Jean pierre serre local algebra

COJlYfighted Material -PI prIng r COJlYfighted Material Springer Berlin Heidelberg New York Barcelona Hong Kong London Milan Paris Si"KapC1r~ 1ilA-yo Jean-Pierre Serre local Algebra Translated from the French by CheeWhye Chin , ~.~ Springer IHn-P;~rr( $~rrr Colltllc dc france rue d'Ulm 75005 Paris france e-mail: serre@dmi.ensJr Tr.n,J.tor: CheeWhye Chin Princeton University Department of Mathematic Princeton, NJ 08544 USA e-mail: cchin@princeton.edu CIP data applied for Ole Deutsche Bibliothek - CIP-Binheitsaufnahme Serre JcaD-Pierre: Local algebra I Jean-Pierre Serre Transl.from the French byCheeWhye Chin.- Berlin; Heidelberg; New York; Barcelona; Hong Kong; London; Milan; Paris; Singapore; Tokyo: Springer, 2000 (Springer monographs in mathematics) Einh,itssachl.: Alg~brelocale «ngl.> ISBN 3-54().66641-9 Mathematics Subject Classification (2000): Bxx ISSN 1439-7382 ISBN 3-540-66641-9 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright AU rishlS are rtserved whelher Ih, whole or pan of Ihe malerial is concerned, sp,cificaUy the rights of translalion, reprinting, reuse of i!lultratlonl recilation broldculing reproduction on microfilm or in any other way and Iorage in dlla blnb Duplicalion oC this publication or parts thereoC is permilled only under the provisions oC tho Germln Copyrighl l.IW of Seplember 9.1965 in ils currenl version and permission ror use musl always be oblained frum Sprinller- Verllg Violltions are liable ror prosecution under the Germln UJpyrishl Llw Sprinller- Verllg i I complny in Ihe OertelimannSpringer publi.hinl group lOGO Print.d In ';.rmlny o Sprinlltr.Verllll Herlln lIeidelbtrll TIw u.e or lIene,.1 du"'lpliv' nlm r'IIi1ltr.d nlm lud.mark, elc.ln Ihis publlcaUon •• n Imply• • v.n In Ih IhMn IIf I •., tfl••11I.ment thlt uch nlme are extmpl f",m Ihe rel lnl p, , I1" Ilw Ind "'UIIIN>II' Ind tn , frea for II'n.rll UN (:ow dellin 1I.IoU'' ' ",, It 1(/" It"., 1)p lln loy In 'fllI.la,,,, In I \Pllll,,, w,\X mac", p",h •• P"" I"" IIII'N"I,,,, ~I'IN 1010'." 111I4k ·HHIO Preface The present book is an English translation of Algebre Locale - Multiplicites published by Springer-Verlag as no 11 of the Lecture Notes seriffi The original text was based on a set of lecturffi, given at the College de France in 1957-1958, and written up by Pierre Gabriel Its aim was to give a short account of Commutative Algebra, with emphasis on the following topics: a) Modules (as opposed to Rings, which were thought to be the only subject of Commutative Algebra, before the emergence of sheaf theory in the 1950s); b) Homological methods, it la Cartan-Eilenberg; c) Intersection multiplicities, viewed as Euler-Poincare characteristics The EngliA(t) : 92 zp(M) : 100, 113 ea(M) , ea(M,p) : 100 - T ori : 102 , i(X, U V, W) : 112, 116 I(X, U V, W) : 112, 116 /.(Z) , f.(W) Z'fy:117 /"(y) : 118 ' : 117 : 70 [...]... means of x = (x l, , x r ) Moreover this complex can be used in other problems of local algebra, for example for the study of the depth of modules over a local ring and of the Cohen-Macaulay modules (those whose Krull dimension coincides with their depth), and also for showing that regular local rings are the only local rings whose homological dimension is finite Once formula (*) is proved, one may...Introduction The intersection multiplicities of algebraic geometry are equal to some "Euler-Poincare characteristics" constructed by means of the Tor functor of Cartan-Eilenberg The main purpose of this course is to prove this result, and to apply it to the fundamental formulae of intersection theory It is necessary to first recall some basic results of local algebra: primary decomposition, Cohen-Seidenberg... Once formula (*) is proved, one may study the Euler-Poincare characteristic constructed by means of Tor When one translates the geometric situation of intersections into the language of local algebra, one obtains a regular local ring A, of dimension n, and two finitely generated Amodules E and F over A, whose tensor product is of finit.e len~t.h over A (t.his means t.hat t.he vll.ril't.ips corrl'spondin,l!;... canonical maps Ap x Mq -+ Mp+q extend to a bilinear map from 11 x M to M; this defines a graded A -algebra structure on A, and a graded 11 -module structure on M [in algebraic geometry, A corresponds to blowing up at the subvariety defined by q , cf e.g [Eis], §5.2] Since q is finitely generated, 11 is an A -algebra generated by a finite number of elements, and thus is in particular a noetherian ring Proposition... characteristic is sufficient for the applications to algebraic geometry (and also to analytic geometry) More specifically, let X be a non-singular variety, let V and W be two irredu(:ible subvarieties of X , and suppose that C = V n W is an irreducible su bvariety of X , with: dim X + dim C = dim V + dim W ("proper" intersection) Let A, A v , Aw be the local rings of X , V and W at C If i(V· W,C;X) denotes... map, one recovers the standard product The commutativity, associativity and projection formulae can be stated and proved for this new product Chapter I Prime Ideals and Localization This chapter summarizes standard results in commutative algebra For more details, see [Bour], Chap II, III, IV 1 Notation and definitions In what follows, all rings are commutative, with a unit element I An ideal I' of a ring... is distinct from A, and maximal among the ideals having this property; it amounts to the same as saying that Aim is a field Such an ideal is prime A ring A is called semilocal if the set of its maximal ideals is finite It is called local if it has one and only one maximal ideal m; one then has A - m = A * , where A * denotes the multiplicative group of invertible elements of A 2 Nakayama'S lemma Let... follows from prop 1, applied to MIN Corollary 2 If A is a local ring, and if M and N are two finitely generated A -modules, then: M ®A N = 0