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.



"Symmetric" Polynomial 4-cocycles
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".


