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### 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 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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