D-modules an d Microlocal Calculu s

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*?&"' *

Translations o f

MATHEMATICAL MONOGRAPHS

Volume 21 7

D-modules an d Microlocal Calculu s

Masaki Kashiwar a

Translated b y Mutsumi Sait o

yj° America n Mathematica l Societ y ? Providence , Rhod e Islan d

10.1090/mmono/217

Editorial Boar d Shoshichi Kobayash i (Chair )

Masamichi Takesak i

DAISU KAISEK I GAIRO N (GENERAL THEOR Y O F ALGEBRAI C ANALYSIS )

by Masak i Kashiwar a Copyright © 200 0 by Masaki Kashiwar a

Originally published i n Japanes e by Iwanami Shoten , Publishers , Tokyo, 2000

Translated fro m th e Japanes e by Mutsumi Sait o

2000 Mathematics Subject Classification. Primar y 32A37 , 32C38, 58J15.

Library o f Congres s Cataloging-in-Publicat io n D a t a

Kashiwara, Masaki , 1947 -[Daisu kaisek i gairon . English ] D-modules an d microloca l calculu s / Masak i Kashiwara ; translate d b y Mut -

sumi Saito . p. cm . - (Translation s o f mathematica l monographs , ISS N 0065-928 2 ;

v. 217 ) Includes bibliographica l reference s an d index . ISBN 0-8218-2766- 9 (pbk . : acid-fre e paper ) 1. D-modules . 2 . Microlocal analysis . I . Title . II . Series . III . Series : Iwanam i

series i n moder n mathematic s

QA614.3.K3813 200 3 516.3'6-dc21 2002027793

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Contents

Preface i x

Preface t o th e Englis h Editio n x i

Introduction

Chapter 1 . Basi c Propertie s o f D-module s 1.1. 1.2. 1.3. 1.4.

D-modules Differential Homomorphism s o f (9x-niodule s A Syste m o f Generator s o f D Left Vx-^aodules an d Righ t P^-module s

Chapter 2 . Characteristi c Varietie s 2.1. 2.2. 2.3. 2.4.

Cotangent Bundle s Characteristic Varietie s Involutivity Codimension Filtratio n

Chapter 3 . Constructio n o f .D-module s 3.1. 3.2. 3.3. 3.4.

Tensor Product s Homological Propertie s o f D-module s Dual Modul e Algebraic Relativ e Cohomolog y

Chapter 4 . Functoria l Propertie s o f D-module s 4.1. 4.2. 4.3. 4.4. 4.5. 4.6. 4.7.

Pull-backs o f D-module s Derived Functor s o f Inverse Image s External Tenso r Produc t Coherence o f Inverse Image s Cauchy-Kovalevskaya Theore m Integrals o f D-module s Coherence o f Integral s

xiii

1 1 2 6 7

11 11 15 21 22

29 29 32 36 39

57 57 59 60 61 68 74 76

V

vi C O N T E N T S

4.8. D-module s Supporte d b y a Submanifol d 8 4 4.9. Adjunctio n Formul a 8 7 4.10. Adjunctio n Formul a (Invers e Form ) 9 3 4.11. Holonomi c System s 9 4

Chapter 5 . Regula r Holonomi c System s 9 9 5.1. Ordinar y Differentia l Equation s

with Regula r Singularitie s 9 9 5.2. Holonomi c Module s wit h Regula r Singularitie s 10 0 5.3. Riemann-Hilber t Correspondenc e 10 2

Chapter 6 . b- functions 10 5 6.1. Motivatio n fo r b- functions 10 5 6.2. £>-modul e Generate d b y f(x) s 10 6 6.3. Rationalit y o f b- functions 10 9 6.4. b- functions i n th e Cas e of

Quasi-homogeneous Isolate d Singularitie s 11 4 6.5. b- function an d Irreducibilit y 11 7

Chapter 7 . Rin g o f Formal Microdifferentia l Operator s 12 3 7.1. Microlocalizatio n 12 3 7.2. Forma l Microdifferentia l Operator s o n Manifold s 12 5 7.3. Microdifferentia l Operator s 12 9 7.4. Algebrai c Propertie s o f £ 13 0 7.5. Relation s betwee n T>x and £x 13 9 7.6. Involutivit y o f the Support s o f Coheren t £ -modules 14 7

Chapter 8 . Microloca l Analysi s o f Holonomi c System s 15 3 8.1. Simpl e £ -modules 15 3 8.2. Quantize d Contac t Transformatio n 15 3 8.3. Subprincipa l Symbol s 15 9 8.4. Preparatio n fo r th e Classificatio n

of Simpl e Holonomi c System s 16 1 8.5. Classificatio n o f Simpl e Holonomi c System s (1 ) 16 8 8.6. Principa l Symbol s o f Simpl e Holonomi c Module s 17 0 8.7. Regula r Holonomi c <?x~niodule s 17 6 8.8. Classificatio n o f Simpl e Holonomi c System s (2 ) 18 1 8.9. Subholonomi c System s 18 6

Chapter 9 . Microloca l Calculu s o f 6-function s 9.1. Microloca l b- functions

191 191

C O N T E N T S vi i

9.2. Existenc e o f Microloca l b- functions 19 6 9.3. Simpl e Intersectio n o f Lagrangian Analyti c Subset s 20 0 9.4. Relatio n o f b- functions o f Two Lagrangia n

Analytic Subset s wit h a Goo d Intersectio n 20 1 9.5. b- functions o f Relativ e Invariant s 20 5

Appendix 21 3 A.l. Finitenes s o f Filtere d Ring s an d Thei r Module s 21 3 A.2. Derive d Categorie s 22 6 A.3. Derive d Functor s o n Categorie s o f Module s 23 9 A.4. Symplecti c Manifold s 24 3

Bibliography 24 7

Index 25 1

Index o f Notation s 253

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Preface

This boo k provide s a n introductio n t o th e genera l theor y o f D-modules, whos e framewor k wa s buil t u p b y my teache r Miki o Sato , Takahiro Kawai , an d me . A s a n applicatio n o f jD-modul e theory , I planned t o writ e ho w th e geometr y o f fla g manifold s an d th e rep -resentation theor y o f semisimpl e Li e algebra s ar e relate d t o eac h other throug h th e theory . Sinc e a n excellen t Japanes e boo k ([HT] ) by Ryosh i Hott a an d Toshiyuk i Tanisak i coverin g thi s topi c ha s bee n already published , I hav e decide d t o write , instead , a boo k o n th e algorithm fo r 6-function s usin g microloca l analysis . Fo r thi s reason , the titl e o f thi s boo k ha s bee n change d fro m th e on e announced. * I apologize fo r an y confusio n thi s ma y hav e caused .

It wa s aroun d 196 9 when I starte d t o stud y D-module s an d mi -crolocal analysi s wit h Sat o an d Kawai . Amon g th e result s I hav e obtained throug h ove r 30 years' study , th e microloca l algorith m fo r b-functions i s one of the mos t beautifu l one s (it s only defec t i s a limite d range o f applications) .

I woul d lik e t o than k Kiyosh i Takeuchi , Toshiyuk i Tanisaki , an d Kenji lohar a fo r pointin g ou t man y mistake s i n preliminar y version s of the book , an d the staf f fro m Iwanam i Shote n fo r thei r cooperation .

Masaki Kashiwar a

Paris, Marc h 200 0

*The origina l Japanes e boo k ha d bee n announce d t o b e publishe d unde r th e title o f "Algebrai c Analysi s an d Representatio n Theory. "

ix

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Preface t o th e Englis h Editio n

This boo k wa s originally writte n i n Japanese , an d wa s publishe d by Iwanam i Shote n o n Marc h 28 , 2000. Th e Japanes e titl e i s "Foun -dation o f Algebrai c Analysis" . I n thi s edition , I change d th e titl e i n order t o expres s th e content s mor e exactly . Th e phras e "microloca l calculus" i n th e presen t titl e i s no t a popula r terminology . A s ex -plained i n the introduction , I intend b y this to sen d th e messag e tha t the microloca l point o f view helps concrete calculations, a s powerfull y as th e Cauch y integra l formul a provide s th e value s o f man y definit e integrals.

Finally, bu t no t th e least , specia l thank s g o t o th e translator , Mutsumi Saito .

Masaki Kashiwar a

Kyoto, Augus t 200 2

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Introduction

The stud y o f D-module s wa s launche d whe n Mikio Sat o gav e a colloquium tal k o n the m a t th e Departmen t o f Mathematics , Uni -versity o f Tokyo , i n 1960 . Le t u s briefl y explai n hi s motivatio n fo r studying D-modules .

The genera l for m o f a syste m o f linea r partia l differentia l equa -tions wit h unknow n function s u\,. .., u p i n x — ( x i , . . ., xn) i s

p

(0.1) ^ P ^ ( x , d ) ^ = 0 ( i = l , . . . , g ) , i= i

where Pij{x 1 d) (i = 1,...,g , j — 1 , . .. ,p ) ar e linea r partia l differen -tial operators , an d eac h o f them i s written a s

£ a*(x)d a, ^ = ( ^ ) • • • ( ^ ) • a = ( a 1 , . . . , o B ) .

The unknow n function s u\ ,..., u p ar e no t intrinsi c i n th e syste m o f equations; they are just dummie s for the purpose of writing the system in a n explici t form . Thi s poin t o f vie w i s a startin g poin t fo r th e introduction o f D-modules . A s a simpl e example , le t u s conside r th e equation

(0.2) ( ^ - A ) « = °<

where A is a comple x number . Le t v(x) — xu(x)\ the n thi s i s trans -formed int o th e equatio n

(0.3) u A _A-lU(ar ) = 0,

because

\JjJU I \ \JbJU

xiv INTRODUCTIO N

Conversely, i f v is a solution t o equatio n (0.3) , then , assumin g tha t A ̂—1 , we see that u = j^-£^v satisfie s equatio n (0.2) , sinc e

dx J dx dx \ dx

Moreover, b y lettin g v = xu(x) fo r a solution u to equatio n (0.2) , we obtain u = JTJ^V, sinc e

1 d \ ( d x . -v — u = —-x — A — 1 ] u = 0. A + 1 dx \ + I \dx

Conversely, b y lettin g u = JT^-^V fo r a solution v to equatio n (0.3) , we obtain v — xu, sinc e

xu — v— I x- A — 1 | v = 0 . A + 1 V dx J

By th e transformation s v = xu and u = ^ + 1 ^ ^ ' w e t n us s e e t n a^ equations (0.2 ) an d (0.3 ) ar e equivalen t t o each other . Henc e the y mean th e same , althoug h the y loo k different . I n othe r words , takin g u o r v as a n unknow n functio n i s quit e artificial . W e ca n formulat e this ide a b y usin g D-modules .

Let u s conside r equatio n (0.1) . Le t D be th e (noncommutative ) ring o f linea r partia l differentia l operators , an d P : D®q —> • D®p th e map give n b y

(Qi,...,Qg)^ \^QiPii,...^QiPip\ •

This ma p P is left D-linear , an d it s cokernel M is the D-modul e corresponding t o equatio n (0.1) . W e can deriv e th e solution s t o (0.1 ) from M as follows : Le t F be a space o f function s i n whic h w e wan t to find th e solutions . Variou s space s ca n b e take n a s F , suc h a s th e space o f C°°-functions, th e spac e o f distributions, o r the spac e of holomorphic functions , dependin g o n th e proble m bein g considered . Since w e ca n appl y differentia l operator s to the function s belongin g to F , w e regard F as a left Z}-module . The n

Hom D (D 0 p , F) = F® p = {(Ul,..., u p); uu . . . , u p e F}.

By a well known propert y (lef t exactness ) o f Hom^ ,

HomD(M, F) = Ker(Hom D(£>ep, F) ^ Hom D (D 0 9 , F))

= {(ixi,.. . ,it p) 6 Fe p ; ^ i , . . . ,u p sa t i s fy (0.1)} .

INTRODUCTION x v

Hence the spac e of solutions to (0.1 ) i n F equal s Hom£>(M, F), whic h depends onl y o n M. Conversely , give n a D-modul e M , fo r eac h iso -morphism

(0.4) M ~ Coker(D 09 ^D® p),

we obtai n equatio n (0.1) . The n (0.1 ) i s considere d a n explici t pre -sentation o f M correspondin g t o th e isomorphis m (0.4) . Ther e ar e many suc h isomorphism s fo r a give n D-module . Eac h isomorphis m gives a differen t explici t presentatio n (0.1) . I n th e previou s example , one D-modul e ha s two distinc t isomorphisms ; i t i s isomorphic t o tw o cokernels:

Coker(£>*^A£>) ~ C o k e r ( D * ^ _ 1 L > ) ,

which lea d t o (0.2 ) an d (0.3 ) respectively . From this point of view, D-modules were introduced, an d the the-

ory ha s grow n int o a rich on e with man y subtheorie s suc h a s charac -teristic varieties , microloca l analysis , holonomi c system s (maximall y overdetermined systems) , etc . W e give an outline of the theory i n thi s book.

Today, beside s th e theor y o f linea r partia l differentia l equations , which i s a root o f D-module theory , th e theor y i s also applied t o rep -resentation theory , conforma l fiel d theory , etc . Fo r application s t o representation theory , [HT ] ha s a detaile d exposition . I n thi s book , we describe applications o f the theory to 6-functions ; i n particular, w e give a n algorith m t o comput e b- functions usin g microloca l analysis . Sato seems to hav e expected tha t th e Z}-modul e theory (i n particula r the theor y o f holonomi c systems ) woul d b e usefu l fo r explici t com -putations. Thi s i s analogous t o Cauchy' s integratio n theorem , whic h has no t onl y theoretica l beaut y an d importance , bu t als o a n applica -tion t o explici t computation s o f definit e integral s throug h Cauchy' s residue formula . Th e theor y o f 6-function s describe d i n thi s boo k i s a goo d exampl e o f applicabilit y o f microloca l analysi s o f holonomi c systems t o explici t computations .

Conventions an d Notation . I n thi s book , a manifol d mean s a nonsingula r comple x analyti c manifold . A morphis m o f manifold s / : X — » Y i s said t o be smooth i f the correspondin g map s o f tangen t spaces T XX — > Tf^Y ar e surjectiv e a t al l x G X.

We abbreviat e th e dimensio n di m X o f a manifold X t o dx , an d write dx/y — dx — dy.

xvi INTRODUCTIO N

We simpl y cal l a shea f o f ring s a ring , an d a shea f o f module s a module.

>—» indicate s injectivity , an d - » surjectivity . For a n analyti c se t Z , w e sa y tha t a propert y (P ) hold s a t a

generic poin t p o n Z i f ther e exist s a nowher e dens e close d analyti c subset Y suc h tha t (P ) hold s a t al l p e Z\Y.

Bibliography

As a Japanes e boo k o n D-modules ,

[HT] R . Hotta , an d T . Tanisaki , D-modules and Algebraic Groups, Springer Moder n Mathematic s Series , Springer-Verla g Tokyo , Tokyo, 199 5 (i n Japanese )

is recommended, whic h contain s a detaile d expositio n particularl y o f thei r applications t o representatio n theory . Othe r book s o n D-module s ar e th e following:

[Bjl] J.-E . Bjork , Rings of Differential Operators, North-Holland , 1979 . [Bj2] J.-E . Bjork , Analytic D-modules and Applications, Mathematic s

and It s Application s 247 , Kluwer Academi c Publ. , 1993 . [Bor] A . Bore l e t al. , Algebraic D-modules, Perspective s i n Math . 2 ,

Academic Press , 1987 . [Kl] M . Kashiwara , Algebraic Study of Systems of Partial Differential

Equations, Memoire s de la S . M. F . 63 , Societe Mathematiqu e d e France, translate d b y A . D'Agnol o an d J.-P . Schneiders , 1995 .

A boo k i n th e sam e serie s a s th e presen t one ,

[T] T . Tanisaki , Rings and Fields 3, Iwanami Courses in Foundation s of Moder n Mathematics , Iwanam i Shoten , 199 8 (i n Japanese) .

contains filtere d ring s an d th e involutivit y o f characteristi c varieties , i n particular th e proo f o f Gabber' s theorem .

For detail s abou t hyperfunction s an d pseudo-differentia l operators , se e

[KKK] M . Kashiwara , T . Kawai , an d T . Kimura , Foundations of Alge-braic Analysis, Kinokuniy a 198 0 (i n Japanese) ; Englis h edition , Princeton Universit y Press , 1986 .

For th e genera l theor y o f pseudo-differentia l operators , se e

[SKK] M . Sato , T . Kawai , an d M . Kashiwara , "Microfunction s an d pseudo-differential equations" , Lectur e Note s i n Math . 287 , Springer-Verlag, 1973 , pp. 264-529.

and

247

248 BIBLIOGRAPHY

[Sc] P . Schapira , Microdifferential Systems in the Complex Domain, Grundlehren de r Mathematische n Wissenschafte n 269 , Springer -Verlag, 1985 .

For ring s o f twiste d differentia l operators , whic h w e di d no t mention , se e

[K2] M . Kashiwara , "Representatio n theor y an d D-module s o n fla g varieties", Orbites Unipotentes et Representations, III, Orbites et Faisceaux Pervers, Asterisqu e 173-174 , Societ e Mathematiqu e de France , 1989 , pp . 55-109.

For detail s abou t th e microloca l analysi s o f b- functions, se e

[SKKO] M . Sato , M . Kashiwara , T . Kimur a an d T . Oshima , Micro-loca l analysis of prehomogeneous vecto r spaces, Invent. Math. 62(1980) , pp. 117-179.

For microloca l computatio n o f th e Fourie r transfor m o f power s o f relativ e invariants, se e

[KM] M . Kashiwara , Microloca l calculu s an d Fourie r transfor m o f rela-tive invariant s o f prehomogeneous vecto r space s (Note s take n b y T. Miwa) , RIM S Kokyurok u 23 8 (1975) , pp . 60-147 (i n Japan -ese); Mathematica l Review s 58:31299 .

For derive d categories , se e th e following , i n additio n t o th e Appendix :

[H] R . Hartshorne , Residues and Duality, Lectur e Note s i n Math. 20 , Springer-Ver lag, 1966 .

[KS] M . Kashiwar a an d P . Schapira , Sheaves on Manifolds, Grund -lehren de r Mathematische n Wissenschafte n 292 , Springer-Verlag , 1990.

The followin g ar e als o cite d i n th e presen t book :

[BBD] A . Beilinson , J . Bernstei n an d P . Deligne , Faisceaux Pervers, Asterisque 100 , Societ e Mathematiqu e d e France , 1983 .

[Berl] J . Bernstein , Module s ove r a rin g o f differentia l operators , Func. Anal, and its Appl. 5(1971) , pp . 89-101.

[Ber2] J . Bernstein , Th e analyti c continuatio n o f generalize d function s with respec t t o a parameter , ibid. 6(1972) , pp . 273-285.

[Ga] O . Gabber , Th e integrabilit y o f the characteristi c variety , Amer. J. Math. 103(1981) , pp . 445-468.

[Gol] V . D . Golovin , Homology of Analytic Sheaves and Duality Theo-rems, Consultant s Bureau , Ne w Yor k an d London , 1989 .

[Hottal] R . Hotta , Introduction to D-modules, Lectur e Not e Serie s 1 , Inst . of Math . Sci. , Madras , 1987 .

[Hotta2] R . Hotta , Rings and Fields 1, Iwanami Course s i n Foundation s of Moder n Mathematics , Iwanam i Shoten , 199 7 (i n Japanese) .

BIBLIOGRAPHY 249

[KF] M . Kashiwara , Systems of Microdifferential Equations (Note s taken b y Teres a Monteir o Fernandes) , Progres s i n Math . 34 , Birkhauser, 1983 .

[KKJ M . Kashiwar a an d T . Kawai , O n holonomi c system s o f microdif -ferential equations . Ill : System s wit h regula r singularities , Publ RIMS, Kyoto Univ. 17(1981) , pp . 813-979.

[K3] M . Kashiwara , Introductio n t o Microloca l Analysis , L'Enseigne-ment Mathematique 32(1986) , pp . 227-259.

[K4] M . Kashiwara , O n the maximall y overdetermine d system s o f dif-ferential operator s I , Publ. RIMS, Kyoto Univ. 10(1975) , pp. 563-579.

[K5] M . Kashiwara , Th e Riemann-Hilber t proble m fo r holonomi c sys -tems, Publ RIMS, Kyoto Univ. 20(1984) , pp . 319-365.

[K6] M . Kashiwara, B-function s an d holonomic systems, Invent. Math. 38(1976), pp . 33-53.

[Ma] B . Malgrange , Integrate s asymptotique s e t monodromie , Ann. Sci. Ecole Norm. Sup.(4) 4(1974) , pp . 405-430.

[SaitoK] K . Saito , Quasihomogen e isoliert e Singularitaten vo n Hyperflach -en, Invent. Math. 14(1971) , pp . 123-142.

[SaitoMl] M . Saito , Module s d e Hodg e polarisables , Publ. RIMS, Kyoto Univ. 24(1988) , pp.'849-995 .

[SaitoM2] M . Saito , Mixe d Hodg e modules , Publ. RIMS, Kyoto Univ. 26(1990), pp . 221-333.

[U] K . Ueno , Algebraic Geometry 1 : From Algebraic Varieties to Schemes, Amer . Math . Soc , Providence , RI , 1999 .

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Index

adjunction formula , 8 7 algebraic relativ e cohomolog y

module, 4 6 Artin-Rees Theorem , 22 4

b- function, 10 5 bounded, 22 7 bounded above , 22 7 bounded below , 22 7

c-soft, 24 3 characteristic variety , 1 7 coherent, 21 4 coherent filtration , 15 , 21 9 coherent ring , 21 4 cohomological dimension , 23 8 complete intersection , 5 1 complex, 22 6 conormal bundle , 6 2 construct ible, 10 2

de Rha m complex , 5 derived category , 23 2 desingularization theorem , 11 1 differential homomorphism , 2 differential operator , 1

- - o f orde r a t mos t ra, 1 direct image , 24 2 directed family , 7 7 distinguished triangle , 230 , 23 4 duality theorem , 9 3

enough injectives , 23 7 exterior differentia l forms , 8 external tenso r product , 60 , 15 4

filtered ^-module , 21 8 filtered module , 21 6 filtered ring , 21 8 filteredly exact , 21 6 filteredly surjective , 21 7 filtration, 1 5 flabby, 24 2 formal adjoint , 1 0 formal microdifferentia l operator ,

124 , $-homogeneous , 16 4

Gabber's theorem , 147 , 15 1 generic point , xv i good X>x - rn°dule, 7 8 good intersection , 18 2 good Ox-module , 7 6

Hamiltonian, 1 4 holonomic £>x- m°dule, 9 4 holonomic system , 15 3 holonomy diagram , 20 9 homogeneous symplecti c manifold ,

243 homogeneous symplecti c

transformation, 15 6 homomorphism o f filtere d modules ,

216

injective, 23 6 inner derivative , 8 integral o f Ai alon g fibers o f / , 7 6 involutive, 22 , 24 4 involutive syste m o f generators , 1 9

251

252 INDEX

irregular singularity , 10 0 isolated singularity , 11 4 isotropic, 24 4

Lagrangian, 244 , 24 5 lattice, 17 8 left derive d functor , 23 8 Lie derivative , 8 locally finitel y generated , 21 3 locally finitel y generate d filtere d

module, 21 9 locally finitel y presented , 21 3

manifold, x v mapping cone , 22 9 microdifferential operator , 12 9 microlocal fe-function, 19 5 Milnor number , 11 7 module, xv i multiplicity, 2 1

Nakayama's Lemma , 13 0 Noetherian module , 21 4 Noetherian ring , 21 4 non-characteristic, 62 , 6 6

octahedral axiom , 23 1 Oka-Cart an Theorem , 21 5 opposite category , 24 1 opposite ring , 7 , 23 9 order, 17 3

perverse sheaf , 10 3 Poisson bracket , 1 4 principal symbol , 13 , 170 , 17 2 projection formula , 82 , 24 3 projective, 23 8 pseudo-coherent, 21 4 purely subholonomic , 18 7

qis, 23 2 quantized contac t transformation ,

156 quasi-homogeneous isolate d

singularity, 11 4 quasi-isomorphism, 23 2

Rees module , 21 6 Rees ring , 21 9 regular holonomic , 101 , 17 7

regular singularity , 10 0 relative d e Rha m complex , 5 8 relative invariant , 20 5 Riemann-Hilbert correspondence ,

102 right derivable , 23 5 ring, xv i

Serre duality , 9 3 simple <£x -m°dule, 15 3 simple intersection , 20 0 simple Lagrangian , 19 2 smooth, x v subholonomic, 107 , 18 6 subprincipal symbol , 15 9 symplectic manifold , 15 , 24 3

i?-finite, 16 4 total symbol , 11 , 12 3 trace morphism , 8 7 triangle, 23 0 triangles

isomorphism of , 23 0 triangulated category , 23 1 truncated complex , 22 8

Index o f Notation s

>—>, xv i

- » , xv i

A0?, 7

Cb(C), 22 7 C(C), 22 6 Ch, 1 7 Ch, 3 9 C~(C), 22 7 C+(C), 22 7

D(A), 23 9 Db(C), 23 6 DS(Cx) , 10 3 ^ c o h ^ x ) , 3 6 Db

9ood(Vx), 7 8 ^ P x ) , 10 1 D(C), 23 2 m , 7 6 D/», 7 6 o r , 60 AT, 5 7 W , 2 D~(C), 23 6 £>+(£), 23 6 D R X , 10 2 D <g>, 3 0 D <g>, 3 6 d x , xv , 74 , 8 7 dx, 1 0

X>x, 1 D x , 3 6 Vx->Y, 5 7 d x / y > xv > 74 , 8 7 Vytx, 7 5

£, 12 9 Sen , 12 4 ^ ( m ) , 12 4 £(m) , 12 9 £ x , 12 8 £ x ( m ) , 12 8 Ext, 24 1 B , 6 0 D B, 60

/ d , 6 1 / A , 19 2 F m ( P x ) , 1 2 A , 6 2

r [ s ] , 4 4 r [x \z ]> 3 9 r [ Z ] , 3 9 r z , 3 9

Hf, 1 4 H ^ , 4 6 Hol(X), 9 5 Ht(M,AT), 22 8

IR(A4), 17 7 i t/, 8

253

254 INDEX O F NOTATION S

K(A), 23 9 Sp , 13 9 K(C), 22 8 Vdx, 16 0

Supp, 3 8 LF(X) , 23 8 £ ® - i , 9 r ^ n M , 22 8 C^-V, 16 0 r < " M , 22 8

239 Tor, 24 0 L^ 5

8 Tr / , 8 7

MEAT, 15 4 M ( / ) , 22 9 M[fc], 22 7 raA, 19 2 Mod(A), 22 6

MA,192 multy, 2 1

i ?^ / 2 , 16 0

^ x , 8

^ x , 7 , 8 « / y , 7 4

^ _ x / y ^ ^ ord(it), 17 3 ordA(w), 17 3 ord P(s ) , 19 6 ordP(s) , 19 6 0T*x(rn), 12 4 O x , 1

Perv(Cx) , 10 3 7TX : T*X - + X, 1 3 + , 6 6 x P* , 10 , 15 9

R e g ( P x ) , 10 3 R/s, 24 3 R/*, 24 2 RF(X), 23 5 MTrsi, 4 6 I H o m , 24 0 RH(X), 10 1 RZF(X), 23 8

<7A(it), 17 3 a(w), 170 , 17 2 Cm, 1 3 So l x , 10 2

Titles i n Thi s Serie s

217 Masak i Kashiwara , D-module s an d microloca l calculus , 200 3

216 G . V . Badalyan , Quasipowe r serie s an d quasianalyti c classe s o f

functions, 200 2

215 Tatsu o Kimura , Introductio n t o prehomogeneou s vecto r spaces , 200 3

214 L . S . Grinblat , Algebra s o f set s an d combinatorics , 200 2

213 V . N . Sachko v an d V . E . Tarakanov , Combinatoric s o f nonnegativ e

matrices, 200 2

212 A . V . Mernikov , S . N . Volkov , an d M . L . Nechaev , Mathematic s o f

financial obligations , 200 2

211 Take o Ohsawa , Analysi s o f severa l comple x variables , 200 2

210 Toshitak e Kohno , Conforma l field theor y an d topology , 200 2

209 Yasumas a Nishiura , Far-from-equilibriu m dynamics , 200 2

208 Yuki o Matsumoto , A n introductio n t o Mors e theory , 200 2

207 Ken'ich i Ohshika , Discret e groups , 200 2

206 Yuj i Shimiz u an d Kenj i Ueno , Advance s i n modul i theory , 200 2

205 Seik i Nishikawa , Variationa l problem s i n geometry , 200 1

204 A . M . Vinogradov , Cohomologica l analysi s o f partia l differentia l

equations an d Secondar y Calculus , 200 1

203 T e Su n Ha n an d King o Kobayashi , Mathematic s o f informatio n an d coding, 200 2

202 V . P . Maslo v an d G . A . Omel'yanov , Geometri c asymptotic s fo r nonlinear PDE . I , 200 1

201 Shigeyuk i Morita , Geometr y o f differentia l forms , 200 1

200 V . V . Prasolo v an d V . M . Tikhomirov , Geometry , 200 1

199 Shigeyuk i Morita , Geometr y o f characteristi c classes , 200 1

198 V . A . Smirnov , Simplicia l an d opera d method s i n algebrai c topology ,

2001

197 Kenj i Ueno , Algebrai c geometr y 2 : Sheave s an d cohomology , 200 1

196 Yu . N . Lin'kov , Asymptoti c statistica l method s fo r stochasti c processes ,

2001

195 Minor u Wakimoto , Infinite-dimensiona l Li e algebras , 200 1

194 Valer y B . Nevzorov , Records : Mathematica l theory , 200 1

193 Toshi o Nishino , Functio n theor y i n severa l comple x variables , 200 1

192 Yu . P . Solovyo v an d E . V , Troitsky , <7*-algebra s an d ellipti c

operators i n differentia l topology , 200 1

191 Shun-ich i Amar i an d Hirosh i Nagaoka , Method s o f informatio n

geometry, 200 0

190 Alexande r N . Starkov , Dynamica l system s o n homogeneou s spaces ,

2000 189 Mitsur u Ikawa , Hyperboli c partia l differentia l equation s an d wav e phenomena, 200 0

TITLES I N THI S SERIE S

188 V . V . Buldygi n an d Yu . V . Kozachenko , Metri c characterizatio n o f random variable s an d rando m processes , 200 0

187 A . V . Fursikov , Optima l contro l o f distribute d systems . Theor y an d

applications, 200 0

186 Kazuy a Kato , Nobushig e Kurokawa , an d Takesh i Saito , Numbe r

theory 1 : Fermat' s dream , 200 0

185 Kenj i U e n o , Algebrai c Geometr y 1 : Pro m algebrai c varietie s t o schemes ,

1999

184 A . V . Mel'nikov , Financia l markets , 199 9

183 Haj im e Sato , Algebrai c topology : a n intuitiv e approach , 199 9

182 I . S . Krasil'shchi k an d A . M . Vinogradov , Editors , Symmetrie s an d

conservation law s fo r differentia l equation s o f mathematica l physics , 199 9

181 Ya . G . Berkovic h an d E . M . Zhmud' , Character s o f finit e groups .

Part 2 , 199 9

180 A . A . Milyut i n an d N . P . Osmolovskii , Calculu s o f variation s an d

optimal control , 199 8

179 V . E . Voskresenskii , Algebrai c group s an d thei r birationa l invariants ,

1998

178 Mitsu o Morimoto , Analyti c functional s o n th e sphere , 199 8

177 Sator u Igari , Rea l analysis—wit h a n introductio n t o wavele t theory , 199 8

176 L . M . Lerma n an d Ya . L . Umanskiy , Four-dimensiona l integrabl e

Hamiltonian system s wit h simpl e singula r point s (topologica l aspects) , 199 8

175 S . K . Godunov , Moder n aspect s o f linea r algebra , 199 8

174 Ya-Zh e Che n an d Lan-Chen g W u , Secon d orde r ellipti c equation s an d

elliptic systems , 199 8

173 Yu . A . Davydov , M . A . Lifshits , an d N . V . Smorodina , Loca l properties o f distribution s o f stochasti c functionals , 199 8 Ya. G . Berkovic h an d E . M . Zhmud' , Character s o f finit e groups . Part 1 , 199 8

E. M . Landis , Secon d orde r equation s o f ellipti c an d paraboli c type , 199 8

Viktor Prasolo v an d Yur i Solovyev , Ellipti c function s an d ellipti c integrals, 199 7 S. K . Godunov , Ordinar y differentia l equation s wit h constan t

coefficient, 199 7

Junjiro Noguchi , Introductio n t o comple x analysis , 199 8

Masaya Yamaguti , Masayosh i Ha t a, an d Ju n Kigami , Mathematic s of fractals , 199 7 Kenji U e n o , A n introductio n t o algebrai c geometry , 199 7

For a complet e lis t o f t i t le s i n thi s series , visi t t h e A M S Bookstor e a t w w w . a m s . o r g / b o o k s t o r e / .