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 \((R \otimes_{ZH} ZG)\). The returned resolution \(S\) has S!.elts:=EltsG. This is a resolution of the \(ZG\)-module \((Z \otimes_{ZH} ZG)\). (Here \(\otimes_{ZH}\) means tensor over \(ZH\).)

Examples:

##### 8.1-2 HomToIntegers

`‣ HomToIntegers`( X ) | ( function )

Inputs either a \(ZG\)-resolution \(X=R\), or an equivariant chain map \(X = (F:R \longrightarrow S)\). 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 \(f:G \longrightarrow GL_n(Z)\) 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 \(f:G \longrightarrow GL_n(Z)\) 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 \(hom:G\rightarrow Q\). 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 \(f: G \longrightarrow G'\) . In this case the induced homomorphism of Lie algebras \(L(f):L(G) \longrightarrow L(G')\) 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 \(X = (F:R \longrightarrow S)\). 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 \(X = (F:R \longrightarrow S)\). It also inputs a function \(rho\colon G\rightarrow \mathbb Z\) where the action of \(g \in G\) on \(\mathbb Z\) 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 \(X = (F:R \longrightarrow S)\), 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 \(X = (F:R \longrightarrow S)\), and a prime \(p\). It also inputs a function \(rho\colon G\rightarrow \mathbb Z\) where the action of \(g \in G\) on \(\mathbb Z\) 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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