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Algèbre commutative: Chapitre 10 by N. Bourbaki

By N. Bourbaki

Algèbre commutative, Chapitre 10

Profondeur, régularité, dualité

Les Éléments de mathématique de Nicolas BOURBAKI ont pour objet une présentation rigoureuse, systématique et sans prérequis des mathématiques depuis leurs fondements.

Ce quantity du Livre d’Algèbre commutative, septième Livre du traité, est los angeles continuation des chapitres antérieurs. Il introduit notamment les notions de profondeur et de lissité, fondamentales en géometrie algébrique. Il se termine par l’introduction des modules dualisants et de los angeles dualité de Grothendieck.

Ce quantity est paru en 1998.

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Additional info for Algèbre commutative: Chapitre 10

Sample text

Riricn,i~ local (le n n - qn De rriêsne, si N est lin B-niocliile, le BqE-rnodillc ( ~ ( p@A ) N) - s'iclcritifie A. 4n ~ ( p@,y ) N,. s lc 2. N, = O . rim, la. i~t le support du B-rnodille ~ ( p @) A -\T est forrni. 11~ prerriiers (le SuppB(N) au-dessus de p. Eri particillicr, pour que le rraohrle et il s l ~ ~ f(/IL'%(! ier de SuppB(N) ~ ( p@) A N soit non. m l , ilfi~,,l~,t au-dessus dc p . I ~ S oN i t p : A + B IL^ homomorph,isrne d'anusea7cn: rroeth,érien,ç et soit N un, B-rrrotlulc qui cst un A-rriodr~lrde type ,fini .

E t ) M/:rM respectivcirrer~t,ct or1 a [me siiite exactc de complexes Soient N lin (A/:cA)-module, ct e' : N + E' iinc r6soliltiori ii'jectivc dc N . d&luit. &-niodulcs 011 AC X 44 $ 3 P H O P O N I ~ F I I K , K ~ C I J L A R I T DIJAIJTÉ ~, Consid6rons la siiite cxacte d'lioniologie associéc à cettc siiitc cxactc. D'aprbs A, X, p. 100, t h . 1 , on a. , El)) = H n l( H o ~ n g r ~(,R , ~( l ) ,El)) + Ext':;/::-,, (Ker(zMl), N) »ri en déduit ilne siiite exacte longuc tic (A/rA)-inodulcs N iinc résolution projcctivc di1 (A/zA-nlotli~lcN .

Par consCyuci~t,lcs foiict,ions p H dii\,,(NP)et p H tlli(Ap) ne sont pas cri g6iiCral srmi-contir~iwsiipérieiirerricnt. - - 3. rien,M (A, X, 3 3, no 6) qu'une rCsoliitjion iiri A-lriotii~lcdc t,vpe fini. Itappelons il^ h1 es1 ilne ré~olutjonprojcctiw 7 n i 1 ~ i m a l (si: ( : I E ~ . (les c I ~niodill(\s L , csl, libre de t,ypc fini. et si le coiilplcxc K* WA L cst ii tliffCreriLielle siirllc. P o i h ~ i l t entrier i 3 0 , on a alors [ E ~ t i ( h fK ,A ) : K A ] = [ ï i ) r f ( ~K, * ) : K,,] (l) rg,\(L,) ; (A, X, p.

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