7.9.4 Bimodules and syzygies and lifts
Let
,...,
be the free algebra.
A free bimodule of rank
over
is
,where
are the generators of the free bimodule.
NOTE: these
are freely non-commutative with respect to
elements of
except constants from the ground field
.
The free bimodule of rank 1
surjects onto the algebra
itself.
A two-sided ideal of the algebra
can be converted to a subbimodule of
.
The syzygy bimodule or even module of bisyzygies
of the given finitely generated subbimodule
is the kernel of the natural homomorphism of
-bimodules
that is
The syzygy bimodule is in general not finitely generated.
Therefore as a bimodule, both the set of generators of the
syzygy bimodule and its Groebner basis
are computed up to a specified length bound.
Given a subbimodule
of a bimodule
, the lift(ing) process
returns a matrix, which encodes the expression of generators
in terms of generators of
like this:
where
are elements from the enveloping algebra
encoded as elements of the free bimodule of rank
,
namely by using the non-commutative generators of the
free bimodule which we call ncgen .
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