An Introduction to the Theory of Algebraic Surfaces: Notes by Oscar Zariski (auth.)

By Oscar Zariski (auth.)

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Additional resources for An Introduction to the Theory of Algebraic Surfaces: Notes by James Cohn, Harvard University, 1957–58

Sample text

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, , . + A r . e % . , r-s. Then clearly tick[ r-s #A. -Z Stj=A Therefore fi ~s+i-l) be the now need not be monic). are linearly independent over ' ~s ~ is a separating Since k(W) is s k(~l, " ' " ~S ) we have . , r-s. Special case. , let W cycle. d. k(W)/k = r - I. uniformizing coordinates of W. r - I Let {~i, "'" Then among the algebraically independent elements.

Is a direct sum and hence a graded ring. Let Yo H' be the integral closure of is transcendental over Ko, R in K. it follows that is a graded ring and ~e can write Rt = ~R" Since RI c where q_>O q IR' q is a direct sum. Rz I + ... + Rz h R' q RT zi is a finite-dimensional vector space over a point U in ordinates of Let A/k and consider Sm. U. and V' A'/k L = k(A) = k(A t). Let A' We say exists a wluation R' = R ' ~ and Hence Kq DI and q are homogeneous. k for each q. , urn) as the coordinates of It is clear that they are strictly homogeneous co- Let respectively.

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' s.

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