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3 Functions for Homological Algebra
 3.1 Resolutions

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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