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4 Presentations of graded algebras
4.1 The `GradedAlgebraPresentation` datatype
4.2 Construction function
4.2-1 GradedAlgebraPresentation construction functions
4.3 Data access functions
4.3-1 BaseRing
4.3-2 CoefficientsRing
4.3-3 IndeterminatesOfGradedAlgebraPresentation
4.3-4 GeneratorsOfPresentationIdeal
4.3-5 PresentationIdeal
4.3-6 IndeterminateDegrees
4.3-7 Example: Constructing and accessing data of a `GradedAlgebraPresentation`
4.4 Other functions
4.4-1 TensorProduct
4.4-2 IsIsomorphicGradedAlgebra
4.4-3 IsAssociatedGradedRing
4.4-4 DegreeOfRepresentative
4.4-5 MaximumDegreeForPresentation
4.4-6 SubspaceDimensionDegree
4.4-7 SubspaceBasisRepsByDegree
4.4-8 CoefficientsOfPoincareSeries
4.4-9 HilbertPoincareSeries
4.4-10 LHSSpectralSequence
4.5 Example: Computing the Lyndon-Hoschild-Serre spectral sequence and mod-p cohomology ring for a small p-group

### 4 Presentations of graded algebras

A graded algebra A is a an algebra that has additional structure, called a grading (see Section 3). Graded algebras of the type found in HAPprime have a presentation as a quotient of a polynomial ring

H^*(G, F) = F[x_1, x_2, ..., x_n] / <I_1, I_2, ..., I_m>

where the polynomial ring indeterminates x_i each have an associated degree d_i and the I_j are relations which together generate an ideal in the ring.

#### 4.1 The `GradedAlgebraPresentation` datatype

For algebras that have a presentation as a quotient of a polynomial ring, the `GradedAlgebraPresentation` datatype stores a quotient R/I where:

- R is a polynomial ring

- I is a set of relations in R that generate an ideal

and it also stores a grading in the form of

- the degree of each indeterminate of R

#### 4.2 Construction function

##### 4.2-1 GradedAlgebraPresentation construction functions

`> GradedAlgebraPresentation`( R, I, degs ) | ( operation )

`> GradedAlgebraPresentationNC`( R, I, degs ) | ( operation )

Returns: `GradedAlgebraPresentation`

Construct a `GradedAlgebraPresentation` object representing a presentation of a graded algebra as the quotient of a polynomial ring R by the ideal I (as a list of relations in R) where the indeterminates of R (as returned by `IndeterminatesOfGradedAlgebraPresentation` (4.3-3) have degrees degs respectively.

The function `GradedAlgebraPresentation` checks that the arguments are compatible, while the `NC` method performs no checks.

#### 4.3 Data access functions

##### 4.3-1 BaseRing

`> BaseRing`( A ) | ( attribute )

Returns: Polynomial ring

Returns the base ring of the graded algebra presentation A.

##### 4.3-2 CoefficientsRing

`> CoefficientsRing`( A ) | ( attribute )

Returns: Ring

Returns the ring of coefficients of the graded algebra presentation A.

##### 4.3-3 IndeterminatesOfGradedAlgebraPresentation

`> IndeterminatesOfGradedAlgebraPresentation`( A ) | ( attribute )

Returns: List

Returns the indeterminates used in the graded algebra presentation A.

##### 4.3-4 GeneratorsOfPresentationIdeal

`> GeneratorsOfPresentationIdeal`( A ) | ( attribute )

Returns: List

Returns the relations in the ring presentation for the graded algebra A. The relations are returned sorted in order of increasing degree, and by indeterminate within each degree.

##### 4.3-5 PresentationIdeal

`> PresentationIdeal`( A ) | ( attribute )

Returns: Ideal

Returns the ideal in the graded algebra presentation A as a GAP ideal Reference: Ideal.

##### 4.3-6 IndeterminateDegrees

`> IndeterminateDegrees`( A ) | ( attribute )

Returns: List

Returns the degrees of the polynomial ring indeterminates in the graded algebra presentation A. The ordering corresponds to the order of the ring indeterminates returned by `IndeterminatesOfGradedAlgebraPresentation` (4.3-3).

##### 4.3-7 Example: Constructing and accessing data of a `GradedAlgebraPresentation`

We demonstrate creating a `GradedAlgebraPresentation` object and reading back its data by creating the graded algebra A with presentation F_2[x_1, x_2, x_3] / (x_1x_2, x_1^3+x_2^3) where x_1 and x_2 have degree 1 and x_3 has degree 4

    gap> R := PolynomialRing(GF(2), 3);;
    gap> A := GradedAlgebraPresentation(R, [R.1*R.2, R.1^3+R.2^3], [1,1,4]);
    Graded algebra GF(2)[ x_1, x_2, x_3 ] / [ x_1*x_2, x_1^3+x_2^3
     ] with indeterminate degrees [ 1, 1, 4 ]
    gap> BaseRing(A);
    GF(2)[x_1,x_2,x_3]
    gap> CoefficientsRing(A);
    GF(2)
    gap> IndeterminatesOfGradedAlgebraPresentation(A);
    [ x_1, x_2, x_3 ]
    gap> PresentationIdeal(A);
    <two-sided ideal in GF(2)[x_1,x_2,x_3], (2 generators)>
    gap> GeneratorsOfPresentationIdeal(A);
    [ x_1*x_2, x_1^3+x_2^3 ]
    gap> IndeterminateDegrees(A);
    [ 1, 1, 4 ]

#### 4.4 Other functions

##### 4.4-1 TensorProduct

`> TensorProduct`( A, B ) | ( operation )

`> TensorProduct`( coll ) | ( operation )

Returns: GradedAlgebraPresentation

Returns a presentation for the graded algebra that is the tensor product of two graded algebras presented by A and B, or of a list of graded algebras.

##### 4.4-2 IsIsomorphicGradedAlgebra

`> IsIsomorphicGradedAlgebra`( A, B ) | ( operation )

Returns `true` if the graded algebras A and B are isomorphic, or `false` otherwise. This function tries all possible ring isomorphisms, so may take a considerable length of time for graded algebras with a large number of dimensions in each degree.

##### 4.4-3 IsAssociatedGradedRing

`> IsAssociatedGradedRing`( A, B ) | ( operation )

Returns `true` if the algebra A is an associated graded ring of the algebra B. This is the case if the additive structure is the same (i.e. the Hilbert-Poincaré series is the same), and the generators for A (and their degrees) are included in the generators for B.

##### 4.4-4 DegreeOfRepresentative

`> DegreeOfRepresentative`( A, p ) | ( operation )

Returns: Integer

Returns the degree of a polynomial representative p from the graded ring presentation A.

##### 4.4-5 MaximumDegreeForPresentation

`> MaximumDegreeForPresentation`( A ) | ( attribute )

Returns: Integer

Returns the maximum degree in generators or relations that is needed to generate the graded algebra presentation A. This is not necessarily the same as the largest degree in any of the relations and generators - some relations may be redundant (for example due to being a Groebner basis), so this routine checks for the largest degree of a required generator, and returns the maximum of this and the generator degrees.

##### 4.4-6 SubspaceDimensionDegree

`> SubspaceDimensionDegree`( A, d ) | ( operation )

`> SubspaceDimensionDegree`( A, degs ) | ( operation )

Returns: Integer or list

Returns the dimension of degree d of the graded algebra A, or a list of dimensions corresponding to the list of degrees degs.

##### 4.4-7 SubspaceBasisRepsByDegree

`> SubspaceBasisRepsByDegree`( A, d ) | ( operation )

`> SubspaceBasisRepsByDegree`( A, degs ) | ( operation )

Returns: List or list of lists

Returns a basis for degree d of the graded algebra A, or a list of bases for the list of degrees degs. Each basis is returned as a list of representatives.

##### 4.4-8 CoefficientsOfPoincareSeries

`> CoefficientsOfPoincareSeries`( A, n ) | ( operation )

Returns: List

Returns the first n coefficients of the Poincaré series for the graded algebra with A. These are equal to the dimensions of degrees 0 to n-1 of the algebra (a fact that is used in the function `SubspaceDimensionDegree` (4.4-6)).

This function uses the singular package.

##### 4.4-9 HilbertPoincareSeries

`> HilbertPoincareSeries`( A ) | ( attribute )

Returns: Rational function

Returns the Poincaré series for the graded algebra A. This is a rational function P(t)/Q(t) which is a is a polynomial whose coefficients are the dimensions of each degree of the algebra.

This function uses the singular package.

##### 4.4-10 LHSSpectralSequence

`> LHSSpectralSequence`( G[, N], n ) | ( operation )

`> LHSSpectralSequenceLastSheet`( G[, N] ) | ( operation )

Returns: `GradedAlgebraPresentation` or list

Computes the Lyndon-Hoschild-Serre spectral sequence for the group extension N -> G -> G/N. If a normal suggroup N is not provided, then the largest central subgroup of G is used, or (if the order of the centre is larger than sqrt|G|) then the central subgroup that leads to the smallest initial sheet size is chosen.

The function `LHSSpectralSequence` returns the first n sheets of the spectral sequence, or all of the sequence up to convergence, if that occurs before the (n+1)th sheet. The Lyndon-Hoschild-Serre spectral sequence starts at the E_2 sheet, so the first element in returned list will always be empty. If n is set to `infinity` then the length of the returned list equals the number of sheets for convergence, and the last sheet in the list is the limiting sheet.

The function `LHSSpectralSequenceLastSheet` returns only the limiting sheet of the spectral sequence. This ring is an associated graded algebra of the mod-p cohomology ring of G, with the same additive structure while not necessarily being isomorphic to it.

There are four options Reference: Options Stack which can be used to guide this algorithm:

- `InitialLHSBicomplexSize` can be used to specify the initial size of the bicomplex (the default is 5). If, in the process of computing the spectral sequence, this is found to be too small then the algorithm restarts with a larger value. Specifying a larger initial value in these cases can save time.

- `LargerLHSBicomplexBreak` if set to `true` will force the calculation to enter a break loop before restarting with a larger bicomplex, should the bicomplex be found to be too small. The user user is prompted to type `return;` before continuing. The default behaviour is `false`, i.e. no prompt.

- `LargerLHSBicomplexFail` if set to `true` will return `fail` should the bicomplex be found to be too small. The default behaviour is `false`, i.e. to either restart or prompt, depending on the setting of the previous option.

- `NoInductiveProof` if set to `true` will not check that the cohomology rings for N and G/N are correct. Instead, it will compute the cohomology rings only up to the degree needed for the bicomplex size (5 by default, or specified by the `InitialLHSBicomplexSize` option).

#### 4.5 Example: Computing the Lyndon-Hoschild-Serre spectral sequence and mod-p cohomology ring for a small p-group

The Lyndon-Hoschild-Serre spectral sequence is relates the cohomologies of a normal subgroup N and a quotient group G/N to the cohomology of the total group G: the limiting sheet of the sequence is an associated graded ring of the cohomology of G.

In this example we calculate the Lyndon-Hoschild-Serre spectral sequence for a group of order 16 using the centre of G as our normal subgroup. By asking for an infinite number of terms, this function calculates enough terms to be sure that the sequence has converged. We compare the dimensions in the first (E_2) and last (E_infty) sheet, we demonstrate that the limiting sheet (the last in the list) is a graded algebra by multiplying some elements, and we calculate the Poincaré series of the last sheet.

    gap> G := SmallGroup(16, 4);;
    gap> SS := LHSSpectralSequence(G, Centre(G), infinity);
    [ , Graded algebra GF(2)[ x_1, x_2, x_3, x_4 ] /
        [  ] with indeterminate degrees [ 1, 1, 1, 1 ],
      Graded algebra GF(2)[ x_1, x_2, x_3, x_4 ] / [ x_2^2, x_1^2+x_1*x_2
         ] with indeterminate degrees [ 1, 1, 2, 2 ],
      Graded algebra GF(2)[ x_1, x_2, x_3, x_4 ] / [ x_2^2, x_1^2+x_1*x_2
         ] with indeterminate degrees [ 1, 1, 2, 2 ] ]
    gap> # i.e. we identify convergence after 3 terms
    gap> #
    gap> # Compare the dimensions of the first and last sheet
    gap> SubspaceDimensionDegree(SS[2], [1..10]);
    [ 4, 10, 20, 35, 56, 84, 120, 165, 220, 286 ]
    gap> SubspaceDimensionDegree(SS[3], [1..10]);
    [ 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 ]
    gap> #
    gap> # Take the two basis elements from degree 1 and check that the
    gap> # product is in degree two
    gap> B := SubspaceBasisRepsByDegree(SS[3], 1);
    [ x_1, x_2 ]
    gap> DegreeOfRepresentative(SS[3], B[1]*B[2]);
    2
    gap> #
    gap> # And find the Poincare series
    gap> HilbertPoincareSeries(SS[3]);
    (1)/(x_1^2-2*x_1+1)

The largest degree in the presentation for the limiting sheet in the Lyndon-Hoschild-Serre spectral sequence for G is the same as the largest degree in the presentation for the mod-p cohomology ring of G. We continue this example by calculating this maximum degree, n, for our group G and then computing the mod-p cohomology ring. We confirm that the cohomology ring is an associated graded ring of the limiting sheet of the spectral sequence, and check whether in this case it is in fact also isomorphic.

    gap> G := SmallGroup(16, 4);;
    gap> Einf := LHSSpectralSequenceLastSheet(G, Centre(G));
    Graded algebra GF(2)[ x_1, x_2, x_3, x_4 ] / [ x_2^2, x_1^2+x_1*x_2
     ] with indeterminate degrees [ 1, 1, 2, 2 ]
    gap> #
    gap> # Find the maximum degree
    gap> n := MaximumDegreeForPresentation(Einf);
    2
    gap> #
    gap> # And calculate the cohomology ring
    gap> H := Mod2CohomologyRingPresentation(G, n);
    Graded algebra GF(2)[ x_1, x_2, x_3, x_4 ] / [ x_1*x_2+x_2^2, x_1^2
     ] with indeterminate degrees [ 1, 1, 2, 2 ]
    gap> #
    gap> # Check for an associated graded ring, and isomorphism
    gap> IsAssociatedGradedRing(H, Einf);
    true
    gap> IsIsomorphicGradedAlgebra(H, Einf);
    true

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