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I Overview of the homalg Package Source Code
 I.1 Rings, Ring Maps, Matrices, Relations, and Generators
 I.2 The Basic Objects
 I.3 The Low Level Algorithms
 I.4 The High Level Algorithms
 I.5 Logical Implications for homalg Objects
 I.6 The subpackage ResidueClassRingForHomalg
 I.7 The homalgTable for GAP4 built-in rings

I Overview of the homalg Package Source Code

The homalg package reached more than 50.000 lines of GAP4 code (excluding the documentation) before the first release was made. To keep this amount of code tracebale, the package was split in several files.

I.1 Rings, Ring Maps, Matrices, Relations, and Generators

Table: The homalg package files
Filename .gd/.gi Content
homalg definitions of the basic GAP4 categories
and some tool functions (e.g. homalgMode)
homalgTable dictionaries between homalg
and the computing engines
HomalgRing internal and external rings
HomalgRingMap ring maps
HomalgMatrix internal and external matrices
HomalgRelations a set of relations
SetsOfRelations several sets of relations
HomalgGenerators a set of generators
SetsOfGenerators several sets of generators

 


I.2 The Basic Objects

Table: The homalg package files (continued)
Filename .gd/.gi Content
HomalgModule modules and submodules allowing several
presentations linked with transition matrices
HomalgMap maps allowing several presentations
of their source and target
HomalgFiltration filtration of a module
HomalgComplex (co)complexes of modules or of (co)complexes
HomalgChainMap chain maps of (co)complexes
consisting of maps or chain maps
HomalgBicomplex bicomplexes of modules or of (co)complexes
HomalgBigradedObject (differential) bigraded modules
HomalgSpectralSequence homological and cohomological
spectral sequences
HomalgFunctor constructors of (multi) functors of
module categories (yet over the same ring),
left derivation of covariant functors,
right derivation of contravariant functors,
left satellites of covariant functors,
right satellites of contravariant functors,
and composition of functors
HomalgDiagram Betti diagrams

 


I.3 The Low Level Algorithms

In the following CAS or CASystem mean computer algebra systems.

Table: The homalg package files (continued)
Filename .gd/.gi Content
Tools the elementary matrix operations that can be
overwritten using the homalgTable
(and hence delegable even to other CASystems)
Service the three operations: basis, reduction, and syzygies;
they can also be overwritten using the homalgTable
(and hence delegable even to other CASystems)
Basic higher level operations for matrices
(cannot be overwritten using the homalgTable)

 


I.4 The High Level Algorithms

Table: The homalg package files (continued)
Filename .gd/.gi Content
Modules subfactors, resolutions, syzygy modules,
parameterizations, intersections, annihilators
Maps resolutions, (co)kernel sequences
Complexes (co)homology, horse shoe lemma, connecting
homomorphisms, Cartan-Eilenberg resolution
ChainMaps (co)homology
SpectralSequences Grothendieck bicomplexes associated to two
composable functors, spectral sequences
of bicomplexes, Grothendieck spectral sequences
Filtrations spectral filtrations, i.e. filtrations induced
by spectral sequences of bicomplexes,
purity filtration
ToolFunctors composition, addition, substraction,
stacking, augmentation, and post dividing maps
BasicFunctors cokernel, image, kernel, tensor product, Hom,
Ext, Tor, RHom, LTensorProduct, HomHom, LHomHom,
BaseChange (preliminary)
OtherFunctors torsion submodule, torsion free factor,
direct sum, pullback, pushout, Auslander dual

 


I.5 Logical Implications for homalg Objects

Table: The homalg package files (continued)
Filename .gd/.gi Content
LIRNG logical implications for rings
LIMAP logical implications for ring maps
LIMAT logical implications for matrices
COLEM clever operations for lazy evaluated matrices
LIMOD logical implications for modules
LIMOR logical implications for morphisms
LICPX logical implications for complexes

 


I.6 The subpackage ResidueClassRingForHomalg

Table: The homalg package files (continued)
Filename .gd/.gi Content
ResidueClassRingForHomalg some global variables
ResidueClassRing residue class rings, their elements, and matrices,
together with their constructors and operations
ResidueClassRingTools the elementary matrix operations for matrices
over residue class rings
ResidueClassRingBasic the three operations: basis, reduction, and syzygies
for matrices over residue class rings

 


I.7 The homalgTable for GAP4 built-in rings

For the purposes of homalg, the ring of integers is, at least up till now, the only ring which is properly supported in GAP4. The GAP4 built-in cababilities for polynomial rings (also univariate) and group rings do not statisfy the minimum requirements of homalg. The GAP4 package Gauss enables GAP to fullfil the homalg requirements for prime fields, and ℤ / p^n.

Table: The homalg package files (continued)
Filename .gi Content
Integers the homalgTable for the ring of integers

 


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