GAP (Modules) - Chapter 12: Exterior Algebra and Koszul Complex

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12 Exterior Algebra and Koszul Complex
12.1 Exterior Algebra: Constructor
12.1-1 ExteriorPower
12.2 Exterior Algebra: Properties and Attributes
12.2-1 IsExteriorPower
12.2-2 ExteriorPowerExponent
12.2-3 ExteriorPowerBaseModule
12.3 Exterior Algebra: Element Properties
12.3-1 IsExteriorPowerElement
12.4 Exterior Algebra: Element Operations
12.4-1 Wedge
12.4-2 ExteriorPowerElementDual
12.4-3 SingleValueOfExteriorPowerElement
12.5 Koszul complex and Cayley determinant
12.5-1 KoszulCocomplex
12.5-2 CayleyDeterminant
12.5-3 Gcd_UsingCayleyDeterminant

### 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 \(\textit{x} \wedge \textit{y}\).

##### 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 \(\textit{x}*\) 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. \([\textit{x}]\) 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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