Effective computation of $\Tor_k(M,N)$
Consejer\'{\i}a de Educaci\'on - Gobierno de Canarias} %
An effective method to compute a presentation of $\Tor _k (M,N)$ for modules on a not necessarily commutative algebra is proposed.
math0412326.tex
%\documentclass{kluwer}
\documentclass[11pt]{article}

\usepackage{amsthm, amsmath}
%\usepackage[english]{babel}
\usepackage{amssymb}
%\usepackage[latin1]{inputenc}
\usepackage[arrow,matrix,cmtip]{xy}
%\usepackage{algorithmic}
%\usepackage{algorithm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  THEOREM-STYLE ENVIRONMENTS
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{example}[theorem]{Example}

\theoremstyle{definition}
\newtheorem{remark}[theorem]{Remark}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  MACROS
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcommand{\expo}{{\rm{exp}\:}}
\newcommand{\im}{{\rm{im}\:}}
\newcommand{\Tor}{{\rm{Tor}}}
\newcommand{\Ext}{{\rm{Ext}}}
\newcommand{\Cen}{{\rm{Cen}}}
\newcommand{\id}{{\rm{Id}}}
\newcommand{\env}{{{R^{\rm{env}}}}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{document}

%\begin{article}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   OPENING
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\begin{opening}
\title{Effective computation of $\Tor_k(M,N)$}
\author{Socorro %\surname{
Garc\'{\i}a Rom\'an
%}\email{sgarcia@ull.es}
\thanks{Supported by DGUI, Consejer\'{\i}a de Educaci\'on - Gobierno de Canarias}
%}
\and
%\author{
Manuel
%\surname{
Garc\'{\i}a Rom\'an
%}\email{mroman@ull.es}
}
%\institute{Universidad de La Laguna}

%\date{}

%\begin{ao}
%Departamento de Matem\'atica Fundamental, Universidad de La Laguna, 38206 La Laguna, %Spain
%\end{ao}

\maketitle

\begin{abstract}
An effective method to compute a presentation of $\Tor _k (M,N)$ for modules on a not necessarily commutative algebra is proposed.
\end{abstract}

%\end{opening}

%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%  INTRODUCTION
%%%%%%%%%%%%%%%%%%%%%%%%%%

\section*{Introduction}

Unlike $\Ext_k(M,N)$ (see~\cite{smith} for an algorithm when the base ring $R$ is a
commutative algebra and~\cite{BCGL01} when $R$ is a PBW algebra), there are not known
effective methods to compute $\Tor_k(M,N)$ for a pair of $R$-modules $M$ and $N$.

In this note,
we propose an algorithm to compute a presentation of $\Tor_k(M,N)$ when $R$ is
a Poincar\'e-Birkhoff-Witt algebra (PBW algebra for short;
see~\cite[et.al.]{BCGL01, libro} for a definition and examples, including the Weyl
algebras, universal enveloping algebras of Lie algebras, the quantum plane, algebras of
quantum matrices and other iterated Ore extensions, etc.).
Since in general $\Tor_k(M,N)$ is just an abelian group when $M$
and $N$ are left $R$-modules, we ask for a two-sided structure on $M$. When $M$ is an
$R$-bimodule then $\Tor_k(M,N)$ is a left $R$-module. We show
that if, in addition, $M$ is finitely generated and $N$ is a finitely generated left
$R$-module, then effective techniques involving G\"robner bases may be used to compute
$\Tor _k(M,N)$ for any $k\ge 0$.

Besides %the left Buchberger algorithm and
the standard algorithm to compute (left) syzygies, the ingredients of our method, are
\begin{itemize}
\item an algorithm for computing a free resolution of the left
$R$-module $N$ given a finite system of generators of $N$. Such an algorithm may be
found in~\cite{libro};
\item a finite presentation of the $R$-bimodule $M$. Using the syzygy bimodule
described in~\cite{gbsb}, we show below a method for computing
such a presentation when $M$ is a centralizing $R$-bimodule and a finite system
of two-sided generators of $M$ is given;
\item theorem~\ref{computeTor};
\end{itemize}
Our algorithm, described in detail in~\ref{observaciongeneral}, follows the lines of~\cite{BCGL01}.

%FALTA AQU\'{I} INTRODUCCI\'{O}N HIST\'{O}RICA
%
%Decir lo siguiente:
%
%- Que Galligo extendi\'{o} el m\'{e}todo que existe para calcular
%$Ext^k(M,N)$ en el anillo de polinomios conmutativo a ciertos
%anillos no-conmutativos como por ejemplo, las \'{a}lgebras de Weyl.
%
%- Que Bueso, Gomez-Torrecillas y Lobillo \cite{BCGL01}
%(parece que sale de la tesis de Lobillo, que fue en el 98)
%calculan el $Ext$.
%
%- Que el m\'{e}todo que aqu\'{\i} se utiliza para calcular el $\Tor^(M,N)$
%"sigue la l\'{\i}nea" de \cite{BCGL01}
%
%**********************************
%
%
%Let $R=k\{x_1,\ldots ,x_n;\ Q, Q', \preceq\}$ be a left PBW ring and
%$N$ a finitely generated left $R$-module. Our starting point is a free resolution
%of $N$ (an algorithm to compute it may be found in~\cite[chapter 6]{libro}).
%\begin{equation}
%%\label{resolucionN}
% \cdots
%%\longrightarrow R^{s_{k+1}}
%\stackrel{\partial_{k+1\ }}{\longrightarrow} R^{s_{k}}
%\stackrel{\partial_{k}}{\longrightarrow} R^{s_{k-1}}
%\longrightarrow \cdots \stackrel{\partial_1}{\longrightarrow}
%R^{s_0} \stackrel{\partial_0}{\longrightarrow} N \longrightarrow 0
%\end{equation}
%whose {\em{deleted complex}} is the resolution
%\begin{equation}
%%\label{resolucionreducidaN}
% \cdots
%\longrightarrow R^{s_{k+1}}
%\stackrel{\partial_{k+1\ }}{\longrightarrow} R^{s_{k}}
%\stackrel{\partial_{k}}{\longrightarrow} R^{s_{k-1}}
%\longrightarrow \cdots \stackrel{\partial_1}{\longrightarrow}
%R^{s_0} \longrightarrow 0
%\end{equation}
%Let $M$ be an $R$-bimodule. Applying the functor $M \otimes_R \_$
%to the resolution {\bf{??}}
%%(\ref{resolucionreducidaN})
%we obtain the
%complex
%\begin{equation}
%%\label{resolucionMotimes}
%\cdots
%%\longrightarrow M \otimes_R R^{s_{k+1}}
%\stackrel{1_M \otimes \partial_{k+1\ }}{\longrightarrow} M
%\otimes_R R^{s_{k}} \stackrel{1_M \otimes
%\partial_{k\ }}{\longrightarrow} M \otimes_R R^{s_{k-1}}
%\longrightarrow \cdots \stackrel{1_M \otimes
%\partial_{1\ }}{\longrightarrow} M \otimes_R R^{s_0}
%%\stackrel{1_M \otimes \partial_{0\ }}{\longrightarrow} M \otimes_R N
%\longrightarrow 0
%\end{equation}
%in $R$-Mod. The functor $\Tor_k(M,-)$ applied to $N$ is given by
%\[
%\Tor_k(M,N)\: =\: \ker(1_M \otimes \partial_{k})\: /\: \im(1_M
%\otimes
%\partial_{k+1}),\ \qquad \forall\: k \geq 0.
%\]

%************ A partir de aqu\'{\i} est\'a corregido

\section{Preliminaries}

This section is devoted to fixing notation and describing some
isomorphisms which will be used later.

%In the following lemma, we just recall some basic results.
%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%  LEMA 1
%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{lemma}\label{lemaisobasicos}
Let $M$ be an $R$-bimodule.

For all $s \geq 1$, the map
%\[
%\begin{array}{rcl}
$\alpha: M^s  \longrightarrow  M \otimes_R R^s ; \
\alpha(f_1,\ldots ,f_s) = \sum_{i=1}^s f_i \otimes e_i $,
%\end{array}
%\]
where $\{e_1,..,e_s\}$ is the canonical basis of $R^s$,
% as an $R$-module.
is an isomorphism of left $R$-modules.

Indeed, $\alpha$ is the composition of the isomorphisms
\begin{equation}\label{isoalpha}
\begin{array}{rcccl}
M^s  & %\stackrel{\phi^s}
{\longrightarrow} &
(M \otimes_R R)^s &
%\stackrel{\psi}
{\longrightarrow} & M \otimes_R R^s \\
(f_1,\ldots ,f_s) & \mapsto & (f_1 \otimes 1,\ldots ,f_s \otimes 1) &
\mapsto & \sum_{i=1}^s f_i \otimes e_i.
\end{array}
\end{equation}


%\item
On the other hand, if $A$ is a subbimodule of $R^m$ and $B$ is a left
submodule of $R^s$, then
\begin{equation}\label{isobeta}
\begin{array}{rcl}
\beta: (R^m \otimes_R R^s) / T &
\longrightarrow & (R^m / A) \otimes_R (R^s / B) \\
(f \otimes g) +T  & \mapsto & (f +A) \otimes (g+B),
\end{array}
\end{equation}
where $T= R^m \otimes B\: +\: A \otimes_R R^s$, is an isomorphism of
left $R$-modules.

Furthermore, if $\{a_1,\ldots ,a_r\}$ is a generator system of $A$
such that $A=Ra_1+\cdots + Ra_r=a_1R+\cdots + a_rR$ and
$\{b_1,\ldots ,b_t\}$ is a generator system of $B$ as a left
$R$-module, then
\[
\{a_i \otimes b_j\: /\ 1 \leq i \leq r,\: 1 \leq j \leq t\}
\]
is a generator system of $A \otimes_R B$ as a left $R$-module, since
for all $a=\sum_{l=l}^r p_la_l \in A$ and
$b=\sum_{j=1}^t p'_jb_j \in B$ with $p_l,p'_j \in R$,
\begin{eqnarray*}
a \otimes b & = & \sum_{j,l} p_la_l \otimes p'_jb_j \\
 & = & \sum_{j,l} p_l(a_lp'_j) \otimes b_j \\
 & = & \sum_{j,l} p_l(\sum_{i=1}^r p^{jl}_ia_i) \otimes b_j
% \qquad \qquad \qquad \qquad (a_lp'_j=\sum_{i=1}^r p^{jl}_ia_i)
 \\
 & = & \sum_{i,j,l} p_lp^{jl}a_i \otimes b_j.
\end{eqnarray*}
%\end{enumerate}
%\end{lemma}




%\begin{proof}
%Since the maps
%$$\begin{array}{rcl}
%\phi: M & \longrightarrow & M \otimes_R R \\
%f & \mapsto & f \otimes 1,
%\end{array}$$
%%es un homommorfismo de $R$-m\'{o}dulos a izquierda;
%and
%$$\begin{array}{rcl}
%\psi: \oplus_{i \in \cal{I}} (M \otimes_R N_i) & \longrightarrow & M \otimes_R %\oplus_{i \in \cal{I}} N_i \\
%(f_i \otimes g_i)_{i \in \cal{I}} & \mapsto & (\sum_{i} f_i
% \otimes q_i(g_i))_{i \in \cal{I}},
%\end{array}$$
%are isomorphisms of left $R$-modules, where for all $j \in
%\cal{I}$,
%$$\begin{array}{rcl} q_j: M
%\otimes_R N_j & \longrightarrow & \oplus_{i \in \cal{I}}
%(M \otimes_R N_i) \\
%(f \otimes g) & \mapsto & (\ldots \ldots .,0,\: \stackrel{j}{f \otimes
%g},\: 0,\ldots \ldots )
%\end{array}$$
%the isomorphism $\alpha$ described in (1) of Lemma
%\ref{lemaisobasicos} is obtained from the composition
%%\[
%%\xymatrix{ M^s  \ar[r]^{\phi^s} &
%%(M \otimes_R R)^s \ar[r]^{\psi} & M \otimes_R R^s \\
%%(m_1,\ldots ,m_s) \ar[r] & (m_1 \otimes 1,\ldots ,m_s \otimes 1) \ar[r] &
%%\sum_{i=1}^s m_i \otimes e_i }
%%\]
%\[
%\begin{array}{rcccl}
%M^s  & \stackrel{\phi^s}{\longrightarrow} &
%(M \otimes_R R)^s & \stackrel{\psi}{\longrightarrow} & M \otimes_R R^s \\
%(f_1,\ldots ,f_s) & \mapsto & (f_1 \otimes 1,\ldots ,f_s \otimes 1) &
%\mapsto & \sum_{i=1}^s f_i \otimes e_i
%\end{array}
%\]
%where $\phi^s = \phi\times \stackrel{s}{\cdots} \times \phi$.
%%denotes the {\em{product}} homomorphism
%%$\phi\times \stackrel{s}{\cdots} \times \phi$.

%Moreover, it can be easily checked that the map $\beta$ defined as
%in (2) of Lemma \ref{lemaisobasicos} is an isomorphism of left
%$R$-modules, and that for all $a=\sum_{l=l}^r p_la_l \in A$,
%$b=\sum_{j=1}^t p'_jb_j \in B$ with $p_l,p'_j \in R$, it follows
%\begin{eqnarray*}
%a \otimes b & = & \sum_{j,l} p_la_l \otimes p'_jb_j \\
% & = & \sum_{j,l} p_l(a_lp'_j) \otimes b_j \\
% & = & \sum_{j,l} p_l(\sum_{i=1}^r p^{jl}_ia_i) \otimes b_j\qquad
% (a_lp'_j=\sum_{i=1}^r p^{jl}_ia_i \in AR=RA) \\
% & = & \sum_{i,j,l} p_lp^{jl}a_i \otimes b_j.
%\end{eqnarray*}
%\end{proof}


%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%  DEFINICI\'{O}N BIM\'{O}DULO DE ARTIN
%%%%%%%%%%%%%%%%%%%%%%%%%%

>From here on, let  $0\rightarrow L \rightarrow R^m \stackrel{p_M}{\rightarrow}
M \rightarrow 0$ be a finite presentation of the $R$-bimodule $M$, and let
$H=\{h_1,\ldots ,h_r\} \subseteq R^m$ be a two-sided generator system of $L$
in such a way that $L=RH=HR$ (e.g., $H$ is a two-sided Gr\"obner basis of $L$).
%; see remark~\ref{bimodulocentralizante} below).

\begin{remark}\label{bimodulocentralizante}
If $M$ is a centralizing bimodule, such a presentation may be computed as follows.
Recall that the $R$-bimodule $M$ is said to be centralizing
%(or an $R$-bimodule in the sense of {\em{Artin}}),
if $M$ is generated as a left $R$-module (or
equivalently, as a right $R$-module) by its centralizer
\[
\Cen_R M=\{m \in M;\, rm=mr, \forall\: r \in R\}.
\]
Let $R$ be a PBW algebra and $M$ an $R$-bimodule.
In~\cite{gbsb} we define the syzygy bimodule of $\{m_1,\ldots ,m_s\}\subseteq M$
as the kernel of the morphism of left $\env $-modules $(\env)^s \longrightarrow M;\ \ e_i \longmapsto m_i$,
where $\env$ is the enveloping algebra $R\otimes R^{\rm{op}}$. We also provide an
algorithm to compute the syzygy bimodule of $\{m_1,\ldots ,m_s\}$ .

If the $R$-bimodule $M$ is centralizing, say generated by
$\{m_1,\ldots ,m_s\}\in \Cen _R M$, then the syzygy bimodule is
$(\mathfrak{m}^s)^{-1} L$, where $\mathfrak{m}^s:(\env )^s \longrightarrow R^s$ is
given by $\mathfrak{m}(f\otimes g)= fg$ and $L$ is the kernel of the epimorphism
$p_M : R^s \longrightarrow M$. In that case, if $G$ is a (left) Gr\"obner
basis of the syzygy bimodule, then
$H= \mathfrak{m}^s (G) \setminus \{ 0 \}$ is a two-sided
Gr\"obner basis of $L$ (cf.~\cite[thm.\ 7]{gbsb}).


%If $R$ is a PBW algebra and $M$ is a $R$-bimodule
%generated by $\{m_1,\ldots ,m_s\}$, then a two-sided Gr\"obner basis $H=\{h_1,\ldots %,h_r\}$ of the kernel bimodule $L$ of the epimorphism $p_M: R^s \longrightarrow M$
%may be computed.
%$M$ is centralizing and $\{m_1,\ldots ,m_s\}\subseteq \Cen_R M$, then $H$ is
%actually a two-sided Gr\"obner basis of the subbimodule $L$ of $R^s$.

%We denote by $\bar{p}_M: R^m/L \longrightarrow M$ such isomorphism
%, that is, $$\bar{p}_M: R^m/L \longrightarrow M$$

\end{remark}


%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%  COROLARIO ISOMORFISMO $R^{sm} / L^s$ y $M \otimes_R R^s$
%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{corollary}\label{isogamma}
If $N$ is a left $R$-module
%presented as $N \cong R^s /B$, or equivalently, by
and
%\[
%\begin{array}{ccccccl}
%0 & \longrightarrow &
$ 0 \rightarrow B
%\stackrel{i}
{\rightarrow}  R^s
%\stackrel{p_N}
{\rightarrow}
 N
\rightarrow 0$
 %\\
% & & & & e_i & \mapsto & g_i=(g_{i1},\ldots ,g_{it}),
%\end{array}
%\]
is a presentation of $N$, then the map
\begin{equation} \label{isogamma}
\begin{array}{rccl} \gamma: & R^{ms} / \alpha^{-1}(T) & \longrightarrow & M \otimes_R N \\ & e_{ij}+ \alpha^{-1}(T) & \longmapsto & p_M\:(e'_j) \otimes p_N(e_i)
\end{array}
\end{equation}
is an isomorphism of left $R$-modules, where
% then the map
%\begin{eqnarray*}
%\gamma: R^{ms} / \alpha^{-1}(T) & \longrightarrow & M \otimes_R N \\
%e_{ij}+ \alpha^{-1}(T) & \mapsto & p_M\:(e'_j) \otimes p_N(e_i)
%\end{eqnarray*}
%is an isomorphism of left $R$-modules, where $\alpha$ is the
%isomorphism described in \ref{isoalpha} and
$T=R^m \otimes_R B \: +\:  L \otimes_R R^s$, and
$\{e_1,\ldots ,e_{s}\}$, resp.\ $\{e'_1,\ldots ,e'_{m}\}$, resp.\
$\{e_{11},\ldots ,e_{1m},\ldots ,e_{s1},\ldots ,$ $e_{sm}\}$ are the canonical bases
of $R^s$, resp.\ $R^m$, resp.\ $R^{ms}=(R^m)^s$, being
%$R^{sm}$ where
$e_{ij}=(0_{R^m},\ldots ,\stackrel{i}{e'_j},\ldots ,0_{R^m})$.

Indeed, $\gamma$ is the composition of isomorphisms
\[
 R^{ms} / \alpha^{-1}(T) \stackrel{\bar{\alpha}}{\longrightarrow}
(R^m \otimes_R R^s) / T \stackrel{\beta}{\longrightarrow}
(R^m / L) \otimes_R (R^s / B)  \stackrel{\bar{p}_M \otimes \bar{p}_N}{\longrightarrow}
 M \otimes_R N %,
\]
where $\bar{\alpha}$, resp.\ $\bar{p_M}$, resp.\ $\bar{p_N}$ is obtained by factoring
$\alpha : R^{ms} \longrightarrow (R^m \otimes_R R^s) / T$, resp.\
$p_M: R^m \longrightarrow M$, resp.\ $p_N: R^s \longrightarrow N$ through
the quotient.


Moreover, if $\{g_1,\ldots ,g_t\}$ is a generator system of $B$ as a
left $R$-module, with $g_k=(g_{k1},..,g_{ks})$, then
\begin{equation} \label{sistgen2}
\{\ (e'_jg_{k1},\ldots ,e'_jg_{ks})\ \}_{1 \leq k \leq t \atop 1 \leq j
\leq m}\ \cup\ \{\ (0_{R^m},\ldots ,\stackrel{i}{h_l},\ldots ,0_{R^m})\ \}_{1 \leq i \leq
s,\atop 1 \leq l\leq r }
\end{equation}
is a generator system of
$\alpha^{-1}(T)= \alpha^{-1}(R^m \otimes_R B) + L^s$ as a left
$R$-module,
%\end{corollary}
%\bigskip
since
\[
\{\ e'_j \otimes g_k\ \}_{ 1 \leq j \leq m,\atop 1 \leq k \leq t } \
\cup\ \{\ h_l \otimes e_i\ \}_{ 1 \leq l\leq r,\atop 1 \leq i \leq s }
\]
is a generator system of $T=R^m \otimes_R B + L \otimes_R R^s$ and
\begin{eqnarray*}
\alpha^{-1}(T) & = & \alpha^{-1}(R^m \otimes_R B)+ \alpha^{-1}(L \otimes_R R^s) \\
 & = & \alpha^{-1}(R^m \otimes_R B) + L^s \\
 & = & _R \langle\ \alpha^{-1}(e'_j \otimes g_k),\: \alpha^{-1}(h_l \otimes e_i)
% l \in \{1,..,r\},\:  i \in \{1,..,s\},\:  j \in \{1,..,m\}
\ \rangle_{i, j, k, l} \\
 & = & \: _R \langle\ (g_{k1}e'_j,\ldots ,g_{ks}e'_j),(0_{R^m},\ldots ,\stackrel{i}{h_l},\ldots ,0_{R^m})
%  l \in \{1,..,r\},\: i \in \{1,..,s\},\:  j \in \{1,..,m\}
\ \rangle_{i, j, k, l}.
\end{eqnarray*}

%\begin{proof}
%Let $\bar{\alpha}$ be the isomorphism obtained from $\alpha$
%\begin{eqnarray*}
%\bar{\alpha}: R^{ms} / \alpha^{-1}(T) & \longrightarrow & (R^m
%\otimes_R R^s) / T
%\\
%e_{ij}+ \alpha^{-1}(T) & \mapsto & (e'_j \otimes e_i) + T,
%\end{eqnarray*}
%and $\bar{p}_M, \bar{p}_N$, the isomorphisms which make the
%following diagrams be commutative.
%\[
%\xymatrix{ R^m \ar[rr]^{p_M} \ar[dr]^{\pi} & & M & R^s \ar[rr]^{p_N} \ar[dr]^{\pi} & & N  \\
% & R^m / L \ar[ru]^{\bar{p}_M} & & & R^s / B \ar[ru]^{\bar{p}_N} &
%}
%\]
%Considering $\alpha$, $\beta$ as in Lemma \ref{lemaisobasicos}, we
%take the composition of isomorphisms
%\[
%\begin{array}{rcccccc}
%\gamma:\  R^{ms} / \alpha^{-1}(L) &
%\stackrel{\bar{\alpha}}{\longrightarrow} & (R^m \otimes_R R^s) / T
%& \stackrel{\beta}{\longrightarrow} & (R^m / L) \otimes_R (R^s /
%B) & \stackrel{\bar{p}_M \otimes \bar{p}_N}{\longrightarrow} & M \otimes_R N, \\
%\end{array}
%\]
%which acts on each element $e_{ij}+\alpha^{-1}(L) \in R^{ms} /
%\alpha^{-1}(L)$ as
%\[ \xymatrix{ e_{ij}+\alpha^{-1}(L)
%\ar[r]^{\bar{\alpha}\ \ \ } & (e'_j \otimes e_i) + T
%\ar[r]^{\beta\ \ \ } & (e'_j+L) \otimes
%(e_i + B) \ar[d]^{\bar{p}_M \otimes \bar{p}_N} \\
% & & p_M(e'_j) \otimes p_N(e_i)}.
%\]
%Furthermore, since $\{g_1,\ldots ,g_t\}$ is a generator system of $B$
%and $\{h_1,\ldots ,h_r\}$ is a two-sided Gr\"obner basis of $L$, from
%\ref{isoalpha} it follows that
%\[
%\{\ e'_j \otimes g_k\: /\: 1 \leq j \leq m,\: 1 \leq k \leq t\ \}\
%\cup\ \{\ h_l \otimes e_i\: /\: 1 \leq l\leq r,\: 1 \leq i \leq s\
%\}
%\]
%is a generator system of $T=R^m \otimes_R B + L \otimes R^s$.
%Hence,
%\begin{eqnarray*}
%\alpha^{-1}(T) & = & \alpha^{-1}(R^m \otimes B)+ \alpha^{-1}(L \otimes R^s) \\
% & = & \alpha^{-1}(R^m \otimes B) + L^s \\
% & = & _R \langle\ \alpha^{-1}(e'_j \otimes g_k),\: \alpha^{-1}(h_l \otimes e_i)\:
% /\: i, j, k, l
%% l \in \{1,..,r\},\:  i \in \{1,..,s\},\:  j \in \{1,..,m\}
% \ \rangle \\
% & = & \: _R \langle\ %(g_{k1}e'_j,\ldots ,g_{ks}e'_j),(0_{R^m},\ldots ,\stackrel{i}{h_l},\ldots ,0_{R^m})\:
% /\: i, j, k, l
%%  l \in \{1,..,r\},\: i \in \{1,..,s\},\:  j \in \{1,..,m\}
%\ \rangle
%\end{eqnarray*}
%\end{proof}
%\bigskip



%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%  SECTION 2
%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Computing $\Tor_k(M,N)$}


Now, let $R=k\{x_1,\ldots ,x_n;\ Q, Q', \preceq\}$ be a left PBW ring and
$N$ a finitely generated left $R$-module. Let
\begin{equation}\label{resolucionN}
 \cdots
%\longrightarrow R^{s_{k+1}}
\stackrel{\partial_{k+1\ }}{\longrightarrow} R^{s_{k}}
\stackrel{\partial_{k}}{\longrightarrow} R^{s_{k-1}}
\longrightarrow \cdots \stackrel{\partial_1}{\longrightarrow}
R^{s_0} \stackrel{\partial_0}{\longrightarrow} N \longrightarrow 0
\end{equation}
be a free resolution of $N$, where $\partial _k$ is the matrix
\[ \left( \begin{array}{c} g_1^k \\ \vdots \\ g_{s_k}^k \end{array} \right)
 \]
with $g_i^k=(g_{i1}^k,\ldots ,g_{is_{k-1}}^k) \in R^{s_{k-1}}$ for $k\geq 1$
and $1\le i \le s_k$ (an algorithm to compute it may be found
in~\cite[chapter 6]{libro}).


In this section we show a method for computing ${\rm{Tor}}_k(M,N)$ when $M$ is a  finitely presented $R$-bimodule, i.e., we show how to compute the $k$-th homology
module of the complex
\begin{equation}\label{resolucionMotimes}
\cdots
%\longrightarrow M \otimes_R R^{s_{k+1}}
\stackrel{1_M \otimes \partial_{k+1\ }}{\longrightarrow} M
\otimes_R R^{s_{k}} \stackrel{1_M \otimes
\partial_{k\ }}{\longrightarrow}
%M \otimes_R R^{s_{k-1}} \longrightarrow
\cdots \stackrel{1_M \otimes \partial_{1\ }}
{\longrightarrow} M \otimes_R R^{s_0}
%\stackrel{1_M \otimes \partial_{0\ }}{\longrightarrow} M \otimes_R N
\longrightarrow 0 .
\end{equation}


%%%%%%%% Caso $k=0
\begin{remark}\label{observacionisoTor}
The case $k=0$ may be treated apart, since $\Tor_0(M, N) \cong M \otimes_R N$. %(see~\cite{rotman}).

We start with the presentations
$0\rightarrow L \rightarrow  R^m {\rightarrow} M \rightarrow 0$ and
$0\rightarrow \ker{\partial_0} \rightarrow  R^{s_0} {\rightarrow} N
\rightarrow 0$ of the $R$-bimodule $M$, resp.\ of the left
$R$-module $N$.

Since  $ \{ g_1^1,\ldots ,g_{s_1}^1 \}$ is a generator system
of $\ker{\partial_0}=\im{\partial_1}$ as a left $R$-module and
\[
\gamma: R^{ms_0} / (\alpha^{-1}(R^m \otimes_R
\ker{\partial_0})+L^{s_0}) \longrightarrow \Tor_0( M , N),
\]
as in~(\ref{isogamma}) is an isomorphism of left $R$-modules, we completely describe
a presentation of $\Tor_0( M , N)$ by giving the generator system
\[
\{\ (e'_jg_{l1}^1,\ldots ,e'_jg_{ls_0}^1)\ \}_ {1 \leq l \leq s_1,\atop 1 \leq j
\leq m}\ \cup\ \{\ (0_{R^m},\ldots ,\stackrel{i}{h_l},\ldots ,0_{R^m})\ \}_{ 1 \leq i
\leq s_0,\atop 1 \leq l\leq r}
\]
of $\alpha^{-1}(R^m \otimes_R \ker{\partial_0})+L^{s_0}$, where
$\{ h_1,\ldots ,h_r\}$ is a two-sided Gr\"obner basis of $M$
(see~(\ref{sistgen2})).
\end{remark}


Let us return to the general case. Consider again the presentation
$0\rightarrow L \rightarrow  R^m \stackrel{p_M}{\rightarrow} M \rightarrow 0$ of the
$R$-bimodule $M$ and the free resolution (\ref{resolucionN}) of $N$.

For all $k\ge 1$, let $d_k=\gamma_{k-1}^{-1}\circ (1_M \otimes \partial_k)\circ
\gamma_k$, where $\gamma_k: R^{ms_k} /L^{s_k}  \longrightarrow  M \otimes_R R^{s_k}$
is the isomorphism defined as in~(\ref{isogamma}) with $N=R^{s_k}$.

Clearly, the complex
\begin{equation}\label{complejodk}
\cdots \longrightarrow  R^{ms_k} / L^{s_k} \stackrel{d_k}{\longrightarrow}
R^{ms_{k-1}} / L^{s_{k-1}} \stackrel{d_{k-1}}{\longrightarrow} \cdots
\stackrel{d_1}{\longrightarrow} R^{ms_0} / L^{s_0}  \longrightarrow  0
\end{equation}
is isomorphic to the one in (\ref{resolucionMotimes}), so
$\Tor_{k}(M,N)$ may be computed as the homology of~(\ref{complejodk}).


By definition, for all $1 \leq i \leq s_k$, $1 \leq j \leq m$,
\begin{equation}\label{homomorfismodk}
\begin{array}{rcl}
d_k(e_{ij}+L^{s_k}) & = &
\gamma_{k-1}^{-1}(1_M \otimes \partial_k)\: \gamma_k(e_{ij}+L^{s_k})  \\
& = & \gamma_{k-1}^{-1}((1_M \otimes \partial_k)\: (p_M(e'_j) \otimes e_i))  \\
& = & \gamma_{k-1}^{-1}(p_M(e'_j) \otimes (g_{i1}^k,\ldots ,g_{is_{k-1}}^k)) \\
% & = & \beta_{k-1}^{-1}(\bar{p}^{\: -1}(F_jg_{i1}^k),\ldots ,
%\bar{p}^{\: -1}(F_jg_{is_{k-1}}^k)) & \\
% & = & \beta_{k-1}^{-1}(\bar{p}^{\: -1}(g_{i1}^kF_j),\ldots ,
%\bar{p}^{\: -1}(g_{is_{k-1}}^kF_j)) & (\: F_j \in \Cen_R(M)\: )\\
% & = & \beta_{k-1}^{-1}(g_{i1}^k\bar{p}^{\: -1}(F_j),\ldots ,
%g_{is_{k-1}}^k\bar{p}^{\: -1}(F_j)) & \\
% & = & \beta_{k-1}^{-1}(g_{i1}^k(e'_j + L),\ldots ,
%g_{is_{k-1}}^k(e'_j + L)) & (\: \bar{p}^{\: -1}(F_j)=e'_j + L\: ) \\
% & = & \beta_{k-1}^{-1}(g_{i1}^ke'_j + L,\ldots ,
%g_{is_{k-1}}^ke'_j + L) &  \\
& = & (g_{i1}^ke'_j,\ldots ,g_{is_{k-1}}^ke'_j)+L^{s_{k-1}}.   \\
\end{array}
\end{equation}
Let $\widetilde{d_k} : R^{ms_k} \longrightarrow R^{ms_{k-1}}$
be the block-built matrix
\[
A_k = \left( \begin{array}{c|c|c}
g_{11}^kI_m & \cdots & g_{1s_{k-1}}^kI_m \\
\hline \vdots & & \vdots \\
\hline g_{s_k1}^kI_m & \cdots & g_{s_ks_{k-1}}^kI_m
\end{array}\right) \in {\cal{M}}_{ms_k \times ms_{k-1}}(R),
\]
where $I_m$ denotes the $m \times m$-identity matrix. Since $A_k$ is
built of blocks which are elements of $R$ times the identity matrix, we have $\widetilde{d_k}(L^{s_k})\subseteq L^{s_{k-1}}$, and the diagram
%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%  LEMA POSTERIOR A LA CONSTRUCCI\'{O}N DE LA MATRIZ
%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{lemma}
%With the previous notation, the homomorphism of left $R$-modules
%defined as
%\begin{eqnarray*}
%\widetilde{d_k}: R^{ms_k} & \longrightarrow & R^{ms_{k-1}}\\
%e_{ij} & \mapsto & e_{ij}A_k+L^{s_{k-1}}
%\end{eqnarray*}
%satisfies $\widetilde{d_k}(L^{s_k}) \subseteq L^{s_{k-1}}$ and it
%makes the following diagram to be commutative
\[\xymatrix{
R^{ms_k} \ar[r]^{\pi_k} \ar[d]_{\widetilde{d_k}} & R^{ms_k} / L^{s_k} \ar[d]^{d_k} \\
R^{ms_{k-1}} \ar[r]_{\pi_{k-1\ \ \ }} & R^{ms_{k-1}} /
L^{s_{k-1}}}
\]
is commutative.
%\end{lemma}
%\smallskip

%\begin{proof}
%%Para todo $F=\sum_{\begin{array}{cc} 1 \leq i \leq s_k \\
%%1 \leq j \leq m \end{array}} F_{ij}e_{ij} \in R^{ms_k}$,
%%\begin{eqnarray*}
%%d_k\pi_k(F) & = & d_k(\sum_{i,j}F_{ij}e_{ij}+L^{s_k}) \\
%% & = & \sum_{i,j} F_{ij}(e_{ij}A_k +L^{s_{k-1}}) \\
%% & = & (\sum_{i,j} F_{ij}e_{ij})(A_k+L^{s_{k-1}}) \\
%% & = & FA_k + L^{s_{k-1}} \\
%% & = & \pi_{k-1}(FA_k) \\
%% & = & \pi_{k-1}\widetilde{d_k}(F).
%%\end{eqnarray*}
%Considering the cube
%%\[
%%\xymatrix{ R^{ms_k} \ar[rrr]^{\pi_k} \ar[ddd]_{\widetilde{d_k}} &
%%& &
%%R^{ms_k}/L^{s_k} \ar[ddd]^{d_k} \\
%% & R^{m} \otimes R^{s_k} \ar[r]^{p\: \otimes 1} \ar[d]_{1 \otimes \partial_k}
%% \ar[ul]_{\alpha_k^{-1}}^{\cong} &
%% M \otimes_R R^{s_k} \ar[d]_{1 \otimes \partial_k} \ar[ur]^{\gamma_k^{-1}}_{\cong} & %\\
%% & R^{m} \otimes R^{s_{k-1}} \ar[r]^{p\: \otimes 1} %\ar[dl]^{\alpha_{k-1}^{-1}}_{\cong} &
%% M \otimes_R R^{s_{k-1}} \ar[rd]_{\gamma_{k-1}^{-1}}^{\cong} & \\
%%R^{ms_{k-1}} \ar[rrr]_{\pi_{k-1}} & & & R^{ms_{k-1}}/L^{s_{k-1}} }
%%\]
%\[
%\xymatrix{ & R^{ms_k} \ar[rr]^{\qquad \pi_k}
%\ar[dd]_{\widetilde{d_k}} &
%& R^{ms_k}/L^{s_k} \ar[dd]^{d_k} \\
% R^{m} \otimes R^{s_k} \ar[rr]^{\qquad \quad p\: \otimes 1} \ar[dd]_{1 \otimes %\partial_k}
% \ar[ur]^{\alpha_k^{-1}}_{\cong} & &
% M \otimes_R R^{s_k} \ar[dd]_{1 \otimes \partial_k} \ar[ur]^{\gamma_k^{-1}}_{\cong} & %\\
% & R^{ms_{k-1}} \ar[rr]^{\qquad \pi_{k-1}} & & R^{ms_{k-1}}/L^{s_{k-1}}
%\\
% R^{m} \otimes R^{s_{k-1}} \ar[rr]^{\qquad \quad p\: \otimes 1}
%\ar[ur]^{\alpha_{k-1}^{-1}}_{\cong} & &
% M \otimes_R R^{s_{k-1}} \ar[ur]^{\gamma_{k-1}^{-1}}_{\cong} &
%}
%\]
%where the isomorphisms $\alpha_k$, $\gamma_k$ are described in
%(\ref{isoalpha}), \ref{observacionisoTor}, resp. Since there
%are two opposite faces linked by isomorphisms and one of them (en
%la que intervienen los homomorfismos $p \otimes 1$) constitutes a
%commutative square, then the square of its opposite face is also
%commutative.
%\end{proof}
%\medskip

The above discussion proves the following result:

\begin{theorem} \label{computeTor}
With the previous notation, for all $k \geq 1$
\begin{enumerate}
\item $\ker d_k=\ker \pi_{k-1}\widetilde{d_k} / L^{s_k}$;
\item $\im d_k=\im \pi_{k-1}\widetilde{d_k} \subseteq R^{ms_{k-1}} / L^{s_{k-1}}$
is generated by
\[
\{ (g_{i1}^ke'_j,\ldots ,g_{is_{k-1}}^ke'_j)+ L^{s_{k-1}}
%\}_{i,j=1}^{s_k,m}$
\}_{1\leq i \leq s_k\atop 1 \leq j \leq m}
\]
as a left $R$-module (note that
$(g_{i1}^ke'_j,\ldots ,g_{is_{k-1}}^ke'_j)$ is the $(j+m(i-1))$-th row of $A_k$);
%meaning by row of $A_k$ each vector
%$(g_{i1}^ke'_j,\ldots ,g_{is_{k-1}}^ke'_j) \in R^{ms_{k-1}}$ con.
\item %From the expression (\ref{ecisoTor}),
%\begin{equation}
$\Tor_k(M,N)  \cong  \ker d_k / \im d_{k+1}
%\\
% & = & (\ker(\pi_{k-1}\widetilde{d_k}) / L^{s_k})\ /\ (_R \langle
%{\rm{filas\ de\ }} A_{k+1} \rangle + L^{s_k}) / L^{s_k} \\
% & \cong &
 = \ker \pi_{k-1}\widetilde{d_k} / (_R \langle
{\rm{rows\ of\ }} A_{k+1} \rangle + L^{s_k})$.
%\end{equation}
\end{enumerate}
\end{theorem}

\begin{remark}\label{observaciongeneral}
To compute $\Tor_k(M,N)$ we start with a finite presentation
$0\rightarrow L \rightarrow R^m \stackrel{p_M}{\rightarrow} M \rightarrow 0$
of the $R$-bimodule $M$ (say, e.g., we have computed a two-sided Gr\"obner basis
$\{h_1,\ldots ,h_r\} \subseteq R^m$ of the $R$ bimodule $L$) and a
free resolution of the left $R$-module $N$ as (\ref{resolucionN}).
%, esto es,
%\[
%L=\:  _R \langle h_1,\ldots ,h_r \rangle = \langle h_1,\ldots ,h_r
%\rangle_R,
%\]

The matrix $A_k$ is block-built as above, using the matrix
\[ \partial _k = \left( \begin{array}{c} g_1^k \\ \vdots \\ g_{s_k}^k \end{array} \right)
. \]

The set $\{\ (0_{R^m},\ldots ,\stackrel{i}{h_l},\ldots ,0_{R^m})\
\}_{i,l=1}^{s_k,r}$
%_{ 1 \leq i \leq s_k,\atop 1 \leq l\leq r }\subseteq R^{ms_k}$
% \rangle \\
% & = & \langle (h_1,\ldots ,0),\ldots ,
%(h_r,\ldots ,0),\ldots , (0,\ldots ,h_1), (0,\ldots ,h_r) \rangle_R.
%\end{eqnarray*}
is a generator system (in fact, it is a two-sided Gr\"obner
basis when $\{h_1,\ldots ,h_r\}$ so is) of $L^{s_k}$ as an $R$-bimodule.

Then, we compute the kernel of $ \pi_{k-1}\widetilde{d_k}: R^{ms_k}  \longrightarrow R^{ms_{k-1}} / L^{s_{k-1}}$ using syzygies.
Indeed (see \cite{BCGL01}), let $H_k$ be the matrix
\[ H_k=\left(\frac{A_k}{A'_k}\right) \in {\cal{M}}_{(ms_k+rs_{k-1})\times
{ms_{k-1}}}(R),\]
where $A'_k$ is the matrix whose rows are the generators of
$L^{s_{k-1}}$ as a left $R$-module. Then, if
\[
Syz^l(H_k)=\: _R \langle p_1,...,p_l \rangle,\ \ {\rm{\ with\ }}
p_i=(p'_i,p''_i) \in R^{ms_k} \times R^{rs_{k-1}},
\]
then $\ker \pi_{k-1}\widetilde{d_k}\ =\ _R \langle p'_1,...,p'_l \rangle$.
\end{remark}

\begin{example}
Let $R=U(\mathfrak{sl}(2))$, the
universal enveloping algebra of the Lie algebra of traceless $2
\times 2$-matrices, where $k=\mathbb{C}$ (or $k=\mathbb{Q}$). We know (see, e.g.,
\cite{libro}) that $R$ is the PBW algebra $k\{x,y,z;\: Q\, \preceq_{\omega}\}$ with
$ Q= \{\: yx-xy+z,\ zx-xz-2x,\ zy-yz+2y\: \}$ and, say, $\omega=(1,2,2)$.

Let $N = R^2 / B$, where $B$ is the left $R$-module generated
by
\[
g_1=(y^3,x),\ g_2=(y,xz),\ g_3=(0,xy^2z-2yz^2+2yz-x).
\]
The left syzygy module $Syz^l(g_1,g_2,g_3)$ is generated by
$g=(1,-y^2,1) \in R^3$, and hence
\begin{equation}\label{resNejemplo}
0  \longrightarrow R \stackrel{\delta_1}{\longrightarrow} R^3
\stackrel{\delta_0}{\longrightarrow} N \longrightarrow 0,
\end{equation}
where $\partial_0 =\left(\begin{array}{c} g_1 \\ g_2 \\ g_3 \end{array} \right)$ and
$\partial_1=\left( g \right)$, is a free resolution of $N$.

Let $L$ be the $R$-bimodule generated by $\{(C,1),(1,C)\}$, where
$C$ is the Casimir element $z^2/2+2xy-z$ (a well known central element of
$U(\mathfrak{sl}(2))$), and let $M=R^2/L$.
%Since the center %$\Cen(R)$
%of $R$ is $k[C]$,
%\[
%\{(C,1),(1,C)\} \subseteq k[C] \times k[C] = \Cen_R(R^2).
%\]
%Hence, $L=R\{(C,1),(1,C)\}=\{(C,1),(1,C)\}R$.

%Let us compute $\Tor_k(M,N)$ for all $k \geq 0$.
For all $k \geq 2$ we have $\Tor_k(M,N)=0$, as the free resolution
(\ref{resNejemplo}) of $N$ has length 2.

For $k=0$, we have (see~\ref{observacionisoTor})
$\Tor_0(M,N) \cong R^6\ /\ (\alpha^{-1}(R^2 \otimes_R B)+L^3)$, and
\begin{eqnarray*}
\lefteqn{\{\  (1,0,-y^2,0,1,0)\ ,\ (0,1,0,-y^2,0,1)\ ,\
(C,1,0,0,0,0)\ ,\ (0,0,C,1,0,0)\ ,}  \\
& & (0,0,0,0,C,1)\ ,\  (1,C,0,0,0,0)\ ,\ (0,0,1,C,0,0)\ , \ (0,0,0,0,1,C) \ \}
\end{eqnarray*}
is a generator system of $\alpha^{-1}(R^2 \otimes_R B)+L^3$ as a left $R$-module.

For $k=1$, we have $\Tor_1(M,N) \cong
\ker(\pi_{0}\widetilde{d_1})\ /\ (_R \langle {\rm{rows\ of\ }} A_{2} \rangle + L) $, but
in this particular example $A_2=0$ since $\partial_2=0$.
%, then $_R \langle {\rm{rows\ of\ }} A_{2} \rangle + L=L$.

As pointed out in~\ref{observaciongeneral}, $\ker \pi_{0}\widetilde{d_1}$ is obtained
from the left syzygy module of the rows of
%\[
%\begin{array}{rcccl}
%R^2 & \stackrel{\widetilde{d_1}}{\longrightarrow} & R^6 &
%\stackrel{\pi_0}{\longrightarrow} & R^6/L^3 \\
%(r_1,r_2) & \longmapsto & (r_1,r_2)A_1 & \longmapsto &
%(r_1,r_2)A_1+ L^3,
%\end{array}
%\]
\[ A_1=\left(\begin{array}{cccccc} 1 & 0 & -y^2 & 0 & 1 & 0
\\ 0 & 1 & 0 & -y^2 & 0 & 1 \end{array}\right) \]
%The left syzygy module of the rows of $A_1$
and the generators of $L^3$. Indeed, by picking up just the first two
components of each element of the generator
system
\[
%_R\left< \
\begin{array}{cll}
\{  (z^2+4xy-2z,2,-2,2y^2,-2,0,0,0)\ ,  &  \\
\   (1,z^2/2+2xy-z,0,0,0,-1,y^2,-1)\ ,  &   \\
    \multicolumn{2}{l}{\ (8xy-4z,-4z^4-4xyz^2+2z^3+4,-4,4y^2,-4,2y^2,} \\
    & -2y^2z^2+16y^2z-32y^2,2z^2) \}
\end{array}
%\right>.
\]
of the left syzygy module, we obtain the generator system
\[ \{ \ (2C,2)\ ,\ (1,C) \ , \ (8xy-4z,-z^4-4xyz^2+2z^3+4)\ \} \]
of $\ker \pi_{0}\widetilde{d_1}$.
Therefore, $\Tor_1(M,N) \cong \ker \pi_{0}\widetilde{d_1} / L = 0 $, since
$(8xy-4z,-z^4-4xyz^2+2z^3+4)\in L$ (one may check this out by dividing the element
by a two-sided Gr\"obner basis of $L$).
%is isomorphic to the
%$R$- submodule of $R^2$ generated by the element $(8xy-4z,-4z^4-4xyz^2+2z^3+4)$.
\end{example}

\section*{Acknowledgement}

Both authors would like to thank Pepe Bueso for his useful suggestions.

%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%  DEFINICION DE
%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%  BIBLIOGRAFIA
%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thebibliography}{99}
%
%\bibitem{gkdim}
%{\sc Bueso, J.L., Castro, F., G\'omez Torrecillas, J.\ y Lobillo, F.J.,}
%{\em Computing the Gelfand Kirillov dimension}. SAC Newsletter, 1 (1996) 39-52.
%
%\bibitem{algpbw}
%{\sc Bueso, J.\ L., Castro, F.\ J., G\'omez Torrecillas, J.\ y Lobillo, F.J.,}
%{\em An introduction to effective calculus in quantum groups}.
%Lect.\ Notes Pure App.\ Math.\ {\bf 197}.
%Rings, Hopf algebras and Brauer groups. (S.\ Caenepeel and A.\
%Verschoren, eds), Marcel Dekker (1998), 55-83.
%
%\bibitem{primalidad}
%{\sc Bueso, J.L., Castro, F., G\'omez Torrecillas, J.\ y Lobillo, F.J.,}
%{\em Test de primalit\'e dans une extension de Ore it\'er\'ee}.
%C. R. Acad. Sci. Paris S\'{e}r. I Math. {\bf 328} (1999), 459-462.
%
\bibitem{BCGL01}
Bueso, J.L.; Castro, F.; G\'omez Torrecillas, J.;
Lobillo, F.J. Homological computations in PBW modules.
Alg.\ Rep.\ Theory \textbf{2001}, 4 (2), 201-218.
%
\bibitem{libro}
Bueso, J.L.; Gomez Torrecillas, J.; Verschoren, A. {\sf
Algorithmic Methods in Noncommutative Algebra: Applications to
Quantum Groups}; Kluwer: Dordrecht, The Netherlands, 2003.
%
%\bibitem{caniglia}
%{\sc Caniglia, L., Galligo, A.\ y Heintz, J.},
%{\em Some new effectivity bounds in computational geometry}.
%Lect.\ Notes in Comput.\ Sci.\ {\bf 357}, (1988), 131-151.
%
%\bibitem{cox}
%{\sc Cox, D., Little, J.\ y O'Shea, D.},
%{\em{Ideals, Varieties, and Algorithms}}. Springer-Verlag, Nueva York, 1992.
%
\bibitem{gbsb} Garc\'{\i}a Rom\'an, S.; Garc\'{\i}a Rom\'an, M.
Gr\"obner bases and syzygies on bimodules, preprint. {\tt{math.RA/0405550}}
%
%\bibitem{lobillo}
%{\sc Lobillo, F.\ J.,}{\em M\'etodos algebraicos y efectivos en
%grupos cu\'anticos}. Tesis doctoral (1998), Universidad de Granada.
%
%\bibitem{mora85}
%{\sc{Mora, T.}}, {\em{Gr\"obner bases for non-commutative polynomial rings}},
%Lect.\ Notes in Comput.\ Sci. {\bf{229}}, 353--362.
%
%\bibitem{rotman}
%{\sc Rotman, J.}, {\em An introduction to homological algebra},
%Academic Press, 1979.
%
\bibitem{smith}
Smith, G. Computing global extension modules.
J.\ Symb.\ Comput.\ \textbf{2000}, 29 (4-5), 729-746.
\end{thebibliography}

%{\sc{Dpto.\ Matem\'atica Fundamental,
%%(\'Area de \'Algebra),
%Fac. de Matem\'aticas,
%Univ. La Laguna}}.
%
%\hspace{1 cm} {\tt{E-mail: mroman@ull.es}}  \hspace{3 cm} {\tt{E-mail: sgarcia@ull.es}}

%\end{article}

\end{document}

math0412326.extracted_bib
\begin{thebibliography}{99}
%
%\bibitem{gkdim}
%{\sc Bueso, J.L., Castro, F., G\'omez Torrecillas, J.\ y Lobillo, F.J.,}
%{\em Computing the Gelfand Kirillov dimension}. SAC Newsletter, 1 (1996) 39-52.
%
%\bibitem{algpbw}
%{\sc Bueso, J.\ L., Castro, F.\ J., G\'omez Torrecillas, J.\ y Lobillo, F.J.,}
%{\em An introduction to effective calculus in quantum groups}.
%Lect.\ Notes Pure App.\ Math.\ {\bf 197}.
%Rings, Hopf algebras and Brauer groups. (S.\ Caenepeel and A.\
%Verschoren, eds), Marcel Dekker (1998), 55-83.
%
%\bibitem{primalidad}
%{\sc Bueso, J.L., Castro, F., G\'omez Torrecillas, J.\ y Lobillo, F.J.,}
%{\em Test de primalit\'e dans une extension de Ore it\'er\'ee}.
%C. R. Acad. Sci. Paris S\'{e}r. I Math. {\bf 328} (1999), 459-462.
%
\bibitem{BCGL01}
Bueso, J.L.; Castro, F.; G\'omez Torrecillas, J.;
Lobillo, F.J. Homological computations in PBW modules.
Alg.\ Rep.\ Theory \textbf{2001}, 4 (2), 201-218.
%
\bibitem{libro}
Bueso, J.L.; Gomez Torrecillas, J.; Verschoren, A. {\sf
Algorithmic Methods in Noncommutative Algebra: Applications to
Quantum Groups}; Kluwer: Dordrecht, The Netherlands, 2003.
%
%\bibitem{caniglia}
%{\sc Caniglia, L., Galligo, A.\ y Heintz, J.},
%{\em Some new effectivity bounds in computational geometry}.
%Lect.\ Notes in Comput.\ Sci.\ {\bf 357}, (1988), 131-151.
%
%\bibitem{cox}
%{\sc Cox, D., Little, J.\ y O'Shea, D.},
%{\em{Ideals, Varieties, and Algorithms}}. Springer-Verlag, Nueva York, 1992.
%
\bibitem{gbsb} Garc\'{\i}a Rom\'an, S.; Garc\'{\i}a Rom\'an, M.
Gr\"obner bases and syzygies on bimodules, preprint. {\tt{math.RA/0405550}}
%
%\bibitem{lobillo}
%{\sc Lobillo, F.\ J.,}{\em M\'etodos algebraicos y efectivos en
%grupos cu\'anticos}. Tesis doctoral (1998), Universidad de Granada.
%
%\bibitem{mora85}
%{\sc{Mora, T.}}, {\em{Gr\"obner bases for non-commutative polynomial rings}},
%Lect.\ Notes in Comput.\ Sci. {\bf{229}}, 353--362.
%
%\bibitem{rotman}
%{\sc Rotman, J.}, {\em An introduction to homological algebra},
%Academic Press, 1979.
%
\bibitem{smith}
Smith, G. Computing global extension modules.
J.\ Symb.\ Comput.\ \textbf{2000}, 29 (4-5), 729-746.
\end{thebibliography}