An Introduction to the Theory of Algebraic Surfaces by Oscar Zariski

By Oscar Zariski

Zariski offers a pretty good creation to this subject in algebra, including his personal insights.

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3(c), can be identified with the space ~ (W) of W derivations of Let GO q k(W)/k. be a q-fold differential, and assume &)q is regular at W. , Dq)~ ~ . , Dq). , DqgO~W, cosets of DI, Def. , on the traces ~i = Tr~i' i = i, D Q@~ q. (~)q be a q-fold differential which is rega]ar at W. , Dq) q>s, we fix define q-s derivations D~ ' "'" D ,.. , s. We now list some simple properties of the trace of a differential: (2) (3) -- If ~ w ( V / k ) , then Trwd ~ = d~ where ~ = TrW ~ . , ~r ) be a set of uniformizing coordinates of W, and let ~)q l'

K(V)/k, ~Or # O; let be a prime divisor of the first kind with respect to be uniformizing coordinates of ~ . Then d~l V; and let "'" d ~r ~ O, ~I"'" and we ~r -316Jr = A d ~ l .. d~r can write Let bJr v~ __at ~ where A6 K. be the valuation defined by (notation" v~ (UOr)) to be ~ . We define the order of v~(A). To see that v~ ( GOr) is independent of the choice of uniformizing coordinates, le~, 171' ""' ~r be another set. Then LJ = B d ~ l "'" d ~ r and B = A "I~ ( ~ ) i 9 ~( ~)I ) Hence v~ (A) = ~ (B) ~ince v~ (I ~~((~.

A we see that ~ ~ + a~ Then we . We have seen that ~ W k(W)-mcdule. By Prop. 2 and Prop. 3(c), can be identified with the space ~ (W) of W derivations of Let GO q k(W)/k. be a q-fold differential, and assume &)q is regular at W. , Dq)~ ~ . , Dq). , DqgO~W, cosets of DI, Def. , on the traces ~i = Tr~i' i = i, D Q@~ q. (~)q be a q-fold differential which is rega]ar at W. , Dq) q>s, we fix define q-s derivations D~ ' "'" D ,.. , s. We now list some simple properties of the trace of a differential: (2) (3) -- If ~ w ( V / k ) , then Trwd ~ = d~ where ~ = TrW ~ .

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