TY - JOUR

T1 - Integrality in codimension one

AU - Thorup, Anders

PY - 2014/12

Y1 - 2014/12

N2 - The paper from 2001 by Simis, Ulrich, and Vasconcelos contained deepresults on codimension, multiplicity and integral extensions. The results and the ideas of the paper led to substantial simplifications in the treatment of the exceptional fiber of a conormal space, considered previously by Kleiman and the present author. In addition, the paper contained the following theorem: Let R ⊆ S be an extension of commutative rings, where R is noetherian, universally catenary, and locally equidimensional. Then the extension R ⊆ S is integral if minimal primes of S contract to minimal primes of R and, for every prime p of height at most 1 in R, the extension Rp ⊆ Sp is integral. Themain purpose of the present note is to give an alternative proof of the theorem, based on standard techniques of projective geometry. In addition, the results on the exceptionalfiber, considered previously by Kleiman and the present author in the complex analytic case, may be based in the algebraic case by a simple key result presented at the end.

AB - The paper from 2001 by Simis, Ulrich, and Vasconcelos contained deepresults on codimension, multiplicity and integral extensions. The results and the ideas of the paper led to substantial simplifications in the treatment of the exceptional fiber of a conormal space, considered previously by Kleiman and the present author. In addition, the paper contained the following theorem: Let R ⊆ S be an extension of commutative rings, where R is noetherian, universally catenary, and locally equidimensional. Then the extension R ⊆ S is integral if minimal primes of S contract to minimal primes of R and, for every prime p of height at most 1 in R, the extension Rp ⊆ Sp is integral. Themain purpose of the present note is to give an alternative proof of the theorem, based on standard techniques of projective geometry. In addition, the results on the exceptionalfiber, considered previously by Kleiman and the present author in the complex analytic case, may be based in the algebraic case by a simple key result presented at the end.

KW - Faculty of Science

KW - Mathematics

U2 - 10.1007/s00574-014-0079-1

DO - 10.1007/s00574-014-0079-1

M3 - Journal article

VL - 45

SP - 865

EP - 870

JO - Bulletin of the Brazilian Mathematical Society, New Series

JF - Bulletin of the Brazilian Mathematical Society, New Series

SN - 1678-7544

IS - 4

ER -