Table Of ContentCAMBRIDGETRACTSINMATHEMATICS
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
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Contents
Introduction 1
Part I. Poincare´ duality quotients 9
I.1 Poincar´eduality,Gorensteinalgebras,andirreducibleideals . 9
I.2 Propertiesofirreducibleideals ........................ 14
I.3 Theancestorideals ................................. 17
I.4 Fundamentalclasses ................................ 18
I.5 Poincar´edualityquotientsofF 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 isaPoincar´edualityalgebrawewritef 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 Poincar´e duality comes from the study of closed manifolds
in algebraic topology, and goes back at least to H. Poincar´e; see e.g. [77]
Section69. Apartfromthegradedcommutationrulesthecohomologyofa
closedorientedmanifoldwithfieldcoefficientsistheprototypicalexample
ofaPoincare´dualityalgebra. However,Poincare´dualityalgebrasalso ap-
pear 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 Poincar´e 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 isaPoincar´edualityquotientofF 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 isaPoincar´eduality
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 isaPoincar´edualityalgebra,
1 n An
buttheringofinvariantsisnotapolynomialalgebra(see[25],Section11,
[83] Section 5, and [93] Section2). It is also not clear that in the modu-
larcase aringofcoinvariantswhich isaPoincar´edualityalgebramust 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 aswellasonwhichPoincar´edualityquotientsofF 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)ofaringofcoinvariantswhichisaPoincar´e
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∗-
Poincar´edualityquotientsofF 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∗-Poincar´edualityalgebrasappearatvariousplaces
inPartsIIIandVI,andindetailinPart V,whereweworkoutanumberof
specialcases.
Thereare manyothermotivationsforstudying Poincar´eduality 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
byconstructingauniversalexamplethatisnolongeraPoincar´edualityal-
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.
