Birational extensions of regular local rings and local UFDs
This paper is a contribution to the problem of investigation of birational extensions of regular local rings in dimension bigger or equal to three. We recall the following results of Abhyankar-Zariski: if \(R\subset S\) are regular local rings of dimension two with the same quotient field then S can be obtained from R by a finite sequence of quadratic transformations. The same problem in dimension three or more is much more complicated, here we can find some partial results and a new point of view to study this problem based in the ``irregular locus'' of an extension. A good account on this problem can be found in the paper by \textit{J. Lipman} in Algebraic geometry and commutative algebra, Pap. in Honor of Masayoshi Nagata, Vol. I, 203-231 (1988; Zbl 0693.13011).
birational extensions of regular local rings
scientific article; zbMATH DE number 4154591
scientific article