#16851: Implement direct sum and tensor products for chain complexes and Koszul
complexes
-------------------------------------+-------------------------------------
Reporter: tscrim | Owner: tscrim
Type: enhancement | Status: needs_review
Priority: major | Milestone: sage-6.4
Component: algebraic | Resolution:
topology | Merged in:
Keywords: | Reviewers:
Authors: Travis Scrimshaw | Work issues:
Report Upstream: N/A | Commit:
Branch: | 990c0dce6e53a005eb2b3cd44709379f5d7bf242
u/tscrim/koszul_complexes | Stopgaps:
Dependencies: |
-------------------------------------+-------------------------------------
Comment (by jhpalmieri):
The tensor product doesn't work if the chain complex is graded over a free
abelian group: computing `(-1)^a` fails in this case, because `a` is a
group element, not an integer. So you need to test for this case and
replace `a` with `sum(a)` or something similar. (See also the
`total_degree` function in my branch at #16508.)
A related problem: if the differential has degree `d`, then I think the
correct formula for the differential on the tensor product `x tensor y`
involves the sign `(-1)^(|x| * |d|)`, where `|d|` is the degree of the
differential on the second factor. With this sign convention (or with your
choice of `(-1)^|x|`), if `C` and `D` are chain complexes with
differentials of even degree, their tensor product will probably not be a
chain complex any more, because the differential won't square to zero. It
would be a good idea to add a warning about this in the documentation. (I
know why you might want to grade over an arbitrary free abelian group, but
I don't know why you would want a differential of even degree, so this is
probably not a big deal.)
Oh, and `cartesian_product` fails if graded over something other than
`ZZ`, because the grading group needs to be explicitly stated if it's not
`ZZ`. So something like this should fix it:
{{{
#!diff
diff --git a/src/sage/homology/chain_complex.py
b/src/sage/homology/chain_complex.py
index 1c6fdfb..29ce6e7 100644
--- a/src/sage/homology/chain_complex.py
+++ b/src/sage/homology/chain_complex.py
@@ -1758,7 +1758,8 @@ class ChainComplex_class(Parent):
ret = {k: matrix.block_diagonal([d.get(k, zero) for d in diffs],
subdivide=subdivide)
for k in keys}
- return ChainComplex(ret, degree_of_differential=deg_diff)
+ return ChainComplex(ret, degree_of_differential=deg_diff,
+ grading_group=self._grading_group)
def tensor(self, *factors, **kwds):
r"""
}}}
You'll need a similar change at the end of the `tensor` method.
--
Ticket URL: <http://trac.sagemath.org/ticket/16851#comment:9>
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