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Given a simplicial abelian group, the alternating sum defines a derivation, making the simplicial abelian group itself into a chain complex.
The derivation is then an endomorphism on that chain complex. But the complex has another point of view because it is still a simplicial set, too and the question is:
Does the alternating-sum-derivation respect the simplicial structure, i. e. does it commute with the face and degeneracy maps? (Maybe not because of the square to zero rule, but anyway …)
If not, is there a derivation respecting the simplicial structure?
Mirco, one question. The title refers to a Moore complex but in the usual description of a Moore complex of a simplicial (abelian) group, $A$, all but one face of the elements at each level is zero which alters the query considerably. You seem to be referring to the chain complex whose group of $n$-chains is $A_n$ with the alternating sum $\Sigma (-1)^{-1}d_i$ as the differential. In both cases, the answer is that the question does not make sense as the dimensions go wrong $\partial :A_2\to A_1$ cannot commute with the faces as, for instance, the equation $\partial d_2(a)= d_2\partial(a)$ for elements $a\in A_2$ does not make sense. (Or have I misunderstood the question? It is quite early here and I have not yet had my second mug of coffee!)
Perhaps if you can sketch some of the background to the question it will be easier to see where you are going.
I refer to the second construction what you declared as $A_n$.
Ok it will fail for the ’highest’ face in each dimension. That’s right. And I must say that I overlooked it (Early here, too) So it can’t respect all structure maps.
Nevertheless for the other faces, the equation $\partial d_j(a) = d_j \partial(a)$ makes sense (for $a \in A_n$ and $j \leq (n-1)$) and the question is, if it is true for any such $a$ and $j$
Now for the degeneracies $\partial s_j(a) = s_j \partial(a)$ the situation is similar as we have to cancel out the highest degeneracy. (Relative to the degree of $a$)
So for short the question is just, if they commute whenever the equation makes sense. The background is that I simply want to know if I can use commutation in an algebraic equation involving $\partial$ and $d_j$ or $s_j$. In a quick try, using the definition and the coherency laws I failed (but that doesn’t mean much)
I just spent the last week and a half mucking around with simplicial abelian groups, and I’m afraid the only nice commutation rule I noticed was the following: if we define $\sigma = \sum_i (-1)^i s_i$ then $\partial \circ \sigma = - \sigma \circ \partial$.
I guess that last equation is because $\sigma$ is a coderivation since it squares to zero, too. But anyway good to know! Thanks
Would it be ok for you to publish the calculation for $\partial \circ \sigma = - \sigma \circ \partial$. I’m always glad if I can avoid those kind of things :-) .
Yes!
@Mirco: It’s just a straightforward if tedious calculation. Just expand and apply the simplicial identities. If it helps, draw a matrix…
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