The package &homalg; is meant to be the first part of a continuously
growing <Alt Only="Text">open source</Alt>
<Alt Not="Text"><URL Text="open
source">http://www.opensource.org/ multi volume book about
<Alt Only="Text">homological</Alt><Alt Not="Text">
<URL Text="homological">http://en.wikipedia.org/wiki/Homological_algebra</URL></Alt>
and <Alt Only="Text">homotopical
algebra</Alt><Alt Not="Text"><URL Text="homotopical
algebra">http://en.wikipedia.org/wiki/Homotopical_algebra.
&homalg; is an attempt to translate as much as possible of homological
algebra, as can be found in books like <Cite Key="ce"/>,
<Cite Key="ML"/>, <Cite Key="HS"/>, <Cite Key="rot"/>,
<Cite Key="weihom"/>, and <Cite Key="GM"/>, into a language that a
computer can directly understand. But just like the aforementioned
books, &homalg; should, to a great extent, be readable by a
mathematician, even without deep programming knowledge. For the
reasons mentioned in (&see; Appendix <Ref Sect="WhyGAP4" Text="Why GAP4?"/>)
&GAP4; was chosen as the language of &homalg;.
The package &homalg; is the foundational part of the project. It
provides procedures to construct basic objects in homological algebra:
<List>
<Item>filtrations of objects</Item>
<Item>complexes (of objects and of complexes)</Item>
<Item>chain morphisms</Item>
<Item>bicomplexes</Item>
<Item>bigraded (differential) objects</Item>
<Item>spectral sequences</Item>
<Item>functors</Item>
</List>
Beside these so-called constructors &homalg;
provides <Alt Only="HTML"><Ref Text="operations"
Sect="operations" BookName="Tutorial"/></Alt>
<Alt Not="HTML"><E>operations</E></Alt> to perform computations with
these objects. The list of operations includes:
<List>
<Item>computation of subfactor objects</Item>
<Item>applying functors (like <C>Ext</C>, <C>Tor</C>, ...) to objects,
morphisms, complexes and chain morphisms</Item>
<Item>derivation and composition of functors</Item>
<Item>horse shoe resolution of short exact sequences of objects</Item>
<Item>connecting homomorphisms and long exact sequences</Item>
<Item>Cartan-Eilenberg resolution of complexes</Item>
<Item>hyper (co)homology</Item>
<Item>spectral sequences of bicomplexes</Item>
<Item>the Grothendieck spectral sequences associated to two
composable functors</Item>
<Item>test if an object is torsion-free, reflexive, projective,
stably free, pure</Item>
<Item>determine the rank, grade, projective dimension, degree
of torsion-freeness, and codegree of purity of an object</Item>
</List>
Using the philosophy of &GAP4;, one or
more <Alt Only="HTML"><Ref Text="methods" Sect="Operations and Methods"
BookName="Tutorial"/></Alt> <Alt Not="HTML">methods</Alt>
are
<Alt Only="HTML"><Ref Text="installed" Sect="Method Installation"
BookName="Reference"/></Alt>
<Alt Not="HTML"><E>installed</E></Alt> for each operation, depending
on <Alt Only="HTML"><Ref Text="properties" Sect="Properties"
BookName="Tutorial"/></Alt> <Alt Not="HTML"><E>properties</E></Alt>
and <Alt Only="HTML"><Ref Text="attributes" Sect="attributes"
BookName="Tutorial"/></Alt> <Alt Not="HTML"><E>attributes</E></Alt> of
these objects. These properties and attributes can themselves be
computed by methods installed for this purpose.
</Subsection>
<Subsection Label="build">
<Heading>Building upon the &homalg; package</Heading>
As mentioned above, the package &homalg; should only be the first and
foundational part of the &homalg; project. On the one hand it is
designed independently of the details of the different matrix
operations, which other packages are meant to provide. Typically,
these packages (like &RingsForHomalg;) heavily rely on existing, well
tested, and optimized systems like &Singular;, &Macaulay2;, or
&MAGMA;. On the other hand other packages can be built upon or extend
the &homalg; package in different ways:
<List>
<Item>add constructors (sheaves, schemes, simplicial sets, ...)</Item>
<Item>add methods for basic operation (Yoneda products, Massey
products, Steenrod operations, ...)</Item>
<Item>add methods to compute sheaf cohomology, local cohomology,
Hochschild (co)homology, cyclic (co)homology...</Item>
<Item>provide algorithms for holonomic <M>D</M>-modules based on the
restriction algorithm: localization, computing tensor
products, <C>Hom</C>, <C>Ext</C>, de Rham cohomology, ...</Item>
<Item>support change of rings, Lyndon/Hochschild-Serre spectral
sequence, base change spectral sequences, ...</Item>
<Item>support perturbation techniques, Serre and Eilenberg-Moore
spectral sequence of simplicial spaces of infinite type,
...</Item>
<Item>...</Item>
</List>
The project will remain open and contributions are highly welcome. The
different packages will be attributed to their respective authors. The
whole project will be attributed to the "&homalg; team", i.e. the
authors and contributers of all packages in the project.
Chapter <Ref Chap="install"/> describes the installation of this
package. The remaining chapters are each devoted to one of the
&homalg; objects (&see; <Ref Label="homalg-provides"/>) with its
constructors, properties, attributes, and operations.
<!-- Finally, Chapter <Ref Chap="examples"/> shows some instructive examples for the usage of this package.
-->
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