The central notion of this work is that of a functor between categories of
finitely presented modules over so-called computable rings, i.e. rings R
where one can algorithmically solve inhomogeneous linear equations with
coefficients in R. The paper describes a way allowing one to realize such
functors, e.g. Hom, tensor product, Ext, Tor, as a
mathematical object in a computer algebra system. Once this is achieved, one
can compose and derive functors and even iterate this process without the
need of any specific knowledge of these functors. These ideas are realized in
the ring independent package homalg. It is designed to extend any computer
algebra software implementing the arithmetics of a computable ring R, as
soon as the latter contains algorithms to solve inhomogeneous linear equations
with coefficients in R. Beside explaining how this suffices, the paper
describes the nature of the extensions provided by homalg.
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A chain of c submodules E =: E0 ≥ E1 ≥ ... ≥ Ec ≥ Ec+1 := 0 gives
rise to c composable 1-cocycles in Ext1(Ei-1/Ei,Ei/Ei+1),
i=1,...,c. In this paper we follow the converse question: When are c
composable 1-cocycles induced by a module E together with a chain of
submodules as above? We call such modules c-extension modules. The case
c=1 is the classical correspondence between 1-extensions and
1-cocycles. For c=2 we prove an existence theorem stating that a
2-extension module exists for two composable 1-cocycles
ηML in Ext1(M,L) and
ηLN in Ext1(L,N), if and only if
their Yoneda product ηML o ηLN
in Ext2(M,N) vanishes. We further prove a modelling theorem
for c=2: In case the set of all such 2-extension modules is non-empty it is an
affine space modelled over the abelian group that we call the first
extension group of 1-cocycles,
Ext1(ηML,ηLN)
:= Ext1(M,N)/(Hom(M,L) o ηLN +
ηML o Hom(L,N)).
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