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CAMBRIDGETRACTSINMATHEMATICS GeneralEditors B.BOLLOBAS,W.FULTON,A.KATOK,F.KIRWAN,P.SARNAK,B.SIMON Poincare´ Duality Algebras, Macaulay’s Dual Systems, and Steenrod Operations Poincare´ Duality Algebras, Macaulay’s Dual Systems, and Steenrod Operations DAGMAR M. MEYER AND LARRY SMITH cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press TheEdinburghBuilding,Cambridgecb22ru,UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521850643 © Cambridge University Press 2005 Thispublicationisincopyright.Subjecttostatutoryexceptionandtotheprovisionof relevantcollectivelicensingagreements,noreproductionofanypartmaytakeplace withoutthewrittenpermissionofCambridgeUniversityPress. Firstpublishedinprintformat 2005 isbn-13 978-0-511-13042-7 eBook (NetLibrary) isbn-10 0-511-13042-2 eBook (NetLibrary) isbn-13 978-0-521-85064-3 hardback isbn-10 0-521-85064-9 hardback CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyofurls forexternalorthird-partyinternetwebsitesreferredtointhispublication,anddoesnot guaranteethatanycontentonsuchwebsitesis,orwillremain,accurateorappropriate. Contents Introduction 1 Part I. Poincare´ duality quotients 9 I.1 Poincaréduality,Gorensteinalgebras,andirreducibleideals . 9 I.2 Propertiesofirreducibleideals ........................ 14 I.3 Theancestorideals ................................. 17 I.4 Fundamentalclasses ................................ 18 I.5 PoincarédualityquotientsofF V ...................... 21 I.6 CountingPoincare´dualityquotientsuptoisomorphism ..... 23 Part II. Macaulay’s dual systems and Frobenius powers 31 II.1 Dividedpoweralgebrasandoperations .................. 32 II.2 Macaulay’sdualprincipalsystems ...................... 35 II.3 Anillustrativeexample .............................. 38 II.4 Relationtotheclassicalformproblem ................... 39 II.5 The K L Paradigm: acomputationaltool .............. 40 II.6 Frobeniuspowers .................................. 44 Part III. Poincare´ duality and the Steenrod algebra 53 III.1 P∗-UnstablePoincare´dualityquotientsofF V .......... 54 q III.2 TheP∗-DoubleAnnihilatorTheorem ................... 58 III.3 Wuclasses ........................................ 60 III.4 P∗-Indecomposables: HitProblems .................... 65 v III.5 AdualinterpretationoftheP∗-DoubleAnnihilatorTheorem . 67 III.6 SteenrodoperationsandFrobeniuspowers ............... 71 III.7 Examples ......................................... 75 Part IV. Dickson, symmetric, and other coinvariants 81 IV.1 Dicksoncoinvariants ................................ 81 IV.2 Wuclassesofalgebrasofcoinvariants ................... 86 IV.3 TheMacaulaydualofDicksoncoinvariantsmod2 ......... 88 IV.4 Symmetriccoinvariants .............................. 89 Part V. The Hit Problem mod 2 93 V.1 PowersofDicksonpolynomialsin2variables ............. 94 V.2 A∗-IndecomposableelementsinF x, y ................ 100 2 V.3 PowersofDicksonpolynomialsin3variables ............. 105 V.4 PowersofDicksonpolynomialsinmanyvariables .......... 109 V.5 Powersofmod2Stiefel–Whitneyclassesin3variables ..... 123 Part VI. Macaulay’s inverse systems and applications 133 VI.1 Macaulay’sinverseprincipalsystems .................... 134 VI.2 CatalecticantMatricesandAncestorIdeals ............... 136 VI.3 Regularideals ..................................... 142 VI.4 Lyingoverforirreducibleideals ....................... 146 VI.5 Changeofringsandinversepolynomials ................ 155 VI.6 InversesystemsandtheSteenrodalgebra ................ 160 VI.7 ChangeofRingsandWuClasses ....................... 165 VI.8 TheHitProblemfortheDicksonandotheralgebras ........ 170 References ........................................... 177 Notation.............................................. 185 Index................................................. 189 vi Introduction T O SET THE STAGE let F be a field (in much of what follows F will be finite,e.g.,F ,thefieldwith2elements)and n Napositiveinteger. 2 Denoteby V =Fn the n-dimensionalvectorspaceoverF,andbyF V the gradedalgebraofhomogeneouspolynomialfunctionsonV. Tobespecific, F V is the symmetric algebra S V∗ on the vector space V∗ dual to V. Sincegradedcommutationrulesplaynoroleherewewillgradethisasan algebraistwould,i.e.,puttingthelinearformsindegree1nomatterwhat the characteristic of the ground field F. The homogeneous component of F V of degree d will be denoted by F V . If we need a notation for a d basis of V∗ we will use z ,..., z ; the correspondingbasis for V will be 1 n denotedbyu ,..., u . Foruptothreevariableswewillmostoftenwrite 1 n x, y, z, respectively u, v, w for the variables and their duals. Recall that agradedvectorspace,algebra,ormoduleissaidtohavefinitetypeifthe homogeneouscomponentsareallfinitedimensionalvectorspaces. DDDDDEEEEEFFFFFIIIIINNNNNIIIIITTTTTIIIIIOOOOONNNNN:LetHbeacommutativegradedconnectedalgebraoffinite typeoverthefieldF.WesaythatHisaPoincare´dualityalgebraofformal dimension dif (i)H =0fori > d, i (ii)dimF Hd =1, (iii)thepairingHi ⊗ Hd−i Hd givenbymultiplicationisnonsingu- lar,i.e.,anelementa H iszeroifandonlyifa·b=0 H for i d allb Hd−i. If H isaPoincarédualityalgebrawewritef dim H fortheformaldimen- sion of H. If the formal dimension is d and H in H is nonzero, then d H is referredto asa fundamental class for H. Fundamental classes are determinedonlyuptomultiplicationbyanonzeroelementofF. 1 2 INTRODUCTION The notion of Poincaré duality comes from the study of closed manifolds in algebraic topology, and goes back at least to H. Poincaré; see e.g. [77] Section69. Apartfromthegradedcommutationrulesthecohomologyofa closedorientedmanifoldwithfieldcoefficientsistheprototypicalexample ofaPoincare´dualityalgebra. However,Poincare´dualityalgebrasalso appear quite naturally in invariant theory as rings of coinvariants of groups whose rings of invariants are polynomial algebras. Indeed, the less than completeunderstanding of the role of Poincaré duality algebras in invari- anttheoryispartofthemotivationforthisstudy. Weexplainthisnext. Incharacteristiczero,ormoregenerallyinthenonmodularcase,i.e.,where the order G of G is invertible in the ground field, it is well known (see e.g. [80] or [87] Section 7.4) that pseudoreflection groups (better said, pseudoreflection representations) are characterized by the fact that their invariantringsarepolynomialalgebras. Thisisknowntofailinthemodu- larcase: theringofinvariantsofareflectiongroupneednotbeapolyno- mialalgebra(seee.g. [100]or[87]Section7.4Example4). If : G GL n,F is a representation of a finite group G over the field F for which the ring of invariants F V G is a polynomial algebra then the ringofcoinvariantsF V isaPoincarédualityquotientofF V ([87]The- G orem 6.5.1). Such a ring of coinvariants is therefore a very special type of Poincare´ duality algebra, viz., a complete intersection. A theorem of R.Steinberg[98](asformulatedbyR.Kane[40],[41]ChapterVII)says: if : G GL n,F is a representation of a finite group over a field F of characteristiczero,thentheringofcoinvariantsF V isaPoincaréduality G algebraif and only if G is apseudoreflectiongroup. Although Steinberg’s proof,aswellasKane’s,makescentraluseofthefactthatthecharacteristic ofFiszeroandnotjustrelativelyprimetotheorderofthegroup,aswould seem more natural, T.-C. Lin [47] has recently removed the need for this extraassumptionandshowntheresultholdsinthenonmodularcase. ItisnotknownwhatthebestextensionofSteinberg’stheoremtofieldsof nonzero characteristic might look like. Several variations of its hypothe- ses and conclusion are possible. There is an ad hoc, characteristic free, proof of the result as originally stated if one restricts to the case of two variables[92]. Examplesinthemodularcaseshowthatinmorevariables theoriginalstatementcanbefalse. Forexample,considerthetautological representationof the alternating group A over a field of characteristic p n lessthanorequalton. ThenF z ,..., z isaPoincarédualityalgebra, 1 n An buttheringofinvariantsisnotapolynomialalgebra(see[25],Section11, [83] Section 5, and [93] Section2). It is also not clear that in the modu- larcase aringofcoinvariantswhich isaPoincarédualityalgebramust be INTRODUCTION 3 a complete intersection, though this is again correct for n =2 by [102]. Other places where Poincare´ duality algebras appear in connection with invarianttheorymaybefoundin[19]and[41]. Animportantfeatureofanalgebraoverafieldofcharacteristicp=0isthe operationofraising anelementtothe p-thpower. Thisislooselyreferred to as the Frobenius map. The Steenrod operations and the Steenrod al- gebrarepresentone wayto organizeinformationhiddenintheFrobenius homomorphism provided the ground field is finite. Let F be the Galois q fieldwith q = p elements, V =Fn,anddefine1 q PPPPP :F V F V q q bytherules (i) PPPPP isF-linear, (ii) PPPPP v = v+vq for v V∗, (iii) PPPPP u·w =PPPPP u ·PPPPP w for u, w F V , q (iv) PPPPP 1 =1. PPPPP isaringhomomorphismofdegree0ifweagree hasdegree 1−q . ByseparatingouthomogeneouscomponentsweobtainF -linearmaps q Pi :F V F V q q bytherequirement PPPPP f = Pi f i. i=0 ThemapsPi maybeassembledintoanalgebracalledtheSteenrodalgebra P∗oftheGaloisfieldF (seee.g. [87]Chapters10and11). Thestrength q of these operations lies in the fact that they commute with the action of GL V onF V andsatisfytheunstabilityconditions q fq i =deg f Pi f = 0 i >deg f forall f F V . Thefirstunstabilityconditionisanontrivialitycondition, q thesecondatrivialitycondition,andtogethertheyimposeaveryrigidre- strictiononF V G aswellasonwhichPoincarédualityquotientsofF V q q canariseasF V forsomerepresentation :G GL n,F . Forexam- q G q ple,usingideasfromalgebraictopologyone candefineinvariantsofsuch quotients called Wu classes. In Appendix B of [60] it is shown that the Wuclasses(seeSection IV.2)ofaringofcoinvariantswhichisaPoincaré duality algebra are always trivial. Not all Poincare´ duality quotients with anunstableactionoftheSteenrodoperationshavethisproperty. 1IfAisaringthenA denotestheringofformalpowerseriesoverAinthevariable . 4 INTRODUCTION Steinberg’spaper[98]containsanumberofotherresultsonringsofcoin- variantsthatarePoincare´dualityalgebras,suchashowtoconstructafun- damentalclass. Again,astheystand,theproofsworkonlyincharacteristic zero. Some of these results have been extendedto nonzero characteristic [91], but by no means all. This unsatisfactory state of affairs suggested that a systematic study of Poincare´ duality quotients of F V admitting q Steenrodoperationssatisfyingtheunstabilityconditionsmightbefruitful. Webeginsuchastudyhere. The first step in our program led us to Macaulay’s theory of irreducible ideals in polynomial algebras. Due to the enormouschanges in terminol- ogy since [49] was written, this theoryis not as accessible as it might be. We present a treatment in current parlance and extend it to encompass the extra structure of a Steenrod algebra action when the ground field is finite. IndoingsoweuncoveraveryunexpectedconnectionbetweenP∗- PoincarédualityquotientsofF V andthesocalledHitProblems,2 e.g.,the q determinationoftheP∗-indecomposableelementsinF V . Thisrelation- q ship is particularly appealing if formulated along the lines of M.C. Crabb and J. R. Hubbuck [16]. Applications to this problem based on what we havelearnedaboutP∗-Poincarédualityalgebrasappearatvariousplaces inPartsIIIandVI,andindetailinPart V,whereweworkoutanumberof specialcases. Thereare manyothermotivationsforstudying Poincaréduality quotients ofF V . Letusjustmentiononemore: ithasitsoriginsinalgebraictopol- ogy and connects up with certain problems in invariant theory which we have not discussed here(see [44]). We paraphrasefrom theintroduction to [1]. “Recently, in studying the coinvariants of reflection groups, I had occasiontoconsidertheformulaeofThomandWu[111]... althoughthese formulaearesimpleandattractive,Ididnotfeelthattheygavemethatcom- pleteunderstandingthatIsought.” In[1]J.F.Adamsprovestheseformulae byconstructingauniversalexamplethatisnolongeraPoincarédualityal- gebra3 butisaninverselimitofringsofcoinvariants(seee.g. [10]or[44]) and verifyingcertain otherformulae in this new object. We would like to understand why the formulae of Thom and Wu are also consequences of 2WeusetheexpressionHitProblem(s)asin[108]Section7:quitegenerally,ifMisagraded moduleoverthepositivelygradedalgebraA overthefieldF,wesaythatu M ishitif thereareelementsu1,..., uk M anda1,..., ak A withdeg a1 ,..., deg ak >0 withu=a1u1+···+akuk.TheelementsofMthatarehitformtheA-submoduleA·M,and thequotientM/A·MthemoduleofA-indecomposableelementsF⊗A M. HitProblem(s) refer to the characterizationof elementsof M that are hit or not, e.g., in the case of the SteenrodalgebraP∗actingonFq z1,..., zn findingconditionsonamonomialthatassure itisP∗-indecomposable. 3Infactitisn’tevenNoetherian.