Witt group of hyperelliptic curves 573

Here (Фуеу^ЧКуЖ (resp. (Qyerff'^ikiyW denotes the subgroup consisting of trace zero elements. The two vertical columns are exact, by ([10], p. 277) and [5]. By the assumption on Y, the two rows are exact. The surjectivity of e„: I„(Y)-*^r(Y, Jf) follows from the surjectivity of the residue map д : Г^\к{Т)) -^(®у,уПКу))У [13], Theorem 5.3).

LEMMA 6.2. Suppose P' and X satisfy PQ{\). Then the sequence

is exact for « ^ 0.

Proof . Since BjA is unramified, by (2.1), (2.2) and (2.3), we have the following commutative diagram:

0

1

IM )

1

ПКТ ) )

•1

e p-\k{y)).

The vertical columns are exact by ([10], p. 277). Exactness of the rows is a consequence of the assumption PQ{\) for X and P* [3]. Exactness of the top row follows from the surjectivity of д :/""ЧМ^^))-^Ф^.€У 1''~\Ку)\ Y' being tained m A^

LEMMA 6.3. Suppose X\ and Г satisfy PQ{2). Then

«1 , -fyWiÄ))nIM)-^ih -fVn-AAy

Proof We assume, by induction, that

«1 , -/>^(^))n/^(v4) =<1, -/>4_^(Л)

for m-^n-X. Let ^ g(<1,-/>Ж(>4)) n/„(v4). By induction, we may write ^ = <1, -fyq^qi G 4_2(^). Since X\ F satisfy PQ(\), and B/A is étale quadratic.

1

I „ iA )

I

< ! . - / >

I " - \k { T ) ) —^ Р{к{Т))

•1 -1

Ф I"-\k(y))^-^ Ф I"-\k(y))

ye Y ye Y

1 1

> FikiX))

•1

xeX