Séminaire d’Algèbre Paul Dubreil Paris 1975–1976 (29ème Année)

194 Pages·1977·3.736 MB·French-English

by  David Eisenbud (auth.), Marie Paule Malliavin (eds.)

Lecture Notes ni
Mathematics
Edited by .A Dold and .B Eckmann
586
Seminaire d'AIg~bre
Paul Dubreil
Paris 1975-1976 (296me Annee)
by Edited .M .P Malliavin
galreV-regnirpS
Berlin-Heidelberq • NewYork 7791
Editor
Marie Paufe Malliavin
Universite Pierre et Marie Curie,
10, rue Saint Louis en I'lle
?5004 Paris, France
AMS Subject Classifications (1970): 10C20,13D20,13E20,13F20,14E20,
14F20, t6-02,17B20,18G20, 20C20, 20E20, 20M20, 32C20
ISBN 3-540-08243-3 Springer-Verlag Berlin • Heidelberg • New York
ISBN 0-38?-08243-3 Springer-Verlag New York • Heidelberg • Berlin
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A
C'est ne erbmevon 1947 qu'Albert ,TELETAHC Professeur ~ la fonda Sorbonne, le
erianim~S" d'Alg~bre et ed Th~orie sed ."serbmoN Jusqu'en 1954, il ne partagea la
direction avec Paul LIERBUD auquel vinrent s'adjoindre, nu na plus tard,
Charles PISOT, puis Marie-Louise NITOCAJ-LIERBUD ne 1957 et ecno~L RUEISEL ne 1962.
A partir ed 1971, ec s~minaire lot ~rcasnoc a uep pros tnemeuqinu ~ l'Alg~bre,
Charles TOSIP ayant fondU, avec Hubert ,EGNALED nu s~minaire ed Th~orie sed .serbmoN
siupeD la retraite ed Paul ,LIERBUD ne 1974, la direction ed ec S~minaire
d'Alg~bre, rattach~ a l'Universit~ Pierre et Marie Curie (Paris Vl) est e~russa
rap Marie-Paule .NIVAILLAM
eL S~minaire s'efforce ed remplir deux r61es principaux : diffuser sed Theories
actuelles encore uep seunnoc et xudonner a alg~bristes l'occasion d'exposer les
progr~s r~cents iuq leur sont dus. nU puoc d'oeil rus la liste etelpmoc sed
Conf6renciers (ci-dessous, rap ordre alphab~tique) permettra ed voir tnemmoc ec
double objectif a 6t~ poursuivi :
ABBOTT J. - ALMEIDA COSTA A, - AMARA M. - AMICE Y.-AMITSUR S,A, -
ANSCOMBRE J.C. - AUBERT K,E, - AYOUB C. - AULT J.
BAER R. - BARANANO E. - BASS H. - BASS J. - BEHANZtN L. - BEHRENS E,A. -
BENABOU J. - BENOtS M. - BENZAGHOU 8. - BENZAKEN C. - BERAN L. - B.ERTIN J, E.-
BERTRAND M. - BERTRANDIAS F. - BIGARD A. - BFRKHO'FF .G.- BLOCH R. - BLYTH T.S.
BOASSON L, - BORUVAK O. - BOULAYE G,- BOUVIER A. - BRAUER R. - BROUE M. -
BRZOZOWSKI D. BUCHSBAUM yD. A. -
CAILLEAUD A. - CALAIS J, - CAZ~NESCU V.E. - CELEYRETTE I. - CHABAUTY. C-,-
CHACRON M. - CHADEYRAS M. - CHAMARD J, Y. - CHAMFY Ch, - CHARLES B ,-
CH~TELET F, - CLIFFORD A. H, - COELHO M. - COHN P.M. - CONRAD P, F.
COURREGE Ph, - CRESTEY M. - CROISOT R. - CURZIO M.
DAVENPORT H. - DECOMPS-GUILLOUX A, - DEHEUVELS R. - DELANGE H. -
DESCOMBES R. - DESQ R. - DtEUDONNE I, - DJABALI M. - DLAB W. - DRESS. F. -
DUBIKAITIS L. - DUBREIL P. - DUBREIL-JACOTIN M.L. -
EGO M. - EISENBUD D, - ENGUEttART M. - EYMARD P.
FAISANT A. - FERRANDON M. ~ FLEISCHER I.- FORT J. - FOSSUM R. M.
FOUQUES A. - FUCHS L.-FARES N.-
GABRIEL P. - GAUTHIER L. - GERENTE A. - GERMA M.CH. - GOLDIE A, W. -
GRANDET-HUGOT Marthe - CRAPPY J. - GRILLET P.A. - GROSSWALD E.-
GUERINDON J. - GUILLEVIN C.
HACQUE M,- HERZ J. C, HOFMANN K.H. - HUDRY A, HUNTER R.P. -
ISHAQ M. ~ JACOBSON N. - JAFFARD P. - IEBLI A. - JOULAIN C. -
KAHANE J, P. - KALOUJNINE L. - KEGEL O H. - KEIMEL K. - KLASA J. - KLOCHIN E.R.
KOLI BIAROKA B. - KOSKAS M.- KOSTANT B. - KRASNER M. - KRISHNAN V, -
KRULL W. - KUNTZMANN J. - KUROS A.G.-
LAFON J.P. - LALLEMENT G. - LAMBEK J, - LAMBOT de FOUGERES D. ~ LAPSCHER F.-
LATSIS D. - LAZARD M. - LECH Ch.- LEDERMANN W. LEFEBVRE P. - LESIEUR L. -
LEVY-BRUHL J. - LEVY-BRUHL P. - LICHTENBAUM S. - LOMBARDO-RADICE L. -
MAC DONALD I.G. - MACLANE S, MC PHERSON R D. - MALEK M.
MALLIAVIN-BRAMERET M.P. MARCHIONNA E. MAROT J. - MATRAS Y. -
MATTENET G. - MAURY G. - MENDES-FRANCE M. - MERLIER Th.- MICALI A. -
MICHEL J. - MILLER D.D. - MOLtNARO I. - MOTAIS de NARBONNE L, MUNN W.D.-
NAGATA M.- NEUMAN B, H. - NICOLAS A. M. - NIVAT M. - NORTHCOTT D.G. -
O CARROLL L, - OCHS A. - OEHMKE R. H, - ORTHEAU J.P. -
PAUGAM M.- PAYAN J, J. - PEINADO R. E, - PERROT J.F. - PETIT J.C. - PERRIN D, -
PETRESCO J. - PETR1CH M. - PIGHAT E.- PICKERT G. - PISOT Ch. - PITTI CH.-
POITOU G. ~ PRESTON G.B. -
RAFFIN R. - RAUZY G. - RAVEl_, J. - READ J.A, * REES D, - RENAULT G. - REVOY Ph.-
RHODES J. - RIABUKHIN J.M.- RIBENBOIM P.- RIGUET J.- ROUX B.- RUEDIN J.-
SAITO T.-vSAKAROVITCH J.- SALLES D.- SAMUEL P,- SCH~JTZENBERGER M. P, -
SCHWARZ B.- SCOTT D. B.- SERRE J. p.- SPRINGER T, A.- STEINFELD O.- STROOKER J.-
SZPIRO L.
IV
TAFT E. J. - TAMARI D. - TAMURA T - TE1SSIER-GUILLEMOT M, - THIBAULT R. -
THIERRIN G,- TISSERON C1.-
VA'N DER WAERDEN B.L.- VAN METER K, M.- VIDAL R. - VIENNOT G.- VINCENT Ph
VORS, G. =
WASCHNITZER G= WILLE R,- WOLFENSTEIN S. -
ZARISKI O,- ZERVOS S.- ZISMAN M,-
Table des Mati~res
DAVID EISENBUD :
Enriched free resolutions and change of rings . . . . . . . . . . . . . .
DAVID EISENBUD :
Solution du probl@me de Serre par Quillen-Suslin . . . . . . . . . . . .
oE MAHARG EVANS Jr. :
A generalized principal ideal theorem with applications to intersection
theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
DRAREG TNEMELLAL :
Sur les produits amalgam@s de monoTdes . . . . . . . . . . . . . . . . . 25
DRAHREG .O MICHLER :
Small projective modules of finite groups . . . . . . . . . . . . . . . . 34
BORIS NOZEHSIOM :
Algebraic surfaces and 4-manifolds . . . . . . . . . . . . . . . . . . . 43
ANNE-MARIE NICOLAS :
Sur les conditions de chaines ascendantes dans des groupes ab@liens et
des modules sans torsion . . . . . . . . . . . . . . . . . . . . . . . . 50
SNARF TROO :
Singularities of coarse moduli schemes . . . . . . . . . . . . . . . . . 61
JULIEN ERREUQ :
Sur les anneaux compl@tement integralement clos . . . . . . . . . . . . . 77
YNOT A. SPRINGER :
Representations de groupes de Weyl et ~l#ments nilpotents d'alg@bres
de Lie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
FLODUR RELHCSTNER :
Comportement de l'application de Dixmier par rapport ~ l'anti automor-
phisme principal pour des alg@bres de Lie compl@tement r@solubles .... 93
J.C. NOSBOR :
Quotient categories and Weyl algebras . . . . . . . . . . . . . . . . I01
J 0 U R N E E S S U R L E S D E M I-G R 0 U P E S
organis#es avec le concours de Jean-Frangois TORREP
J.M. EIWOH :
Sur les demi-groupes engendr#s par des idempotents . . . . . . . . . . II0
DRAREG JACOB :
Semi-groupes lin~aires de rang born#. D~cidabilit# de la finitude . . 116
DRAREG TNEMELLAL :
Presentations de mono#des et probl~m~algorithmiques . . . . . . . . . 136
F. PASTIJN et .H STREANYER :
Semilattices of modules . . . . . . . . . . . . . . . . . . . . . . . . 145
DANIEL PERRIN :
La representation ergodique des codes biprefixes . . . . . . . . . . . 156
J. SAKAROVITCH :
Sur les groupes infinis, consid6r~s comme monoTdes syntaxiques de langages
formels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
H.J. SHYR et G. THIERRIN :
Codes and binary relations . . . . . . . . . . . . . . . . . . . . . . . 180
ENRICHED FREE RESOLUTIONS AND CHANGE OF RINGS
by David EISENBUD
The author is grateful for the support of the Alfred P. Sloan foundation
and the hospitality of the Institut des Hautes Etudes Scientifiques during the
preparation of this work.
Let A be a commutative local ring ang let B = A/I be a factor ring.
In [B-El it is shown that any A-free resolution of B possesses the structure of
a homotopy-associative, commutative differential graded algebra. This algebra
structure, which generalizes the exterior algebra structure of the Koszul complex,
can be quite useful in the analysis of free resolutions (The results of ~B-E~ are
one example). The idea of its construction is as follows : If
: .... -_~ 2 F ~ I F _._~ A
is an A-free resolution of B , then $2(~) , the symmetric square of ~ , inherits
the structure of a complex from F ~ F . The first few terms of $2(~ ) are as
follows :
.-. > / o / o F4 ~ A 2 F3 ~ A • 2 @ F A I > @ F A ~ S2(A)
7 ° , I
°. • ~ 3 ~ F F] > 2 ~ F I F ^2F I A
J o /
~ s2(F )2
...
The natural isomorphism F. ~ A ~ F. extends, uniquely up to homotopy,
i l
to a map of complexes $2(~ ) ~ , and this map, combined with the natural map :
acts as the multiplication map for the algebra structure on ~ .
What about the resolution of a module ? If M is an A-module annihilated
by 1-that is, if M is a B = A/I -module- and if @ is a free resolution of M,
then the natural map B ~A M M > extends, ~iquely up to homotopy, to a map :
~A ~ ..... > ~
which makes ~ into a homotopy associative, differential graded P-module.
In [B-El we conjectured that the comparaison map $2(~ ) ~ ~ , above,
could be chosen in such a way that the algebra structure on ~ is associative
(not just up to homotopy). It seems reasonable to extend this and to conjecture
that, with a suitable associative algebra structure on ~ , the resolution ~ can
be made into an associative ~-module.
Consider now the case in which I is generated by a regular sequence, so
that • is the Koszul complex, which is naturally an (associative) differential
graded commutative algebra (in fact the underlying algebra is an exterior algebra).
In this case it is easy to say what an F-module structure on ~ looks like
(see Proposition I), and in certain cases-for example when the module M resolved
by ~ is just the residue class field-it is clear that one exists. Noreover, if
one examines the construction, due to Tare, of a B-free resolution of the residue
class field from an A-free resolution, one sees that the data required by Tare to
make the construction is equivalent to the data involved in constructing an F-module
structure on ~
One problem in extending Tate's idea to resolution ~ of an arbitrary
B-module N is the possible non-associativity of ~ . In this paper we will show
(Theorem 2) how an analogue of Tate's construction can be carried through, for
arbitrary ~ , (but always under the assumption that ~ is a Koszul complex) by
considering not only the algebra structure on ~ but also a sort of "higher homo-
topy associatlvmty . . ), that ~ satisfies even if it is not associative (Theorem I).
It would be interesting to know the right generalization of this "higher
homotopy associativity" for more general (perhaps non-associative) ~[ , and to know
how, in general, to derive resolutions over B from resolutions over A (The
evidence in [LevJ and [ArJ seems to point to the introduction of matric Massey
products for this purpose).
From now on, suppose that I = (x I ,..., x n) where I x ,...) Xn form an
A-sequence. The minimal free resolution of B = A/I is then the Koszul complex :
_ft- 0 .~ ~ n A > ...__~ 3 n A =~ ~ A2 An $ > n A ~ A . .
Choose a basis I£ ''''' ~ can so that ~: .~ ~ .x 6A . The following is
n l l
immediate from the definition of a differential graded module :
Proposition 1 - If
: "" ~> kG ~> ~k-1 @Z
""
is a differential ~rade~ &-mo&~le , th~q an assoc&#t~v8 ~ifferential ~raded A-module
structure on ~ is equivalent to ~ set pf n maps 9f graded A-modules f~_o
degree + I :
s 1 ,..., Sn : ~ ~ ~
satisfying :
(1) S.~ + ~S. = x. o 1
1 1 1
2
(2) s. s =- s s. , s = o .
m 0 j i m
Now if ~ is an A-free resolution of a B-medule M , then multiplication
by x. is 0 in M , so x..1 : ~ >~ is homotopie to 0 ; to say that
i I
s. : @---~ ~ is a homotopy for x..1 is exactly condition (I) of the proposition,
l 1
so maps satisfying condition (I) do indeed exist (Alternatively, these maps could
be constructed by choosing a map of complexes ~ ^ @___>@ and regarding the induced
map An ~ ~ --9 ~ as giving n maps ~ ~ @ of degree I). Unfortuneatly there
is no reason why an arbitrarily choosen set of maps s. should satisfy (2).
l
On the other hand, the homotepy-uniqueness of the s. implies that
l
s s. - s.s. is at least homotopic to 0 . Theorem I extends this remark and tells
13 31
us what we may expect from a random choice of homotopies.
We first introduce some multi-index conventions. A multi-index (of length n)
is a sequence :
4= <~I ''''' 4n ~
where each ~. is an integer ~ 0 . We write :
1
0 = <0 ,..., 0
The order of a multi-index is :
n
|~I = ..7 ,~,.
i=I x
The sum 4 + ~ is defined as <~I + ~I ..... 4n + ~n TM "
Theorem I - Let A be a riu~ and let M ~ ~ which is ~hi]ate~ by
elements I x ,..., Xn£ A . SuDnose the ideal (x I ,..., Xn) contains a non zero
divisor. If ~ is a free resolution of M , then there are endomorohisms s~ of
deeTee 2 ~| -I of ~ as a a~raded module, $or each multi-index ~ , satisfvin~ :
i) s is the differential of ~ ;
o
th
ii) f~__I ~ is the multi-inde x 0 < ,..., 0,1,0 ,..., 0 > I( in the D
place) then the map
SoS ~ + S~ ° S
is multiplication by x :
J
iii) If ~ is a multi-index with J~J~1 , then
Z s %=0
Proof of theorem I. Beginning with the definition s ='~ , we will construct the
o
s by induction on J~l •
Condition ii) is the assertion that s~ , for ~=<0 ,..., 0,1,0 ,..., O~
(with I in the jth-place) is a hemotopy for multiplication by x.. Since x.N = O,
3 0