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.
‣ ExteriorPower( k, M ) | ( operation ) |
Returns: a homalg module
Construct the k-th exterior power of module M.
‣ IsExteriorPower( M ) | ( property ) |
Returns: true or false
Marks a module as an exterior power of another module.
‣ ExteriorPowerExponent( M ) | ( attribute ) |
Returns: an integer
The exponent of the exterior power.
‣ ExteriorPowerBaseModule( M ) | ( attribute ) |
Returns: a homalg module
The module that M is an exterior power of.
‣ IsExteriorPowerElement( x ) | ( property ) |
Returns: true or false
Checks if the element x is from an exterior power.
‣ Wedge( x, y ) | ( operation ) |
Returns: an element of an exterior power
Calculate \(\textit{x} \wedge \textit{y}\).
‣ 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 \(\textit{x}*\) in the (n-q)-th exterior power, as defined in [CQ11].
‣ SingleValueOfExteriorPowerElement( x ) | ( operation ) |
Returns: a ring element
For x in a highest exterior power, returns its single coordinate in the canonical basis; i.e. \([\textit{x}]\) as defined in [CQ11].
‣ KoszulCocomplex( a, E ) | ( operation ) |
Returns: a homalg cocomplex
Calculate the E-valued Koszul complex of a.
‣ CayleyDeterminant( C ) | ( operation ) |
Returns: a ring element
Calculate the Cayley determinant of the complex C, as defined in [CQ11].
‣ Gcd_UsingCayleyDeterminant( x, y[, ...] ) | ( function ) |
Returns: a ring element
Returns the greatest common divisor of the given ring elements, calculated using the Cayley determinant.
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