By Oscar Zariski
Zariski offers an effective advent to this subject in algebra, including his personal insights.
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Additional info for An Introduction to the Theory of Algebraic Surfaces
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 ~~((~. ) ) = o (by ~op. 2, this r Jacobian determinant is a unit in the quotient ring Prop.
Since D ~ ~W' D~i6~. we have D ~r on k(~), Let D then ~ = O Ai = D ~ i ~ on then /v~ which shows that is a derivation of k(W)/k To such k(~). Let ~ = Try, D g W r D D = E A. Since Ai~ ~ ~ = 0 on k(W). -38- Since (a) holds, we can assume ~ ~ I ' transcendence basis of k(W)/k. , tr_s is a polynomaal in all its am~, r-s. are uniformizing parameters of W. O'I/i~). We know More we shall show that (this means that the ~ 2-residues of the ~ Eam. Hence ) ~- ~ . j:l separable algebraic over ~-s~" ti~k[~]~ ~ ] / ~ N~, i = I, , .
It is clear that H is a multipllcatlvely closed s;~t and that ~ i = ~[Yl]H 9 Since ~[yl ] is noetherian, it follows that ~ I noetherian and hence ~ l is a local ring. To prove ~ l is is regular we must show dimk MI/M12 = 2. It is clear that d ~ To show that ~m~ ~ 2 Ml~12_<2 = 2, where ~ , ~l,qEMU; then A,B 6 ~ . We can write i = ~ / ~ q $ M n+l and Y = 0 is not tangent to (q). Then ~x 2 + ~I y = O. Since % x 26M n+2, y E N that ~i E M n+l, Therefore B is a non-unit in Furthermore, since Y = 0 is a non-unit in ~l" generate M1.
An Introduction to the Theory of Algebraic Surfaces by Oscar Zariski