It is an old theorem of Heaton's (based on work of Eilenberg and MacLane), that a polynomial 3-cocycle $f(x,y,z)$ which is "symmetric," in the sense that $f(x,y,z)-f(x,z,y)+f(z,x,y)=0$, is always a coboundary. Does anyone know of a similar theorem for polynomial 4-cocycles? The proof of the theorem for 3-cocycles doesn't seem to be written down in one clear piece anywhere, so I've had a tough time just generalizing it, thus a good reference for a modern treatment of even that case would be much appreciated as well.
$\begingroup$
$\endgroup$
3
asked Jul 12, 2013 at 4:09
-
1$\begingroup$ What is this cocycle living on? (I don't do any topology, but I have done a fair bit of naive Hochschild cohomology.) I guess from your tags that this is a cocycle on a not-necessarily abelian-group taking values in the augmentation module, but what is a polynomial cocycle? $\endgroup$– Yemon ChoiCommented Jul 12, 2013 at 8:04
-
$\begingroup$ It comes up in connection with formal group laws. For example, Lazard's lemma for symmetric polynomial $2$-cocycles. Definition: For an abelian group $A$, a symmetric 2-cocycle is a “polynomial” $P(x,y) \in A[x, y] = A \otimes_{\mathbb{Z}} \mathbb{Z}[x, y]$ with $P(x, y) = P(y,x)$ and $P(x, y+z) + P(y, z) = P(x,y) + P(x+y, z)$. $\endgroup$– Dietrich BurdeCommented Jul 12, 2013 at 8:45
-
$\begingroup$ @YemonChoi So, it's living in something like the cobar construction on $R[x]$, or you can think of it as being in $H^*(\mathbb{G}_a,\mathbb{G}_a)$ where the first is a monoid acting trivially on the module on the right, as described by Demazure and Gabriel. I don't really know of another way to place polynomial cocycles inside of some Hochschild cohomology. $\endgroup$– Jonathan BeardsleyCommented Jul 12, 2013 at 14:17
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
1
Heaton showed that if the field $K$ has characteristic zero, then all polynomial cocycles are coboundaries. In characteristic $p$ this is not always the case, but the following result holds (which you may know already).
THEOREM: For $n> 2$ and $K$ a field of characteristic $p$, every polynomial $n$-cocycle over $K$ of degree less than $p$ in the first indeterminate and less than $p-1$ in the second is a coboundary.
It is not possible to relax the requirements given in the above theorem on the degrees; there are counterexamples, see R. Heaton: A Note on Polynomial Cocycles, Math. Zeitschr. 76, 2357-239 (1961).
For a modern application, see for example chapter $1$ on formal groups (and symmetric polynomial cocycles) in the lecture notes on Shimura Varieties (class of Robert Kottwitz), see http://www.math.ou.edu/~sspallone/ under "Other Writings".
answered Jul 12, 2013 at 8:33
-
1$\begingroup$ Thanks @Dietrich, so you're not aware of any other results regarding polynomials which are "killed" by certain permutations? I sort of wonder if this is connected to the Hodge decomposition and filtration that Loday gives for Hochschild (co)homology. $\endgroup$ Commented Jul 12, 2013 at 14:23