7.9.4 Bimodules and syzygies and lifts
Let
411#411
300#300,...,
301#301 be the free algebra.
A free bimodule of rank
299#299 over
190#190 is
412#412,where
413#413 are the generators of the free bimodule.
NOTE: these
413#413 are freely non-commutative with respect to
elements of
190#190 except constants from the ground field
50#50.
The free bimodule of rank 1
414#414 surjects onto the algebra
190#190 itself.
A two-sided ideal of the algebra
190#190 can be converted to a subbimodule of
414#414.
The syzygy bimodule or even module of bisyzygies
of the given finitely generated subbimodule
415#415is the kernel of the natural homomorphism of
190#190-bimodules
416#416that is
417#417
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
418#418 of a bimodule
13#13, the lift(ing) process
returns a matrix, which encodes the expression of generators
419#419
in terms of generators of
420#420 like this:
421#421
where
422#422 are elements from the enveloping algebra
423#423encoded as elements of the free bimodule of rank
295#295,
namely by using the non-commutative generators of the
free bimodule which we call ncgen .
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