# Correction to “On the purity of the branch locus”

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## Abstract

Correction to “On the purity of the branch locus” COMPOSITIO MATHEMATICA ALLENB. ALTMAN STEVEN L. KLEIMAN Correction to “On the purity of the branch locus” Compositio Mathematica, tome 26, no 3 (1973), p. 175-180. <http://www.numdam.org/item?id=CM_1973__26_3_175_0> © Foundation Compositio Mathematica, 1973, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http:// http://www.compositio.nl/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commer- ciale ou impression systématique est constitutive d’une infraction pé- nale. Toute copie ou impression de ce fichier doit contenir la pré- sente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ 175 CORRECTION TO ’ON THE PURITY OF THE BRANCH LOCUS’ by Allen B. Altman and Steven L. Kleiman COMPOSITIO MATHEMATICA, Vol. 26, Fasc. 3, 1973, pag. 175-180 Noordhoff Intemationaal Publishing Printed in the Netherlands The proof ([2, p. 464]) fails because the algebra of principal parts P’(A) is not a finitely generated A-module. However, the proof does go through if we replace Pm(A) by the algebra of topological principal parts tpm( A), defined below. We check this after proving two preliminary results of independent interest. PROPOSITION. Let R be a ring, q an ideal of R, and M a finitely generated R-module. Consider the following separated completions: Assume R is noetherian and q is finitely generated. Then there is a canon- ical isomorphism, PROOF. With no finiteness assumptions on R and q, the canonical map, is surjective (the proof is straightforward, see [3, p. 108]). Therefore, since q is finitely generated, we have an equality, for each positive integer n. So, since qn is obviously equal to (qR)n, we obtain an equality, Consequently, (GD II, 1.10), there is a canonical isomorphism, Hence, R is equal to lim (/()n); in ot

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