GAP (HAP commands) - Chapter 8: Functors

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8 Functors
8.1
8.1-1 ExtendScalars
8.1-2 HomToIntegers
8.1-3 HomToIntegersModP
8.1-4 HomToIntegralModule
8.1-5 TensorWithIntegralModule
8.1-6 HomToGModule
8.1-7 InduceScalars
8.1-8 LowerCentralSeriesLieAlgebra
8.1-9 TensorWithIntegers
8.1-10 FilteredTensorWithIntegers
8.1-11 TensorWithTwistedIntegers
8.1-12 TensorWithIntegersModP
8.1-13 TensorWithTwistedIntegersModP
8.1-14 TensorWithRationals

### 8 Functors

#### 8.1

##### 8.1-1 ExtendScalars

`‣ ExtendScalars`( R, G, EltsG ) | ( function )

Inputs a ZH-resolution R, a group G containing H as a subgroup, and a list EltsG of elements of G. It returns the free ZG-resolution (R ⊗_ZH ZG). The returned resolution S has S!.elts:=EltsG. This is a resolution of the ZG-module (Z ⊗_ZH ZG). (Here ⊗_ZH means tensor over ZH.)

Examples:

##### 8.1-2 HomToIntegers

`‣ HomToIntegers`( X ) | ( function )

Inputs either a ZG-resolution X=R, or an equivariant chain map X = (F:R ⟶ S). It returns the cochain complex or cochain map obtained by applying HomZG( _ , Z) where Z is the trivial module of integers (characteristic 0).

Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9

##### 8.1-3 HomToIntegersModP

`‣ HomToIntegersModP`( R ) | ( function )

Inputs a ZG-resolution R and returns the cochain complex obtained by applying HomZG( _ , Z_p) where Z_p is the trivial module of integers mod p. (At present this functor does not handle equivariant chain maps.)

Examples: 1 , 2 , 3 , 4

##### 8.1-4 HomToIntegralModule

`‣ HomToIntegralModule`( R, f ) | ( function )

Inputs a ZG-resolution R and a group homomorphism f:G ⟶ GL_n(Z) to the group of n×n invertible integer matrices. Here Z must have characteristic 0. It returns the cochain complex obtained by applying HomZG( _ , A) where A is the ZG-module Z^n with G action via f. (At present this function does not handle equivariant chain maps.)

Examples: 1 , 2 , 3 , 4

##### 8.1-5 TensorWithIntegralModule

`‣ TensorWithIntegralModule`( R, f ) | ( function )

Inputs a ZG-resolution R and a group homomorphism f:G ⟶ GL_n(Z) to the group of n×n invertible integer matrices. Here Z must have characteristic 0. It returns the chain complex obtained by tensoring over ZG with the ZG-module A=Z^n with G action via f. (At present this function does not handle equivariant chain maps.)

Examples: 1

##### 8.1-6 HomToGModule

`‣ HomToGModule`( R, A ) | ( function )

Inputs a ZG-resolution R and an abelian G-outer group A. It returns the G-cocomplex obtained by applying HomZG( _ , A). (At present this function does not handle equivariant chain maps.)

Examples: 1 , 2 , 3 , 4

##### 8.1-7 InduceScalars

`‣ InduceScalars`( R, hom ) | ( function )

Inputs a ZQ-resolution R and a surjective group homomorphism hom:G→ Q. It returns the unduced non-free ZG-resolution.

Examples:

##### 8.1-8 LowerCentralSeriesLieAlgebra

`‣ LowerCentralSeriesLieAlgebra`( G ) | ( function )

`‣ LowerCentralSeriesLieAlgebra`( f ) | ( function )

Inputs a pcp group G. If each quotient G_c/G_c+1 of the lower central series is free abelian or p-elementary abelian (for fixed prime p) then a Lie algebra L(G) is returned. The abelian group underlying L(G) is the direct sum of the quotients G_c/G_c+1 . The Lie bracket on L(G) is induced by the commutator in G. (Here G_1=G, G_c+1=[G_c,G] .)

The function can also be applied to a group homomorphism f: G ⟶ G' . In this case the induced homomorphism of Lie algebras L(f):L(G) ⟶ L(G') is returned.

If the quotients of the lower central series are not all free or p-elementary abelian then the function returns fail.

This function was written by Pablo Fernandez Ascariz

Examples: 1 , 2 , 3

##### 8.1-9 TensorWithIntegers

`‣ TensorWithIntegers`( X ) | ( function )

Inputs either a ZG-resolution X=R, or an equivariant chain map X = (F:R ⟶ S). It returns the chain complex or chain map obtained by tensoring with the trivial module of integers (characteristic 0).

Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30

##### 8.1-10 FilteredTensorWithIntegers

`‣ FilteredTensorWithIntegers`( R ) | ( function )

Inputs a ZG-resolution R for which "filteredDimension" lies in NamesOfComponents(R). (Such a resolution can be produced using TwisterTensorProduct(), ResolutionNormalSubgroups() or FreeGResolution().) It returns the filtered chain complex obtained by tensoring with the trivial module of integers (characteristic 0).

Examples: 1 , 2

##### 8.1-11 TensorWithTwistedIntegers

`‣ TensorWithTwistedIntegers`( X, rho ) | ( function )

Inputs either a ZG-resolution X=R, or an equivariant chain map X = (F:R ⟶ S). It also inputs a function rho: G→ Z where the action of g ∈ G on Z is such that g.1 = rho(g). It returns the chain complex or chain map obtained by tensoring with the (twisted) module of integers (characteristic 0).

Examples: 1 , 2 , 3 , 4

##### 8.1-12 TensorWithIntegersModP

`‣ TensorWithIntegersModP`( X, p ) | ( function )

Inputs either a ZG-resolution X=R, or a characteristics 0 chain complex, or an equivariant chain map X = (F:R ⟶ S), or a chain map between characteristic 0 chain complexes, together with a prime p. It returns the chain complex or chain map obtained by tensoring with the trivial module of integers modulo p.

Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9

##### 8.1-13 TensorWithTwistedIntegersModP

`‣ TensorWithTwistedIntegersModP`( X, p, rho ) | ( function )

Inputs either a ZG-resolution X=R, or an equivariant chain map X = (F:R ⟶ S), and a prime p. It also inputs a function rho: G→ Z where the action of g ∈ G on Z is such that g.1 = rho(g). It returns the chain complex or chain map obtained by tensoring with the trivial module of integers modulo p.

Examples: 1

##### 8.1-14 TensorWithRationals

`‣ TensorWithRationals`( R ) | ( function )

Inputs a ZG-resolution R and returns the chain complex obtained by tensoring with the trivial module of rational numbers.

Examples: 1 , 2 , 3

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