#19634: Implement Hochschild (co)homology
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Reporter: tscrim | Owner: tscrim
Type: enhancement | Status: new
Priority: major | Milestone: sage-6.10
Component: categories | Resolution:
Keywords: | Merged in:
Authors: Travis Scrimshaw | Reviewers:
Report Upstream: N/A | Work issues:
Branch: | Commit:
public/homology/hochschild-19634 | cf8cf9e2c7a320b361cdca0c0542c2eedbb96338
Dependencies: #19609, #19608 | Stopgaps:
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Comment (by jhpalmieri):
Hi Travis, I'm finally getting around to looking at this. I have some
issues with the documentation. I don't have much experience with
Hochschild homology, so some of these may just be my own ignorance.
- If you're working over a base ring ''R'' which is not a field, do things
go wrong if ''M'' is not projective as an ''R''-module?
- I think the face maps ''d_i'' would be better without signs, and then
the boundary map would be their alternating sum (as the Wikipedia page
says). The point is that the unsigned face maps would be part of a
simplicial structure.
- When you mention Tor and Ext, they should be over the ring ''A^e^'', the
tensor product of ''A'' and ''A^op^'', instead of ''A''.
Looking at the code, I wonder if it would be better to use Sage's
`ChainComplex` class, at least when computing homology: to compute
homology in dimension ''d'', construct the chain complex by
{{{
C = ChainComplex({d: self.boundary(d).matrix(), d+1:
self.boundary(d+1).matrix()}, degree_of_differential=-1)
}}}
Then compute `C.homology(d)` and lift the answer back to the original
Hochschild complex. I didn't try any kind of lifting, but it seemed that
constructing `C` and then computing `C.homology(d)` was faster than
directly computing `HH.homology(d)`, at least in the cases I tried. One
possible advantage is that if some day we speed up computations with
general chain complexes, then this class would benefit.
--
Ticket URL: <http://trac.sagemath.org/ticket/19634#comment:2>
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