GeneralizedHibiringsandHibiideals - PDF document

1 GeneralizedHibiringsandHibiideals J¨urge
GeneralizedHibiringsandHibiideals J¨urgenHerzogUniversit¨atDuisburg-EssenEllwangen,March2011 OutlineHibirings HibiidealsGeneralizedHibiringsandHibiideals Outline Hibirings Hibiideals GeneralizedHibiringsandHibiideals Outline HibiringsHibiideals GeneralizedHibiringsa

2 ndHibiideals Hibirings In1985Hibiintrodu
ndHibiideals Hibirings In1985Hibiintroducedaclassofalgebraswhichnowadaysarecalled Hibirings .Theyaresemigroupringsattachedtoniteposets,andmaybeviewedasnaturalgeneralizationsofpolynomialrings. Hibirings In1985Hibiintroducedaclassofalgebraswhichnowadaysarecalled Hibi

3 rings .Theyaresemigroupringsattachedton
rings .Theyaresemigroupringsattachedtoniteposets,andmaybeviewedasnaturalgeneralizationsofpolynomialrings.Indeed,apolynomialringinnvariablesoveraeldKisjusttheHibiringoftheposet [n1]=f1;2;:::;n1g . Hibirings In1985Hibiintroducedaclassofalgebraswhichnowadaysarecall

4 ed Hibirings .Theyaresemigroupringsattac
ed Hibirings .Theyaresemigroupringsattachedtoniteposets,andmaybeviewedasnaturalgeneralizationsofpolynomialrings.Indeed,apolynomialringinnvariablesoveraeldKisjusttheHibiringoftheposet [n1]=f1;2;:::;n1g .Let P=fp1;:::;png beaniteposet.A posetideal IofPisasubsetof

5 Pwhichsatisesthefollowingcondition:fore
Pwhichsatisesthefollowingcondition:forevery p2I; and q2P with qp ,itfollows q2I . Hibirings In1985Hibiintroducedaclassofalgebraswhichnowadaysarecalled Hibirings .Theyaresemigroupringsattachedtoniteposets,andmaybeviewedasnaturalgeneralizationsofpolynomialrings.Ind

6 eed,apolynomialringinnvariablesoveraeld
eed,apolynomialringinnvariablesoveraeldKisjusttheHibiringoftheposet [n1]=f1;2;:::;n1g .Let P=fp1;:::;png beaniteposet.A posetideal IofPisasubsetofPwhichsatisesthefollowingcondition:forevery p2I; and q2P with qp ,itfollows q2I .Let I(P) bethesetoftheposetideals

7 ofP.ThenI(P)isasublatticeofthepowersetof
ofP.ThenI(P)isasublatticeofthepowersetofP;andhenceitisadistributivelattice. ByBirkhoff'stheoremanynitedistributivelatticearisesinthisway. ByBirkhoff'stheoremanynitedistributivelatticearisesinthisway.LetKbeaeld.ThentheHibiringoverKattachedtoPisthetoricring K[I(P)]

8 K[x1;:::;xn;y1;:::;yn] generatedbythese
K[x1;:::;xn;y1;:::;yn] generatedbythesetofmonomials fuI:I2I(P)g where uI=Qpi2IxiQpi62Iyi . ByBirkhoff'stheoremanynitedistributivelatticearisesinthisway.LetKbeaeld.ThentheHibiringoverKattachedtoPisthetoricring K[I(P)]K[x1;:::;xn;y1;:::;yn] generatedbythesetofmono

9 mials fuI:I2I(P)g where uI=Qpi2IxiQpi62I
mials fuI:I2I(P)g where uI=Qpi2IxiQpi62Iyi .Let T=K[ftI:tI2I(P)g] bethepolynomialringinthevariablestIoverK,and 'T!K[I(P)] theK-algebrahomomorphismwithtI7!uI. ByBirkhoff'stheoremanynitedistributivelatticearisesinthisway.LetKbeaeld.ThentheHibiringoverKattachedtoPist

10 hetoricring K[I(P)]K[x1;:::;xn;y1;:::;y
hetoricring K[I(P)]K[x1;:::;xn;y1;:::;yn] generatedbythesetofmonomials fuI:I2I(P)g where uI=Qpi2IxiQpi62Iyi .Let T=K[ftI:tI2I(P)g] bethepolynomialringinthevariablestIoverK,and 'T!K[I(P)] theK-algebrahomomorphismwithtI7!uI.OnefundamentalresultconcerningHibiringsisth

11 atthetoricidealLP=Ker'hasareducedGr¨obne
atthetoricidealLP=Ker'hasareducedGr¨obnerbasisconsistingoftheso-called Hibirelations : tItJtI\JtI[JwithI6JandJ6I: HibishowedthatanyHibiringisanormalCohen–Macaulaydomainofdimension 1+jPj ,andthatitisGorensteinifandonlyiftheattachedposetPisgraded,thatis,allmaximalc

12 hainsofPhavethesamecardinality. Hibishow
hainsofPhavethesamecardinality. HibishowedthatanyHibiringisanormalCohen–Macaulaydomainofdimension 1+jPj ,andthatitisGorensteinifandonlyiftheattachedposetPisgraded,thatis,allmaximalchainsofPhavethesamecardinality.Moregenerally,foranynitelattice L ,notnecessarilydist

13 ributive,onemayconsidertheKalgebra K[L]
ributive,onemayconsidertheKalgebra K[L] withgeneratorsy,2L,andrelations yy=y^y_ where^and_denotemeetandjoininL.HibishowedthatK[L]isadomainifandonlyifLisdistributive,inotherwords,ifLisanideallatticeofaposet. LetKbeaeldand X=(xij)i=1;:::;mj=1;:::;n amatrixofi

14 ndeterminates.WedenotebyK[X]thepolynomia
ndeterminates.WedenotebyK[X]thepolynomialringoverKwiththeindeterminatesxij,andbyAtheK-subalgebraofK[X]generatedbyallmaximalminorsofX. LetKbeaeldand X=(xij)i=1;:::;mj=1;:::;n amatrixofindeterminates.WedenotebyK[X]thepolynomialringoverKwiththeindeterminatesxij,andbyA

15 theK-subalgebraofK[X]generatedbyallmaxim
theK-subalgebraofK[X]generatedbyallmaximalminorsofX.TheK-algebra AK[X] isthecoordinateringofthe Grassmannian ofthem-dimensionalvectorK-subspacesofKn. LetKbeaeldand X=(xij)i=1;:::;mj=1;:::;n amatrixofindeterminates.WedenotebyK[X]thepolynomialringoverKwiththeindeter

16 minatesxij,andbyAtheK-subalgebraofK[X]ge
minatesxij,andbyAtheK-subalgebraofK[X]generatedbyallmaximalminorsofX.TheK-algebra AK[X] isthecoordinateringofthe Grassmannian ofthem-dimensionalvectorK-subspacesofKn.LetbethelexicographicorderonK[X]inducedby x11�x12�


�x1n�x21�x2

17 2�


�xm1�xm2�
2�


�xm1�xm2�


�xmn: Wedenoteby =[a1;a2;:::;am] themaximalminorofXwithcolumnsa1a2


am.Then in()=x1;a1x2;a2


xm;am isthe`diagonal'of. Let S=K[x1;:::;xn] beapolynomialring,amonomialorderonSand AS aK-subalgebra. Let S=K[x1;:::

18 ;xn] beapolynomialring,amonomialorderonS
;xn] beapolynomialring,amonomialorderonSand AS aK-subalgebra.ThentheK-algebra in(A) generatedbyallmonomials in(f) with f2A iscalledthe initialalgebra ofAwithrespectto. Let S=K[x1;:::;xn] beapolynomialring,amonomialorderonSand AS aK-subalgebra.ThentheK-algebra in(A

19 ) generatedbyallmonomials in(f) with f2A
) generatedbyallmonomials in(f) with f2A iscalledthe initialalgebra ofAwithrespectto.Ingeneralin(A)isnotnitelygenerated.Asubset SA iscalleda Sagbibases ofAwithrespectto,iftheelementsf2SgenerateAoverK.ThisconcepthasbeenintroducedbyRobbianoandSweedlerandindependentl

20 ybyKapurandMadlener. Let S=K[x1;:::;xn]
ybyKapurandMadlener. Let S=K[x1;:::;xn] beapolynomialring,amonomialorderonSand AS aK-subalgebra.ThentheK-algebra in(A) generatedbyallmonomials in(f) with f2A iscalledthe initialalgebra ofAwithrespectto.Ingeneralin(A)isnotnitelygenerated.Asubset SA iscalleda Sagbi

21 bases ofAwithrespectto,iftheelementsf2Sg
bases ofAwithrespectto,iftheelementsf2SgenerateAoverK.ThisconcepthasbeenintroducedbyRobbianoandSweedlerandindependentlybyKapurandMadlener.TheoremThemaximalminorsofXformaSagbibasesoftheGrassmannianalgebraA. Let S=K[x1;:::;xn] beapolynomialring,amonomialorderonSand A

22 S aK-subalgebra.ThentheK-algebra in(A) g
S aK-subalgebra.ThentheK-algebra in(A) generatedbyallmonomials in(f) with f2A iscalledthe initialalgebra ofAwithrespectto.Ingeneralin(A)isnotnitelygenerated.Asubset SA iscalleda Sagbibases ofAwithrespectto,iftheelementsf2SgenerateAoverK.Thisconcepthasbeenintroduce

23 dbyRobbianoandSweedlerandindependentlyby
dbyRobbianoandSweedlerandindependentlybyKapurandMadlener.TheoremThemaximalminorsofXformaSagbibasesoftheGrassmannianalgebraA.Whatistheuseofthistheorem? WedeneapartialonthesetLofmaximalminorsofX: [a1;a2;:::;am][b1;b2;:::;bm],aibiforalli ThesetLwiththispartialorderi

24 sadistributivelattice. Wedeneapartialon
sadistributivelattice. WedeneapartialonthesetLofmaximalminorsofX: [a1;a2;:::;am][b1;b2;:::;bm],aibiforalli ThesetLwiththispartialorderisadistributivelattice.Theorem in(A) isisomorphictotheHibiring K[L] ofthelattice L . WedeneapartialonthesetLofmaximalminorsofX:

25 [a1;a2;:::;am][b1;b2;:::;bm],aibiforal
[a1;a2;:::;am][b1;b2;:::;bm],aibiforalli ThesetLwiththispartialorderisadistributivelattice.Theorem in(A) isisomorphictotheHibiring K[L] ofthelattice L .Indeed,letTbethepolynomialringoverKinthevariablestwith2L,andlet :T!in(A)betheK-algebrahomomorphismwith (t)=in

26 ().OneshowsthattheHibirelations t1t2
().OneshowsthattheHibirelations t1t2t1_2t1^2;1;12L generateKer . WedeneapartialonthesetLofmaximalminorsofX: [a1;a2;:::;am][b1;b2;:::;bm],aibiforalli ThesetLwiththispartialorderisadistributivelattice.Theorem in(A) isisomorphictotheHibiring K[L] ofthelatt

27 ice L .Indeed,letTbethepolynomialringove
ice L .Indeed,letTbethepolynomialringoverKinthevariablestwith2L,andlet :T!in(A)betheK-algebrahomomorphismwith (t)=in().OneshowsthattheHibirelations t1t2t1_2t1^2;1;12L generateKer .CorollaryThecoordinateringAoftheGrassmannianofm-dimensionalK-subspacesofK

28 nisaGorensteinringofdimension m(nm)+1 .
nisaGorensteinringofdimension m(nm)+1 . Hibiideals LetPbeaniteposet.Theideal HPK[x1;:::;xn;y1;:::;yn] whichhisgeneratedbythemonomials uI=Yp2IxpYp62Iyq;II(P) iscalledthe Hibiideal ofP. Hibiideals LetPbeaniteposet.Theideal HPK[x1;:::;xn;y1;:::;yn] whichhisgenera

29 tedbythemonomials uI=Yp2IxpYp62Iyq;II(P
tedbythemonomials uI=Yp2IxpYp62Iyq;II(P) iscalledthe Hibiideal ofP.Theorem(a) HP hasalinearresolution.(b) HP=Tpq(xp;yq) . Hibiideals LetPbeaniteposet.Theideal HPK[x1;:::;xn;y1;:::;yn] whichhisgeneratedbythemonomials uI=Yp2IxpYp62Iyq;II(P) iscalledthe Hibiideal

30 ofP.Theorem(a) HP hasalinearresolution.(
ofP.Theorem(a) HP hasalinearresolution.(b) HP=Tpq(xp;yq) .Application:Let G beanitesimplegraphonthevertexset[n].Onedenesthe edgeideal IG ofGasthemonomialidealinK[x1;:::;xn]withsetofgenerators fxixj:fi;jg2E(G)g . Hibiideals LetPbeaniteposet.Theideal HPK[x1;:::;x

31 n;y1;:::;yn] whichhisgeneratedbythemonom
n;y1;:::;yn] whichhisgeneratedbythemonomials uI=Yp2IxpYp62Iyq;II(P) iscalledthe Hibiideal ofP.Theorem(a) HP hasalinearresolution.(b) HP=Tpq(xp;yq) .Application:Let G beanitesimplegraphonthevertexset[n].Onedenesthe edgeideal IG ofGasthemonomialidealinK[x1;:::;xn]

32 withsetofgenerators fxixj:fi;jg2E(G)g .F
withsetofgenerators fxixj:fi;jg2E(G)g .ForwhichgraphsisIGCohen–Macaulay? Theorem(H-Hibi)LetGbeabipartitegraphwithvertexpartitionV[V0.Thenthefollowingconditionsareequivalent: (a) GisaCohen–Macaulaygraph; (b) jVj=jV0jandtheverticesV=fx1;:::;xngandV0=fy1;:::;yngcanbela

33 belledsuchthat: (i) fxi;yigareedgesfori=
belledsuchthat: (i) fxi;yigareedgesfori=1;:::;n; (ii) iffxi;yjgisanedge,thenij; (iii) iffxi;yjgandfxj;ykgareedges,thenfxi;ykgisanedge. P1234 y1y2y3y4x1x2x3x4G(P) TheAlexanderdual:letIbeasquarefreemonomialideal.Then I=r\j=1PFj; whereforasubsetF[n]weset

34 PF=(fxi:i2Fg) . TheAlexanderdual:letIbea
PF=(fxi:i2Fg) . TheAlexanderdual:letIbeasquarefreemonomialideal.Then I=r\j=1PFj; whereforasubsetF[n]weset PF=(fxi:i2Fg) .Theideal I_=(xF1;:::;xFr) iscalledthe Alexanderdual ofI.HereforasubsetF[n]weset xF=Qi2Fxi . TheAlexanderdual:letIbeasquarefreemonomialideal.The

35 n I=r\j=1PFj; whereforasubsetF[n]weset
n I=r\j=1PFj; whereforasubsetF[n]weset PF=(fxi:i2Fg) .Theideal I_=(xF1;:::;xFr) iscalledthe Alexanderdual ofI.HereforasubsetF[n]weset xF=Qi2Fxi .Example: I=(x1x4;x1x5;x2x5;x3x5)=(x1;x2;x3)\(x1;x5)\(x4;x5) . TheAlexanderdual:letIbeasquarefreemonomialideal.Then I=r\

36 j=1PFj; whereforasubsetF[n]weset PF=(fx
j=1PFj; whereforasubsetF[n]weset PF=(fxi:i2Fg) .Theideal I_=(xF1;:::;xFr) iscalledthe Alexanderdual ofI.HereforasubsetF[n]weset xF=Qi2Fxi .Example: I=(x1x4;x1x5;x2x5;x3x5)=(x1;x2;x3)\(x1;x5)\(x4;x5) .Therefore I_=(x1x2x3;x1x5;x4x5) . Theorem(Eagon-Reiner)LetISbea

37 squarefreemonomialideal.ThenI_isCohen-Ma
squarefreemonomialideal.ThenI_isCohen-Macaulay,ifandonlyifIhasalinearresolution. Theorem(Eagon-Reiner)LetISbeasquarefreemonomialideal.ThenI_isCohen-Macaulay,ifandonlyifIhasalinearresolution.Since HP=Tpq(xp;yq) andhasalinearresolution,theAlexanderdual H_P isCohen–M

38 acaulaybytheEagon–ReinerTheorem. Theorem
acaulaybytheEagon–ReinerTheorem. Theorem(Eagon-Reiner)LetISbeasquarefreemonomialideal.ThenI_isCohen-Macaulay,ifandonlyifIhasalinearresolution.Since HP=Tpq(xp;yq) andhasalinearresolution,theAlexanderdual H_P isCohen–MacaulaybytheEagon–ReinerTheorem.But H_P=(fxpyq:p

39 qg) istheedgeidealofabipartitegraphsati
qg) istheedgeidealofabipartitegraphsatisfyingtheconditions(i),(ii)and(ii).ThisprovesonedirectionoftheclassicationtheoremofCohen-Macaulaybipartitegraphs. GeneralizedHibiidealsandHibirings LetPbeaniteposetand I(P) thesetofposetidealsofP.Anr-multichainofI(P)isachain

40 ofposetidealsoflengthr, I:I1I2


Ir=P
ofposetidealsoflengthr, I:I1I2


Ir=P: GeneralizedHibiidealsandHibirings LetPbeaniteposetand I(P) thesetofposetidealsofP.Anr-multichainofI(P)isachainofposetidealsoflengthr, I:I1I2


Ir=P: Wedeneapartialorderontheset Ir(P) ofallr-multichainsofI(P)bysetting I

41 I0 if IkI0k fork=1;:::;r. GeneralizedHi
I0 if IkI0k fork=1;:::;r. GeneralizedHibiidealsandHibirings LetPbeaniteposetand I(P) thesetofposetidealsofP.Anr-multichainofI(P)isachainofposetidealsoflengthr, I:I1I2


Ir=P: Wedeneapartialorderontheset Ir(P) ofallr-multichainsofI(P)bysetting II0 if IkI0k fo

42 rk=1;:::;r.ThepartiallyorderedsetIr(P)is
rk=1;:::;r.ThepartiallyorderedsetIr(P)isadistributivelattice,ifwedenethemeetof I:I1


Ir and I0:I01


I0r as I\I0 where (I\I0)k=Ik\I0k fork=1;:::;r,andthejoinas I[I0 where (I[I0)k=Ik[I0k fork=1;:::;r. Witheachr-multichainIofIr(P)weassociateamonomial uI inthepol

43 ynomialring S=K[fxij:1ir;1jng] inrni
ynomialring S=K[fxij:1ir;1jng] inrnindeterminateswhichisdenedas uI=x1J1x2J2


xrJr; where xkJk=Qp`2Jkxk` and Jk=IknIk1 fork=1;:::;r. Witheachr-multichainIofIr(P)weassociateamonomial uI inthepolynomialring S=K[fxij:1ir;1jng] inrnindeterminateswhichisdeneda

44 s uI=x1J1x2J2


xrJr; where xkJk=Qp`2Jkx
s uI=x1J1x2J2


xrJr; where xkJk=Qp`2Jkxk` and Jk=IknIk1 fork=1;:::;r.Wedenoteby Hr;P themonomialidealinSgeneratedbythesemonomialsandby Rr(P) theK-subalgebrageneratedbythemonomialgeneratorsofHr;P. Witheachr-multichainIofIr(P)weassociateamonomial uI inthepolynomialr

45 ing S=K[fxij:1ir;1jng] inrnindetermi
ing S=K[fxij:1ir;1jng] inrnindeterminateswhichisdenedas uI=x1J1x2J2


xrJr; where xkJk=Qp`2Jkxk` and Jk=IknIk1 fork=1;:::;r.Wedenoteby Hr;P themonomialidealinSgeneratedbythesemonomialsandby Rr(P) theK-subalgebrageneratedbythemonomialgeneratorsofHr;P.Forr=2thei

46 dealHr;PisjusttheclassicalHibiideal,andR
dealHr;PisjusttheclassicalHibiideal,andRr(P)theHibiringoftheideallatticeI(P)ofP. Let T bethepolynomialringoverKinthesetofindeterminates ftI:I2Ir(P)g . Let T bethepolynomialringoverKinthesetofindeterminates ftI:I2Ir(P)g .Furthermorelet ':T!Rr(P) bethesurjectiveK-alge

47 brahomomorphismwith '(tI)=uI forall I2Ir
brahomomorphismwith '(tI)=uI forall I2Ir(P) . Let T bethepolynomialringoverKinthesetofindeterminates ftI:I2Ir(P)g .Furthermorelet ':T!Rr(P) bethesurjectiveK-algebrahomomorphismwith '(tI)=uI forall I2Ir(P) .TheoremTheset =ftItI0tI[I0tI\I02T:I;I02Ir(P)incomparableg

48 isareducedGr¨obnerbasisoftheideal Lr=Ker
isareducedGr¨obnerbasisoftheideal Lr=Ker' withrespecttothereverselexicographicorder. Let T bethepolynomialringoverKinthesetofindeterminates ftI:I2Ir(P)g .Furthermorelet ':T!Rr(P) bethesurjectiveK-algebrahomomorphismwith '(tI)=uI forall I2Ir(P) .TheoremTheset =ftItI

49 0tI[I0tI\I02T:I;I02Ir(P)incomparableg i
0tI[I0tI\I02T:I;I02Ir(P)incomparableg isareducedGr¨obnerbasisoftheideal Lr=Ker' withrespecttothereverselexicographicorder.CorollaryRr(P)isanormalCohen–Macaulaydomainofdimension n(r1)+1 CorollaryLetPbeaniteposet.Thefollowingconditionsareequivalent: I Rr(P)isGorens

50 tein. I R2(P)isGorenstein. I Pisgraded.
tein. I R2(P)isGorenstein. I Pisgraded. CorollaryLetPbeaniteposet.Thefollowingconditionsareequivalent: I Rr(P)isGorenstein. I R2(P)isGorenstein. I Pisgraded.Proof:Oneshowsthat Rr(P)=R2(P[r1]) . CorollaryLetPbeaniteposet.Thefollowingconditionsareequivalent: I Rr

51 (P)isGorenstein. I R2(P)isGorenstein. I
(P)isGorenstein. I R2(P)isGorenstein. I Pisgraded.Proof:Oneshowsthat Rr(P)=R2(P[r1]) .FinallyweconsiderthegeneralizedHibiidealHr;PanditsAlexanderdual. Let CP amultichainoflengthr,i.e., C=fp1;p2;:::;prg with p1p2


pr .Let C bethesetofallmultichainsoflengthrof

52 P. Let CP amultichainoflengthr,i.e., C=
P. Let CP amultichainoflengthr,i.e., C=fp1;p2;:::;prg with p1p2


pr .Let C bethesetofallmultichainsoflengthrofP.Wedenethemonomial uC=Qri=1xi;pi andlet Ir;P=(fuC:C2Cg) . Let CP amultichainoflengthr,i.e., C=fp1;p2;:::;prg with p1p2


pr .Let C bethesetofallm

53 ultichainsoflengthrofP.Wedenethemonomia
ultichainsoflengthrofP.Wedenethemonomial uC=Qri=1xi;pi andlet Ir;P=(fuC:C2Cg) .TheidealsIr;Pmaybeinterpretedasfacetidealsofacompletelybalancedsimplicialcomplexes,asintroducedbyStanley. Let CP amultichainoflengthr,i.e., C=fp1;p2;:::;prg with p1p2


pr .Let C bet

54 hesetofallmultichainsoflengthrofP.Weden
hesetofallmultichainsoflengthrofP.Wedenethemonomial uC=Qri=1xi;pi andlet Ir;P=(fuC:C2Cg) .TheidealsIr;Pmaybeinterpretedasfacetidealsofacompletelybalancedsimplicialcomplexes,asintroducedbyStanley.Theorem(a) Hr;P hasalinearresolution.(b) H_r;P=Ir;P . Let CP amultich

55 ainoflengthr,i.e., C=fp1;p2;:::;prg with
ainoflengthr,i.e., C=fp1;p2;:::;prg with p1p2


pr .Let C bethesetofallmultichainsoflengthrofP.Wedenethemonomial uC=Qri=1xi;pi andlet Ir;P=(fuC:C2Cg) .TheidealsIr;Pmaybeinterpretedasfacetidealsofacompletelybalancedsimplicialcomplexes,asintroducedbyStanley.Theore

56 m(a) Hr;P hasalinearresolution.(b) H_r;P
m(a) Hr;P hasalinearresolution.(b) H_r;P=Ir;P .CorollaryThefacetidealofacompletelybalancedsimplicialcomplexarisingfromaposetisCohen–Macaulay. J.A.EagonandV.Reiner,ResolutionsofStanley-ReisnerringsandAlexanderduality.J.ofPureandAppl.Algebra,130,265–275(1998). V.Ene,J

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58 0). T.Hibi,Distributivelattices,afnesem
0). T.Hibi,Distributivelattices,afnesemigroupringsandalgebraswithstraighteninglaws,in“CommutativeAlgebraandCombinatorics”(M.NagataandH.Matsumura,eds.)Adv.Stud.PureMath.11,North-Holland,Amsterdam,93–109(1987). R.Stanley,BalancedCohen-Macaulaycomplexes.Trans.Amer.Mat