Hi Trevor,
thank you for reporting this bug. This is indeed a problem, as the
implementation is written only with the commutative case in mind. We should
make sure that it is called in that case only.
As for your proposed fix, I am afraid I am not so familiar with the
non-commutative theory myself, so I would value another opinion on this or
a reference to some literature that covers this specifically.
Trevor Karn schrieb am Montag, 18. Oktober 2021 um 19:28:39 UTC+2:
> 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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