I forgot to add that Sage does recognize that f(a1*b2) is nonzero:
sage: f(a1*a2)
b1*b2
On Monday, October 18, 2021 at 12:21:47 PM UTC-5 Trevor Karn wrote:
> Hi all,
>
> It looks like I have found a bug in the `.kernel()` method of a ring
> homomorphism from one `GradedCommutativeAlgebra` to another. I think I
> have identified the issue, but was hoping to post here for confirmation
> that my thinking makes sense before opening a trac ticket and working on a
> fix.
>
> *The bug:* let A, B be `GradedCommutativeAlgebra`s each generated by two
> elements (a_1, a_2 and b_1, b_2 respectively) of degree-1 (so A,B are
> exterior algebras). Define a homomorphism f: A -> B taking a_1 -> b_1 and
> a_2 -> b_1 + b_2.
>
> Then f(a_1a_2) = f(a_1)f(a_2) = b_1(b_1+b_2) = b_1^2 + b_1b_2 = 0 + b_1b2
> =/= 0.
>
> When I perform the same computation in Sage, I get:
>
> sage: A = GradedCommutativeAlgebra(QQ,['a1','a2'], (1,1))
> sage: B = GradedCommutativeAlgebra(QQ,['b1','b2'], (1,1))
> sage: A.inject_variables();
> sage: B.inject_variables();
> sage: f = A.hom([b1, b1+b2], codomain=B)
> sage: f.kernel()
> Twosided Ideal (0, a1*a2, 0) of Graded Commutative Algebra with generators
> ('a1', 'a2') in degrees (1, 1) over Rational Field
>
> which I believe to be incorrect by my computation above.
>
> *What I think is going wrong:*
> When computing the kernel, the graph ideal is computed as an ideal in the
> tensor product ring of the domain tensored with the codomain. In
> constructing this it uses a (commutative) polynomial ring and takes a
> quotient. In creating the commutative polynomial ring the
> `.defining_ideal().gens()` method is called on the domain. The
> `.defining_ideal()` for a noncommutative ring has generators and relations,
> but calling the `.gens()` method accesses the generators only. For example:
>
> sage: A.defining_ideal()
> Twosided Ideal (a1^2, a2^2) of Noncommutative Multivariate Polynomial Ring
> in a1, a2 over Rational Field, nc-relations: {a2*a1: -a1*a2}
> sage: A.defining_ideal().gens()
> (a1^2, a2^2)
>
> And so the relation that a2*a1 = - a1*a2 is forgotten in the tensor
> product ring.
>
> *My proposed fix:*
> Add a function in sage.rings.morphism that computes the tensor product
> ring when the two rings are noncommutative, then add a check inside of
> graph_ideal to call _tensor_product_ring_nc as opposed to
> _tensor_product_ring.
>
> Does this seem like a reasonable plan, or is there a better approach?
>
> Thanks!
>
> -Trevor
>
>
>
>
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