7.10 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

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.
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