Vol. 306, No. 2, 2020

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Cohomological kernels of purely inseparable field extensions

Roberto Aravire, Bill Jacob and Manuel O’Ryan

Vol. 306 (2020), No. 2, 385–419
Abstract

Let $F$ be a field of characteristic $p$ and $L$ a finite purely inseparable extension. The kernel ${H}_{p}^{n+1}\left(L∕F\right)=ker\left({H}_{p}^{n+1}F\to {H}_{p}^{n+1}L\right)$ has been described by Sobiech, and independently when $p=2$ by Aravire, Laghribi, and O’Ryan. When $L$ has exponent $1$, the kernel is the sum of the kernels of the simple subextensions, but when $L$ has larger exponent it is significantly more complex. This paper determines ${H}_{{p}^{m}}^{n+1}\left(L∕F\right)$ for $L=F\left(\sqrt[{p}^{e}]{x}\right)$ and all $m,n,e\ge 1$. Whereas the results when $m=1$ used the theory of differential forms, the results for $m>1$ require the de Rham Witt complex. The $m>1$ case clarifies why the “messy” generators in the $m=1$ case arose, as they emerge from relations in the de Rham Witt complex. As a corollary, when $L$ is modular over $F$ and $m$ exceeds the exponent of $L$, the kernel ${H}_{{p}^{m}}^{2}\left(L∕F\right)$ is the sum of the kernels of the simple subextensions.

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