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Lecture Notes in Mathematics 1327 Edited by A. Dold and B. Eckmann Winfried Bruns Udo Vetter Determinantal Rings Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Authors Winfried Bruns FB Mathematik/Informatik Universita(cid:127)t Osnabru(cid:127)ck 49069Osnabru(cid:127)ck Germany Udo Vetter FB Mathematik Universita(cid:127)t Oldenburg 26111Oldenburg Germany This book is now out of print. The authors are grateful to Springer-Verlag for the permission to make this postscript (cid:12)le accessible. ISBN 3-540-19468-1Springer-VerlagBerlin Heidelberg New York ISBN 0-387-19468-1Springer-VerlagNew York Berlin Heidelberg (cid:13)c Springer-VerlagBerlin Heidelberg 1988 Preface Determinantal rings and varieties have been a central topic of commutative algebra and algebraic geometry. Their study has attracted many prominent researchers and has motivated the creation of theories which may now be considered part of general commutativeringtheory. Acoherenttreatmentofdeterminantalringsislackinghowever. Wearealgebraists,andthereforethe subjectwillbetreatedfromanalgebraicpoint of view. Our main approach is via the theory of algebras with straightening law. Its axiomsconstituteaconvenientsystematicframework,andthestandardmonomialtheory on which it is based yields computationally e(cid:11)ective results. This approach suggests (andissimpli(cid:12)edby)thesimultaneoustreatmentofthe coordinateringsoftheSchubert subvarieties of Grassmannians, a program carried out very strictly. Other methods have not been neglected. Principal radical systems are discussed in detail, and one section each is devoted to invariant and representation theory. However, free resolutions are (almost) only covered for the \classical"case of maximal minors. OurpersonalviewofthesubjectismostvisiblyexpressedbytheinclusionofSections 13{15 in which we discuss linear algebra over determinantal rings. In particular the technical details of Section 15 (and perhaps only these) are somewhat demanding. Thebibliographycontainsseveraltitleswhichhavenotbeencitedinthe text. They mainly covertopics not discussed: geometricmethods and ideals generated byminors of symmetric matrices and Pfa(cid:14)ans of alternating ones. We have tried hard to keep the text as self-contained as possible. The basics of commutative algebra supplied by Part I of Matsumura’s book [Mt] (and some additions given in Section 16) su(cid:14)ce as a foundation for Sections 3{7, 9, 10, and 12. Whenever necessary to draw upon notions and results not covered by [Mt], for example divisor class groups and canonical modules in Section 8, precise references have been provided. It is no surprise that multilinear algebra plays a role in a book on determinantal rings, and in Sections 2 and 13{15 we expect the reader not to be frightened by exterior and symmetric powers. Even Section 11 which connects our subject and the representation theory of the general linear groups, does not need an extensive preparation; the linear reductivity of these groups is the only essential fact to be imported. The rudiments on Ext and Tor contained in every introduction to homological algebra will be used freely, though rarely,andsomefamiliaritywith a(cid:14)neandprojectivevarieties, asdeveloppedin Chapter I of Hartshorne’s book [Ha.2], is helpful. We hope this text will serve as a reference. It may be useful for seminars following a course in commutative ring theory. A vast number of notions, results, and techniques canbeillustratedsigni(cid:12)cantlybyapplyingthemtodeterminantalrings,anditmayeven bepossibletoreversetheusualsequenceof\theory"and\application": tolearnabstract commutative algebra through the explorationof the special classwhich is the subject of this book. Each section contains a subsection \Comments and References"where we have col- lected the information on our sources. The references given should not be considered iv Preface assignments of priority too seriously; they rather re(cid:13)ect the authors’ history in learning thesubjectandgivecredittothecolleaguesinwhoseworkswehaveparticipated. While it is impossible to mention all of them here, it may be fair to say that we could not havewrittenthistextwithoutthefundamentalcontributionsofBuchsbaum,deConcini, Eagon, Eisenbud, Hochster, Northcott, and Procesi. The(cid:12)rstauthorgaveaseriesof lecturesondeterminantalringsatthe Universidade federal de Pernambuco, Recife, Brazil, in March and April 1985. We are indebted to Aron Simis who suggested to write an extended version for the IMPA subseries of the Lecture Notes in Mathematics. (By now it has become a very extended version). Finally we thank Petra Du(cid:127)vel, Werner Lohmann and Matthias Varelmann for their help in the production of this book. We are grateful to the sta(cid:11) of the Computing Center of our university, in particular ThomasHaarmann, for generouscooperationand providing excellent printing facilities. Vechta, January 1988 Winfried Bruns Udo Vetter Contents 1. Preliminaries . . . . . . . . . . . . . . . . . . . . 1 A. Notations and Conventions . . . . . . . . . . . . . . . 1 B. Minors and Determinantal Ideals . . . . . . . . . . . . . 3 C. Determinantal Rings and Varieties . . . . . . . . . . . . 4 D. Schubert Varieties and Schubert Cycles . . . . . . . . . . . 6 E. Comments and References . . . . . . . . . . . . . . . 9 2. Ideals of Maximal Minors . . . . . . . . . . . . . . . 10 A. Classical Results on Height and Grade . . . . . . . . . . . 10 B. The Perfection of I (X) and Some Consequences . . . . . . . . 13 m C. The Eagon-Northcott Complex . . . . . . . . . . . . . 16 D. The Complex of Gulliksen and Neg(cid:23)ard . . . . . . . . . . . 22 E. Comments and References . . . . . . . . . . . . . . . 25 3. Generically Perfect Ideals . . . . . . . . . . . . . . . 27 A. The Transfer of Perfection . . . . . . . . . . . . . . . 27 B. The Substitution of Indeterminates by a Regular Sequence . . . . . 30 C. The Transfer of Integrity and Related Properties . . . . . . . . 34 D. The Bound for the Height of Specializations . . . . . . . . . 36 E. Comments and References . . . . . . . . . . . . . . . 36 4. Algebras with Straightening Law on Posets of Minors . . . . . 38 A. Algebras with Straightening Law . . . . . . . . . . . . . 38 B. G(X) as an ASL . . . . . . . . . . . . . . . . . . 40 C. The Linear Independence of the Standard Monomials in G(X) . . . . 43 D. B[X] as an ASL . . . . . . . . . . . . . . . . . . 45 E. Comments and References . . . . . . . . . . . . . . . 48 5. The Structure of an ASL . . . . . . . . . . . . . . . 50 A. ASL Structures on Residue Class Rings . . . . . . . . . . . 50 B. Syzygies and the Straightening Law . . . . . . . . . . . . 53 C. Nilpotents, Regular Elements and Dimension . . . . . . . . . 54 D. Wonderful Posets and the Cohen-Macaulay Property . . . . . . . 58 E. The Arithmetical Rank of Certain Ideals . . . . . . . . . . 61 F. Comments and References . . . . . . . . . . . . . . . 63 6. Integrity and Normality. The Singular Locus . . . . . . . . 64 A. Integrity and Normality . . . . . . . . . . . . . . . . 64 B. The Singular Locus . . . . . . . . . . . . . . . . . 67 C. Comments and References . . . . . . . . . . . . . . . 72 vi Contents 7. Generic Points and Invariant Theory . . . . . . . . . . . 73 A. A Generic Point for R (X) . . . . . . . . . . . . . . 73 r+1 B. Invariants and Absolute Invariants . . . . . . . . . . . . 74 C. The Main Theorem of Invariant Theory for GL and SL . . . . . . 76 D. Remarks on Invariant Theory . . . . . . . . . . . . . . 80 E. The Classical Generic Point for G(X;(cid:13)) . . . . . . . . . . 82 F. G(X;(cid:13)) and R(X;(cid:14)) as Rings of Invariants . . . . . . . . . . 84 G. The Depth of Modules of Invariants . . . . . . . . . . . . 88 H. Comments and References . . . . . . . . . . . . . . . 91 8. The Divisor Class Group and the Canonical Class . . . . . . . 93 A. The Divisor Class Group . . . . . . . . . . . . . . . 93 B. The Canonical Class of R (X) . . . . . . . . . . . . . 97 r+1 C. The General Case . . . . . . . . . . . . . . . . . . 100 D. Comments and References . . . . . . . . . . . . . . . 104 9. Powers of Ideals of Maximal Minors . . . . . . . . . . . 105 A. Ideals and Subalgebras of Maximal Minors . . . . . . . . . . 105 B. ASL Structures on Graded Algebras Derived from an Ideal . . . . . 108 C. Graded Algebras with Respect to Ideals of Maximal Minors . . . . 112 D. The Depth of Powersof Ideals of Maximal Minors . . . . . . . 117 E. Comments and References . . . . . . . . . . . . . . . 120 10. Primary Decomposition . . . . . . . . . . . . . . . . 122 A. Symbolic Powersof Determinantal Ideals . . . . . . . . . . 122 B. The Symbolic Graded Ring . . . . . . . . . . . . . . . 124 C. PrimaryDecomposition of Products of Determinantal Ideals . . . . 126 D. Comments and References . . . . . . . . . . . . . . . 133 11. Representation Theory . . . . . . . . . . . . . . . . 135 A. The Filtration of K[X] by the Intersections of Symbolic Powers . . . 135 B. Bitableaux and the Straightening Law Revisited . . . . . . . . 137 C. The Decomposition of K[X] into Irreducible G-Submodules . . . . 141 D. G-Invariant Ideals . . . . . . . . . . . . . . . . . 145 E. U-Invariants and Algebras Generated by Minors . . . . . . . . 149 F. Comments and References . . . . . . . . . . . . . . . 152 12. Principal Radical Systems . . . . . . . . . . . . . . . 153 A. A Propedeutic Example. Principal Radical Systems . . . . . . . 153 B. A Principal Radical System for the Determinantal Ideals . . . . . 155 C. The Perfection of Determinantal Ideals . . . . . . . . . . . 158 D. Comments and References . . . . . . . . . . . . . . . 160 13. Generic Modules . . . . . . . . . . . . . . . . . . 162 A. The Perfection of the Image of a Generic Map . . . . . . . . . 162 B. The Perfection of a Generic Module . . . . . . . . . . . . 165 C. HomologicalProperties of Generic Modules . . . . . . . . . . 171 D. Comments and References . . . . . . . . . . . . . . . 173 Contents vii 14. The Module of K(cid:127)ahler Di(cid:11)erentials . . . . . . . . . . . 174 A. Perfection and Syzygies of Some Determinantal Ideals . . . . . . 175 B. The Lower Bound for the Depth of the Di(cid:11)erential Module . . . . . 177 C. The Syzygetic Behaviour of the Di(cid:11)erential Module . . . . . . . 181 D. Comments and References . . . . . . . . . . . . . . . 183 15. Derivations and Rigidity . . . . . . . . . . . . . . . 184 A. The LowerBound for the Depth of the Module of Derivations . . . . 184 B. The Perfection of the Module of Derivations . . . . . . . . . 189 C. Syzygetic Behaviourand Rigidity . . . . . . . . . . . . . 197 D. Comments and References . . . . . . . . . . . . . . . 201 16. Appendix . . . . . . . . . . . . . . . . . . . . . 202 A. Determinants and Modules. Rank . . . . . . . . . . . . 202 B. Grade and Acyclicity . . . . . . . . . . . . . . . . 206 C. Perfection and the Cohen-MacaulayProperty . . . . . . . . . 209 D. Dehomogenization . . . . . . . . . . . . . . . . . 211 E. How to Compare \Torsionfree" . . . . . . . . . . . . . 213 F. The Theorem of Hilbert-Burch . . . . . . . . . . . . . . 217 G. Comments and References . . . . . . . . . . . . . . . 218 Bibliography . . . . . . . . . . . . . . . . . . . . . 219 Index of Notations . . . . . . . . . . . . . . . . . . . 229 Subject Index . . . . . . . . . . . . . . . . . . . . 232 1. Preliminaries This section serves two purposes. Its Subsections A and B list the ubiquitous basic notations. In C and D we introduce the principal objects of our investigationand relate them to their geometric counterparts. A. Notations and Conventions Generally we will use the terminology of [Mt] which seems to be rather standard now. In some inessential details our notations di(cid:11)er from those of [Mt]; for example we try to saveparentheses whenever they seem dispensable. A main di(cid:11)erence is the use of the attributes \local" and \normal": for us they always include the property of being noetherian. In the following we explain some notations and list the few conventions the reader is asked to keep in mind throughout. All rings and algebras are commutative and have an element 1. Nevertheless we will sometimes list \commutative"amongthe hypotheses of a proposition ortheorem in order to signalize that the ring under consideration is only supposed to be an arbitrary commutative ring. A reduced ring has no nilpotent elements. The spectrum of a ring A, SpecA for short, is the set of its prime ideals endowed with the Zariski topology. The radical of an ideal I is denoted RadI. The dimension of A is denoted dimA, and the height of I is abbreviated htI. All the modules M considered will be unitary, i.e. 1x=x for all x2M. AnnM is the annihilator of M, and the support of M is given by SuppM =fP 2SpecA: M 6=0g: P We use the notion of associated prime ideals only for (cid:12)nitely generated modules over noetherian rings: AssM =fP 2SpecA: depthM =0g: P Thedepth ofamoduleM overalocalringisthe lengthof amaximalM-sequenceinthe maximal ideal. The projective dimension of a module is denoted pdM. We remind the reader of the equation of Auslander and Buchsbaum for (cid:12)nitely generated modules over local rings A: pdM +depthM =depthA if pdM <1 (cf.[Mt],p. 114,Exercise4). Ifamodulecanbeconsideredamoduleoverdi(cid:11)erentrings (in a natural way), an index will indicate the ring with respect to which an invariant is formed: For example, Ann M is the annihilator of M as an A-module. Instead of A Matsumura’s depth (M) we use grade(I;M) and call it, needless to say, the grade of I I with respect to M;cf.16.Bforadiscussionof grade. Therank rkF ofafreemoduleF is the numberof elementsof oneof itsbases. We discussamoregeneralconcept ofrankin 16.A: M has rank r if M (cid:10)Q is a free Q-module of rank r, Q denoting the total ring of 2 1. Preliminaries fractionsof A. Therankofalinearmapisthe rankofitsimage. Thelength of amodule M is indicated by (cid:21)(M). ThenotationsofhomologicalalgebraconcerningHom,(cid:10),andtheirderivedfunctors seem to be completely standardized; for them we refer to [Rt]. Let A be a ring, M and N A-modules, and f: M !N a homomorphism. We put M(cid:3) =Hom (M;A) A and f(cid:3) =Hom (f;A): N(cid:3) !M(cid:3): A M(cid:3) and f(cid:3) are called the duals of M and f. For the symmetric and exterior powers of M (cf. [Bo.1] for multilinear algebra) we use the symbols i M and S (M) j ^ i i resp. Sometimes we shall have to refer to bases of F(cid:3), F and F(cid:3), given a basis e ;:::;e ofthefreemoduleF. Thebasis ofF(cid:3)dualtoe ;:::;e isdenotedbye(cid:3);:::;e(cid:3). 1 n 1 V n V 1 n For I =(i ;:::;i ) the notation e is used as an abbreviation of e ^(cid:1)(cid:1)(cid:1)^e , whereas 1 k I i1 ik e(cid:3) expands into e(cid:3) ^(cid:1)(cid:1)(cid:1)^e(cid:3) . (The notation e will be naturally extended to arbitrary I i1 ik I families of elements of a module.) Weneedsomecombinatorialnotations. AsubsetI (cid:26)Zalsorepresentsthesequence of its elements in ascending order. For subsets I ;:::;I (cid:26)Z we let 1 n (cid:27)(I ;:::;I ) 1 n denotethesignumofthepermutationI :::I (givenbyiuxtaposition)ofI [:::[I rela- 1 n 1 n tivetoitsnaturalorder,providedtheI arepairwisedisjoint;otherwise(cid:27)(I ;:::;I )=0. i 1 n A useful formula: (cid:27)(I ;:::;I )=(cid:27)(I ;:::;I )(cid:27)(I [:::[I ;I ): 1 n 1 n(cid:0)1 1 n(cid:0)1 n For elements i ;:::;i 2Z we de(cid:12)ne 1 n (cid:27)(i ;:::;i )=(cid:27)(fi g;:::;fi g): 1 n 1 n The cardinality of a set I is denoted jIj. For a set I we let S(m;I)=fJ: J (cid:26)I;jJj=mg: Last, not least, by 1;:::;i;:::;n we indicate that i is to be omitted from the sequence 1;:::;n. b