On May 4, 2008, at 12:49 PM, David Kohel wrote:
> Hi,
>
> Tensor products (of commutative rings) are "necessary" for
> representing the
> coordinate rings of a product of [affine] schemes.
>
> For commutative rings, a new tensor product class imay not be needed
> or desirable, rather what is missing currently is the homomorphisms.
>
> Given algebras A and B, I would return C and the two maps m1: A -> C
> and m2: B -> C:
>
> (C, m1, m2) = A.tensor_product(B,ring=R)
I might prefer to have something like
C = A.tensor_product(B, ring=R)
and then one can do
C.coerce_map_from(A)
C.coerce_map_from(B)
to get the maps. One would rarely need the maps themselves, as one
could do C(a) and C(b), or even c + a, and the coercion model would
handle it all well (here a, b, and c live in A, B, and C resp. of
course). C would probably be implemented one of David Roe's wrapper
classes, so Q tensor Z[x] would be Q[x], but would still know its
factors.
> The algebra C would be an algebra which represents the tensor product
> and whose class could depend on A and B. If the ring is not specified,
> then A and B should have the same base_ring.
Or perhaps a reasonable default could be found. (e.g. if A is over Z
and B over Q, then the tensor makes sense over Z).
> Tensor products of polynomial rings and their quotients would be the
> first natural cases to implement.
Certainly.
There seems to be a lot of talk about having a tensor product
structure in Sage lately, and coincidentally I wrote up a simple
tensor product implementation just a few weeks ago. It doesn't really
do anything other than just store elements as sums of pairs (a,b),
and has 0% doctest coverage (in fact, it's just sitting in a big
notebook cell) but perhaps I could check it in if people are
interested. I am not sure what operations (if any) one can do for
completely generic tensor products, but we should still have them.
- Robert
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