For any hyperelliptic curve X, we give an explicit basis of the first de-Rham
cohomology of X in terms of \v{C}ech cohomology. We use this to produce a
family of curves in characteristic p>2 for which the Hodge-de-Rham short exact
sequence does not split equivariantly; this generalises a result of Hortsch.
Further, we use our basis to show that the hyperelliptic involution acts on the
first de-Rham cohomology by multiplication by -1, i.e., acts as the identity
when p=2.Comment: 23 pages; v2: references added; v3: comments of referees addressed,
to appear in Res. Number Theor
