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 with the relations

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
.
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
,
is admissible. Hence, SINGULAR can compute standard
bases in (Loc
)
,
in particular in
the Weyl algebra
and in the
``local Weyl algebra''
.
Moreover, we have implemented standard basis and syzygies of
modules over the algebra (Loc
)
with the relations

in particular over the tensor product of with the exterior algebra.