Singular

Hello Ibar,

the computations you request (and indeed, much more) may be done using Singular, by its non-commutative extension Plural. Please go to the Online Manual -> Plural, there you will see the complete documentation of the non-commutative part of Singular.

First, defining the exterior algebra in Singular. We represent it as a factor-algebra of an anticommutative ring (that is, a ring with
the generators X_1,...,X_N and relations Xj*Xi = -Xi*Xj, for all i<j) modulo the two-sided ideal, generated by squares of the generators, that is {Xi^2 | 1<=i<=N}.

The automatic way to set an exterior algebra up is the function 'Exterior()' from the library nctools.lib, which requires from you only to set up a commutative ring, the rest will be done automatically along the lines described above. Take a look on the documentation and example of this function (e.g. [url]http://www.singular.uni-kl.de/Manual/3-0-0/sing_445.htm#SEC497[/url]).

Of course, you can define an exterior algebra manually (like described e.g. in [url]http://www.singular.uni-kl.de/Manual/3-0-0/sing_367.htm#SEC407[/url]).

After you have defined exterior algebra (note, that it is of type qring), you can use 'std' for Groebner bases, 'syz' for syzygies, 'nres' or 'mres' for free resolutions, 'betti' for graded Betti numbers and so on - please consult with the documentation.

Good luck! Please inform us on your progress.

Viktor Levandovskyy

Hello,

I would like to make some GB-computations (such as generic initial ideal) over an external algebra.
Is it possible to use Singular for such purposes?
Is Singular able to deal with skew-commutative rings?