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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