Singular Manual: Bimodules and syzygies and lifts

| ### 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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