Chain Complexes

• Complexes of Modules
  • Creation
    • Complex(L, d): List, RngIntElt → ModCpx
    • Complex(f, d): Map, RngIntElt → ModCpx
    • ZeroComplex(A, m, n): AlgBas, RngIntElt, RngIntElt → ModCpx
    • Dual(C): ModCpx → ModCpx
    • Dual(C, n): ModCpx, RngIntElt → ModCpx
  • Subcomplexes and Quotient Complexes
    • sub< C | Q >: ModCpx, SeqEnum[ModAlg] → ModCpx, MapChn
    • sub< C | L >: ModCpx, List → ModCpx, MapChn
    • RandomSubcomplex(C, Q): ModCpx, SeqEnum → ModCpx, MapChn
    • quo< C | D >: ModCpx, ModCpx → ModCpx
    • quo< C | S >: ModCpx, SeqEnum[ModAlg] → ModCpx
    • quo< C | L >: ModCpx, List → ModCpx
  • Access Functions
    • Degrees(C): ModCpx → RngIntElt
    • Algebra(C): ModCpx → AlgBas
    • BoundaryMap(C, n): ModCpx, RngIntElt → ModMatRngElt
    • BoundaryMaps(C): ModCpx → List
    • DimensionsOfHomology(C): ModCpx → SeqEnum
    • DimensionOfHomology(C, n): ModCpx, RngIntElt → RngIntElt
    • DimensionsOfTerms(C): ModCpx → SeqEnum
    • Term(C, n): ModCpx, RngIntElt → ModAlg
    • Terms(C): ModCpx → SeqEnum
  • Elementary Operations
    • DirectSum(C, D): ModCpx, ModCpx → ModCpx
    • Homology(C): ModCpx → SeqEnum
    • HomologyOfChainComplex(C): ModCpx → SeqEnum
    • Homology(C, n): ModCpx, RngIntElt → SeqEnum
    • Prune(C): ModCpx → ModCpx
    • Prune(C,n): ModCpx, RngIntElt → ModCpx
    • Preprune(C): ModCpx → ModCpx
    • Preprune(C,n): ModCpx, RngIntElt → ModCpx
    • Shift(C, n): ModCpx, RngIntElt → ModCpx
    • ShiftToDegreeZero(C): ModCpx → ModCpx
    • Splice(C, D): ModCpx, ModCpx → ModCpx
    • Splice(C, D, f): ModCpx, ModCpx, ModMatRngElt → ModCpx
  • Extensions
    • LeftExactExtension(C): ModCpx → ModCpx
    • RightExactExtension(C): ModCpx → ModCpx
    • ExactExtension(C): ModCpx → ModCpx
    • LeftZeroExtension(C): ModCpx → ModCpx
    • LeftZeroExtension(C, n): ModCpx, RngIntElt → ModCpx
    • RightZeroExtension(C): ModCpx → ModCpx
    • RightZeroExtension(C, n): ModCpx, RngIntElt → ModCpx
    • ZeroExtension(C): ModCpx → ModCpx
    • EqualizeDegrees(C, D): ModCpx, ModCpx → ModCpx, ModCpx
    • EqualizeDegrees(Q): SeqEnum → SeqEnum
    • EqualizeDegrees(C, D, n): ModCpx, ModCpx, RngIntElt → ModCpx, ModCpx
  • Predicates
    • IsExact(C): ModComplex → BoolElt
    • IsShortExactSequence(C): ModCpx → BoolElt, RngIntElt
    • IsExact(C, n): ModCpx, RngIntElt → BoolElt
    • IsZeroComplex(C): ModCpx → BoolElt
    • IsZeroMap(C, n): ModCpx, RngIntElt → BoolElt
    • IsZeroTerm(C, n): ModCpx, RngIntElt → BoolElt
    • Example: Complexes
• Chain Maps
  • Creation
    • ChainMap(Q, C, D, n): SeqEnum, ModCpx, ModCpx, RngIntElt → MapChn
    • ZeroChainMap(C, D): ModCpx, ModCpx → MapChn
  • Access Functions
    • Degree(f): MapChn → RngIntElt
    • ModuleMap(f, n): MapChn, RngIntElt → ModMatRngElt
    • Kernel(f): MapChn → ModCpx, MapChn
    • Cokernel(f): MapChn → ModCpx, MapChn
    • Image(f): MapChn → ModCpx, MapChn, MapChn
  • Elementary Operations
    • f + g: MapChn, MapChn → MapChn
    • a * g: RngElt, MapChn → MapChn
    • f * g: MapChn, MapChn → MapChn
  • Predicates
    • IsSurjective(f): MapChn → BoolElt
    • IsInjective(f): MapChn → BoolElt
    • IsZero(f): MapChn → BoolElt
    • IsIsomorphism(f): MapChn → BoolElt
    • IsShortExactSequence(f, g): MapChn, MapChn → BoolElt
    • IsChainMap(L, C, D, n): List, ModCpx, ModCpx, RngIntElt → BoolElt
    • IsChainMap(f): MapChn → BoolElt
    • IsProperChainMap(f): MapChn → BoolElt
    • HasDefinedModuleMap(C,n): ModCpx, RngIntElt → BoolElt
    • Example: Chainmaps
  • Maps on Homology
    • InducedMapOnHomology(f, n): MapChn, RngIntElt → ModTupFldElt
    • ConnectingHomomorphism(f, g, n): MapChn, MapChn, RngIntElt → ModMatRngElt
    • LongExactSequenceOnHomology(f, g): MapChn, MapChn → ModCpx
    • Example: Long Exact Sequence