrincipalIdealRing}}{39}{subsection.4.3.40}
\contentsline {subsection}{\numberline {4.3.41}\leavevmode {\color {Chapter }IsRightPrincipalIdealRing}}{39}{subsection.4.3.41}
\contentsline {subsection}{\numberline {4.3.42}\leavevmode {\color {Chapter }IsRegular}}{39}{subsection.4.3.42}
\contentsline {subsection}{\numberline {4.3.43}\leavevmode {\color {Chapter }IsFiniteFreePresentationRing}}{40}{subsection.4.3.43}
\contentsline {subsection}{\numberline {4.3.44}\leavevmode {\color {Chapter }IsLeftFiniteFreePresentationRing}}{40}{subsection.4.3.44}
\contentsline {subsection}{\numberline {4.3.45}\leavevmode {\color {Chapter }IsRightFiniteFreePresentationRing}}{40}{subsection.4.3.45}
\contentsline {subsection}{\numberline {4.3.46}\leavevmode {\color {Chapter }IsSimpleRing}}{40}{subsection.4.3.46}
\contentsline {subsection}{\numberline {4.3.47}\leavevmode {\color {Chapter }IsSemiSimpleRing}}{40}{subsection.4.3.47}
\contentsline {subsection}{\numberline {4.3.48}\leavevmode {\color {Chapter }BasisAlgorithmRespectsPrincipalIdeals}}{40}{subsection.4.3.48}
\contentsline {subsection}{\numberline {4.3.49}\leavevmode {\color {Chapter }IsMinusOne}}{40}{subsection.4.3.49}
\contentsline {section}{\numberline {4.4}\leavevmode {\color {Chapter }Rings: Attributes}}{40}{section.4.4}
\contentsline {subsection}{\numberline {4.4.1}\leavevmode {\color {Chapter }Zero (for homalg ring elements)}}{40}{subsection.4.4.1}
\contentsline {subsection}{\numberline {4.4.2}\leavevmode {\color {Chapter }One (for homalg ring elements)}}{41}{subsection.4.4.2}
\contentsline {subsection}{\numberline {4.4.3}\leavevmode {\color {Chapter }homalgTable}}{41}{subsection.4.4.3}
\contentsline {subsection}{\numberline {4.4.4}\leavevmode {\color {Chapter }RingElementConstructor}}{41}{subsection.4.4.4}
\contentsline {subsection}{\numberline {4.4.5}\leavevmode {\color {Chapter }TypeOfHomalgMatrix}}{41}{subsection.4.4.5}
\contentsline {subsection}{\numberline {4.4.6}\leavevmode {\color {Chapter }ConstructorForHomalgMatrices}}{41}{subsection.4.4.6}
\contentsline {subsection}{\numberline {4.4.7}\leavevmode {\color {Chapter }Zero (for homalg rings)}}{41}{subsection.4.4.7}
\contentsline {subsection}{\numberline {4.4.8}\leavevmode {\color {Chapter }One (for homalg rings)}}{41}{subsection.4.4.8}
\contentsline {subsection}{\numberline {4.4.9}\leavevmode {\color {Chapter }MinusOne}}{41}{subsection.4.4.9}
\contentsline {subsection}{\numberline {4.4.10}\leavevmode {\color {Chapter }IndeterminatesOfPolynomialRing}}{42}{subsection.4.4.10}
\contentsline {subsection}{\numberline {4.4.11}\leavevmode {\color {Chapter }IndeterminateCoordinatesOfRingOfDerivations}}{42}{subsection.4.4.11}
\contentsline {subsection}{\numberline {4.4.12}\leavevmode {\color {Chapter }IndeterminateDerivationsOfRingOfDerivations}}{42}{subsection.4.4.12}
\contentsline {subsection}{\numberline {4.4.13}\leavevmode {\color {Chapter }IndeterminateAntiCommutingVariablesOfExteriorRing}}{42}{subsection.4.4.13}
\contentsline {subsection}{\numberline {4.4.14}\leavevmode {\color {Chapter }IndeterminatesOfExteriorRing}}{42}{subsection.4.4.14}
\contentsline {subsection}{\numberline {4.4.15}\leavevmode {\color {Chapter }WeightsOfIndeterminates}}{42}{subsection.4.4.15}
\contentsline {subsection}{\numberline {4.4.16}\leavevmode {\color {Chapter }MatrixOfWeightsOfIndeterminates}}{42}{subsection.4.4.16}
\contentsline {subsection}{\numberline {4.4.17}\leavevmode {\color {Chapter }CoefficientsRing}}{42}{subsection.4.4.17}
\contentsline {subsection}{\numberline {4.4.18}\leavevmode {\color {Chapter }KrullDimension}}{43}{subsection.4.4.18}
\contentsline {subsection}{\numberline {4.4.19}\leavevmode {\color {Chapter }LeftGlobalDimension}}{43}{subsection.4.4.19}
\contentsline {subsection}{\numberline {4.4.20}\leavevmode {\color {Chapter }RightGlobalDimension}}{43}{subsection.4.4.20}
\contentsline {subsection}{\numberline {4.4.21}\leavevmode {\color {Chapter }GlobalDimension}}{43}{subsection.4.4.21}
\contentsline {subsection}{\numberline {4.4.22}\leavevmode {\color {Chapter }GeneralLinearRank}}{43}{subsection.4.4.22}
\contentsline {subsection}{\numberline {4.4.23}\leavevmode {\color {Chapter }ElementaryRank}}{43}{subsection.4.4.23}
\contentsline {subsection}{\numberline {4.4.24}\leavevmode {\color {Chapter }StableRank}}{43}{subsection.4.4.24}
\contentsline {section}{\numberline {4.5}\leavevmode {\color {Chapter }Rings: Operations and Functions}}{43}{section.4.5}
\contentsline {chapter}{\numberline {5}\leavevmode {\color {Chapter }Ring Maps}}{44}{chapter.5}
\contentsline {section}{\numberline {5.1}\leavevmode {\color {Chapter }Ring Maps: Category and Representations}}{44}{section.5.1}
\contentsline {subsection}{\numberline {5.1.1}\leavevmode {\color {Chapter }IsHomalgRingMap}}{44}{subsection.5.1.1}
\contentsline {subsection}{\numberline {5.1.2}\leavevmode {\color {Chapter }IsHomalgRingSelfMap}}{44}{subsection.5.1.2}
\contentsline {subsection}{\numberline {5.1.3}\leavevmode {\color {Chapter }IsHomalgRingMapRep}}{44}{subsection.5.1.3}
\contentsline {section}{\numberline {5.2}\leavevmode {\color {Chapter }Ring Maps: Constructors}}{44}{section.5.2}
\contentsline {subsection}{\numberline {5.2.1}\leavevmode {\color {Chapter }RingMap (constructor for ring maps)}}{44}{subsection.5.2.1}
\contentsline {section}{\numberline {5.3}\leavevmode {\color {Chapter }Ring Maps: Properties}}{45}{section.5.3}
\contentsline {subsection}{\numberline {5.3.1}\leavevmode {\color {Chapter }IsMorphism (for ring maps)}}{45}{subsection.5.3.1}
\contentsline {subsection}{\numberline {5.3.2}\leavevmode {\color {Chapter }IsIdentityMorphism (for ring maps)}}{45}{subsection.5.3.2}
\contentsline {subsection}{\numberline {5.3.3}\leavevmode {\color {Chapter }IsMonomorphism (for ring maps)}}{45}{subsection.5.3.3}
\contentsline {subsection}{\numberline {5.3.4}\leavevmode {\color {Chapter }IsEpimorphism (for ring maps)}}{45}{subsection.5.3.4}
\contentsline {subsection}{\numberline {5.3.5}\leavevmode {\color {Chapter }IsIsomorphism (for ring maps)}}{45}{subsection.5.3.5}
\contentsline {subsection}{\numberline {5.3.6}\leavevmode {\color {Chapter }IsAutomorphism (for ring maps)}}{45}{subsection.5.3.6}
\contentsline {section}{\numberline {5.4}\leavevmode {\color {Chapter }Ring Maps: Attributes}}{46}{section.5.4}
\contentsline {subsection}{\numberline {5.4.1}\leavevmode {\color {Chapter }Source (for ring maps)}}{46}{subsection.5.4.1}
\contentsline {subsection}{\numberline {5.4.2}\leavevmode {\color {Chapter }Range (for ring maps)}}{46}{subsection.5.4.2}
\contentsline {subsection}{\numberline {5.4.3}\leavevmode {\color {Chapter }DegreeOfMorphism (for ring maps)}}{46}{subsection.5.4.3}
\contentsline {subsection}{\numberline {5.4.4}\leavevmode {\color {Chapter }CoordinateRingOfGraph (for ring maps)}}{46}{subsection.5.4.4}
\contentsline {subsection}{\numberline {5.4.5}\leavevmode {\color {Chapter }KernelSubmodule (for ring maps)}}{46}{subsection.5.4.5}
\contentsline {subsection}{\numberline {5.4.6}\leavevmode {\color {Chapter }KernelEmb (for ring maps)}}{46}{subsection.5.4.6}
\contentsline {section}{\numberline {5.5}\leavevmode {\color {Chapter }Ring Maps: Operations and Functions}}{46}{section.5.5}
\contentsline {subsection}{\numberline {5.5.1}\leavevmode {\color {Chapter }Kernel (for ring maps)}}{46}{subsection.5.5.1}
\contentsline {chapter}{\numberline {6}\leavevmode {\color {Chapter }Matrices}}{47}{chapter.6}
\contentsline {section}{\numberline {6.1}\leavevmode {\color {Chapter }Matrices: Category and Representations}}{47}{section.6.1}
\contentsline {subsection}{\numberline {6.1.1}\leavevmode {\color {Chapter }IsHomalgMatrix}}{47}{subsection.6.1.1}
\contentsline {subsection}{\numberline {6.1.2}\leavevmode {\color {Chapter }IsHomalgInternalMatrixRep}}{47}{subsection.6.1.2}
\contentsline {section}{\numberline {6.2}\leavevmode {\color {Chapter }Matrices: Constructors}}{47}{section.6.2}
\contentsline {subsection}{\numberline {6.2.1}\leavevmode {\color {Chapter }HomalgInitialMatrix (constructor for initial matrices filled with zeros)}}{47}{subsection.6.2.1}
\contentsline {subsection}{\numberline {6.2.2}\leavevmode {\color {Chapter }HomalgInitialIdentityMatrix (constructor for initial quadratic matrices with ones on the diagonal)}}{48}{subsection.6.2.2}
\contentsline {subsection}{\numberline {6.2.3}\leavevmode {\color {Chapter }HomalgZeroMatrix (constructor for zero matrices)}}{49}{subsection.6.2.3}
\contentsline {subsection}{\numberline {6.2.4}\leavevmode {\color {Chapter }HomalgIdentityMatrix (constructor for identity matrices)}}{49}{subsection.6.2.4}
\contentsline {subsection}{\numberline {6.2.5}\leavevmode {\color {Chapter }HomalgVoidMatrix (constructor for void matrices)}}{49}{subsection.6.2.5}
\contentsline {subsection}{\numberline {6.2.6}\leavevmode {\color {Chapter }HomalgMatrix (constructor for matrices using a listlist)}}{49}{subsection.6.2.6}
\contentsline {subsection}{\numberline {6.2.7}\leavevmode {\color {Chapter }HomalgDiagonalMatrix (constructor for diagonal matrices)}}{51}{subsection.6.2.7}
\contentsline {subsection}{\numberline {6.2.8}\leavevmode {\color {Chapter }\texttt {\char 92\relax }* (copy a matrix over a different ring)}}{51}{subsection.6.2.8}
\contentsline {section}{\numberline {6.3}\leavevmode {\color {Chapter }Matrices: Properties}}{52}{section.6.3}
\contentsline {subsection}{\numberline {6.3.1}\leavevmode {\color {Chapter }IsZero (for matrices)}}{52}{subsection.6.3.1}
\contentsline {subsection}{\numberline {6.3.2}\leavevmode {\color {Chapter }IsIdentityMatrix}}{52}{subsection.6.3.2}
\contentsline {subsection}{\numberline {6.3.3}\leavevmode {\color {Chapter }IsPermutationMatrix}}{52}{subsection.6.3.3}
\contentsline {subsection}{\numberline {6.3.4}\leavevmode {\color {Chapter }IsSpecialSubidentityMatrix}}{52}{subsection.6.3.4}
\contentsline {subsection}{\numberline {6.3.5}\leavevmode {\color {Chapter }IsSubidentityMatrix}}{53}{subsection.6.3.5}
\contentsline {subsection}{\numberline {6.3.6}\leavevmode {\color {Chapter }IsLeftRegularMatrix}}{53}{subsection.6.3.6}
\contentsline {subsection}{\numberline {6.3.7}\leavevmode {\color {Chapter }IsRightRegularMatrix}}{53}{subsection.6.3.7}
\contentsline {subsection}{\numberline {6.3.8}\leavevmode {\color {Chapter }IsInvertibleMatrix}}{53}{subsection.6.3.8}
\contentsline {subsection}{\numberline {6.3.9}\leavevmode {\color {Chapter }IsLeftInvertibleMatrix}}{53}{subsection.6.3.9}
\contentsline {subsection}{\numberline {6.3.10}\leavevmode {\color {Chapter }IsRightInvertibleMatrix}}{53}{subsection.6.3.10}
\contentsline {subsection}{\numberline {6.3.11}\leavevmode {\color {Chapter }IsEmptyMatrix}}{53}{subsection.6.3.11}
\contentsline {subsection}{\numberline {6.3.12}\leavevmode {\color {Chapter }IsDiagonalMatrix}}{53}{subsection.6.3.12}
\contentsline {subsection}{\numberline {6.3.13}\leavevmode {\color {Chapter }IsScalarlMatrix}}{54}{subsection.6.3.13}
\contentsline {subsection}{\numberline {6.3.14}\leavevmode {\color {Chapter }IsUpperTriangularMatrix}}{54}{subsection.6.3.14}
\contentsline {subsection}{\numberline {6.3.15}\leavevmode {\color {Chapter }IsLowerTriangularMatrix}}{54}{subsection.6.3.15}
\contentsline {subsection}{\numberline {6.3.16}\leavevmode {\color {Chapter }IsStrictUpperTriangularMatrix}}{54}{subsection.6.3.16}
\contentsline {subsection}{\numberline {6.3.17}\leavevmode {\color {Chapter }IsStrictLowerTriangularMatrix}}{54}{subsection.6.3.17}
\contentsline {subsection}{\numberline {6.3.18}\leavevmode {\color {Chapter }IsUpperStairCaseMatrix}}{54}{subsection.6.3.18}
\contentsline {subsection}{\numberline {6.3.19}\leavevmode {\color {Chapter }IsLowerStairCaseMatrix}}{54}{subsection.6.3.19}
\contentsline {subsection}{\numberline {6.3.20}\leavevmode {\color {Chapter }IsTriangularMatrix}}{54}{subsection.6.3.20}
\contentsline {subsection}{\numberline {6.3.21}\leavevmode {\color {Chapter }IsBasisOfRowsMatrix}}{55}{subsection.6.3.21}
\contentsline {subsection}{\numberline {6.3.22}\leavevmode {\color {Chapter }IsBasisOfColumnsMatrix}}{55}{subsection.6.3.22}
\contentsline {subsection}{\numberline {6.3.23}\leavevmode {\color {Chapter }IsReducedBasisOfRowsMatrix}}{55}{subsection.6.3.23}
\contentsline {subsection}{\numberline {6.3.24}\leavevmode {\color {Chapter }IsReducedBasisOfColumnsMatrix}}{55}{subsection.6.3.24}
\contentsline {subsection}{\numberline {6.3.25}\leavevmode {\color {Chapter }IsMutableMatrix}}{55}{subsection.6.3.25}
\contentsline {subsection}{\numberline {6.3.26}\leavevmode {\color {Chapter }IsInitialMatrix}}{55}{subsection.6.3.26}
\contentsline {subsection}{\numberline {6.3.27}\leavevmode {\color {Chapter }IsInitialIdentityMatrix}}{55}{subsection.6.3.27}
\contentsline {subsection}{\numberline {6.3.28}\leavevmode {\color {Chapter }IsVoidMatrix}}{55}{subsection.6.3.28}
\contentsline {section}{\numberline {6.4}\leavevmode {\color {Chapter }Matrices: Attributes}}{56}{section.6.4}
\contentsline {subsection}{\numberline {6.4.1}\leavevmode {\color {Chapter }NrRows}}{56}{subsection.6.4.1}
\contentsline {subsection}{\numberline {6.4.2}\leavevmode {\color {Chapter }NrColumns}}{56}{subsection.6.4.2}
\contentsline {subsection}{\numberline {6.4.3}\leavevmode {\color {Chapter }DeterminantMat}}{56}{subsection.6.4.3}
\contentsline {subsection}{\numberline {6.4.4}\leavevmode {\color {Chapter }ZeroRows}}{56}{subsection.6.4.4}
\contentsline {subsection}{\numberline {6.4.5}\leavevmode {\color {Chapter }ZeroColumns}}{56}{subsection.6.4.5}
\contentsline {subsection}{\numberline {6.4.6}\leavevmode {\color {Chapter }NonZeroRows}}{56}{subsection.6.4.6}
\contentsline {subsection}{\numberline {6.4.7}\leavevmode {\color {Chapter }NonZeroColumns}}{56}{subsection.6.4.7}
\contentsline {subsection}{\numberline {6.4.8}\leavevmode {\color {Chapter }PositionOfFirstNonZeroEntryPerRow}}{57}{subsection.6.4.8}
\contentsline {subsection}{\numberline {6.4.9}\leavevmode {\color {Chapter }PositionOfFirstNonZeroEntryPerColumn}}{57}{subsection.6.4.9}
\contentsline {subsection}{\numberline {6.4.10}\leavevmode {\color {Chapter }DegreesOfEntries}}{57}{subsection.6.4.10}
\contentsline {subsection}{\numberline {6.4.11}\leavevmode {\color {Chapter }RowRankOfMatrix}}{57}{subsection.6.4.11}
\contentsline {subsection}{\numberline {6.4.12}\leavevmode {\color {Chapter }ColumnRankOfMatrix}}{57}{subsection.6.4.12}
\contentsline {subsection}{\numberline {6.4.13}\leavevmode {\color {Chapter }LeftInverse (for matrices)}}{57}{subsection.6.4.13}
\contentsline {subsection}{\numberline {6.4.14}\leavevmode {\color {Chapter }RightInverse (for matrices)}}{57}{subsection.6.4.14}
\contentsline {section}{\numberline {6.5}\leavevmode {\color {Chapter }Matrices: Operations and Functions}}{58}{section.6.5}
\contentsline {subsection}{\numberline {6.5.1}\leavevmode {\color {Chapter }HomalgRing (for matrices)}}{58}{subsection.6.5.1}
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\contentsline {subsection}{\numberline {6.5.3}\leavevmode {\color {Chapter }CertainRows (for matrices)}}{58}{subsection.6.5.3}
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\contentsline {subsection}{\numberline {6.5.15}\leavevmode {\color {Chapter }GetRowIndependentUnitPositions (for matrices)}}{60}{subsection.6.5.15}
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\contentsline {subsection}{\numberline {6.5.22}\leavevmode {\color {Chapter }SyzygiesGeneratorsOfColumns (for matrices)}}{62}{subsection.6.5.22}
\contentsline {subsection}{\numberline {6.5.23}\leavevmode {\color {Chapter }SyzygiesGeneratorsOfRows (for pairs of matrices)}}{62}{subsection.6.5.23}
\contentsline {subsection}{\numberline {6.5.24}\leavevmode {\color {Chapter }SyzygiesGeneratorsOfColumns (for pairs of matrices)}}{62}{subsection.6.5.24}
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\contentsline {subsection}{\numberline {6.5.27}\leavevmode {\color {Chapter }ReducedSyzygiesGeneratorsOfRows (for matrices)}}{63}{subsection.6.5.27}
\contentsline {subsection}{\numberline {6.5.28}\leavevmode {\color {Chapter }ReducedSyzygiesGeneratorsOfColumns (for matrices)}}{63}{subsection.6.5.28}
\contentsline {subsection}{\numberline {6.5.29}\leavevmode {\color {Chapter }BasisOfRowsCoeff (for matrices)}}{63}{subsection.6.5.29}
\contentsline {subsection}{\numberline {6.5.30}\leavevmode {\color {Chapter }BasisOfColumnsCoeff (for matrices)}}{63}{subsection.6.5.30}
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\contentsline {subsection}{\numberline {6.5.32}\leavevmode {\color {Chapter }DecideZeroColumnsEffectively (for pairs of matrices)}}{64}{subsection.6.5.32}
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\contentsline {subsection}{\numberline {6.5.37}\leavevmode {\color {Chapter }SyzygiesOfColumns (for matrices)}}{65}{subsection.6.5.37}
\contentsline {subsection}{\numberline {6.5.38}\leavevmode {\color {Chapter }ReducedSyzygiesOfRows (for matrices)}}{65}{subsection.6.5.38}
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\contentsline {subsection}{\numberline {8.1.3}\leavevmode {\color {Chapter }IsHomalgGeneratorsOfRightModule}}{72}{subsection.8.1.3}
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\contentsline {chapter}{\numberline {9}\leavevmode {\color {Chapter }Modules}}{74}{chapter.9}
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\contentsline {subsection}{\numberline {9.2.1}\leavevmode {\color {Chapter }LeftPresentation (constructor for left modules)}}{75}{subsection.9.2.1}
\contentsline {subsection}{\numberline {9.2.2}\leavevmode {\color {Chapter }RightPresentation (constructor for right modules)}}{76}{subsection.9.2.2}
\contentsline {subsection}{\numberline {9.2.3}\leavevmode {\color {Chapter }HomalgFreeLeftModule (constructor for free left modules)}}{76}{subsection.9.2.3}
\contentsline {subsection}{\numberline {9.2.4}\leavevmode {\color {Chapter }HomalgFreeRightModule (constructor for free right modules)}}{76}{subsection.9.2.4}
\contentsline {subsection}{\numberline {9.2.5}\leavevmode {\color {Chapter }HomalgZeroLeftModule (constructor for zero left modules)}}{77}{subsection.9.2.5}
\contentsline {subsection}{\numberline {9.2.6}\leavevmode {\color {Chapter }HomalgZeroRightModule (constructor for zero right modules)}}{77}{subsection.9.2.6}
\contentsline {subsection}{\numberline {9.2.7}\leavevmode {\color {Chapter }\texttt {\char 92\relax }* (transfer a module over a different ring)}}{77}{subsection.9.2.7}
\contentsline {subsection}{\numberline {9.2.8}\leavevmode {\color {Chapter }Subobject (constructor for submodules using maps)}}{79}{subsection.9.2.8}
\contentsline {subsection}{\numberline {9.2.9}\leavevmode {\color {Chapter }Subobject (constructor for submodules using matrices)}}{79}{subsection.9.2.9}
\contentsline {subsection}{\numberline {9.2.10}\leavevmode {\color {Chapter }Subobject (constructor for submodules using a list of ring elements)}}{79}{subsection.9.2.10}
\contentsline {subsection}{\numberline {9.2.11}\leavevmode {\color {Chapter }LeftSubmodule (constructor for left submodules)}}{79}{subsection.9.2.11}
\contentsline {subsection}{\numberline {9.2.12}\leavevmode {\color {Chapter }RightSubmodule (constructor for right submodules)}}{80}{subsection.9.2.12}
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\contentsline {section}{\numberline {9.4}\leavevmode {\color {Chapter }Modules: Attributes}}{83}{section.9.4}
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\contentsline {subsection}{\numberline {9.4.2}\leavevmode {\color {Chapter }TheIdentityMorphism}}{83}{subsection.9.4.2}
\contentsline {subsection}{\numberline {9.4.3}\leavevmode {\color {Chapter }FullSubmodule}}{83}{subsection.9.4.3}
\contentsline {subsection}{\numberline {9.4.4}\leavevmode {\color {Chapter }EmbeddingInSuperObject}}{83}{subsection.9.4.4}
\contentsline {subsection}{\numberline {9.4.5}\leavevmode {\color {Chapter }FactorObject}}{83}{subsection.9.4.5}
\contentsline {subsection}{\numberline {9.4.6}\leavevmode {\color {Chapter }ResidueClassRing}}{83}{subsection.9.4.6}
\contentsline {subsection}{\numberline {9.4.7}\leavevmode {\color {Chapter }UnderlyingSubobject}}{83}{subsection.9.4.7}
\contentsline {subsection}{\numberline {9.4.8}\leavevmode {\color {Chapter }NatTrIdToHomHom{\textunderscore }R (for maps)}}{84}{subsection.9.4.8}
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\contentsline {section}{\numberline {9.5}\leavevmode {\color {Chapter }Modules: Operations and Functions}}{85}{section.9.5}
\contentsline {subsection}{\numberline {9.5.1}\leavevmode {\color {Chapter }HomalgRing (for modules)}}{85}{subsection.9.5.1}
\contentsline {subsection}{\numberline {9.5.2}\leavevmode {\color {Chapter }ByASmallerPresentation (for modules)}}{85}{subsection.9.5.2}
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\contentsline {subsection}{\numberline {9.5.4}\leavevmode {\color {Chapter }SuperObject (for submodules)}}{86}{subsection.9.5.4}
\contentsline {subsection}{\numberline {9.5.5}\leavevmode {\color {Chapter }\texttt {\char 92\relax }* (constructor for ideal multiples)}}{86}{subsection.9.5.5}
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\contentsline {chapter}{\numberline {10}\leavevmode {\color {Chapter }Maps}}{89}{chapter.10}
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\contentsline {subsection}{\numberline {10.2.1}\leavevmode {\color {Chapter }HomalgMap (constructor for maps)}}{90}{subsection.10.2.1}
\contentsline {subsection}{\numberline {10.2.2}\leavevmode {\color {Chapter }HomalgZeroMap (constructor for zero maps)}}{91}{subsection.10.2.2}
\contentsline {subsection}{\numberline {10.2.3}\leavevmode {\color {Chapter }HomalgIdentityMap (constructor for identity maps)}}{92}{subsection.10.2.3}
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\contentsline {subsection}{\numberline {10.3.1}\leavevmode {\color {Chapter }IsMorphism (for maps)}}{92}{subsection.10.3.1}
\contentsline {subsection}{\numberline {10.3.2}\leavevmode {\color {Chapter }IsGeneralizedMorphism (for maps)}}{92}{subsection.10.3.2}
\contentsline {subsection}{\numberline {10.3.3}\leavevmode {\color {Chapter }IsGeneralizedEpimorphism (for maps)}}{92}{subsection.10.3.3}
\contentsline {subsection}{\numberline {10.3.4}\leavevmode {\color {Chapter }IsGeneralizedMonomorphism (for maps)}}{93}{subsection.10.3.4}
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\contentsline {subsection}{\numberline {10.3.6}\leavevmode {\color {Chapter }IsIdentityMorphism (for maps)}}{93}{subsection.10.3.6}
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\contentsline {subsection}{\numberline {10.4.3}\leavevmode {\color {Chapter }DegreeOfMorphism (for maps)}}{94}{subsection.10.4.3}
\contentsline {subsection}{\numberline {10.4.4}\leavevmode {\color {Chapter }CokernelEpi (for maps)}}{94}{subsection.10.4.4}
\contentsline {subsection}{\numberline {10.4.5}\leavevmode {\color {Chapter }CokernelNaturalGeneralizedIsomorphism (for maps)}}{94}{subsection.10.4.5}
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\contentsline {chapter}{\numberline {11}\leavevmode {\color {Chapter }Complexes}}{98}{chapter.11}
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\contentsline {subsection}{\numberline {11.1.3}\leavevmode {\color {Chapter }IsCocomplexOfFinitelyPresentedObjectsRep}}{98}{subsection.11.1.3}
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\contentsline {subsection}{\numberline {11.2.2}\leavevmode {\color {Chapter }HomalgCocomplex (constructor for cocomplexes given a module)}}{100}{subsection.11.2.2}
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\contentsline {subsection}{\numberline {11.3.6}\leavevmode {\color {Chapter }IsGradedObject}}{101}{subsection.11.3.6}
\contentsline {subsection}{\numberline {11.3.7}\leavevmode {\color {Chapter }IsExactSequence}}{101}{subsection.11.3.7}
\contentsline {subsection}{\numberline {11.3.8}\leavevmode {\color {Chapter }IsShortExactSequence}}{102}{subsection.11.3.8}
\contentsline {subsection}{\numberline {11.3.9}\leavevmode {\color {Chapter }IsSplitShortExactSequence}}{102}{subsection.11.3.9}
\contentsline {subsection}{\numberline {11.3.10}\leavevmode {\color {Chapter }IsTriangle}}{102}{subsection.11.3.10}
\contentsline {subsection}{\numberline {11.3.11}\leavevmode {\color {Chapter }IsExactTriangle}}{102}{subsection.11.3.11}
\contentsline {section}{\numberline {11.4}\leavevmode {\color {Chapter }Complexes: Attributes}}{102}{section.11.4}
\contentsline {subsection}{\numberline {11.4.1}\leavevmode {\color {Chapter }BettiDiagram (for complexes)}}{102}{subsection.11.4.1}
\contentsline {section}{\numberline {11.5}\leavevmode {\color {Chapter }Complexes: Operations and Functions}}{102}{section.11.5}
\contentsline {subsection}{\numberline {11.5.1}\leavevmode {\color {Chapter }Add (to complexes given a map)}}{102}{subsection.11.5.1}
\contentsline {subsection}{\numberline {11.5.2}\leavevmode {\color {Chapter }ByASmallerPresentation (for complexes)}}{103}{subsection.11.5.2}
\contentsline {chapter}{\numberline {12}\leavevmode {\color {Chapter }Chain Maps}}{106}{chapter.12}
\contentsline {section}{\numberline {12.1}\leavevmode {\color {Chapter }ChainMaps: Categories and Representations}}{106}{section.12.1}
\contentsline {subsection}{\numberline {12.1.1}\leavevmode {\color {Chapter }IsHomalgChainMap}}{106}{subsection.12.1.1}
\contentsline {subsection}{\numberline {12.1.2}\leavevmode {\color {Chapter }IsHomalgChainSelfMap}}{106}{subsection.12.1.2}
\contentsline {subsection}{\numberline {12.1.3}\leavevmode {\color {Chapter }IsChainMapOfFinitelyPresentedObjectsRep}}{106}{subsection.12.1.3}
\contentsline {subsection}{\numberline {12.1.4}\leavevmode {\color {Chapter }IsCochainMapOfFinitelyPresentedObjectsRep}}{106}{subsection.12.1.4}
\contentsline {section}{\numberline {12.2}\leavevmode {\color {Chapter }Chain Maps: Constructors}}{107}{section.12.2}
\contentsline {subsection}{\numberline {12.2.1}\leavevmode {\color {Chapter }HomalgChainMap (constructor for chain maps given a map)}}{107}{subsection.12.2.1}
\contentsline {section}{\numberline {12.3}\leavevmode {\color {Chapter }Chain Maps: Properties}}{108}{section.12.3}
\contentsline {subsection}{\numberline {12.3.1}\leavevmode {\color {Chapter }IsMorphism (for chain maps)}}{108}{subsection.12.3.1}
\contentsline {subsection}{\numberline {12.3.2}\leavevmode {\color {Chapter }IsGeneralizedMorphism (for chain maps)}}{108}{subsection.12.3.2}
\contentsline {subsection}{\numberline {12.3.3}\leavevmode {\color {Chapter }IsGeneralizedEpimorphism (for chain maps)}}{108}{subsection.12.3.3}
\contentsline {subsection}{\numberline {12.3.4}\leavevmode {\color {Chapter }IsGeneralizedMonomorphism (for chain maps)}}{108}{subsection.12.3.4}
\contentsline {subsection}{\numberline {12.3.5}\leavevmode {\color {Chapter }IsGeneralizedIsomorphism (for chain maps)}}{108}{subsection.12.3.5}
\contentsline {subsection}{\numberline {12.3.6}\leavevmode {\color {Chapter }IsIdentityMorphism (for chain maps)}}{108}{subsection.12.3.6}
\contentsline {subsection}{\numberline {12.3.7}\leavevmode {\color {Chapter }IsMonomorphism (for chain maps)}}{108}{subsection.12.3.7}
\contentsline {subsection}{\numberline {12.3.8}\leavevmode {\color {Chapter }IsEpimorphism (for chain maps)}}{109}{subsection.12.3.8}
\contentsline {subsection}{\numberline {12.3.9}\leavevmode {\color {Chapter }IsSplitMonomorphism (for chain maps)}}{109}{subsection.12.3.9}
\contentsline {subsection}{\numberline {12.3.10}\leavevmode {\color {Chapter }IsSplitEpimorphism (for chain maps)}}{109}{subsection.12.3.10}
\contentsline {subsection}{\numberline {12.3.11}\leavevmode {\color {Chapter }IsIsomorphism (for chain maps)}}{109}{subsection.12.3.11}
\contentsline {subsection}{\numberline {12.3.12}\leavevmode {\color {Chapter }IsAutomorphism (for chain maps)}}{109}{subsection.12.3.12}
\contentsline {subsection}{\numberline {12.3.13}\leavevmode {\color {Chapter }IsGradedMorphism (for chain maps)}}{109}{subsection.12.3.13}
\contentsline {subsection}{\numberline {12.3.14}\leavevmode {\color {Chapter }IsQuasiIsomorphism (for chain maps)}}{109}{subsection.12.3.14}
\contentsline {section}{\numberline {12.4}\leavevmode {\color {Chapter }Chain Maps: Attributes}}{110}{section.12.4}
\contentsline {subsection}{\numberline {12.4.1}\leavevmode {\color {Chapter }Source (for chain maps)}}{110}{subsection.12.4.1}
\contentsline {subsection}{\numberline {12.4.2}\leavevmode {\color {Chapter }Range (for chain maps)}}{110}{subsection.12.4.2}
\contentsline {section}{\numberline {12.5}\leavevmode {\color {Chapter }Chain Maps: Operations and Functions}}{110}{section.12.5}
\contentsline {subsection}{\numberline {12.5.1}\leavevmode {\color {Chapter }ByASmallerPresentation (for chain maps)}}{110}{subsection.12.5.1}
\contentsline {chapter}{\numberline {13}\leavevmode {\color {Chapter }Bicomplexes}}{111}{chapter.13}
\contentsline {section}{\numberline {13.1}\leavevmode {\color {Chapter }Bicomplexes: Category and Representations}}{111}{section.13.1}
\contentsline {subsection}{\numberline {13.1.1}\leavevmode {\color {Chapter }IsHomalgBicomplex}}{111}{subsection.13.1.1}
\contentsline {subsection}{\numberline {13.1.2}\leavevmode {\color {Chapter }IsBicomplexOfFinitelyPresentedObjectsRep}}{111}{subsection.13.1.2}
\contentsline {subsection}{\numberline {13.1.3}\leavevmode {\color {Chapter }IsBicocomplexOfFinitelyPresentedObjectsRep}}{111}{subsection.13.1.3}
\contentsline {section}{\numberline {13.2}\leavevmode {\color {Chapter }Bicomplexes: Constructors}}{112}{section.13.2}
\contentsline {subsection}{\numberline {13.2.1}\leavevmode {\color {Chapter }HomalgBicomplex (constructor for bicomplexes given a complex of complexes)}}{112}{subsection.13.2.1}
\contentsline {section}{\numberline {13.3}\leavevmode {\color {Chapter }Bicomplexes: Properties}}{113}{section.13.3}
\contentsline {subsection}{\numberline {13.3.1}\leavevmode {\color {Chapter }IsBisequence}}{113}{subsection.13.3.1}
\contentsline {subsection}{\numberline {13.3.2}\leavevmode {\color {Chapter }IsBicomplex}}{113}{subsection.13.3.2}
\contentsline {subsection}{\numberline {13.3.3}\leavevmode {\color {Chapter }IsTransposedWRTTheAssociatedComplex}}{113}{subsection.13.3.3}
\contentsline {section}{\numberline {13.4}\leavevmode {\color {Chapter }Bicomplexes: Attributes}}{113}{section.13.4}
\contentsline {subsection}{\numberline {13.4.1}\leavevmode {\color {Chapter }TotalComplex}}{113}{subsection.13.4.1}
\contentsline {subsection}{\numberline {13.4.2}\leavevmode {\color {Chapter }SpectralSequence (for bicomplexes)}}{113}{subsection.13.4.2}
\contentsline {section}{\numberline {13.5}\leavevmode {\color {Chapter }Bicomplexes: Operations and Functions}}{113}{section.13.5}
\contentsline {subsection}{\numberline {13.5.1}\leavevmode {\color {Chapter }UnderlyingComplex}}{113}{subsection.13.5.1}
\contentsline {subsection}{\numberline {13.5.2}\leavevmode {\color {Chapter }ByASmallerPresentation (for bicomplexes)}}{113}{subsection.13.5.2}
\contentsline {chapter}{\numberline {14}\leavevmode {\color {Chapter }Bigraded Objects}}{115}{chapter.14}
\contentsline {section}{\numberline {14.1}\leavevmode {\color {Chapter }BigradedObjects: Categories and Representations}}{115}{section.14.1}
\contentsline {subsection}{\numberline {14.1.1}\leavevmode {\color {Chapter }IsHomalgBigradedObject}}{115}{subsection.14.1.1}
\contentsline {subsection}{\numberline {14.1.2}\leavevmode {\color {Chapter }IsHomalgBigradedObjectAssociatedToAnExactCouple}}{115}{subsection.14.1.2}
\contentsline {subsection}{\numberline {14.1.3}\leavevmode {\color {Chapter }IsHomalgBigradedObjectAssociatedToAFilteredComplex}}{115}{subsection.14.1.3}
\contentsline {subsection}{\numberline {14.1.4}\leavevmode {\color {Chapter }IsHomalgBigradedObjectAssociatedToABicomplex}}{115}{subsection.14.1.4}
\contentsline {subsection}{\numberline {14.1.5}\leavevmode {\color {Chapter }IsBigradedObjectOfFinitelyPresentedObjectsRep}}{116}{subsection.14.1.5}
\contentsline {section}{\numberline {14.2}\leavevmode {\color {Chapter }Bigraded Objects: Constructors}}{116}{section.14.2}
\contentsline {subsection}{\numberline {14.2.1}\leavevmode {\color {Chapter }HomalgBigradedObject (constructor for bigraded objects given a bicomplex)}}{116}{subsection.14.2.1}
\contentsline {subsection}{\numberline {14.2.2}\leavevmode {\color {Chapter }AsDifferentialObject (for homalg bigraded objects stemming from a bicomplex)}}{116}{subsection.14.2.2}
\contentsline {subsection}{\numberline {14.2.3}\leavevmode {\color {Chapter }DefectOfExactness (for homalg differential bigraded objects)}}{117}{subsection.14.2.3}
\contentsline {section}{\numberline {14.3}\leavevmode {\color {Chapter }Bigraded Objects: Properties}}{119}{section.14.3}
\contentsline {subsection}{\numberline {14.3.1}\leavevmode {\color {Chapter }IsEndowedWithDifferential}}{119}{subsection.14.3.1}
\contentsline {subsection}{\numberline {14.3.2}\leavevmode {\color {Chapter }IsStableSheet}}{119}{subsection.14.3.2}
\contentsline {section}{\numberline {14.4}\leavevmode {\color {Chapter }Bigraded Objects: Operations and Functions}}{119}{section.14.4}
\contentsline {subsection}{\numberline {14.4.1}\leavevmode {\color {Chapter }ByASmallerPresentation (for bigraded objects)}}{119}{subsection.14.4.1}
\contentsline {chapter}{\numberline {15}\leavevmode {\color {Chapter }Spectral Sequences}}{120}{chapter.15}
\contentsline {section}{\numberline {15.1}\leavevmode {\color {Chapter }SpectralSequences: Categorie and Representations}}{120}{section.15.1}
\contentsline {subsection}{\numberline {15.1.1}\leavevmode {\color {Chapter }IsHomalgSpectralSequence}}{120}{subsection.15.1.1}
\contentsline {subsection}{\numberline {15.1.2}\leavevmode {\color {Chapter }IsHomalgSpectralSequenceAssociatedToAnExactCouple}}{120}{subsection.15.1.2}
\contentsline {subsection}{\numberline {15.1.3}\leavevmode {\color {Chapter }IsHomalgSpectralSequenceAssociatedToAFilteredComplex}}{120}{subsection.15.1.3}
\contentsline {subsection}{\numberline {15.1.4}\leavevmode {\color {Chapter }IsHomalgSpectralSequenceAssociatedToABicomplex}}{121}{subsection.15.1.4}
\contentsline {subsection}{\numberline {15.1.5}\leavevmode {\color {Chapter }IsSpectralSequenceOfFinitelyPresentedObjectsRep}}{121}{subsection.15.1.5}
\contentsline {subsection}{\numberline {15.1.6}\leavevmode {\color {Chapter }IsSpectralCosequenceOfFinitelyPresentedObjectsRep}}{121}{subsection.15.1.6}
\contentsline {section}{\numberline {15.2}\leavevmode {\color {Chapter }Spectral Sequences: Constructors}}{121}{section.15.2}
\contentsline {subsection}{\numberline {15.2.1}\leavevmode {\color {Chapter }HomalgSpectralSequence (constructor for spectral sequences given a bicomplex)}}{121}{subsection.15.2.1}
\contentsline {section}{\numberline {15.3}\leavevmode {\color {Chapter }Spectral Sequences: Attributes}}{123}{section.15.3}
\contentsline {subsection}{\numberline {15.3.1}\leavevmode {\color {Chapter }GeneralizedEmbeddingsInTotalObjects}}{123}{subsection.15.3.1}
\contentsline {subsection}{\numberline {15.3.2}\leavevmode {\color {Chapter }GeneralizedEmbeddingsInTotalDefects}}{123}{subsection.15.3.2}
\contentsline {section}{\numberline {15.4}\leavevmode {\color {Chapter }Spectral Sequences: Operations and Functions}}{123}{section.15.4}
\contentsline {subsection}{\numberline {15.4.1}\leavevmode {\color {Chapter }ByASmallerPresentation (for spectral sequences)}}{123}{subsection.15.4.1}
\contentsline {chapter}{\numberline {16}\leavevmode {\color {Chapter }Functors}}{125}{chapter.16}
\contentsline {section}{\numberline {16.1}\leavevmode {\color {Chapter }Functors: Category and Representations}}{126}{section.16.1}
\contentsline {subsection}{\numberline {16.1.1}\leavevmode {\color {Chapter }IsHomalgFunctor}}{126}{subsection.16.1.1}
\contentsline {subsection}{\numberline {16.1.2}\leavevmode {\color {Chapter }IsHomalgFunctorRep}}{126}{subsection.16.1.2}
\contentsline {section}{\numberline {16.2}\leavevmode {\color {Chapter }Functors: Constructors}}{126}{section.16.2}
\contentsline {subsection}{\numberline {16.2.1}\leavevmode {\color {Chapter }CreateHomalgFunctor (constructor for functors)}}{126}{subsection.16.2.1}
\contentsline {subsection}{\numberline {16.2.2}\leavevmode {\color {Chapter }InsertObjectInMultiFunctor (constructor for functors given a multi-functor and an object)}}{126}{subsection.16.2.2}
\contentsline {subsection}{\numberline {16.2.3}\leavevmode {\color {Chapter }RightSatelliteOfCofunctor (constructor of the right satellite of a contravariant functor)}}{127}{subsection.16.2.3}
\contentsline {subsection}{\numberline {16.2.4}\leavevmode {\color {Chapter }LeftSatelliteOfFunctor (constructor of the left satellite of a covariant functor)}}{127}{subsection.16.2.4}
\contentsline {subsection}{\numberline {16.2.5}\leavevmode {\color {Chapter }RightDerivedCofunctor (constructor of the right derived functor of a contravariant functor)}}{128}{subsection.16.2.5}
\contentsline {subsection}{\numberline {16.2.6}\leavevmode {\color {Chapter }LeftDerivedFunctor (constructor of the left derived functor of a covariant functor)}}{128}{subsection.16.2.6}
\contentsline {subsection}{\numberline {16.2.7}\leavevmode {\color {Chapter }ComposeFunctors (constructor for functors given two functors)}}{128}{subsection.16.2.7}
\contentsline {section}{\numberline {16.3}\leavevmode {\color {Chapter }Functors: Attributes}}{129}{section.16.3}
\contentsline {subsection}{\numberline {16.3.1}\leavevmode {\color {Chapter }Genesis}}{129}{subsection.16.3.1}
\contentsline {section}{\numberline {16.4}\leavevmode {\color {Chapter }Basic Functors}}{130}{section.16.4}
\contentsline {subsection}{\numberline {16.4.1}\leavevmode {\color {Chapter }functor{\textunderscore }Cokernel}}{130}{subsection.16.4.1}
\contentsline {subsection}{\numberline {16.4.2}\leavevmode {\color {Chapter }Cokernel}}{131}{subsection.16.4.2}
\contentsline {subsection}{\numberline {16.4.3}\leavevmode {\color {Chapter }functor{\textunderscore }ImageModule}}{132}{subsection.16.4.3}
\contentsline {subsection}{\numberline {16.4.4}\leavevmode {\color {Chapter }ImageModule}}{132}{subsection.16.4.4}
\contentsline {subsection}{\numberline {16.4.5}\leavevmode {\color {Chapter }functor{\textunderscore }Kernel}}{133}{subsection.16.4.5}
\contentsline {subsection}{\numberline {16.4.6}\leavevmode {\color {Chapter }Kernel (for maps)}}{133}{subsection.16.4.6}
\contentsline {subsection}{\numberline {16.4.7}\leavevmode {\color {Chapter }functor{\textunderscore }DefectOfExactness}}{134}{subsection.16.4.7}
\contentsline {subsection}{\numberline {16.4.8}\leavevmode {\color {Chapter }DefectOfExactness}}{134}{subsection.16.4.8}
\contentsline {subsection}{\numberline {16.4.9}\leavevmode {\color {Chapter }Functor{\textunderscore }Hom}}{135}{subsection.16.4.9}
\contentsline {subsection}{\numberline {16.4.10}\leavevmode {\color {Chapter }Hom}}{136}{subsection.16.4.10}
\contentsline {subsection}{\numberline {16.4.11}\leavevmode {\color {Chapter }Functor{\textunderscore }TensorProduct}}{139}{subsection.16.4.11}
\contentsline {subsection}{\numberline {16.4.12}\leavevmode {\color {Chapter }TensorProduct}}{140}{subsection.16.4.12}
\contentsline {subsection}{\numberline {16.4.13}\leavevmode {\color {Chapter }Functor{\textunderscore }Ext}}{142}{subsection.16.4.13}
\contentsline {subsection}{\numberline {16.4.14}\leavevmode {\color {Chapter }Ext}}{142}{subsection.16.4.14}
\contentsline {subsection}{\numberline {16.4.15}\leavevmode {\color {Chapter }Functor{\textunderscore }Tor}}{143}{subsection.16.4.15}
\contentsline {subsection}{\numberline {16.4.16}\leavevmode {\color {Chapter }Tor}}{143}{subsection.16.4.16}
\contentsline {subsection}{\numberline {16.4.17}\leavevmode {\color {Chapter }Functor{\textunderscore }RHom}}{144}{subsection.16.4.17}
\contentsline {subsection}{\numberline {16.4.18}\leavevmode {\color {Chapter }RHom}}{144}{subsection.16.4.18}
\contentsline {subsection}{\numberline {16.4.19}\leavevmode {\color {Chapter }Functor{\textunderscore }LTensorProduct}}{147}{subsection.16.4.19}
\contentsline {subsection}{\numberline {16.4.20}\leavevmode {\color {Chapter }LTensorProduct}}{147}{subsection.16.4.20}
\contentsline {subsection}{\numberline {16.4.21}\leavevmode {\color {Chapter }Functor{\textunderscore }HomHom}}{150}{subsection.16.4.21}
\contentsline {subsection}{\numberline {16.4.22}\leavevmode {\color {Chapter }Functor{\textunderscore }LHomHom}}{150}{subsection.16.4.22}
\contentsline {section}{\numberline {16.5}\leavevmode {\color {Chapter }Tool Functors}}{150}{section.16.5}
\contentsline {section}{\numberline {16.6}\leavevmode {\color {Chapter }Other Functors}}{150}{section.16.6}
\contentsline {section}{\numberline {16.7}\leavevmode {\color {Chapter }Functors: Operations and Functions}}{150}{section.16.7}
\contentsline {subsection}{\numberline {16.7.1}\leavevmode {\color {Chapter }NameOfFunctor}}{150}{subsection.16.7.1}
\contentsline {subsection}{\numberline {16.7.2}\leavevmode {\color {Chapter }InstallFunctor}}{151}{subsection.16.7.2}
\contentsline {subsection}{\numberline {16.7.3}\leavevmode {\color {Chapter }InstallDeltaFunctor}}{151}{subsection.16.7.3}
\contentsline {chapter}{\numberline {17}\leavevmode {\color {Chapter }Examples}}{153}{chapter.17}
\contentsline {section}{\numberline {17.1}\leavevmode {\color {Chapter }ExtExt}}{153}{section.17.1}
\contentsline {section}{\numberline {17.2}\leavevmode {\color {Chapter }Purity}}{154}{section.17.2}
\contentsline {section}{\numberline {17.3}\leavevmode {\color {Chapter }TorExt-Grothendieck}}{156}{section.17.3}
\contentsline {section}{\numberline {17.4}\leavevmode {\color {Chapter }TorExt}}{158}{section.17.4}
\contentsline {chapter}{\numberline {A}\leavevmode {\color {Chapter }Development}}{160}{appendix.A}
\contentsline {section}{\numberline {A.1}\leavevmode {\color {Chapter }Why was \textsf {homalg} discontinued in \href {http://www.maplesoft.com/} {Maple}?}}{160}{section.A.1}
\contentsline {section}{\numberline {A.2}\leavevmode {\color {Chapter }Why \href {http://www.gap-system.org/} {GAP4}?}}{160}{section.A.2}
\contentsline {subsection}{\numberline {A.2.1}\leavevmode {\color {Chapter }\textsf {GAP} is free and open software}}{160}{subsection.A.2.1}
\contentsline {subsection}{\numberline {A.2.2}\leavevmode {\color {Chapter }\textsf {GAP} has an area of expertise}}{161}{subsection.A.2.2}
\contentsline {subsection}{\numberline {A.2.3}\leavevmode {\color {Chapter }\textsf {GAP4} can communicate}}{161}{subsection.A.2.3}
\contentsline {subsection}{\numberline {A.2.4}\leavevmode {\color {Chapter }\textsf {GAP4} is a \emph {mathematical} object-oriented programming language}}{161}{subsection.A.2.4}
\contentsline {subsection}{\numberline {A.2.5}\leavevmode {\color {Chapter }\textsf {GAP4} packages are easily extendible}}{162}{subsection.A.2.5}
\contentsline {section}{\numberline {A.3}\leavevmode {\color {Chapter }Why not \href {http://www.sagemath.org/} {Sage}?}}{162}{section.A.3}
\contentsline {section}{\numberline {A.4}\leavevmode {\color {Chapter }How does \textsf {homalg} compare to \textsf {Sage}?}}{162}{section.A.4}
\contentsline {subsection}{\numberline {A.4.1}\leavevmode {\color {Chapter }They differ in objectives and scale}}{162}{subsection.A.4.1}
\contentsline {subsection}{\numberline {A.4.2}\leavevmode {\color {Chapter }They differ in the programming language}}{162}{subsection.A.4.2}
\contentsline {subsection}{\numberline {A.4.3}\leavevmode {\color {Chapter }They differ in the way they communicate with the outer world}}{162}{subsection.A.4.3}
\contentsline {chapter}{\numberline {B}\leavevmode {\color {Chapter }The Mathematical Idea behind \textsf {homalg}}}{164}{appendix.B}
\contentsline {chapter}{\numberline {C}\leavevmode {\color {Chapter }The Basic Matrix Operations}}{165}{appendix.C}
\contentsline {section}{\numberline {C.1}\leavevmode {\color {Chapter }Main}}{165}{section.C.1}
\contentsline {section}{\numberline {C.2}\leavevmode {\color {Chapter }Effective}}{165}{section.C.2}
\contentsline {section}{\numberline {C.3}\leavevmode {\color {Chapter }Relative}}{165}{section.C.3}
\contentsline {section}{\numberline {C.4}\leavevmode {\color {Chapter }Reduced}}{166}{section.C.4}
\contentsline {chapter}{\numberline {D}\leavevmode {\color {Chapter }The Matrix Tool Operations}}{167}{appendix.D}
\contentsline {section}{\numberline {D.1}\leavevmode {\color {Chapter }The Tool Operations \emph {without} a Fallback Method}}{167}{section.D.1}
\contentsline {subsection}{\numberline {D.1.1}\leavevmode {\color {Chapter }ZeroMatrix (homalgTable entry for initial matrices)}}{167}{subsection.D.1.1}
\contentsline {subsection}{\numberline {D.1.2}\leavevmode {\color {Chapter }IdentityMatrix (homalgTable entry for initial identity matrices)}}{167}{subsection.D.1.2}
\contentsline {subsection}{\numberline {D.1.3}\leavevmode {\color {Chapter }ZeroMatrix (homalgTable entry)}}{167}{subsection.D.1.3}
\contentsline {subsection}{\numberline {D.1.4}\leavevmode {\color {Chapter }IdentityMatrix (homalgTable entry)}}{168}{subsection.D.1.4}
\contentsline {subsection}{\numberline {D.1.5}\leavevmode {\color {Chapter }Involution (homalgTable entry)}}{168}{subsection.D.1.5}
\contentsline {subsection}{\numberline {D.1.6}\leavevmode {\color {Chapter }CertainRows (homalgTable entry)}}{168}{subsection.D.1.6}
\contentsline {subsection}{\numberline {D.1.7}\leavevmode {\color {Chapter }CertainColumns (homalgTable entry)}}{168}{subsection.D.1.7}
\contentsline {subsection}{\numberline {D.1.8}\leavevmode {\color {Chapter }UnionOfRows (homalgTable entry)}}{168}{subsection.D.1.8}
\contentsline {subsection}{\numberline {D.1.9}\leavevmode {\color {Chapter }UnionOfColumns (homalgTable entry)}}{168}{subsection.D.1.9}
\contentsline {subsection}{\numberline {D.1.10}\leavevmode {\color {Chapter }DiagMat (homalgTable entry)}}{169}{subsection.D.1.10}
\contentsline {subsection}{\numberline {D.1.11}\leavevmode {\color {Chapter }KroneckerMat (homalgTable entry)}}{169}{subsection.D.1.11}
\contentsline {subsection}{\numberline {D.1.12}\leavevmode {\color {Chapter }MulMat (homalgTable entry)}}{169}{subsection.D.1.12}
\contentsline {subsection}{\numberline {D.1.13}\leavevmode {\color {Chapter }AddMat (homalgTable entry)}}{169}{subsection.D.1.13}
\contentsline {subsection}{\numberline {D.1.14}\leavevmode {\color {Chapter }SubMat (homalgTable entry)}}{169}{subsection.D.1.14}
\contentsline {subsection}{\numberline {D.1.15}\leavevmode {\color {Chapter }Compose (homalgTable entry)}}{169}{subsection.D.1.15}
\contentsline {subsection}{\numberline {D.1.16}\leavevmode {\color {Chapter }IsZeroMatrix (homalgTable entry)}}{170}{subsection.D.1.16}
\contentsline {subsection}{\numberline {D.1.17}\leavevmode {\color {Chapter }NrRows (homalgTable entry)}}{170}{subsection.D.1.17}
\contentsline {subsection}{\numberline {D.1.18}\leavevmode {\color {Chapter }NrColumns (homalgTable entry)}}{171}{subsection.D.1.18}
\contentsline {subsection}{\numberline {D.1.19}\leavevmode {\color {Chapter }Determinant (homalgTable entry)}}{172}{subsection.D.1.19}
\contentsline {section}{\numberline {D.2}\leavevmode {\color {Chapter }The Tool Operations with a Fallback Method}}{173}{section.D.2}
\contentsline {subsection}{\numberline {D.2.1}\leavevmode {\color {Chapter }AreEqualMatrices (homalgTable entry)}}{173}{subsection.D.2.1}
\contentsline {subsection}{\numberline {D.2.2}\leavevmode {\color {Chapter }IsIdentityMatrix (homalgTable entry)}}{173}{subsection.D.2.2}
\contentsline {subsection}{\numberline {D.2.3}\leavevmode {\color {Chapter }IsDiagonalMatrix (homalgTable entry)}}{174}{subsection.D.2.3}
\contentsline {subsection}{\numberline {D.2.4}\leavevmode {\color {Chapter }ZeroRows (homalgTable entry)}}{175}{subsection.D.2.4}
\contentsline {subsection}{\numberline {D.2.5}\leavevmode {\color {Chapter }ZeroColumns (homalgTable entry)}}{175}{subsection.D.2.5}
\contentsline {subsection}{\numberline {D.2.6}\leavevmode {\color {Chapter }GetColumnIndependentUnitPositions (homalgTable entry)}}{176}{subsection.D.2.6}
\contentsline {subsection}{\numberline {D.2.7}\leavevmode {\color {Chapter }GetRowIndependentUnitPositions (homalgTable entry)}}{177}{subsection.D.2.7}
\contentsline {subsection}{\numberline {D.2.8}\leavevmode {\color {Chapter }GetUnitPosition (homalgTable entry)}}{178}{subsection.D.2.8}
\contentsline {subsection}{\numberline {D.2.9}\leavevmode {\color {Chapter }DegreesOfEntries (homalgTable entry)}}{179}{subsection.D.2.9}
\contentsline {chapter}{\numberline {E}\leavevmode {\color {Chapter }Logic Subpackages}}{181}{appendix.E}
\contentsline {section}{\numberline {E.1}\leavevmode {\color {Chapter }\textsf {LIRNG}: Logical Implications for Rings}}{181}{section.E.1}
\contentsline {section}{\numberline {E.2}\leavevmode {\color {Chapter }\textsf {LIMAP}: Logical Implications for Ring Maps}}{181}{section.E.2}
\contentsline {section}{\numberline {E.3}\leavevmode {\color {Chapter }\textsf {LIMAT}: Logical Implications for Matrices}}{181}{section.E.3}
\contentsline {section}{\numberline {E.4}\leavevmode {\color {Chapter }\textsf {COLEM}: Clever Operations for Lazy Evaluated Matrices}}{181}{section.E.4}
\contentsline {subsection}{\numberline {E.4.1}\leavevmode {\color {Chapter }Eval (for matrices created with HomalgInitialMatrix)}}{181}{subsection.E.4.1}
\contentsline {subsection}{\numberline {E.4.2}\leavevmode {\color {Chapter }Eval (for matrices created with HomalgInitialIdentityMatrix)}}{182}{subsection.E.4.2}
\contentsline {subsection}{\numberline {E.4.3}\leavevmode {\color {Chapter }Eval (for matrices created with HomalgZeroMatrix)}}{183}{subsection.E.4.3}
\contentsline {subsection}{\numberline {E.4.4}\leavevmode {\color {Chapter }Eval (for matrices created with HomalgIdentityMatrix)}}{184}{subsection.E.4.4}
\contentsline {subsection}{\numberline {E.4.5}\leavevmode {\color {Chapter }Eval (for matrices created with Involution)}}{185}{subsection.E.4.5}
\contentsline {subsection}{\numberline {E.4.6}\leavevmode {\color {Chapter }Eval (for matrices created with CertainRows)}}{185}{subsection.E.4.6}
\contentsline {subsection}{\numberline {E.4.7}\leavevmode {\color {Chapter }Eval (for matrices created with CertainColumns)}}{186}{subsection.E.4.7}
\contentsline {subsection}{\numberline {E.4.8}\leavevmode {\color {Chapter }Eval (for matrices created with UnionOfRows)}}{187}{subsection.E.4.8}
\contentsline {subsection}{\numberline {E.4.9}\leavevmode {\color {Chapter }Eval (for matrices created with UnionOfColumns)}}{188}{subsection.E.4.9}
\contentsline {subsection}{\numberline {E.4.10}\leavevmode {\color {Chapter }Eval (for matrices created with DiagMat)}}{189}{subsection.E.4.10}
\contentsline {subsection}{\numberline {E.4.11}\leavevmode {\color {Chapter }Eval (for matrices created with KroneckerMat)}}{190}{subsection.E.4.11}
\contentsline {subsection}{\numberline {E.4.12}\leavevmode {\color {Chapter }Eval (for matrices created with MulMat)}}{191}{subsection.E.4.12}
\contentsline {subsection}{\numberline {E.4.13}\leavevmode {\color {Chapter }Eval (for matrices created with AddMat)}}{191}{subsection.E.4.13}
\contentsline {subsection}{\numberline {E.4.14}\leavevmode {\color {Chapter }Eval (for matrices created with SubMat)}}{192}{subsection.E.4.14}
\contentsline {subsection}{\numberline {E.4.15}\leavevmode {\color {Chapter }Eval (for matrices created with Compose)}}{193}{subsection.E.4.15}
\contentsline {subsection}{\numberline {E.4.16}\leavevmode {\color {Chapter }Eval (for matrices created with LeftInverse)}}{194}{subsection.E.4.16}
\contentsline {subsection}{\numberline {E.4.17}\leavevmode {\color {Chapter }Eval (for matrices created with RightInverse)}}{194}{subsection.E.4.17}
\contentsline {section}{\numberline {E.5}\leavevmode {\color {Chapter }\textsf {LIMOD}: Logical Implications for Modules}}{195}{section.E.5}
\contentsline {section}{\numberline {E.6}\leavevmode {\color {Chapter }\textsf {LIMOR}: Logical Implications for Morphisms}}{195}{section.E.6}
\contentsline {section}{\numberline {E.7}\leavevmode {\color {Chapter }\textsf {LICPX}: Logical Implications for Complexes}}{195}{section.E.7}
\contentsline {chapter}{\numberline {F}\leavevmode {\color {Chapter }The subpackage \textsf {ResidueClassRingForHomalg} as a sample ring package}}{196}{appendix.F}
\contentsline {section}{\numberline {F.1}\leavevmode {\color {Chapter }The Mandatory Basic Operations}}{196}{section.F.1}
\contentsline {subsection}{\numberline {F.1.1}\leavevmode {\color {Chapter }BasisOfRowModule (ResidueClassRing)}}{196}{subsection.F.1.1}
\contentsline {subsection}{\numberline {F.1.2}\leavevmode {\color {Chapter }BasisOfColumnModule (ResidueClassRing)}}{196}{subsection.F.1.2}
\contentsline {subsection}{\numberline {F.1.3}\leavevmode {\color {Chapter }DecideZeroRows (ResidueClassRing)}}{197}{subsection.F.1.3}
\contentsline {subsection}{\numberline {F.1.4}\leavevmode {\color {Chapter }DecideZeroColumns (ResidueClassRing)}}{197}{subsection.F.1.4}
\contentsline {subsection}{\numberline {F.1.5}\leavevmode {\color {Chapter }SyzygiesGeneratorsOfRows (ResidueClassRing)}}{197}{subsection.F.1.5}
\contentsline {subsection}{\numberline {F.1.6}\leavevmode {\color {Chapter }SyzygiesGeneratorsOfColumns (ResidueClassRing)}}{198}{subsection.F.1.6}
\contentsline {subsection}{\numberline {F.1.7}\leavevmode {\color {Chapter }BasisOfRowsCoeff (ResidueClassRing)}}{199}{subsection.F.1.7}
\contentsline {subsection}{\numberline {F.1.8}\leavevmode {\color {Chapter }BasisOfColumnsCoeff (ResidueClassRing)}}{199}{subsection.F.1.8}
\contentsline {subsection}{\numberline {F.1.9}\leavevmode {\color {Chapter }DecideZeroRowsEffectively (ResidueClassRing)}}{200}{subsection.F.1.9}
\contentsline {subsection}{\numberline {F.1.10}\leavevmode {\color {Chapter }DecideZeroColumnsEffectively (ResidueClassRing)}}{200}{subsection.F.1.10}
\contentsline {subsection}{\numberline {F.1.11}\leavevmode {\color {Chapter }RelativeSyzygiesGeneratorsOfRows (ResidueClassRing)}}{201}{subsection.F.1.11}
\contentsline {subsection}{\numberline {F.1.12}\leavevmode {\color {Chapter }RelativeSyzygiesGeneratorsOfColumns (ResidueClassRing)}}{201}{subsection.F.1.12}
\contentsline {section}{\numberline {F.2}\leavevmode {\color {Chapter }The Mandatory Tool Operations}}{202}{section.F.2}
\contentsline {subsection}{\numberline {F.2.1}\leavevmode {\color {Chapter }ZeroMatrix (ResidueClassRing)}}{202}{subsection.F.2.1}
\contentsline {subsection}{\numberline {F.2.2}\leavevmode {\color {Chapter }IdentityMatrix (ResidueClassRing)}}{202}{subsection.F.2.2}
\contentsline {subsection}{\numberline {F.2.3}\leavevmode {\color {Chapter }Involution (ResidueClassRing)}}{202}{subsection.F.2.3}
\contentsline {subsection}{\numberline {F.2.4}\leavevmode {\color {Chapter }CertainRows (ResidueClassRing)}}{203}{subsection.F.2.4}
\contentsline {subsection}{\numberline {F.2.5}\leavevmode {\color {Chapter }CertainColumns (ResidueClassRing)}}{203}{subsection.F.2.5}
\contentsline {subsection}{\numberline {F.2.6}\leavevmode {\color {Chapter }UnionOfRows (ResidueClassRing)}}{204}{subsection.F.2.6}
\contentsline {subsection}{\numberline {F.2.7}\leavevmode {\color {Chapter }UnionOfColumns (ResidueClassRing)}}{204}{subsection.F.2.7}
\contentsline {subsection}{\numberline {F.2.8}\leavevmode {\color {Chapter }DiagMat (ResidueClassRing)}}{205}{subsection.F.2.8}
\contentsline {subsection}{\numberline {F.2.9}\leavevmode {\color {Chapter }KroneckerMat (ResidueClassRing)}}{205}{subsection.F.2.9}
\contentsline {subsection}{\numberline {F.2.10}\leavevmode {\color {Chapter }MulMat (ResidueClassRing)}}{206}{subsection.F.2.10}
\contentsline {subsection}{\numberline {F.2.11}\leavevmode {\color {Chapter }AddMat (ResidueClassRing)}}{206}{subsection.F.2.11}
\contentsline {subsection}{\numberline {F.2.12}\leavevmode {\color {Chapter }SubMat (ResidueClassRing)}}{206}{subsection.F.2.12}
\contentsline {subsection}{\numberline {F.2.13}\leavevmode {\color {Chapter }Compose (ResidueClassRing)}}{206}{subsection.F.2.13}
\contentsline {subsection}{\numberline {F.2.14}\leavevmode {\color {Chapter }IsZeroMatrix (ResidueClassRing)}}{207}{subsection.F.2.14}
\contentsline {subsection}{\numberline {F.2.15}\leavevmode {\color {Chapter }NrRows (ResidueClassRing)}}{207}{subsection.F.2.15}
\contentsline {subsection}{\numberline {F.2.16}\leavevmode {\color {Chapter }NrColumns (ResidueClassRing)}}{207}{subsection.F.2.16}
\contentsline {subsection}{\numberline {F.2.17}\leavevmode {\color {Chapter }Determinant (ResidueClassRing)}}{207}{subsection.F.2.17}
\contentsline {section}{\numberline {F.3}\leavevmode {\color {Chapter }Some of the Recommended Tool Operations}}{207}{section.F.3}
\contentsline {subsection}{\numberline {F.3.1}\leavevmode {\color {Chapter }AreEqualMatrices (ResidueClassRing)}}{207}{subsection.F.3.1}
\contentsline {subsection}{\numberline {F.3.2}\leavevmode {\color {Chapter }IsIdentityMatrix (ResidueClassRing)}}{208}{subsection.F.3.2}
\contentsline {subsection}{\numberline {F.3.3}\leavevmode {\color {Chapter }IsDiagonalMatrix (ResidueClassRing)}}{208}{subsection.F.3.3}
\contentsline {subsection}{\numberline {F.3.4}\leavevmode {\color {Chapter }ZeroRows (ResidueClassRing)}}{208}{subsection.F.3.4}
\contentsline {subsection}{\numberline {F.3.5}\leavevmode {\color {Chapter }ZeroColumns (ResidueClassRing)}}{208}{subsection.F.3.5}
\contentsline {chapter}{\numberline {G}\leavevmode {\color {Chapter }Debugging \textsf {homalg}}}{209}{appendix.G}
\contentsline {section}{\numberline {G.1}\leavevmode {\color {Chapter }Increase the assertion level}}{209}{section.G.1}
\contentsline {section}{\numberline {G.2}\leavevmode {\color {Chapter }Use \texttt {homalgMode}}}{211}{section.G.2}
\contentsline {subsection}{\numberline {G.2.1}\leavevmode {\color {Chapter }homalgMode}}{211}{subsection.G.2.1}
\contentsline {chapter}{\numberline {H}\leavevmode {\color {Chapter }The Core Packages and the Idea behind their Splitting}}{213}{appendix.H}
\contentsline {section}{\numberline {H.1}\leavevmode {\color {Chapter }The 5=1+4 split}}{213}{section.H.1}
\contentsline {subsection}{\numberline {H.1.1}\leavevmode {\color {Chapter }Logically independent}}{213}{subsection.H.1.1}
\contentsline {subsection}{\numberline {H.1.2}\leavevmode {\color {Chapter }Black boxes}}{213}{subsection.H.1.2}
\contentsline {subsection}{\numberline {H.1.3}\leavevmode {\color {Chapter }Summing up}}{214}{subsection.H.1.3}
\contentsline {section}{\numberline {H.2}\leavevmode {\color {Chapter }The 4=1+1+1+1 split}}{214}{section.H.2}
\contentsline {subsection}{\numberline {H.2.1}\leavevmode {\color {Chapter }\textsf {HomalgToCAS}}}{214}{subsection.H.2.1}
\contentsline {subsection}{\numberline {H.2.2}\leavevmode {\color {Chapter }\textsf {IO{\textunderscore }ForHomalg} and Alternatives}}{214}{subsection.H.2.2}
\contentsline {subsection}{\numberline {H.2.3}\leavevmode {\color {Chapter }\textsf {RingsForHomalg}}}{214}{subsection.H.2.3}
\contentsline {subsection}{\numberline {H.2.4}\leavevmode {\color {Chapter }Your own \textsf {RingsForHomalg}}}{215}{subsection.H.2.4}
\contentsline {subsection}{\numberline {H.2.5}\leavevmode {\color {Chapter }\textsf {ExamplesForHomalg}}}{215}{subsection.H.2.5}
\contentsline {subsection}{\numberline {H.2.6}\leavevmode {\color {Chapter }Documentation}}{215}{subsection.H.2.6}
\contentsline {subsection}{\numberline {H.2.7}\leavevmode {\color {Chapter }Crediting}}{215}{subsection.H.2.7}
\contentsline {subsection}{\numberline {H.2.8}\leavevmode {\color {Chapter }Stability}}{215}{subsection.H.2.8}
\contentsline {chapter}{\numberline {I}\leavevmode {\color {Chapter }Overview of the \textsf {homalg} Package Source Code}}{216}{appendix.I}
\contentsline {section}{\numberline {I.1}\leavevmode {\color {Chapter }Rings, Ring Maps, Matrices, Relations, and Generators}}{216}{section.I.1}
\contentsline {section}{\numberline {I.2}\leavevmode {\color {Chapter }The Basic Objects}}{217}{section.I.2}
\contentsline {section}{\numberline {I.3}\leavevmode {\color {Chapter }The Low Level Algorithms}}{217}{section.I.3}
\contentsline {section}{\numberline {I.4}\leavevmode {\color {Chapter }The High Level Algorithms}}{218}{section.I.4}
\contentsline {section}{\numberline {I.5}\leavevmode {\color {Chapter }Logical Implications for \textsf {homalg} Objects}}{219}{section.I.5}
\contentsline {section}{\numberline {I.6}\leavevmode {\color {Chapter }The subpackage \textsf {ResidueClassRingForHomalg}}}{219}{section.I.6}
\contentsline {section}{\numberline {I.7}\leavevmode {\color {Chapter }The homalgTable for \textsf {GAP4} built-in rings}}{219}{section.I.7}
