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7.9.4 Bimodules and syzygies and lifts

Let $A = K$ $<x_1$,..., $x_n >$ be the free algebra. A free bimodule of rank $r$ over $A$ is $A e_1 A \oplus \ldots \oplus A e_r A$,where $e_i$ are the generators of the free bimodule.

NOTE: these $e_i$ are freely non-commutative with respect to elements of $A$ except constants from the ground field $K$.

The free bimodule of rank 1 $AeA$ surjects onto the algebra $A$ itself. A two-sided ideal of the algebra $A$ can be converted to a subbimodule of $AeA$.

The syzygy bimodule or even module of bisyzygies of the given finitely generated subbimodule $N = \langle g_1,\ldots,g_m \rangle \subset \bigoplus_{i=1}^r A e_i A$is the kernel of the natural homomorphism of $A$-bimodules $ \bigoplus_{j=1}^m A \epsilon_j A \longrightarrow \bigoplus_{i=1}^r A e_i A, \; \epsilon_j \mapsto g_j, $that is $\sum_{j=1}^m \sum_k \ell_{jk} \epsilon_j r_{jk} \mapsto \sum_{j=1}^m \sum_k \ell_{jk} g_j r_{jk}. $

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 $N$ of a bimodule $M$, the lift(ing) process returns a matrix, which encodes the expression of generators $N_1, \ldots, N_s$

in terms of generators of $M_1, \ldots, M_m$ like this: $N_i = \sum_{j=1}^m \sum_k \ell_{jk} M_j r_{jk} = \sum_{j=1}^m T_{ij} M_j,$

where $T_ij$ are elements from the enveloping algebra $R \langle X \rangle \otimes R \langle X \rangle,$encoded as elements of the free bimodule of rank $m$, namely by using the non-commutative generators of the free bimodule which we call ncgen.