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12 Exterior Algebra and Koszul Complex
 12.1 Exterior Algebra: Constructor
 12.2 Exterior Algebra: Properties and Attributes
 12.3 Exterior Algebra: Element Properties
 12.4 Exterior Algebra: Element Operations
 12.5 Koszul complex and Cayley determinant

12 Exterior Algebra and Koszul Complex

What follows are several operations related to the exterior algebra of a free module:

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 xy.

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.

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