Resolution of Singularities of Embedded Algebraic Surfaces by Shreeram Shankar Abhyankar

By Shreeram Shankar Abhyankar

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RESOLVERSAND 45 PRINCIPALIZERS principalizer ( R , , I, , Si), such that R, is an iterated monoidal transform of R. We say that R is principalixable if: given any iterated monoidal transform R' of R, any nonzero nonprincipal ideal I' in R', and any valuation ring V of the quotient field of R such that V dominates R', there exists a finite principalizer [(Ri ,Ii Si)"'i. 9 (R,n 9 In,)] such that (Ro9 I") = (R',1')and I/' dominates R,,, . 9 52. 1). Let R be a regular local domain, let J be a nonzero principal ideal in R such that ( R , 1)is unresolved, and let V be a valuation ring of the quotient field of R such that V dominates R.

1). 1 ) we know that H’-l( Ker H”) = Ker H and Ker H‘ # {O}. Consequently it suffices to find distinct prime ideals P,, C P,,+, C ... C P, in B with P, = Ker H“ because then we can take P; = W-l(PJ for m j n. , h’(x,))T for m < j n, and Pi,= {O}. Then Pk C P;n+lC ... C Pk are distinct prime ideals in T and Ph = M ( T ) . Let h;: T -+ T/P,’ be the canonical epimorphism and let Pi = PjB. (u) for all u E T; clearly Ker H i = Pi and hence Pj is a prime ideal in B and P i n T = Pj’. Therefore P,, C P,+, C ...

It suffices to take j = p. 2). Let R be a regular local domain. Let I be a nonzero principal ideal in R, and let S E %(R)such that ( S ,I ) has a normal crossing at R. Let z E R n M ( S )such that ordRz = 1 and (zR,I ) has a quasinormal crossing at R. Then ( S ,X I )has a normal crossing at R. PROOF. 5. I). 3). Let J be a nonzero principal ideal in a regular local domain R. 3)] we know that S is regular). Then ( R , J ) is resolved. PROOF. If J = R then we have nothing to show. So assume that J # R.

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