6. The Weyl algebra and the exterior algebra

For applications to D-modules and to vector bundles over projective spaces the computation of standard bases and syzygies in two special non-commutative algebras is also implemented in SINGULAR. The advantage of SINGULAR having almost all possible monomial orderings implemented allows us to compute in the local and in the global case.

SINGULAR can compute the standard basis of modules over the (non-commutative) algebra $K\langle D_1, \ldots, D_n,\; X_1, \ldots, X_n\rangle$ with the relations

$$\begin{eqnarray*}X_iX_j & = & X_jX_i \mbox{ for all } i,j\\ D_iD_j & = & D_jD_i... ... = & D_jX_i \mbox{ for all } i \not= j\\ X_iD_i+1 & = & D_iX_i. \end{eqnarray*}$$

The D_i may be considered as differential operators.

The only restriction we have to make to the ordering is the assumption that L([m, m']) < L(m, m') for all monomials m, m' with $[m, m'] \not= 0$. Especially a product ordering with the property that monomials in the D_i are always greater than monomials in the X_i and which is a wellordering on $K[D_i, \ldots, D_n]$, is admissible. Hence, SINGULAR can compute standard bases in (Loc $_< K[X_1, \ldots, X_n]$) $[D_1, \ldots, D_n]$, in particular in the Weyl algebra $K[X_1, \ldots, X_n]$ $[D_1, \ldots, D_n]$ and in the ``local Weyl algebra'' $K[X_1, \ldots, X_n]_{(X_1, \ldots, X_n)}$ $[D_1, \ldots, D_n]$. Moreover, we have implemented standard basis and syzygies of modules over the algebra (Loc $_< K[X_1, \ldots, X_n]$) $\langle Y_1, \ldots, Y_m\rangle$ with the relations

$$\begin{eqnarray*}Y_i^2 & = & 0,\\ Y_iY_j & = & -Y_jY_i, \end{eqnarray*}$$

in particular over the tensor product of $K[X_1, \ldots, X_n]$ with the exterior algebra.