Quotients of exterior algebras

asked 2021-03-19 04:30:05 +0100

tkarn
78 ●9

I am trying to take a quotient of an exterior algebra by a two-sided ideal, and having some trouble. As an easy case to understand the syntax, I tried to compute $\Lambda^\mathbb{Q}[x,y,z]/(x-y)$, where I would expect $\overline{x} = \overline{y}$. However, it seems to not consider xbar and ybar the same.

sage: E.<x,y,z> = algebras.Exterior(QQ);
sage: I = E.ideal(x-y);
sage: Q = E.quo(I);
sage: xbar,ybar,zbar = Q.gens();
sage: xbar == ybar
False

I thought maybe that was fine because xbar and ybar are generators of the quotient so maybe there is some funkiness going on there. So then I tried to take the image of x and y under the quotient map and check if they are equal, but it still said they were not.

sage: q = Q.cover()
sage: q(x) == q(y)
False

Am I misunderstanding something in Sage? In the algebra of it? Both?

answered 2021-03-21 23:00:30 +0100

tkarn
78 ●9

updated 2021-03-21 23:01:25 +0100

Ok it looks like this is the current (Sage 9.3beta8) behavior.

In the source code for Ideal_nc, which is the type of I, there is no implementation of a .reduce() method. Since Ideal_nc is a subclass of Ideal_generic it defaults to the Ideal_generic.reduce() method. But when Ideal_generic.reduce() is called it defaults to the identity:

    def reduce(self, f):
        r"""
        Return the reduction of the element of `f` modulo ``self``.

        This is an element of `R` that is equivalent modulo `I` to `f` where
        `I` is ``self``.

        EXAMPLES::

            sage: ZZ.ideal(5).reduce(17)
            2
            sage: parent(ZZ.ideal(5).reduce(17))
            Integer Ring
        """
        return f       # default

So this is asking if xbar and ybar are literally the same, which they are not.

Thanks to Travis Scrimshaw for pointing this out to me in private communication.

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