GAP (HAPprime) - Chapter 3: Functions for Homological Algebra

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3 Functions for Homological Algebra
3.1 Resolutions
3.1-1 ResolutionPrimePowerGroup
3.1-2 ExtendResolutionPrimePowerGroup

### 3 Functions for Homological Algebra

#### 3.1 Resolutions

##### 3.1-1 ResolutionPrimePowerGroup

`> ResolutionPrimePowerGroupRadical`( G, n ) | ( operation )

`> ResolutionPrimePowerGroupGF`( G, n ) | ( operation )

`> ResolutionPrimePowerGroupAutoMem`( G, n ) | ( operation )

`> ResolutionPrimePowerGroupGF2`( G, n ) | ( operation )

`> ResolutionPrimePowerGroupRadical`( M, n ) | ( operation )

`> ResolutionPrimePowerGroupGF`( M, n ) | ( operation )

`> ResolutionPrimePowerGroupAutoMem`( M, n ) | ( operation )

`> ResolutionPrimePowerGroupGF2`( M, n ) | ( operation )

Returns: `HAPResolution`

Returns n terms of a minimal free FG-resolution for either the ground ring of a prime power group G or of a module M. For the module version, M must be passed as an `FpGModuleGF` object - see HAPprime Datatypes: FG-modules in the HAPprime datatypes reference manual.

Three versions of this function are provided:

`ResolutionPrimePowerGroupRadical`

  uses the same resolution-building method as the HAP function `ResolutionPrimePowerGroup`, but stores the resolution in a different format that takes only about half the memory of the HAP version.

`ResolutionPrimePowerGroupGF`

  calculates the resolution using HAPprime's G-generator form of modules, which reduces memory use by around a factor of two over `ResolutionPrimePowerGroupRadical`, but is slower by an order of magnitude.

`ResolutionPrimePowerGroupAutoMem`

  automatically switches between the two previous versions based on the available memory. It uses the `Radical` version until it gets close to the limit of the available memory, and then switches to the `GF` version.

`ResolutionPrimePowerGroupGF2`

  calculates the resolution by FG-matrix partitioning. The amount of partitioning is governed by the Reference: Options Stack option `MaxFGExpansionSize`. The default value means that until the boundary map takes about 128Mb, the method is equivalent to `ResolutionPrimePowerGroupRadical`, and then it tends towards `ResolutionPrimePowerGroupGF` in terms of time, but saves less memory.

See the HAPprime datatypes reference manual for details of the different algorithms, in particular the chapters on the G-generator form of FG-modules HAPprime Datatypes: FG-modules and FG-module homomorphisms HAPprime Datatypes: FG-module homomorphisms and on resolutions HAPprime Datatypes: Resolutions.

##### 3.1-2 ExtendResolutionPrimePowerGroup

`> ExtendResolutionPrimePowerGroupRadical`( R ) | ( operation )

`> ExtendResolutionPrimePowerGroupGF`( R ) | ( operation )

`> ExtendResolutionPrimePowerGroupAutoMem`( R ) | ( operation )

`> ExtendResolutionPrimePowerGroupGF2`( R ) | ( operation )

Returns: `HAPResolution`

Returns the resolution R extended by one term. The three variants offer a choice between memory and speed, and correspond to the different versions of `ResolutionPrimePowerGroup` in HAPprime. See the documentation (above) for those functions for a description of the different variants.

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