7.9.4 Bimodules and syzygies and lifts
Let
413#413
302#302,...,
303#303 be the free algebra.
A free bimodule of rank
301#301 over
191#191 is
414#414,where
415#415 are the generators of the free bimodule.
NOTE: these
415#415 are freely non-commutative with respect to
elements of
191#191 except constants from the ground field
50#50.
The free bimodule of rank 1
416#416 surjects onto the algebra
191#191 itself.
A two-sided ideal of the algebra
191#191 can be converted to a subbimodule of
416#416.
The syzygy bimodule or even module of bisyzygies
of the given finitely generated subbimodule
417#417is the kernel of the natural homomorphism of
191#191-bimodules
418#418that is
419#419
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
420#420 of a bimodule
13#13, the lift(ing) process
returns a matrix, which encodes the expression of generators
421#421
in terms of generators of
422#422 like this:
423#423
where
424#424 are elements from the enveloping algebra
425#425encoded as elements of the free bimodule of rank
297#297,
namely by using the non-commutative generators of the
free bimodule which we call ncgen .
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