chap12.txt

  
  12 Exterior Algebra and Koszul Complex
  
  What  follows  are  several  operations related to the exterior algebra of a
  free module:
  
      A  constructor  for the graded parts of the exterior algebra (exterior
        powers)
  
      Several Operations on elements of these exterior powers
  
      A constructor for the Koszul complex
  
      An  implementation  of  the  Cayley  determinant as defined in [CQ11],
        which  allows  calculating  greatest  common divisors from finite free
        resolutions.
  
  
  12.1 Exterior Algebra: Constructor
  
  12.1-1 ExteriorPower
  
  ExteriorPower( k, M )  operation
  Returns:  a homalg module
  
  Construct the k-th exterior power of module M.
  
  
  12.2 Exterior Algebra: Properties and Attributes
  
  12.2-1 IsExteriorPower
  
  IsExteriorPower( M )  property
  Returns:  true or false
  
  Marks a module as an exterior power of another module.
  
  12.2-2 ExteriorPowerExponent
  
  ExteriorPowerExponent( M )  attribute
  Returns:  an integer
  
  The exponent of the exterior power.
  
  12.2-3 ExteriorPowerBaseModule
  
  ExteriorPowerBaseModule( M )  attribute
  Returns:  a homalg module
  
  The module that M is an exterior power of.
  
  
  12.3 Exterior Algebra: Element Properties
  
  12.3-1 IsExteriorPowerElement
  
  IsExteriorPowerElement( x )  property
  Returns:  true or false
  
  Checks if the element x is from an exterior power.
  
  
  12.4 Exterior Algebra: Element Operations
  
  12.4-1 Wedge
  
  Wedge( x, y )  operation
  Returns:  an element of an exterior power
  
  Calculate x ∧ y.
  
  12.4-2 ExteriorPowerElementDual
  
  ExteriorPowerElementDual( x )  operation
  Returns:  an element of an exterior power
  
  For  x in a q-th exterior power of a free module of rank n, return x* in the
  (n-q)-th exterior power, as defined in [CQ11].
  
  12.4-3 SingleValueOfExteriorPowerElement
  
  SingleValueOfExteriorPowerElement( x )  operation
  Returns:  a ring element
  
  For  x  in  a  highest  exterior power, returns its single coordinate in the
  canonical basis; i.e. [x] as defined in [CQ11].
  
  
  12.5 Koszul complex and Cayley determinant
  
  12.5-1 KoszulCocomplex
  
  KoszulCocomplex( a, E )  operation
  Returns:  a homalg cocomplex
  
  Calculate the E-valued Koszul complex of a.
  
  12.5-2 CayleyDeterminant
  
  CayleyDeterminant( C )  operation
  Returns:  a ring element
  
  Calculate the Cayley determinant of the complex C, as defined in [CQ11].
  
  12.5-3 Gcd_UsingCayleyDeterminant
  
  Gcd_UsingCayleyDeterminant( x, y[, ...] )  function
  Returns:  a ring element
  
  Returns  the  greatest common divisor of the given ring elements, calculated
  using the Cayley determinant.