New York Journal of Mathematics
Volume 21 (2015) 73-91


Alan Koch

Scaffolds and integral Hopf Galois module structure on purely inseparable extensions

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Published: February 17, 2015
Keywords: Hopf Galois extensions, integral Galois module theory, scaffolds
Subject: Primary 16T05. Secondary 11R33, 11S15, 12F15

Let p be prime. Let L/K be a finite, totally ramified, purely inseparable extension of local fields, [L:K]=pn, n≧2. It is known that L/K is Hopf Galois for numerous Hopf algebras H, each of which can act on the extension in numerous ways. For a certain collection of such H we construct "Hopf Galois scaffolds'' which allow us to obtain a Hopf analogue to the Normal Basis Theorem for L/K. The existence of a scaffold structure depends on the chosen action of H on L. We apply the theory of scaffolds to describe when the fractional ideals of L are free over their associated orders in H.

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Department of Mathematics, Agnes Scott College, 141 E. College Ave., Decatur, GA 30030, USA