×

A norm functor. (Un foncteur norme.) (French) Zbl 1017.13005

Summary: All rings in the paper are commutative and unitary. Let \(S\) be a finite and locally free \(R\)-algebra, and let norm: \(S\to R\) be the usual norm map. The author constructs a functor \(N:S\text{-Mod}\to R\text{-Mod}\) which extends both the “ corestriction” introduced by C. Riehm [Invent. Math. 11, 73-98 (1970; Zbl 0199.34904)] when \(S/R\) is a finite separable fields extension, and its generalisation by M. A. Knus and M. Ojanguren [Math. Z. 142, 33-45 (1975; Zbl 0297.13009)] in case where \(S\) is étale over \(R\).
Unlike these, the definition the author gives does not rely upon descent methods, but it rather uses a universal property: \(N(F)\) is equipped with a \(R\)-polynomial law \(\nu_F: F\to N(F)\) satisfying the relations \(\nu_F(sx)= \text{norm} (s)\nu_F(x)\), for \(s\in S\) and \(x\in F\), and the couple \((N(F), \nu_F)\) is universal for these properties. We thus get a well defined functor even if \(S\) is ramified over \(R\), but then the image of a projective \(S\)-module may fail to be projective over \(R\). Nevertheless, the norm of an invertible \(S\)-module is always invertible and for these models our construction gives the classical one. Moreover, if \(S\) is locally of the form \(R[X]/(P)\), then \(N(F)\) is projective over \(R\) for any projective \(S\)-module (of finite type); but that fails to be true if \(R\to S\) is only supposed to be a complete intersection morphism. These points are discussed in some details before the author focusses on the étale case in order to emphasize the isomorphism between the “Weil restriction” and the norm functor (when applied to commutative algebras), and the intricate relations between \(N(S\otimes E)\) and \(E^{\otimes d}\) for an \(R\)-module \(E\).

MSC:

13B40 Étale and flat extensions; Henselization; Artin approximation
14E22 Ramification problems in algebraic geometry
14F20 Étale and other Grothendieck topologies and (co)homologies
55R12 Transfer for fiber spaces and bundles in algebraic topology
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML

References:

[1] ADAMS (J.-F.) . - Infinite Loop Spaces , Annals of Math. Studies, Study 90, Princeton Univ. Press, 1978 . MR 80d:55001 | Zbl 0398.55008 · Zbl 0398.55008
[2] BOSCH (S.) , LÜTKEBOHMERT (W.) , RAYNAUD (M.) . - Néron Models . - Springer-Verlag, 1990 . Zbl 0705.14001 · Zbl 0705.14001
[3] BOURBAKI (N.) . - Algèbre , ch. 4 à 7. - Masson, Paris, 1981 . Zbl 0498.12001 · Zbl 0498.12001
[4] BREEN (L.) . - Bitorseurs et cohomologie non abélienne , in The Grothendieck Festschrift, vol. I, Birkhäuser, 1990 . MR 92m:18019 | Zbl 0743.14034 · Zbl 0743.14034
[5] DELIGNE (P.) . - Cohomologie à supports propres , exposé XVII de SGA 4, Théorie des topos et cohomologie étale des schémas, Lecture Notes in Math. 305, Springer-Verlag, 1973 . MR 50 #7132 | Zbl 0255.14011 · Zbl 0255.14011
[6] DEMAZURE (M.) , GABRIEL (P.) . - Groupes algébriques . - Masson, 1970 .
[7] DRAXL (P.) . - Skew Fields . - London Math. Soc., Lecture Note Series 81, Cambridge Univ. Press, 1983 . MR 85a:16022 | Zbl 0498.16015 · Zbl 0498.16015
[8] FULTON (W.) , HARRIS (J.) . - Representation Theory . - Grad. Texts in Math. 129, Springer-Verlag, 1991 . MR 93a:20069 | Zbl 0744.22001 · Zbl 0744.22001
[9] GROTHENDIECK (A.) . - Technique de descente et théorèmes d’existence en géométrie algébrique , II ; Le théorème d’existence en théorie formelle des modules, Sém. Bourbaki, 1959 / 1960 , exposé 195. Numdam | Zbl 0234.14007 · Zbl 0234.14007
[10] GROTHENDIECK (A.) . - Éléments de géométrie algébrique (EGA) , rédigés avec la collaboration de J. Dieudonné, Publ. Math. I.H.E.S., Paris 1960 - 1967 . Numdam
[11] HAILE (D.) . - On Central Simple Algebras of Given Exponent , J. Algebra, t. 57, 1979 , p. 449-465. MR 80i:16011 | Zbl 0408.16016 · Zbl 0408.16016
[12] IVERSEN (B.) . - Linear Determinants with Applications to the Picard Scheme of a Family of Algebraic Curves . - Lecture Notes in Math. 174, Springer-Verlag, 1970 . MR 45 #1917 | Zbl 0205.50802 · Zbl 0205.50802
[13] KNUS (M.-A.) , OJANGUREN (M.) . - Théorie de la Descente et Algèbres d’Azumaya . - Lecture Notes in Math. 389, Springer-Verlag, 1974 . MR 54 #5209 | Zbl 0284.13002 · Zbl 0284.13002
[14] KNUS (M.-A.) , OJANGUREN (M.) . - A Norm for Modules and Algebras , Math.Z., t. 142, 1975 , p. 34-45. Article | MR 51 #3139 | Zbl 0297.13009 · Zbl 0297.13009
[15] OESTERLÉ (M.) . - Nombres de Tamagawa et groupes unipotents en caractéristique p , Invent. Math., t. 78, 1984 , p. 13-88. MR 86i:11016 | Zbl 0542.20024 · Zbl 0542.20024
[16] RIEHM (C.) . - The Corestriction of Algebraic Structures , Invent. Math., t. 11, 1970 , p. 73-98. MR 45 #8736 | Zbl 0199.34904 · Zbl 0199.34904
[17] ROBY (N.) . - Lois polynômes et lois formelles en théorie des modules , Ann. scient. Éc. Norm. Sup., t. 80, 1963 , p. 213-348. Numdam | MR 28 #5091 | Zbl 0117.02302 · Zbl 0117.02302
[18] ROBY (N.) . - Lois polynômes multiplicatives universelles , C. R. Acad. Sc. Paris, t. 290, 1980 , p. 869-871. MR 81j:13011 | Zbl 0471.13008 · Zbl 0471.13008
[19] SERRE (J.-P.) . - Corps locaux . - Hermann, Paris, 1962 . MR 27 #133 | Zbl 0137.02601 · Zbl 0137.02601
[20] SERRE (J.-P.) . - Cohomologie Galoisienne , 5e éd. - Lecture Notes in Math. 5, Springer-Verlag, 1994 . MR 96b:12010 | Zbl 0812.12002 · Zbl 0812.12002
[21] TIGNOL (J.-P.) . - On the Corestriction of Central Simple Algebras , Math. Z., t. 194, 1987 , p. 267-274. MR 88f:12005 | Zbl 0595.16012 · Zbl 0595.16012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.