%!PS-Adobe-3.0 %%Title: dvialw dev.dvi %%CreationDate: Mon Jul 12 15:10:34 1999 %%Creator: dvialw version 3.0.154 [27-Feb-1998] (TeX82 DVI Translator %%+ for PostScript) %%For: Paul Roberts %%BoundingBox: 0 0 612 792 %%DocumentData: Clean7Bit %%DocumentMedia: A 612 792 0 () () %%DocumentFonts: (atend) %%DocumentNeededResources: (atend) %%DocumentSuppliedResources: (atend) %%Orientation: Portrait %%Requirements: numcopies(1) %%PageOrder: Special %%Pages: (atend) %%EndComments %======================================================================= %%BeginProlog %%BeginResource: procset /usr/local/lib/tex/inputs/dvialw.ps 0 0 /DVIALWdict 200 dict def DVIALWdict begin /setpacking where {/currpack currentpacking def pop true setpacking} if /B { /ht exch def /wd exch def gsave currentpoint newpath moveto wd 1 sub 0 rlineto 0 ht 1 sub rlineto wd 1 sub neg 0 rlineto closepath fill grestore } def() pop /BB { save /F0 NF 1 /F0 SF } def() pop /BE { restore } def() pop /BOP { PageWidth PageHeight gt { PageHeight 0 translate 90 rotate } if 1 PXtoBP dup scale } def() pop /BPtoPX { 72 div Resolution mul } def() pop /ch-image {ch-data 0 get} def /ch-xoff {ch-data 1 get} def /ch-yoff {ch-data 2 get} def /ch-width {ch-data 3 get} def /ch-height {ch-data 4 get} def /ch-tfmw {ch-data 5 get} def /CharBuilder { /ch-code exch def /font-dict exch def /ch-data font-dict /BitMaps get ch-code get def ch-data null eq not { ch-tfmw 0 ch-xoff neg ch-yoff ch-height sub ch-width ch-xoff sub ch-yoff setcachedevice 0 0 SNAP translate ch-width ch-height true [ 1 0 0 -1 ch-xoff ch-yoff ] {ch-image} imagemask } if } def() pop /D { /ch-code exch def /ch-data exch def currentfont /BitMaps get ch-code ch-data put currentfont /Encoding get ch-code dup 3 string cvs cvn put } def() pop /EOP { showpage } def() pop /ExtendFont { /x-scale exch def /the-font exch [x-scale 0 0 1 0 0] TransFont } def /FRAME { dup currentpoint newpath moveto 0 exch rlineto exch 0 rlineto neg 0 exch rlineto closepath 2 setlinewidth [3] 0 setdash stroke [] 0 setdash } def() pop /M { moveto } def() pop /NF { /newname exch def newname 7 dict def newname load begin /BitMaps 256 array def /BuildChar { CharBuilder } def /Encoding 256 array def 0 1 255 { Encoding exch /.notdef put } for /FontBBox [ 0 0 1 1 ] def /FontMatrix [ 1 0 0 1 0 0 ] def /FontType 3 def end newname newname load definefont pop } def() pop /ObliqueSlant { dup sin exch cos div neg } def /P { show } def() pop /PXtoBP { Resolution div 72 mul } def() pop /Q { moveto wd ht B } def() pop /R { 0 rmoveto show } def() pop /ReEncodeFont { /TempEncoding exch def findfont dup length dict begin {1 index /FID ne 2 index /UniqueID ne and {def} {pop pop} ifelse} forall /Encoding TempEncoding def currentdict end /TempFontName exch definefont pop /TempFontName } def /RESIZE { /PictureHeight exch def /PictureWidth exch def /Wscale PictureWidth BoxWidth div def /Hscale PictureHeight BoxHeight div def 0 BoxHeight translate 0 PictureHeight neg translate Wscale Hscale lt { 0 PictureHeight Wscale BoxHeight mul sub 2 div translate Wscale dup scale } if Wscale Hscale gt { PictureWidth Hscale BoxWidth mul sub 2 div 0 translate Hscale dup scale } if Wscale Hscale eq { Hscale dup scale } if } def() pop /Resolution 300 def /RF { exch findfont exch 7200 7227 div mul BPtoPX scalefont definefont pop } def() pop /S { moveto show } def() pop /SB { /-save0- save def 1 BPtoPX dup scale /showpage {} def CurrentX CurrentY translate } def() pop /SCALE { dup neg 1 add 0 exch BoxHeight mul translate scale } def() pop /SE { -save0- restore } def() pop /SF { findfont exch scalefont setfont } def() pop /SlantFont { /the-slant exch def /the-font exch [1 0 the-slant 1 0 0] TransFont } def /SNAP { transform 2 {floor 0.5 add exch} repeat itransform } def() pop /T { currentpoint exch pop moveto show } def() pop /TB { save } def() pop /TE { restore } def() pop /TransFont { exch findfont exch makefont dup length dict /nn exch def {1 index /FID ne {nn 3 1 roll put} {pop pop} ifelse } forall dup nn definefont pop } def /setpacking where {pop currpack setpacking} if /X { dup /delta exch def 0 rmoveto show } bind def() pop /a { -6 0 rmoveto show } bind def() pop /b { -5 0 rmoveto show } bind def() pop /c { -4 0 rmoveto show } bind def() pop /d { -3 0 rmoveto show } bind def() pop /e { -2 0 rmoveto show } bind def() pop /f { -1 0 rmoveto show } bind def() pop /g { show } bind def() pop /h { 1 0 rmoveto show } bind def() pop /i { 2 0 rmoveto show } bind def() pop /j { 3 0 rmoveto show } bind def() pop /k { 4 0 rmoveto show } bind def() pop /l { 5 0 rmoveto show } bind def() pop /m { 6 0 rmoveto show } bind def() pop /n { -6 delta add 0 rmoveto show } def() pop /o { -5 delta add 0 rmoveto show } def() pop /p { -4 delta add 0 rmoveto show } def() pop /q { -3 delta add 0 rmoveto show } def() pop /r { -2 delta add 0 rmoveto show } def() pop /s { -1 delta add 0 rmoveto show } def() pop /t { delta 0 rmoveto show } def() pop /u { 1 delta add 0 rmoveto show } def() pop /v { 2 delta add 0 rmoveto show } def() pop /w { 3 delta add 0 rmoveto show } def() pop /x { 4 delta add 0 rmoveto show } def() pop /y { 5 delta add 0 rmoveto show } def() pop /z { 6 delta add 0 rmoveto show } def() pop /DX { xpos add /xpos exch def } def() pop /DY { dup /dy exch def ypos add /ypos exch def } def() pop /A { /ypos exch def /xpos exch def N } def() pop /C { neg DY neg DX N } def() pop /E { neg DX N } def() pop /F { DY neg DX N } def() pop /G { neg DY N } def() pop /H { dy I } def() pop /I { DY N } def() pop /J { neg DY DX N } def() pop /K { DX N } def() pop /L { DY DX N } def() pop /N { xpos ypos moveto } def() pop /O { dy neg I } def() pop /U { wd ht B } def() pop 0 1 255 { =string cvs cvn pop } for /oldsetlinewidth systemdict begin /setlinewidth load end def /setlinewidth { LineWidthScale mul oldsetlinewidth } def /LineWidthScale 1 def /setlinewidthscale { /LineWidthScale exch def } def /max { dup 3 2 roll dup 4 1 roll exch gt {pop} {exch pop} ifelse } def /min { dup 3 2 roll dup 4 1 roll exch lt {pop} {exch pop} ifelse } def /sp { 65536 div 72.27 div 72 mul } def /BBox { /BoxURY exch def /BoxURX exch def /BoxLLY exch def /BoxLLX exch def /BoxWidth BoxURX BoxLLX sub def /BoxHeight BoxURY BoxLLY sub def } def /EPSBBox { /EPSFileURY exch def /EPSFileURX exch def /EPSFileLLY exch def /EPSFileLLX exch def /EPSFileHeight EPSFileURY EPSFileLLY sub def /EPSFileWidth EPSFileURX EPSFileLLX sub def } def /EPSFileBBoxPath { newpath EPSFileLLX EPSFileLLY moveto EPSFileLLX EPSFileURY lineto EPSFileURX EPSFileURY lineto EPSFileURX EPSFileLLY lineto closepath } def /EPSFileClip { EPSFileBBoxPath clip newpath } def /EPSWH { /FigureHeight exch def /FigureWidth exch def } def /FigureHeight { EPSFileHeight } def /FigureWidth { EPSFileWidth } def /GB { userdict /TeXDict known { TeXDict begin /VResolution { Resolution } def /DVImag 1000 def /normalscale { } def DVIALWdict /BoxLLX known { BoxLLX BoxLLY moveto } if } if } def /GE { userdict /TeXDict known { end } if } def /SB { 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(not)o(use)p(reduction)o(to)p(the)o H(diagonal.)g 125 99 J(The)g(v)20 X (anishing)b(conjecture)t(w)s(as)e(pro)s(v)e(en)d(ab)t(out)i(ten)t(y)s (ears)e(ago)s(\050Rob)t(erts)i(\1336\135,)t(Gillet-)u 125 100 C(Soul)g (\023)e(e)-40 X(\1333\135\051)30 X(using)r F9(K)t Fa(-theoretic)l(metho) t(ds.)h(The)42 X(pro)27 X(of)k(in)t(\1336\135)x(uses)t(the)x(theory)t (of)x(lo)t(cal)k H(Chern)g(c)39 X(haracters,)e(while)w(that)t(in)u (\1333\135)t(uses)u(the)t(theory)u(of)t(Adams)u(op)t(erations)j(on)t 99 G(Grothendiec)g(k)e(groups)27 X(of)u(complexes.)t 125 100 J(The)g(main) 38 X(topic)t(of)t(these)t(notes)t(is)t(the)t(recen)t(t)e(pro)t(of)i(of) t(Gabb)t(er)i(of)t(the)t(non-)t 125 99 C(negativit)g(y)e(conjecture,)o (in)q(the)o(course)o(of)o(whic)o(h)e(he)o(also)o(giv)o(es)e(a)o(new)o (pro)o(of)i(of)o(the)o 100 G(v)g(anishing)b(conjecture.)30 X(In)41 X (addition,)30 X(w)s(e)e(discuss)s(some)s(questions)s(whic)s(h)e(arise)s (when)s H(attempting)g(to)v(extend)u(these)v(ideas)u(to)v(pro)u(v)e(e)d (the)v(p)u(ositivit)j(y)d(conjecture.)v(First,)49 X(w)33 X(e)d 99 G (recall)g(a)s(sp)s(ectral)i(sequence)s(argumen)s(t)e(used)s(b)s(y)e (Serre)s(in)s(his)s(pro)s(of)i(and)s(whic)s(h)e(w)s(as)e 100 G(extended) g(b)w(y)d(Gabb)w(er)i(to)w(reduce)v(these)w(questions)v(on)w(mo)v(dules) j(o)v(v)e(er)d(regular)w(lo)v(cal)j H(rings)g(to)o(questions)n(on)o(lo) n(cally)j(free)n(shea)o(v)d(es)e(on)n(pro)o(jectiv)k(e)e(space.)n 1411 428 J(2)g EOP end %%PageTrailer %======================================================================= %%Page: 3 3 %%BeginPageSetup DVIALWdict begin %%EndPageSetup BOP F4 1123 5385 A(2)g(The)137 X(Serre)46 X(sp)t(ectral)k(sequence)t Fa 182 G(The)g(main)34 X(theorem)u(of)t(this)u(section)t(relates)u(the)t(Euler) u(c)t(haracteristic)e(of)t(a)u(Koszul)t 100 G(complex)g(on)v(a)u(mo)v (dule)i(to)v(the)u(Sam)v(uel)d(m)v(ultiplicit)d(y)e(of)u(the)v(mo)u (dule.)j(Let)60 X F9(A)37 X Fa(b)q(e)k(a)q H(lo)g(cal)i(ring,)27 X(and) t(let)t F9(M)t Fa(b)36 X(e)i(a)27 X(\014nitely)t(generated)t F9(A)t Fa (-mo)g(dule)i(of)t(dimension)t(at)t(most)t F9(k)t Fa(.)j H(Let)g F1(a)y Fa(b)y(e)i(an)y(ideal)y(of)y F9(A)y Fa(suc)y(h)e(that)y F9(M)y(=)k F1(a) g F9(M)g Fa(has)41 X(\014nite)32 X(length.)t(W)50 X(e)-7 X(recall)32 X (that)t(the)t H(asso)g(ciated)i(Hilb)x(ert-Sam)i(uel)e(p)w(olynomial)j F9(P)w F1 1488 30 L(a)g Fc 12 53 C(M)g Fa 74 23 L(\050)g F9(n)g Fa(\051) g(is)w(de\014ned)x(to)w(b)x(e)i(the)x(p)w(olynomial)j 1550 100 C(for)g (whic)p(h)d F9 959 100 J(P)g F1 65 34 L(a)g Fc 12 54 C(M)g Fa 74 20 L (\050)g F9(n)g Fa(\051)g(=)23 X(length)t(\050)g F9(M)g(=)k F1(a)g Fc 653 34 L(n)g F9 45 34 J(M)g Fa(\051)9 X 1784 142 C(for)g(large)21 X F9 (n)t Fa(.)g(If)35 X(the)21 X(dimension)t(of)t F9(M)t Fa(is)30 X(at)21 X (most)t F9(k)t Fa(,)j(w)u(e)e(de\014ne)t(the)t(Sam)t(uel)e(m)t (ultiplicit)e(y)d F9 99 G(e)g Fc 39 12 J(k)g Fa 41 12 L(\050)g F1(a)g F9(;)g(M)14 X Fa(\051)o(to)26 X(b)t(e)i F9(k)t Fa(!)i(times)t(the)t(co) t(e\016cien)i(t)e(of)t F9(n)t Fc 1455 30 L(k)g Fa 67 30 J(in)g F9(P)t F1 159 30 L(a)g Fc 11 53 C(M)g Fa 74 23 L(\050)g F9(n)g Fa(\051)g (\050if)t(the)t(dimension)t(of)t F9(M)t Fa(is)35 X 1824 100 C(less)g (than)28 X F9(k)s Fa(,)j F9(e)s Fc 484 12 J(k)g Fa 41 12 L(\050)g F1(a) g F9(;)g(M)14 X Fa(\051)o(will)28 X(b)s(e)j(zero\051.)s F6 525 170 C (Theorem)g(1)x F7(With)41 X(notation)n(as)34 X(ab)u(ove,)b(let)v F9(x)t Fd 1579 12 J(1)g F9 37 12 L(;)g(:)14 X(:)t(:)t(;)t(x)t Fc 232 12 J(k)g F7 75 12 L(b)g(e)c(a)34 X(se)u(quenc)b(e)d(of)s(elements)v 1923 100 C (of)g F9(A)t F7(,)g(and)u(let)s F1(a)t F7(b)s(e)c(the)s(ide)t(al)b (gener)t(ate)b(d)d(by)r F9(x)u Fd 1562 12 J(1)g F9 37 12 L(;)g(:)14 X(:) t(:)t(;)t(x)t Fc 232 12 J(k)g F7 41 12 L(.)g(Assume)50 X(that)34 X F9(M) s(=)l F1(a)f F9(M)h F7(is)42 X(a)34 X 1872 99 C(mo)g(dule)c(of)o (\014nite)p(length.)o(L)y(et)b F9(K)p Fb 1086 12 J(\017)g F7 68 12 L(b) g(e)c(the)o(Koszul)p(c)o(omplex)c(on)o F9(x)p Fd 990 12 J(1)g F9 37 12 L(;)g(:)14 X(:)t(:)t(;)t(x)t Fc 232 12 J(k)g F7 41 12 L(,)g(and)30 X (let)s F9 1826 247 C(\037)g Fa(\050)g F9(K)g Fb 155 12 J(\017)g F8 57 12 L(\012)g F9(M)18 X Fa(\051)10 X(=)22 X Fc 358 104 L(k)g F2 42 25 C(X) g Fc 6 177 J(i)g Fd(=0)g Fa 114 98 L(\050)g F8(\000)g Fa(1\051)g Fc 171 34 L(i)g F7 28 34 J(length)g Fa(\050)g F9(H)g Fc 314 12 J(i)g Fa 28 12 L(\050)g F9(K)g Fb 103 12 J(\017)g F8 57 12 L(\012)g F9(M)p Fa(\051\051) 10 X F9(:)f F7 1977 243 C(Then)g F9 1001 100 J(\037)g Fa(\050)g F9(K)g Fb 155 12 J(\017)g F8 57 12 L(\012)g F9(M)18 X Fa(\051)10 X(=)22 X F9(e) v Fc 355 12 J(k)g Fa 41 12 L(\050)g F1(a)g F9(;)g(M)14 X Fa(\051)o F9(:) g Fa 1484 170 C(W)g(e)-7 X(sk)34 X(etc)d(h)e(the)s(argumen)t(t)d(used)t (to)s(pro)t(v)d(e)e(this)s(theorem.)t(The)54 X(main)34 X(idea)s(is)t(to) s 125 99 C(examine)g(the)22 X(sp)s(ectral)j(sequence)s(de\014ned)t(b)s (y)e(the)s(\014ltration)t(on)s F9(K)t Fb 2147 12 J(\017)g Fa 60 12 L (induced)g(b)t(y)d(p)t(o)i(w)e(ers)d 2207 100 C(of)g F1(a)x Fa(.)g(F)36 X(or)-7 X(eac)26 X(h)e F9(n)t F8(\025)q Fa(0)q(and)t(for)t(eac)t(h)e F9 (i)t Fa(w)t(e)e(consider)t(the)t(quotien)t(t)e F1(a)t Fc 2232 30 L(n)g F9 45 30 J(K)g Fc 71 12 J(i)g F9 27 12 L(=)g F1(a)g Fc 83 30 L(n)g Fd (+1)g F9 127 30 J(K)g Fc 71 12 J(i)g Fa 28 12 L(.)g(F)36 X(or)-7 X 2684 100 C(eac)g(h)e F9(r)27 X F8(\025)s Fa(0)o(w)v(e)c(then)v(tak)s(e)f(the) s(sub)v(quotien)h(t)f F9(E)s Fc 1563 30 L(r)g 5 51 C(i;n)g Fa 116 21 L (of)g(this)u(mo)t(dule)j(de\014ned)t(b)u(y)d F9 1337 235 C(E)g Fc 66 34 L(r)g 5 55 C(i;n)g Fa 111 21 L(=)g F8 294 56 L(f)g F9(k)g Fc 85 12 J(i) g F8 51 12 L(2)g F1(a)p Fc 120 30 L(n)g F9 45 30 J(K)g Fc 71 12 J(i)g F8 28 12 L(j)g F9(d)g Fc 66 12 J(i)g Fa 28 12 L(\050)g F9(k)g Fc 75 12 J(i)g Fa 28 12 L(\051)g F8(2)p F1(a)p Fc 175 30 L(n)g Fd(+)g Fc(r)g F9 128 30 J(K)g Fc 71 12 J(i)g Fb(\000)g Fd(1)g F8 113 12 L(g)g Fa(+)18 X F1(a)u Fc 185 30 L(n)g Fd(+1)g F9 128 30 J(K)g 1593 37 C 1887 4 B Fc 1664 25 L(i)g Fa 1664 101 C(\050)g F8(f)g F9(d)g Fc 117 12 J(i)g Fd(+1) g Fa 112 12 L(\050)g F9(k)g Fc 75 12 J(i)g Fd(+1)g Fa 112 12 L(\051)g F8(j)g F9(k)g Fc 98 12 J(i)g Fd(+1)g F8 135 12 L(2)g F1(a)y Fc 120 24 L (n)g Fb(\000)g Fc(r)g Fd(+1)i F9 215 24 J(K)g Fc 71 12 J(i)g Fd(+1)g F8 112 12 L(g)g(\\)t F1(a)u Fc 175 24 L(n)g F9 45 24 J(K)g Fc 71 12 J(i)g Fa 28 12 L(\051)g(+)t F1(a)u Fc 175 24 L(n)g Fd(+1)g F9 128 24 J(K)g Fc 71 12 J(i)g F9 37 69 L(:)g Fa 2379 219 C(The)g F9(E)u Fc 228 31 L(r)g 5 52 C(i;n)g Fa 108 21 L(de\014ne)g(a)u(sp)v(ectral)i(sequence)v(\050the) u(usual)v(sp)u(ectral)j(sequence)u(asso)v(ciated)i(to)v 456 99 C(a)g (\014ltered)v(complex\051.)w(While)34 X(the)21 X(precise)s(de\014nition) t(is)s(necessarily)t(quite)s(complicated,)t 100 G(the)g(idea)29 X(is)t (that)t F9(E)t Fc 653 30 L(r)g 5 51 C(i;n)g Fa 117 21 L(is)g(the)t(sub) t(quotien)i(t)e(of)t F1(a)t Fc 823 30 L(n)g F9 45 30 J(K)g Fc 71 12 J(i) g F9 28 12 L(=)g F1(a)g Fc 84 30 L(n)g Fd(+1)g F9 128 30 J(K)g Fc 71 12 J(i)g Fa 57 12 L(consisting)g(of)t(elemen)t(ts)e 2072 100 C(whose)g(b)n (oundaries)i(lie)n F9(r)22 X Fa(steps)w(further)t(do)u(wn)d(in)u(the)t (\014ltration)u(mo)t(dulo)j(b)t(oundaries)j(of)t 99 G(elemen)g(ts)e (whic)s(h)e(lie)s(at)s(most)s F9(r)s F8(\000)7 X Fa(1)l(steps)21 X (further)t(up)t(in)t(the)t(\014ltration.)t(As)35 X F9(r)21 X Fa(gets)v (large,)t 100 G(this)g(sub)33 X(quotien)i(t)e(approac)t(hes)e(the)t (submo)t(dule)i(of)t(elemen)t(ts)e(whose)t(b)t(oundaries)i(are)t 99 G (zero)g(mo)r(dulo)i(the)r(submo)q(dule)j(consisting)q(of)r(all)q(of)r (the)q(b)r(oundaries.)i(In)46 X(fact,)31 X(it)t(can)t 100 G(b)g(e)i(sho) o(wn)d(using)o(the)n(Artin-Rees)o(lemma)n(\050see)o(Serre)n (\1337\135\051)o(that)n(the)o(sp)n(ectral)j(sequence)n H(do)g(es)i(in)q (fact)p(con)q(v)d(erge)e(to)p(the)q F1(a)p Fa(-adic)h(\014ltration)p(on) q(the)p(homology)q(of)p F9(K)q Fb 2423 12 J(\017)g Fa 38 12 L(.)g 2336 99 C(P)g(art)e(of)18 X(the)u(general)t(theory)u(of)t(sp)u(ectral)i (sequences,)u(whic)v(h)e(can)t(b)u(e)i(v)u(eri\014ed)d(directly)u 125 100 C(in)g(this)v(case)u(from)v(the)u(ab)v(o)i(v)e(e)d(de\014nition,)v (is)w(that)v(the)u(b)v(oundary)i(map)v F9(d)u Fc 2315 12 J(i)g Fa 47 12 L(on)g F9(K)v Fb 179 12 J(\017)g Fa 57 12 L(induces)g 2598 100 C(a)g (map)28 X F9(d)s Fc 298 31 L(r)g 52 G(i;n)g Fa 116 21 L(from)g F9(E)t Fc 263 31 L(r)g 5 52 C(i;n)g Fa 116 21 L(to)g F9(E)t Fc 168 31 L(r)g 5 52 C(i)g Fb(\000)g Fd(1)g Fc(;n)g Fd(+)g Fc(r)g Fa 283 21 L(for)g(eac)t (h)d F9(i;)t(n)14 X Fa(and)28 X F9(r)s Fa(,)j(and)s(that)t(w)s(e)e(ha)s (v)e(e)d F9 380 184 C(E)g Fc 66 36 L(r)g Fd(+1)i Fc 5 59 C(i;n)g Fa 149 23 L(=)g(Ker)o(\050)g F9(d)g Fc 297 34 L(r)g 55 G(i;n)g Fa 88 21 L(\051) g F9(=)g Fa(Im\050)g F9(d)g Fc 248 34 L(r)g 55 G(i)g Fd(+1)g Fc(;n)g Fb (\000)g Fc(r)g Fa 258 21 L(\051)g F9(:)g Fa 544 248 C(3)g EOP end %%PageTrailer %======================================================================= %%Page: 4 4 %%BeginPageSetup DVIALWdict begin %%EndPageSetup BOP Fa 1123 5385 A(Th)g(us)e(the)25 X(mo)u(dules)i(at)u(the)t F9(r)u Fa(+) 17 X(1)q(stage)26 X(can)s(b)t(e)i(computed)t(as)s(the)t(homology)s(of)t (those)s 112 G(at)g(the)v F9(r)u Fa 284 30 L(th)g 111 30 J(stage)g (under)v(maps)u(induced)v(b)u(y)e(the)u(b)v(oundary)i(maps)v(of)u F9(K) v Fb 1990 12 J(\017)g Fa 38 12 L(.)g 2298 100 C(W)g(e)-7 X(next)28 X (examine)s(the)t(complexes)s(de\014ned)t(b)s(y)e F9(E)s Fd 1662 30 L(0) g Fc 5 52 C(i;n)g Fa 116 22 L(and)g F9(E)t Fd 227 30 L(1)g Fc 5 52 C (i;n)g Fa 88 22 L(.)g 2083 100 C(If)g(w)x(e)e(let)x F9(r)y Fa(=)x(0)x (in)w(the)z(ab)w(o)k(v)c(e)f(de\014nition)w(of)z F9(E)w Fc 1605 30 L(r) g 5 52 C(i;n)g Fa 88 22 L(,)g(the)y(condition)y(that)x F9(d)y Fc 800 12 J(i)g Fa 28 12 L(\050)g F9(k)g Fc 75 12 J(i)g Fa 28 12 L(\051)g F8(2)w F1 2744 100 C(a)g Fc 42 30 L(n)g Fd(+)g Fc(r)g F9 128 30 J(K)g Fc 71 12 J(i)g Fb(\000)g Fd(1)g Fa 139 12 L(states)g(that)r F9(d)s Fc 456 12 J(i) g Fa 28 12 L(\050)g F9(k)g Fc 75 12 J(i)g Fa 28 12 L(\051)g F8(2)o F1(a) o Fc 175 30 L(n)g F9 45 30 J(K)g Fc 71 12 J(i)g Fb(\000)g Fd(1)g Fa 113 12 L(,)g(whic)s(h)d(is)s(alw)r(a)e(ys)d(true)s(since)r F9(k)s Fc 1053 12 J(i)g Fa 54 12 L(is)g(assumed)r 2478 100 C(to)g(b)39 X(e)j(in)t F1(a) u Fc 387 30 L(n)g F9 45 30 J(K)g Fc 71 12 J(i)g Fa 67 12 L(and)g F9(d)t Fc 216 12 J(i)g Fa 67 12 L(is)g(a)t(mo)u(dule)i(homomorphism.)u (Similarly)72 X(,)-7 X(the)42 X(condition)r 853 100 C(that)g F9(k)33 X Fc 228 12 J(i)g Fd(+1)g F8 144 12 L(2)g F1(a)s Fc 129 30 L(n)g Fd(+1)g F9 130 30 J(K)g Fc 71 12 J(i)g Fd(+1)g Fa 145 12 L(implies)g(that)t F9 (d)t Fc 515 12 J(i)g Fd(+1)g Fa 112 12 L(\050)g F9(k)g Fc 75 12 J(i)g Fd(+1)g Fa 112 12 L(\051)g F8(2)s F1(a)s Fc 193 30 L(n)g Fd(+1)g F9 130 30 J(K)g Fc 71 12 J(i)g Fa 28 12 L(,)g(so)u(that)u(when)s F9(r)u Fa(=)u (0)t 2083 100 C(the)g(denominator)y(in)y(the)y(ab)y(o)j(v)d(e)e (de\014nition)y(of)y F9(E)y Fc 1706 30 L(r)g 5 52 C(i;n)g Fa 127 22 L (is)g(just)y F1(a)y Fc 310 30 L(n)g Fd(+1)g F9 130 30 J(K)g Fc 71 12 J (i)g Fa 28 12 L(.)g(Hence)69 X F9(E)38 X Fd 415 30 L(0)g Fc 5 52 C(i;n) g Fa 2777 87 C(is)g(simply)n F1(a)n Fc 400 30 L(n)g F9 45 30 J(K)g Fc 71 12 J(i)g F9 28 12 L(=)g F1(a)g Fc 84 30 L(n)g Fd(+1)g F9 128 30 J(K) g Fc 71 12 J(i)g Fa 28 12 L(.)g(F)51 X(urthermore,)-7 X(since)33 X F9(K) t Fb 863 12 J(\017)g Fa 70 12 L(is)g(the)s(Koszul)s(complex)s(on)s(the) s 1788 100 C(generators)g(of)t F1(a)t Fa(,)g(the)u(maps)t F9(d)t Fc 1021 12 J(i)g Fa 60 12 L(are)g(all)t(zero)t(mo)t(dulo)i F1(a)t Fa(,)g (and)u(the)t(maps)t(induced)t(b)t(y)d 1081 99 C(the)g(b)q(oundary)i (maps)q F9(d)q Fc 785 12 J(i)g Fa 58 12 L(on)g F9(E)q Fd 184 30 L(0)g Fc 5 52 C(i;n)g Fa 119 22 L(are)g(zero.)q(It)45 X(then)30 X(follo)t(ws) e(that)t F9(E)t Fd 1164 30 L(1)g Fc 5 52 C(i;n)g Fa 119 22 L(is)g(also) t(equal)t 2419 109 C(to)g F1(a)r Fc 143 30 L(n)g F9 45 30 J(K)g Fc 71 12 J(i)g F9 28 12 L(=)g F1(a)g Fc 84 30 L(n)g Fd(+1)g F9 128 30 J(K)g Fc 71 12 J(i)g Fa 28 12 L(.)g 473 100 C(W)g(e)-7 X(next)29 X(consider)t (the)t(maps)t F9(d)t Fd 1064 30 L(1)g Fc 51 G(i;n)g Fa 117 21 L(induced) g(b)t(y)e F9(d)t Fc 470 12 J(i)g Fa 57 12 L(on)g F9(E)t Fd 183 30 L(1)g Fc 5 51 C(i;n)g Fa 88 21 L(;)g(w)u(e)d(denote)u(this)s 2099 47 C 44 4 B 2708 47 L(map)g F9 2708 115 C(d)g Fc 43 12 J(i)g Fa 28 12 L(.)g(Since) 64 X F9(K)37 X Fb 384 12 J(\017)g Fa 75 12 L(is)g(the)t(Koszul)t (complex)t(on)t F9(x)t Fd 1035 12 J(1)g F9 37 12 L(;)g(:)14 X(:)t(:)t(;) t(x)t Fc 232 12 J(k)g Fa 41 12 L(,)g(the)39 X(map)r F9(d)r Fc 451 12 J (i)g Fa 65 12 L(is)g(de\014ned)r(b)r(y)e 2391 100 C(a)g(matrix)q(with)r F8(\006)q F9(x)g Fc 666 12 J(i)g Fa 64 12 L(in)g(certain)q(p)r(ositions) i(and)r(zeros)q(in)r(the)q(remaining)r(p)q(ositions.)j 730 100 C(Th)g 211 68 L U 107 68 C(us)g F9(d)28 X Fc 150 12 J(i)g Fa 55 12 L(is)g (de\014ned)t(b)s(y)e(the)s(same)t(matrix)s(in)t(whic)s(h)e F9(x)s Fc 1488 12 J(i)g Fa 55 12 L(is)g(considered)t(as)s(a)t(map)s(from)t F1 1852 99 C(a)g Fc 42 30 L(n)g F9 45 30 J(K)g Fc 71 12 J(i)g F9 28 12 L(=) g F1(a)g Fc 84 30 L(n)g Fd(+1)g F9 127 30 J(K)g Fc 71 12 J(i)g Fa 53 12 L(to)g F1(a)r Fc 141 30 L(n)g Fd(+1)g F9 129 30 J(K)g Fc 71 12 J(i)g Fb (\000)g Fd(1)g F9 113 12 L(=)g F1(a)g Fc 84 30 L(n)g Fd(+2)g F9 128 30 J(K)g Fc 71 12 J(i)g Fb(\000)g Fd(1)g Fa 138 12 L(for)g(eac)r(h)d F9(n) r Fa(.)g 567 67 L 77 4 B 147 67 C(Let)g F9(K)r Fc 223 12 J(i)g Fa 51 12 L(denote)g(the)r(asso)q(ciated)j 2090 100 C(graded)g(mo)36 X(dule)j(of) t F9(K)u Fc 757 12 J(i)g Fa 64 12 L(under)g(the)t(\014ltration)u(b)t(y) e(p)t(o)j(w)d(ers)e(of)t F1(a)u Fa(.)f 1615 68 L 44 4 B 225 68 C(Then)g F9(d)t Fc 269 12 J(i)g Fa 63 12 L(de\014nes)g(a)t 2543 99 C(map)g(of)28 X(degree)t(one)t 890 67 L 77 4 B 197 67 C(from)g F9(K)t Fc 274 12 J(i)g 156 79 L U Fa 102 67 C(to)g F9(K)t Fc 179 12 J(i)g Fb(\000)g Fd(1)g Fa 112 12 L(,)g(and)t(the)t(ab)t(o)i(v)e(e)d(description)u(sho)s(ws)f(that) s(the)u 1312 100 C(resulting)g(complex)36 X(is)t(the)t(Koszul)t(complex) t 1665 46 L 48 4 B 123 46 C(on)g F9(x)t Fd 170 12 J(1)g F9 37 12 L(;)g (:)14 X(:)t(:)t 185 46 L U 37 46 C(;)g(x)t Fc 84 12 J(k)g Fa 41 12 L(,) g 310 46 L U 249 46 C(where)g F9(x)36 X Fc 296 12 J(i)g Fa 64 12 L (denotes)g(the)t 2443 100 C(image)g(of)27 X F9(x)t Fc 381 12 J(i)g Fa 55 12 L(in)g(the)t(comp)t(onen)i(t)e(of)t(degree)t(1)t(of)t(the)t(asso) t(ciated)i(graded)t(ring)t F9(g)t(r)j F1 2234 12 J(a)g Fa 46 12 L(\050) g F9(A)g Fa(\051.)g 2591 99 C(Th)g(us)e(w)t(e)e(ha)t(v)e(e)d(sho)u(wn)d (that)u(if)t(for)u(eac)t(h)e F9(i)t Fa(w)u(e)d 616 116 J 77 4 B 910 116 L(let)g F9 910 183 C(K)g Fc 76 12 J(i)g Fa 51 12 L(=)g F8(\010)p Fc 153 12 J(n)g Fb(\025)g Fd(0)g F9 130 12 L(E)g Fd 66 34 L(1)g Fc 5 54 C(i;n) g Fa 111 20 L(=)g F8(\010)p Fc 153 12 J(n)g Fb(\025)g Fd(0)g F1 130 12 L(a)g Fc 42 34 L(n)g F9 44 34 J(K)g Fc 71 12 J(i)g F9 28 12 L(=)g F1(a) g Fc 84 34 L(n)g Fd(+1)g F9 127 34 J(K)g Fc 71 12 J(i)g F9 28 12 L(;)g Fa 2101 196 C(the)g(maps)u F9(d)v Fd 404 30 L(1)g Fc 52 G(i;n)g Fa 117 22 L(induced)g(b)u(y)e F9(d)u Fc 468 12 J(i)g Fa 56 12 L(de\014ne)g(a)u 637 67 L U 326 67 C(complex)g F9(K)u Fb 403 12 J(\017)g Fa 68 12 L(whic) g(h)e(is)u(the)v(Koszul)u(complex)v 1711 54 C 48 4 B 116 46 C(on)g F9(x) u Fd 163 12 J(1)g F9 37 12 L(;)g(:)14 X(:)t(:)t 185 46 L U 37 46 C(;)g (x)t Fc 84 12 J(k)g Fa 69 12 L(o)g(v)e(er)d F9(g)28 X(r)j F1 259 12 J(a) g Fa 46 12 L(\050)g F9(A)g Fa(\051.)g 681 100 C(Up)g(to)u(no)t(w)e(w)t (e)e(ha)t(v)e(e)d(considered)u(the)t(\014ltration)u(on)t F9(K)u Fb 1817 12 J(\017)g Fa 67 12 L(without)g(men)u(tioning)d(the)u 2009 99 C(mo)g (dule)i F9(M)35 X Fa(.)9 X(Ho)58 X(w)e(ev)d(er,)e(exactly)36 X(the)s (same)r(argumen)s(t)d(holds)s(for)r F9(K)s Fb 2305 12 J(\017)g F8 61 12 L(\012)g F9(M)23 X Fa(,)9 X(and)37 X(w)q(e)e 2366 100 C(obtain)g(a)n(sp) 30 X(ectral)j(sequence)t F9(E)u Fc 1063 30 L(r)g 5 52 C(i;n)g Fa 88 22 L(\050)g F9(M)g Fa(\051)9 X(whic)31 X(h)d(con)t(v)d(erges)e(to)s(the)t (homology)s(of)t F9(K)s Fb 1597 12 J(\017)g F8 59 12 L(\012)g F9 2802 109 C(M)g Fa(and)44 X(suc)35 X(h)d(that)t(the)s(mo)t(dules)i F9(E)t Fd 1225 30 L(1)g Fc 5 51 C(i;n)g Fa 88 21 L(\050)g F9(M)g Fa(\051)9 X(form) 35 X(the)s(Koszul)t(complex)s(induced)t(b)s 1308 53 C U 2822 53 L(y)g F9 2822 99 C(x)g Fd 47 12 J(1)g F9 38 12 L(;)g(:)14 X(:)t(:)t 185 46 L U 37 46 C(;)g(x)t Fc 84 12 J(k)g Fa 70 12 L(on)g(the)28 X(asso)u(ciated) i(graded)u(mo)t(dule)j F9(g)t(r)j F1 1302 12 J(a)g Fa 46 12 L(\050)g F9 (M)g Fa(\051.)9 X(This)39 X(Koszul)28 X(complex)u(can)t 1735 100 C(also) g(b)n(e)j(expressed)n 734 67 L 77 4 B 97 67 C(as)g F9(K)n Fb 174 12 J (\017)g F8 46 12 L(\012)g F9(g)8 X(r)j F1 152 12 J(a)g Fa 46 12 L(\050) g F9(M)g Fa(\051.)u(Since)35 X(w)22 X(e)e(are)t(assuming)u(that)t F9(M) u(=)k F1(a)g F9(M)g Fa(has)31 X(\014nite)23 X 1055 100 C(length,)g(the) 31 X(homology)s(of)s(the)s(Koszul)s(complex)s(induced)s(b)s 2080 46 L 48 4 B 74 46 C(y)g F9(x)s Fd 122 12 J(1)g F9 37 12 L(;)g(:)14 X(:)t(:)t 185 46 L U 37 46 C(;)g(x)t Fc 84 12 J(k)g Fa 71 12 L(on)g F9(g)30 X(r)j F1 198 12 J(a)g Fa 46 12 L(\050)g F9(M)g Fa(\051)9 X 2712 99 C(also)g (has)34 X(\014nite)s(length.)t(Th)55 X(us,)e(since)35 X(the)s F9(r)r Fa (+)25 X(1)q(stage)34 X(of)s(the)t(sp)s(ectral)j(sequence)s(is)t 112 G (obtained)g(from)y(the)y F9(r)y Fa 754 30 L(th)g 122 30 J(stage)g(b)y(y) e(taking)y(homology)y(,)-7 X(the)42 X(Euler)p(c)r(haracteristic)d(is)r 876 100 C(preserv)g(ed)e(from)23 X(eac)v(h)c(stage)v(of)t(the)v(sp)t (ectral)j(sequence)t(to)v(the)t(next.)v(Hence,)34 X(since)26 X(the)p H (sp)g(ectral)i(sequence)v(con)u(v)e(erges)d(to)v(the)u(homology)v(of)u F9(K)v Fb 1816 12 J(\017)g F8 56 12 L(\012)g F9(M)18 X Fa(,)10 X(w)27 X (e)e(ha)t(v)e(e)d F9 1176 182 C(\037)g Fa(\050)g F9(K)g Fb 155 12 J (\017)g F8 57 12 L(\012)g Fc 65 12 J(A)g F9 73 12 L(M)g Fa(\051)9 X(=) 23 X F9(\037)t 337 67 L 77 4 B F2 38 E(\000)g F9 38 67 J(K)g Fb 76 12 J (\017)g F8 55 12 L(\012)g Fc 65 15 J(g)g(r)i F1 65 19 J(a)g Fd 46 19 L (\050)g Fc(A)g Fd(\051)g F9 124 15 L(g)g(r)j F1 80 12 J(a)g Fa 46 12 L (\050)g F9(M)g Fa(\051)9 X F2 154 67 L(\001)g F9 51 67 J(:)g Fa 2020 183 C(T)g(o)-7 X(complete)27 X(the)t(pro)t(of)j(that)t(this)t(Euler)t(c) t(haracteristic)e(is)t(equal)t(to)t F9(k)t Fa(!)j(times)t(the)t 125 100 C(Sam)g(uel)e(m)r(ultiplicit)e(y)d(,)a(w)r(e)f(in)q(terpret)f(the)q 1670 67 L U 323 67 C(complex)g F9(K)r Fb 400 12 J(\017)g F8 52 12 L (\012)g Fc 65 15 J(g)g(r)i F1 65 19 J(a)g Fd 46 19 L(\050)g Fc(A)g Fd (\051)g F9 120 15 L(g)g(r)j F1 80 12 J(a)g Fa 46 12 L(\050)g F9(M)g Fa (\051)9 X(as)25 X(a)u(complex)t 810 330 C(4)g EOP end %%PageTrailer %======================================================================= %%Page: 5 5 %%BeginPageSetup DVIALWdict begin %%EndPageSetup BOP BB [<000000000F000000001F000000003E000000007C00000000F800000001F00000000 3E000000007C00000000FC00000001F800000001F000000003E000000007E00000000FC0 0000001F800000001F000000003F000000007E000000007C00000000FC00000001F80000 0001F800000003F000000003E000000007E00000000FC00000000FC00000001F80000000 1F800000003F000000003F000000007E000000007E00000000FE00000000FC00000000FC 00000001F800000001F800000003F800000003F000000003F000000007F000000007E000 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s(Hilb)s(ert)i 2388 100 C(p)g(olynomial)i F9(P)s Fc 483 12 J(i)g Fa 55 12 L(suc)g(h)e(that)r F9 422 247 J(P)g Fc 53 12 J(i)g Fa 28 12 L(\050)g F9(n)g Fa(\051)g(=)n Fc 225 104 L(n)g Fb(\000)g Fd(1)g F2 3 25 J(X)g Fc 3 177 J(j)g Fd(=0)j 392 165 L U Fa 258 67 C(length\050)g F9(K)g Fc 334 53 L(M)g 73 G(i)g Fa 74 20 L(\051)g Fc 32 12 J(j)g F9 35 12 L(;)g Fa 1881 291 C(where)g 276 67 L U 32 67 C(\050)g F9(K)g Fc 108 53 L(M)g 73 G(i)g Fa 74 20 L(\051)g Fc 32 12 J(j)g Fa 66 12 L(denotes)g(the)v(comp) u(onen)j(t)d 976 67 L U 98 67 C(of)g F9(K)v Fc 174 53 L(M)g 73 G(i)g Fa 103 20 L(of)g(degree)v F9(j)u Fa(.)l(Ho)46 X(w)e(ev)d(er,)e 1037 67 L U 206 67 C(since)g F9(K)31 X Fc 282 53 L(M)g Fb 73 G(\017)g Fa 2792 80 C (is)g(a)q(Koszul)p(complex)q(on)p(the)q(asso)p(ciated)j(graded)p(mo)q (dule)i(of)q F9(M)p Fa(,)10 X(w)27 X(e)e(also)t(ha)u(v)d(e)e F9 1009 218 J(P)g Fc 53 12 J(i)g Fa 28 12 L(\050)g F9(n)g Fa(\051)g(=)p F9 286 56 L(k)g 9 113 J(i)g F9 98 57 L(P)g F1 65 34 L(a)g Fc 12 55 C(M)g Fa 74 21 L(\050)g F9(n)g F8(\000)18 X F9(i)u Fa(\051)f 1610 252 C(for)g(all) 38 X F9(i)t Fa(,)g(where)v F9(P)t F1 672 30 L(a)g Fc 12 53 C(M)g Fa 112 23 L(is)g(the)t(Hilb)t(ert)i(p)t(olynomial)i(of)t F9(M)t Fa(.)9 X(The) 67 X(shift)38 X(b)t(y)e F9(i)t 1944 67 L U Fa 107 67 C(in)g F9(K)t Fc 183 53 L(M)g Fb 73 G(\017)g Fa 2792 80 C(is)g(necessary)u(so)v(that)u (the)v(b)u(oundary)j(maps)u(will)v(b)u(e)j(maps)u(of)v(graded)u(mo)v (dules)i(of)v 100 G(degree)g(zero.)q(By)59 X(the)35 X(additivit)t(y)e (of)t(Hilb)t(ert)i(p)t(olynomials,)i F2 2063 62 L(P)g Fc 87 20 J(k)g 67 G(i)g Fd(=0)g Fa 110 25 L(\050)g F8(\000)g Fa(1\051)g Fc 171 30 L(i)g F9 28 30 J(P)g Fc 53 12 J(i)g Fa 28 12 L(\050)g F9(n)g Fa(\051)g(giv)t (es)e 2540 126 C(the)g(Hilb)25 X(ert)j(p)t(olynomial)j(de\014ned)t(b)u (y)d(the)u(homology)t 1854 67 L U 92 67 C(of)g F9(K)t Fc 169 53 L(M)g Fb 73 G(\017)g Fa 74 20 L(,)g(whic)u(h)e(is)t(constan)u(t)d(with)u 2005 121 C(v)g(alue)b F9(\037)p 292 67 L U Fa 32 67 C(\050)g F9(K)g Fc 108 53 L(M)g Fb 73 G(\017)g Fa 74 20 L(\051.)g(But)35 X(a)21 X(direct)t (computation)t(\050w)t(e)e(pro)t(v)e(e)d(a)u(more)s(general)u(v)s (ersion)f(of)s(this)u 442 111 C(in)g(a)x(later)x(section\051)x(sho)x(ws) e(that)x F2 1074 62 L(P)g Fc 88 20 J(k)g 67 G(i)g Fd(=0)g Fa 110 25 L (\050)g F8(\000)g Fa(1\051)g Fc 171 30 L(i)g F2 28 37 L(\000)g Fc 38 30 J(k)g 6 65 J(i)g F2 30 95 L(\001)g F9 38 67 J(P)g F1 65 30 L(a)g Fc 12 53 C(M)g Fa 74 23 L(\050)g F9(n)g F8(\000)13 X F9(i)u Fa(\051)f(is)26 X F9(k)r Fa(!)k(times)r(the)t(co)r(e\016cien)j(t)d 1710 100 C(of)g F9(n)v Fc 145 30 L(k)g Fa 69 30 J(in)g F9(P)v F1 162 30 L(a)g Fc 12 53 C(M)g Fa 74 23 L(\050)g F9(n)g F8(\000)18 X F9(i)u Fa(\051,)f(whic)29 X(h)c (pro)t(v)c(es)f(the)q(result.)t 313 100 C(The)g(p)x(oin)i(t)e(of)w(this) x(computation)w(is)x(that)w(it)x(transforms)w(questions)x(ab)w(out)j (Euler)w 125 99 C(c)g(haracteristics)e(in)x(to)e(questions)x(ab)z(out)h (Hilb)z(ert)h(p)z(olynomials,)h(whic)37 X(h)c(are)r(often)p(eas-)r 100 G(ier)g(to)u(deal)v(with.)u(W)69 X(e)-7 X(consider)38 X(one)u (particularly)t(imp)u(ortan)i(t)e(case.)t(Let)69 X F9(R)38 X Fa(b)u(e)i (a)u H(regular)g(lo)t(cal)i(ring,)t(and)w(let)t F1(p)t Fa(and)t F1(q)t Fa(b)t(e)i(ideals)t(of)t F9(R)t Fa(suc)u(h)d(that)u F9(R)s(=)i F1(p)f F8(\012)26 X F9(R)s(=)h F1(q)g Fa(has)38 X 99 G(\014nite)g(length.)p (Supp)55 X(ose)j F1(q)33 X Fa(is)v(generated)t(b)v(y)d(a)v(regular)t (sequence)v F9(x)t Fd 2288 12 J(1)g F9 37 12 L(;)g(:)14 X(:)t(:)t(;)t(x) t Fc 232 12 J(k)g Fa 41 12 L(.)g(Then)56 X 2598 100 C(dim\050)g F9(R)g (=)h F1(q)g Fa(\051)g(=)23 X(dim\050)t F9(R)g Fa(\051)h F8(\000)16 X F9 (k)u Fa(,)i(so)28 X(that)r(w)s(e)d(ha)s(v)d(e)e(dim)r(\050)h F9(R)f(=)i F1(p)f Fa(\051)h F8(\024)n Fa(dim\050)p F9(R)f Fa(\051)i F8(\000)15 X Fa(dim\050)w F9(R)f(=)i F1(q)f Fa(\051)h(=)22 X F9(k)v Fa(,)i H(and)g (dim)42 X(\050)g F9(R)g(=)h F1(p)g Fa(\051)g(+)28 X(dim\050)t F9(R)g(=) h F1(q)g Fa(\051)g(=)47 X(dim)u(\050)f F9(R)h Fa(\051)g(if)p(and)n(only) p(if)n(dim)p(\050)f F9(R)h(=)g F1(p)h Fa(\051)f(=)u F9(k)t Fa(.)j(Since) 80 X F9 99 G(x)g Fd 47 12 J(1)g F9 37 12 L(;)g(:)14 X(:)t(:)t(;)t(x)t Fc 232 12 J(k)g Fa 70 12 L(is)g(a)29 X(regular)t(sequence,)t(the)t (Koszul)t(complex)t F9(K)t Fb 1624 12 J(\017)g Fa 67 12 L(on)g F9(x)t Fd 163 12 J(1)g F9 37 12 L(;)g(:)14 X(:)t(:)t(;)t(x)t Fc 232 12 J(k)g Fa 70 12 L(is)g(a)29 X(free)t 2579 100 C(resolution)g(of)v F9(R)v(=)h F1(q)g Fa(.)g(Th)47 X(us)e(T)31 X(or)-7 X Fc 1044 36 L(R)g 2 56 C(i)g Fa 55 20 L(\050)g F9(R)g(=)h F1(q)g F9(;)g(R)14 X(=)g F1(p)h Fa(\051)f (is)32 X(the)r(homology)t F9(H)r Fc 1109 12 J(i)g Fa 28 12 L(\050)g F9 (K)g Fb 102 12 J(\017)g F8 59 12 L(\012)g F9(R=)21 X F1(p)g Fa(\051.)g (Ap-)47 X 2395 99 C(plying)g(the)28 X(ab)s(o)j(v)d(e)e(theorem)s(with)t F9(M)s Fa(=)y F9(R)n(=)i F1(p)f Fa(,)h(w)s(e)e(deduce)s(that)t F9 958 172 J(\037)g Fa(\050)g F9(R)g(=)h F1(q)g F9(;)g(R)14 X(=)g F1(p)h Fa (\051)f(=)24 X F9(e)r Fc 597 12 J(k)g Fa 41 12 L(\050)g F1(q)g F9(;)g(R) 14 X(=)g F1(p)h Fa(\051)f F9(:)h Fa 1596 172 C(Since)g(the)29 X(Sam)t (uel)e(m)t(ultiplicit)e(y)d F9(e)u Fc 1137 12 J(k)g Fa 41 12 L(\050)g F1(q)g F9(;)g(R)14 X(=)g F1(p)h Fa(\051)f(is)30 X(alw)r(a)f(ys)c (non-negativ)u(e)c(and)u(is)q(p)u(ositiv)g(e)g 1178 100 C(if)g(and)r (only)q(if)r(the)q(dimension)r(of)q F9(R)r(=)g F1(p)h Fa(is)q(equal)r (to)q F9(k)r Fa(,)i(this)r(pro)q(v)e(es)d(the)r(conjectures)q(in)r H (this)g(case.)r 125 100 J(Serre's)g(pro)38 X(of)i(of)t(the)t(m)t (ultiplicit)e(y)d(conjectures)u(in)s(the)u(equic)s(haracteristic)f(case) s 125 100 C(pro)g(ceeded)i(b)27 X(y)d(reducing)t(to)s(the)t(case)s(of)t (a)s(regular)t(sequence)s(b)t(y)d(reduction)t(to)s(the)t(di-)s 99 G (agonal.)g(If)36 X F9(R)26 X Fa(is)t(a)t(p)s(o)j(w)d(er)e(series)s(ring) t F9(k)s Fa(\133\133)j F9(X)g Fd 1405 12 J(1)g F9 37 12 L(;)g(:)14 X(:) t(:)t(;)t(X)t Fc 253 12 J(d)g Fa 39 12 L(\135\135)g(and)25 X F9(M)u Fa (and)34 X F9(N)26 X Fa(are)34 X F9(R)26 X Fa(-mo)g(dules)j 1734 100 C (with)g F9(M)o F8(\012)15 X Fc 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L(,)g(and)t(letting)s F8(T)t F9 (or)21 X Fb 624 37 L(O)g Ff 53 8 J(W)g Fc 55 52 C(i)g Fa 122 23 L(\050) g F8(M)g F9(;)g F8(N)14 X Fa(\051)r(b)29 X(e)i(the)t F9(i)s Fa 597 30 L (th)g 111 30 J(homology)g 2517 100 C(of)g F8(F)u Fb 157 12 J(\017)g F8 58 12 L(\012)g(N)20 X Fa(.)13 X(Usually)44 X(w)32 X(e)c(de\014ne)t F8(F) q Fb 963 12 J(\017)g Fa 69 12 L(b)g(y)e(de\014ning)r(a)s(complex)r(of)s (graded)r(mo)s(dules)i F9(F)s Fc 1588 12 J(i)g Fa 2835 87 C(whic)g(h)e (de\014ne)21 X(lo)v(cally)h(free)v(shea)t(v)e(es)d(and)u(whic)u(h)d(giv) u(e)d(a)u(resolution)u(of)u F9(M)u Fa(\050or)31 X F9(N)22 X Fa(\051.)9 X(In)35 X(the)22 X 100 G(case)g(where)y F9(W)y Fa(=)35 X F9(P)23 X Fa(,) 12 X(a)27 X(b)t(ounded)i(resolution)t(can)t(b)t(e)i(constructed)t(using) t(direct)t(sums)t 99 G(of)g(copies)r(of)q F8(O)r Fc 493 12 J(P)g Fa 55 12 L(\050)g F9(n)g Fa(\051,)g(so)r(this)r(pro)q(cess)j(is)q(quite)r (easy)q(to)r(carry)q(out.)r(W)35 X(e)a(also)24 X(de\014ne)u(the)t 548 100 C(complex)g F8(T)y F9(or)q Fb 473 36 L(O)g Ff 53 8 J(W)g Fa 68 28 J (\050)g F8(M)g F9(;)g F8(N)14 X Fa(\051)r(to)29 X(b)t(e)i(the)t(complex) t F8(F)t Fb 1058 12 J(\017)g F8 58 12 L(\012)g(N)19 X Fa(.)n(This)40 X (complex)30 X(is)r(of)t(course)r 1710 100 C(not)g(w)r(ell-de\014ned)d (as)r(a)q(complex,)r(but)q(it)r(is)q(w)r(ell-de\014ned)d(up)r(to)q (quasi-isomorphism.)r 125 99 J(The)g(last)23 X(ingredien)u(t)d(in)u(the) t(generalization)u(to)t(pro)u(jectiv)k(e)e(space)t(is)u(the)t(pushdo)u (wn)d 125 100 C(of)g(complexes)30 X(from)t F9(P)t Fa(to)41 X F9(X)30 X Fa(b)z(y)e(the)t(map)t F9(\036)t Fa(,)g(whic)t(h)e(w)t(e)e(denote)t F9 (\036)t Fb 2233 12 J(\003)g Fa 38 12 L(.)g(In)43 X(general)30 X(this)t 2271 100 C(functor)g(is)w(the)w(deriv)w(ed)e(functor)w(of)w(the)w (global)w(section)w(functor)w(on)w(shea)w(v)e(es,)d(but)y(in)v 99 G(the) g(case)t(of)t(pro)t(jectiv)l(e)d(space)u(o)s(v)f(er)c F9(R)v Fa(it)s(is) v(not)r(di\016cult)v(to)r(giv)v(e)c(a)v(direct)r(de\014nition)v 100 G (using)g(Cec)n(h)e(cohomology)n(.)-7 X(Let)36 X F9(A)24 X Fa(=)s F9(R)s Fa(\133)h F9(X)g Fd 1370 12 J(0)g F9 37 12 L(;)g(:)14 X(:)t(:)t(;)t(X)t Fc 253 12 J(n)g Fa 45 12 L(\135,)g(and)25 X(let)s F9(P)s Fa(=)35 X(Pro) 23 X(j\050)l F9(A)g Fa(\051)g(as)u(ab)u(o)i(v)e(e.)d 1705 100 C(Let)g F9(C)y Fb 214 31 L(\017)g Fa 66 31 J(b)g(e)i(the)y(complex)x 232 156 J (0)g F8(!)t F2 171 79 L(Y)g F9 120 79 J(A)g Fc 62 12 J(X)g Ff 54 8 J(i) g F8 54 20 L(!)g F2 106 79 L(Y)g F9 120 79 J(A)g Fc 62 12 J(X)g Ff 54 8 J(i)g Fc 26 8 L(X)g Ff 54 8 J(j)g F8 60 20 L(!)g F9(:)t(:)14 X(:)t F8(!) 23 X F9(A)t Fc 394 12 J(X)g Fg 54 8 J(0)g Fc 32 8 L(X)g Fg 54 8 J(1)g Fb 32 8 L(\001\001\001)g Fc(X)g Ff 114 8 J(n)g F8 69 20 L(!)g Fa(0)t 2204 157 C(where)g(for)35 X(an)s(y)e(elemen)s(t)e F9(Y)s F8(2)54 X F9(A) 34 X Fa(,)h F9(A)u Fc 1234 12 J(Y)g Fa 92 12 L(denotes)g(the)u(lo)t (calization)j(of)t F9(A)u Fa(obtained)t(b)u(y)d 1326 99 C(in)g(v)e (erting)d F9(Y)22 X Fa(.)o(If)36 X F8(M)20 X Fb 643 12 J(\017)g Fa 59 12 L(is)g(a)u(b)u(ounded)i(complex)u(of)u(coheren)u(t)e(shea)u(v)e(es)d (o)v(v)d(er)e F9(P)u Fa(represen)32 X(ted)e 702 100 C(b)g(y)e(a)q (complex)q(of)q(graded)q(mo)q(dules)i F9(M)q Fb 1294 12 J(\017)g Fa 38 12 L(,)g(w)r(e)d(then)r(de\014ne)p F9(\036)r Fb 656 12 J(\003)g Fa 38 12 L(\050)g F8(M)g Fb 132 12 J(\017)g Fa 38 12 L(\051)g(to)q(b)q(e)i (the)q(graded)q 2196 99 C(part)g(of)z(degree)z(zero)z(of)z(the)z (complex)z F9(C)z Fb 1410 30 L(\017)g F8 63 30 J(\012)g Fc 65 12 J(A)g F9 79 12 L(M)g Fb 81 12 J(\017)g Fa 38 12 L(.)g(Then)68 X F9(\036)37 X Fb 367 12 J(\003)g Fa 38 12 L(\050)g F8(M)g Fb 132 12 J(\017)g Fa 38 12 L(\051)g(is)u(a)u(b)u(ounded)i 2311 100 C(complex)g(of)29 X F9(R)t Fa (-mo)h(dules)i(with)t(\014nitely)t(generated)t(homology)t(and)t(is)t(w) t(ell-de\014ned)e(up)t H(to)g(quasi-isomorphism.)s 125 99 J(No)g(w)e (supp)n(ose)j(that)n F8(M)p Fa(and)n F8(N)p Fa(are)z(coheren)p(t)c(shea) p(v)c(es)f(on)22 X F9(P)x Fa(suc)33 X(h)g(that)21 X F8(M)z(\012)7 X Fb 2546 12 J(O)g Ff 52 8 J(P)g F8 61 20 L(N)g Fa 2784 100 C(has)g(supp)35 X(ort)i(whic)t(h)e(lies)t(o)t(v)e(er)d(the)u(closed)s(p)u(oin)h(t)f(of) s F9(R)u Fa(,)g(whic)w(h)d(w)u(e)d(denote)u F9(s)s Fa(.)h(Then)58 X F8 H(T)g F9(or)21 X Fb 146 37 L(O)g Ff 53 8 J(P)g Fc 53 52 C(i)g Fa 105 23 L(\050)g F8(M)g F9(;)g F8(N)14 X Fa(\051)r(has)44 X(supp)s(ort)j(lying) s(o)t(v)d(er)e F9(s)s Fa(for)t(all)s F9(i)t Fa(,)g(so)w(that)t(the)s (homology)t(of)s F9 251 108 C(\036)g Fb 49 12 J(\003)g Fa 38 12 L(\050) g F8(T)g F9(or)21 X Fb 178 37 L(O)g Ff 53 8 J(P)g Fc 52 52 C(i)g Fa 105 23 L(\050)g F8(M)g F9(;)g F8(N)14 X Fa(\051\051)r(is)23 X(supp)s(orted) j(at)s(the)t(maximal)s(ideal)t(and)s(th)t(us)d(has)t(\014nite)s(length.) t 371 99 C(Hence)g(w)y(e)d(can)y(de\014ne)x F9 464 145 J(\037)g Fa(\050) g F8(M)g F9(;)g F8(N)14 X Fa(\051)r(=)23 X F2 444 79 L(X)g Fa 120 79 J (\050)g F8(\000)g Fa(1\051)g Fc 171 34 L(i)g Fa 28 34 J(length)g F2 240 67 L(\000)g F9 38 67 J(H)g Fc 76 34 L(i)g Fa 27 34 J(\050)g F9(\036)g Fb 81 12 J(\003)g Fa 38 12 L(\050)g F8(T)g F9(or)r Fb 178 36 L(O)g Ff 53 8 J(P)g Fa 53 28 J(\050)g F8(M)g F9(;)g F8(N)14 X Fa(\051\051)r F2 313 67 L(\001)g F9 53 67 J(:)g Fa 966 259 C(7)g EOP end %%PageTrailer %======================================================================= %%Page: 8 8 %%BeginPageSetup DVIALWdict begin %%EndPageSetup BOP Fa 1248 5385 A(The)g(\014rst)43 X(part)t(of)t(the)t(reduction)t(is)t(to) t(sho)t(w)e(that)t(it)t(su\016ces)t(to)t(sho)t(w)e(that)t 125 100 C(the) g(Euler)29 X(c)u(haracteristic)d F9(\037)u Fa(\050)f F8(O)h Fc 1033 12 J(Z)g Fe 49 17 L(0)g F9 27 5 J(;)g F8(O)14 X Fc 103 12 J(Y)g Fe 53 17 L (0)g Fa 27 5 J(\051)g(is)29 X(non-negativ)u(e)d(and)u(is)t(zero)u(if)t (dim)u(\050)f F9(R)h(=)g F1(p)h Fa(\051)f(+)21 X 1292 100 C(dim\050)g F9(R)g(=)h F1(q)g Fa(\051)g F9(\074)31 X Fa(dim\050)s F9(R)h Fa(\051.)g (The)51 X(p)31 X(oin)j(t)d(is)v(that)t(the)v(assumptions)t(on)v F9(\036) t Fa(imply)v(that)t(this)v H(new)g(Euler)19 X(c)s(haracteristic)e(is)s (closely)t(related)s(to)t(the)s(Euler)t(c)s(haracteristic)e F9(\037)s Fa(\050)h F9(R)f(=)i F1(p)f F9(;)h(R)n(=)h F1(q)g Fa(\051)g H(de\014ned) g(earlier.)31 X(Let)49 X F9(G)31 X Fb 805 12 J(\017)g Fa 70 12 L(b)g(e) i(a)u(\014nite)t(free)u(resolution)t(of)u F9(R)t(=)h F1(p)g Fa(o)t(v)e (er)d F9(R)u Fa(.)g(Let)49 X F8(F)31 X Fb 1762 12 J(\017)g Fa 70 12 L(b) g(e)i(a)u 2707 100 C(\014nite)g(lo)23 X(cally)j(free)t(resolution)t(of) t F8(O)t Fc 1157 12 J(Z)g Fe 49 17 L(0)g Fa 50 5 J(as)g(ab)t(o)j(v)d(e,) e(and)u(let)t F8(F)t Fb 685 12 J(\017)g Fa 61 12 L(b)g(e)i(de\014ned)u (b)s(y)f(a)s(complex)u F9 2002 100 C(F)g Fb 53 12 J(\017)g Fa 66 12 L (of)g(graded)y(mo)x(dules.)j(W)36 X(e)a(then)27 X(ha)u(v)d(e)e(the)t (follo)u(wing)d(\\pro)u(jection)k(form)u(ula":)d F9 715 169 J(\036)g Fb 49 12 J(\003)g Fa 38 12 L(\050)g F8(F)g Fb 92 12 J(\017)g Fa 38 12 L (\051)g F8(\012)18 X F9(G)u Fb 198 12 J(\017)g F8 61 34 L(\030)g Fa 26 G(=)g F9 88 4 L(\036)g Fb 49 12 J(\003)g Fa 39 12 L(\050)g F8(F)g Fb 92 12 J(\017)g F8 57 12 L(\012)g F9(\036)t Fb 132 34 L(\003)g Fa 38 34 J (\050)g F9(G)g Fb 97 12 J(\017)g Fa 38 12 L(\051\051)g F9(:)g Fa 1815 169 C(T)g(o)-7 X(pro)31 X(v)e(e)d(this)u(form)t(ula,)e(w)u(e)e(use)t (the)u(de\014nition)t(of)u F9(\036)t Fb 1791 12 J(\003)g Fa 70 12 L(in) g(terms)t(of)u(the)t(complex)u F9 1986 99 C(C)g Fb 65 30 L(\017)g Fa 65 30 J(de\014ned)g(ab)o(o)j(v)d(e.)e(The)y(asso)p(ciativit)i(y)e(of)o(the) p(tensor)o(pro)p(duct)i(implies)p(that)o(w)p(e)d(ha)p(v)d(e)e 130 100 C (isomorphisms)g(of)q(complexes:)p 659 199 J(\050)g F9(C)g Fb 97 34 L (\017)g F8 57 34 J(\012)g Fc 65 12 J(A)g F9 73 12 L(F)g Fb 53 12 J(\017) g Fa 38 12 L(\051)g F8(\012)18 X Fc 115 12 J(R)g F9 73 12 L(G)g Fb 65 12 J(\017)g F8 61 34 L(\030)g Fa 26 G(=)g F9 88 4 L(C)g Fb 65 34 L(\017) g F8 57 34 J(\012)g Fc 65 12 J(A)g Fa 73 12 L(\050)g F9(F)g Fb 85 12 J (\017)g F8 57 12 L(\012)g Fc 65 12 J(R)g F9 73 12 L(G)g Fb 65 12 J(\017) g Fa 38 12 L(\051)g F8 55 22 L(\030)g Fa 26 G(=)g F8 1244 115 C(\030)g Fa 26 G(=)g F9 88 4 L(C)g Fb 65 34 L(\017)g F8 57 34 J(\012)g Fc 65 12 J(A)g Fa 73 12 L(\050)g F9(F)g Fb 85 12 J(\017)g F8 57 12 L(\012)g Fc 65 12 J(A)g Fa 73 12 L(\050)g F9(A)g F8(\012)t Fc 177 12 J(R)g F9 73 12 L(G)g Fb 65 12 J(\017)g Fa 38 12 L(\051\051)g F9(:)g Fa 1879 141 C (Since)g F9(A)w F8(\012)k Fc 341 12 J(R)g F9 59 12 L(G)g Fb 65 12 J (\017)g Fa 59 12 L(de\014nes)g(a)w(lo)v(cally)j(free)v(resolution)w(of) v F8(O)w Fc 1267 12 J(Y)g Fe 53 17 L(0)g Fa 27 5 J(,)g(it)x(is)w(clear) v(that)w(the)v(complex)w F9 1871 100 C(C)g Fb 65 30 L(\017)g F8 51 30 J (\012)g Fc 65 12 J(A)g Fa 67 12 L(\050)g F9(F)g Fb 85 12 J(\017)g F8 52 12 L(\012)g Fc 65 12 J(A)g Fa 67 12 L(\050)g F9(A)g F8(\012)o Fc 172 12 J(R)g F9 68 12 L(G)g Fb 65 12 J(\017)g Fa 38 12 L(\051\051)g(represen) 25 X(ts)e F9(\036)t Fb 528 12 J(\003)g Fa 38 12 L(\050)g F8(F)g Fb 92 12 J(\017)g F8 52 12 L(\012)g F9(\036)13 X Fb 127 30 L(\003)g Fa 38 30 J(\050)g F9(G)g Fb 97 12 J(\017)g Fa 38 12 L(\051\051.)g(Since)36 X (\050)25 X F9(C)g Fb 436 30 L(\017)g F8 51 30 J(\012)g Fc 65 12 J(A)g F9 67 12 L(F)g Fb 54 12 J(\017)g Fa 38 12 L(\051)g F8(\012)13 X Fc 110 12 J(R)g F9 68 12 L(G)g Fb 65 12 J(\017)g Fa 2824 88 C(represen)g(ts)e F9(\036)27 X Fb 442 12 J(\003)g Fa 38 12 L(\050)g F8(F)g Fa(\051)8 X F8 (\012)19 X F9(G)s Fb 298 12 J(\017)g Fa 38 12 L(,)g(this)28 X(pro)s(v)e (es)d(the)t(ab)s(o)j(v)d(e)e(isomorphism.)s 691 99 C(T)g(o)-7 X (complete)24 X(the)t(pro)t(of)i(of)t(the)t(fact)t(that)t(it)t(su\016ces) t(to)t(pro)t(v)e(e)d(non-negativit)u(y)d(and)u 125 100 C(v)g(anishing)b (for)z F9(\037)z Fa(\050)g F8(O)g Fc 650 12 J(Z)g Fe 49 17 L(0)g F9 27 5 J(;)g F8(O)14 X Fc 103 12 J(Y)g Fe 53 17 L(0)g Fa 27 5 J(\051,)g(w)30 X(e)e(use)t(induction)t(on)t(the)t(dimension)t(of)t F9(R)t(=)g F1(q)h Fa(together)s 909 100 C(with)g(the)z(assumption)z(that)z(the)z(map)z (induced)z(b)z(y)d F9(\036)z Fa(from)z F9(Z)z Fb 2122 30 L(0)g Fa 59 30 J(to)g F9(Z)z Fa(is)42 X(generically)n 2181 99 C(\014nite.)g(Let)65 X F8(F)38 X Fb 490 12 J(\017)g Fa 75 12 L(b)g(e)i(a)s(lo)s(cally)i(free)s (resolution)s(of)s F8(O)s Fc 1207 12 J(Z)g Fe 49 17 L(0)g Fa 64 5 J(on) g F9(P)s Fa(.)12 X(If)65 X(w)38 X(e)d(lo)t(calize)h(at)t F1(q)r Fa(,)h 1885 100 C(the)g(generic)27 X(\014niteness)u(of)t F9(\036)u Fa(implies) t(that)u(the)t(resulting)u(map)t(from)u(Pro)t(j)l(\050\050)g F9(A=I)g Fa(\051)7 X F1 2684 12 J(q)g Fa 45 12 L(\051)g(to)27 X 2729 100 C(Sp)g (ec\050\050)i F9(R)g(=)h F1(q)g Fa(\051)g F1(q)g Fa(\051)g(is)z (de\014ned)z(b)z(y)e(a)z(\014nite)z(\014eld)z(extension)z(of)z(a)z(giv) z(en)e(degree)z(whic)z(h)e(w)z(e)e 99 G(denote)g F9(n)o Fa(.)g(Th)35 X (us)e F9(\036)21 X Fb 624 12 J(\003)g Fa 38 12 L(\050)g F8(F)g Fb 92 12 J(\017)g Fa 38 12 L(\051)g(lo)u(calized)i(at)u F1(q)t Fa(is)u (isomorphic)t(to)u(\050\050)t F9(R)h(=)g F1(q)h Fa(\051)f F1 1376 12 J (q)g Fa 45 12 L(\051)g Fc 32 30 L(n)g Fa 45 30 J(,)g(so)v(the)t(complex) u F9 2290 100 C(\036)g Fb 49 12 J(\003)g Fa 38 12 L(\050)g F8(F)g Fb 92 12 J(\017)g Fa 38 12 L(\051)g(is)29 X(isomorphic)t(to)t(the)t(mo)t(dule) i(\050)t F9(R)g(=)h F1(q)g Fa(\051)g Fc 1321 30 L(n)g Fa 74 30 J(up)g (to)t(a)t(complex)t(with)t(homology)t(of)t 1612 99 C(dimension)g (strictly)s(less)r(than)s(the)r(dimension)s(of)r F9(R=)s F1(q)g Fa(.)g 125 100 J(By)g(the)s(pro)r(jection)l(form)r(ula,)e(w)r(e)e(ha)r(v)e(e)d (that)s F9 709 169 J(\036)g Fb 49 12 J(\003)g Fa 38 12 L(\050)g F8(F)g Fb 92 12 J(\017)g Fa 38 12 L(\051)g F8(\012)18 X F9(G)u Fb 198 12 J (\017)g F8 61 34 L(\030)g Fa 26 G(=)g F9 88 4 L(\036)g Fb 49 12 J(\003) g Fa 39 12 L(\050)g F8(F)g Fb 92 12 J(\017)g F8 57 12 L(\012)g F9(\036) t Fb 132 34 L(\003)g Fa 38 34 J(\050)g F9(G)g Fb 97 12 J(\017)g Fa 38 12 L(\051\051)g F9(;)g Fa 1940 169 C(where)g F9(G)25 X Fb 303 12 J(\017) g Fa 63 12 L(is)g(a)t(free)t(resolution)t(of)t F9(R)t(=)h F1(p)g Fa(.)g (Since)36 X F8(F)25 X Fb 1258 12 J(\017)g Fa 63 12 L(is)g(a)t(lo)t (cally)i(free)t(resolution)t(of)t F8(O)t Fc 1103 12 J(Z)g Fe 49 17 L(0) g Fa 2839 104 C(and)g F9(\036)w Fb 211 30 L(\003)g Fa 38 30 J(\050)g F9 (G)g Fb 97 12 J(\017)g Fa 38 12 L(\051)g(is)w(a)x(lo)w(cally)j(free)w (resolution)x(of)w F8(O)x Fc 1185 12 J(Y)g Fe 53 17 L(0)g Fa 27 5 J(,)g (the)x(complex)w F9(\036)x Fb 570 12 J(\003)g Fa 38 12 L(\050)g F8(F)g Fb 93 12 J(\017)g F8 57 12 L(\012)g F9(\036)n Fb 133 30 L(\003)g Fa 38 30 J(\050)g F9(G)g Fb 97 12 J(\017)g Fa 38 12 L(\051\051)g(is)w 2713 100 C(quasi-isomorphic)g(to)z F9(\036)z Fb 794 12 J(\003)g Fa 38 12 L (\050)g F8(T)g F9(or)p Fb 177 30 L(O)g Ff 53 8 J(P)g Fa 52 22 J(\050)g F8(O)g Fc 98 12 J(Z)g Fe 49 17 L(0)g F9 27 5 J(;)g F8(O)14 X Fc 103 12 J(Y)g Fe 53 17 L(0)g Fa 27 5 J(\051\051)g F9(:)g Fa(Hence,)31 X(taking) u(Euler)t(c)t(haracteristics)e 1471 100 C(and)g(using)o(the)o(ab)o(o)i (v)e(e)d(isomorphism,)p(w)n(e)f(ha)n(v)f(e)c F9(\037)q Fa(\050)e F9 (\036)i Fb 1701 12 J(\003)g Fa 38 12 L(\050)g F8(F)g Fb 92 12 J(\017)g Fa 38 12 L(\051)g F9(;)g(R)14 X(=)g F1(p)h Fa(\051)f(=)24 X F9(\037)r Fa(\050)h F8(O)f Fc 510 12 J(Z)g Fe 49 17 L(0)g F9 27 5 J(;)g F8(O)14 X Fc 103 12 J(Y)g Fe 53 17 L(0)g Fa 27 5 J(\051.)g(Ap-)36 X 2638 99 C (plying)g(the)n(induction)n(h)n(yp)d(othesis,)j(w)n(e)e(ha)29 X(v)e(e)d (that)u F9(\037)u Fa(\050)g F9(M)g(;)k(R)14 X(=)g F1(p)h Fa(\051)f(is) 31 X(zero)r(whenev)t(er)c(the)t 100 G(dimension)g(of)r F9(M)s Fa(is)38 X(less)30 X(than)s(the)t(dimension)s(of)t F9(R)s(=)h F1(q)g Fa(.)g(Th) 42 X(us,)e(since)29 X F9(\036)u Fb 2380 12 J(\003)g Fa 38 12 L(\050)g F8(O)g Fc 98 12 J(Z)g Fe 49 17 L(0)g Fa 27 5 J(\051)g(is)t(iso-)u 2592 99 C(morphic)g(to)s(\050)t F9(R)f(=)i F1(q)f Fa(\051)h Fc 637 30 L(n)g Fa 73 30 J(up)g(to)s(something)t(dimension)s(strictly)t(less)s(than)t (the)s(dimension)t 710 100 C(of)g F9(R)s(=)g F1(q)h Fa(,)f(w)t(e)c(ha)t (v)c(e)f(that)q F9 201 169 J(\037)g Fa(\050)g F8(O)g Fc 150 12 J(Z)g Fe 49 17 L(0)g F9 27 5 J(;)g F8(O)14 X Fc 103 12 J(Y)g Fe 53 17 L(0)g Fa 27 5 J(\051)g(=)23 X F9(\037)t Fa(\050)g F9(\036)g Fb 276 12 J(\003)g Fa 38 12 L(\050)g F8(F)g Fb 92 12 J(\017)g Fa 38 12 L(\051)g F9(;)g(R) 14 X(=)g F1(p)h Fa(\051)f(=)24 X F9(\037)r Fa(\050\050)h F9(R)f(=)i F1 (q)f Fa(\051)h Fc 655 34 L(n)g F9 45 34 J(;)g(R)14 X(=)g F1(p)h Fa(\051) f(=)24 X F9(n)r Fa(\050)h F9(\037)f Fa(\050)h F9(R)f(=)i F1(q)f F9(;)h (R)13 X(=)h F1(p)g Fa(\051\051)g F9(:)g Fa 1754 169 C(Th)g(us)e(the)22 X(v)u(anishing,)b(non-negativit)v(y)d(,)a(and)t(p)v(ositivit)h(y)f(of)s F9(\037)v Fa(\050)e F8(O)i Fc 2051 12 J(Z)g Fe 49 17 L(0)g F9 27 5 J(;) g F8(O)14 X Fc 103 12 J(Y)g Fe 53 17 L(0)g Fa 27 5 J(\051)g(are)22 X (equiv)u(alen)b(t)e 2310 99 C(to)g(the)z(corresp)y(onding)j(prop)y (erties)j(of)y F9(\037)z Fa(\050)g F9(R)g(=)h F1(q)g F9(;)g(R)14 X(=)g F1(p)h Fa(\051.)f 1411 247 J(8)g EOP end %%PageTrailer %======================================================================= %%Page: 9 9 %%BeginPageSetup DVIALWdict begin %%EndPageSetup BOP Fa 1248 5385 A(Th)g(us)e(w)27 X(e)e(ha)t(v)e(e)d(reduced)u(the)t(m)u (ultiplicit)d(y)e(conjectures)t(to)u(corresp)t(onding)j(conjec-)t 125 100 C(tures)g(on)36 X(Euler)s(c)t(haracteristics)d(de\014ned)t(b)s(y)e (subsc)s(hemes)e F9(Y)s Fb 2056 30 L(0)g Fa 59 30 J(and)g F9(Z)t Fb 232 30 L(0)g Fa 59 30 J(of)g(pro)t(jectiv)k(e)e 2406 100 C(space)g(o)v(v)d (er)e F9(R)u Fa(,)h(where)x F9(Z)v Fb 862 30 L(0)g Fa 61 30 J(is)g (regular)v(and)v F9(Y)v Fb 623 30 L(0)g Fa 61 30 J(is)g(the)v(pullbac)v (k)d(of)v(a)v(subsc)v(heme)d(of)v 1607 100 C(Sp)g(ec\050)i F9(R)g Fa (\051.)h(In)t(particular,)28 X(the)s(ideal)t F9(I)s Fa(de\014ning)z F9 (Z)s Fb 1652 30 L(0)g Fa 50 30 J(is)g(lo)s(cally)j(generated)s(b)t(y)d (a)t(regular)s 1702 100 C(sequence,)g(and)w(this)v(fact)w(mak)v(es)e(it) v(p)w(ossible)i(to)w(use)v(the)w(Serre)v(sp)w(ectral)i(sequence)w(to)v H(reduce)g(to)t(the)s(case)t(of)s(asso)t(ciated)i(graded)t(rings.)s 125 100 J(Let)g F9(g)y(r)j Fc 234 12 J(I)g Fa 38 12 L(\050)g F9(A)g Fa(\051) g(=)w F9(A=I)w F8(\010)t F9(I)n(=I)m Fd 636 30 L(2)g F8 58 30 J(\010)g F9(:)n(:)14 X(:)t Fa(b)32 X(e)j(the)t(asso)u(ciated)i(graded)u(ring)t (of)u F9(I)t Fa(.)7 X(Let)51 X F9(B)33 X Fa 1091 100 C(denote)g F9(A=) 22 X F1(p)g F9(A)g Fa(,)g(and)u(let)s F9(g)t(r)j Fc 866 12 J(I)g Fa 38 12 L(\050)g F9(B)g Fa(\051)k(denote)t(the)s(asso)t(ciated)i(graded)t (ring)s(of)t(the)s(image)t(of)s F9(I)t Fa 904 99 C(in)g F9(B)r Fa(.)l(W) 34 X(e)a(note)20 X(that)u(b)t(oth)j F9(g)t(r)j Fc 967 12 J(I)g Fa 38 12 L(\050)g F9(A)g Fa(\051)g(and)t F9(g)u(r)i Fc 381 12 J(I)g Fa 38 12 L (\050)g F9(B)g Fa(\051)k(are)u(bigraded)t(rings,)u(with)u(one)u(grading) t 1424 100 C(induced)g(b)29 X(y)e(the)t(grading)t(on)t F9(A)t Fa(and)t (the)t(other)t(corresp)t(onding)i(to)t(p)t(o)i(w)e(ers)d(of)u F9(I)s Fa (.)7 X(The)40 X H(ring)g F9(g)20 X(r)j Fc 244 12 J(I)g Fa 38 12 L(\050) g F9(B)g Fa(\051)k(is)t(also)u(a)t(bigraded)u(mo)t(dule)j(o)t(v)e(er)d F9(g)u(r)i Fc 1321 12 J(I)g Fa 38 12 L(\050)g F9(A)g Fa(\051.)g(W)34 X (e)a(mak)19 X(e)f(the)t(con)w(v)c(en)f(tion)c(that)w 1641 99 C(the)g F9 (i;)32 X(j)14 X Fa(comp)36 X(onen)j(t)d(of)q F9(g)o(r)k Fc 892 12 J(I)g Fa 38 12 L(\050)g F9(A)g Fa(\051)g(is)p(the)p(comp)p(onen)i(t)e(of)p F9 (I)p Fc 964 30 L(j)g F9 34 30 J(=I)g Fc 85 30 L(j)g Fd(+1)j Fa 150 30 J (of)g(degree)p F9(i)p Fa(.)g(W)49 X(e)-7 X(let)32 X F9 2163 100 C(E)g Fa(denote)v(the)r(sc)q(heme)e(Pro)q(j)l(\050)g F9(g)g(r)j Fc 1069 12 J (I)g Fa 38 12 L(\050)g F9(A)g Fa(\051\051,)g(where)r F9(g)r(r)j Fc 534 12 J(I)g Fa 38 12 L(\050)g F9(A)g Fa(\051)g(is)r(considered)q(to)r(b)q (e)j(a)q(graded)r 1679 100 C(mo)g(dule)i(b)23 X(y)d(the)t(grading)r(in) t(the)r(\014rst)t(comp)r(onen)k(t)c(\050the)u(grading)q(induced)u(from) q(that)u(on)q F9 99 G(A)g Fa(\051.)g(Then)35 X F9(E)23 X Fa(can)x(b)t(e) i(de\014ned)t(lo)s(cally)j(as)s(follo)t(ws:)d(if)35 X F9(U)22 X Fa(is) 32 X(an)22 X(a\016ne)u(op)t(en)j(set)t(in)u F9(Z)t Fb 2687 30 L(0)g Fa 46 30 J(and)g F8 2733 100 C(O)g Fc 66 12 J(U)g Fa 86 12 L(is)g(the)30 X (ring)t(suc)t(h)e(that)t F9(U)t Fa(=)z(Sp)q(ec\050)i F8(O)g Fc 1228 12 J(U)g Fa 56 12 L(\051,)g(then)u(the)t(\014b)t(er)i(of)t F9(E)t Fa(o)y(v) d(er)e F9(U)t Fa(is)39 X(de\014ned)30 X 1436 99 C(to)g(b)t(e)i(Sp)t (ec\050)i F9(C)g Fa(\051,)m(where)t F9(C)t Fa(is)z(the)t(asso)t(ciated) i(graded)t(ring)t(of)t F8(O)t Fc 2148 12 J(U)g Fa 85 12 L(b)g(y)e(the)t (restriction)t 2233 100 C(of)g F9(I)z Fa(to)42 X F9(U)n Fa(.)9 X(Since) 61 X F9(I)35 X Fa(is)43 X(lo)35 X(cally)j(generated)t(b)u(y)d(a)u (regular)t(sequence,)u F9(C)v Fa(is)42 X(lo)35 X(cally)j(a)t H(p)g (olynomial)i(ring)24 X(o)s(v)e(er)d F8(O)t Fc 832 12 J(U)g Fa 56 12 L(.) g(W)35 X(e)a(note)22 X(that)w F8(O)t Fc 617 12 J(Z)g Fe 49 17 L(0)g Fa 50 5 J(is)g(a)u(quotien)v(t)d(of)v(b)u(oth)j F9(A)u Fa(and)v F9(g)u(r)j Fc 1070 12 J(I)g Fa 38 12 L(\050)g F9(A)g Fa(\051.)g 2712 99 C(Let)g F8 (M)z Fa(denote)y(the)z(sheaf)y(on)z F9(E)y Fa(de\014ned)33 X(b)n(y)e (the)n(graded)o(mo)n(dule)j F9(g)n(r)k Fc 2297 12 J(I)g Fa 38 12 L(\050) g F9(B)g Fa(\051.)k 2210 100 C(W)g(e)-7 X(next)40 X(sho)t(w)e(that)t (the)t(Serre)t(sp)t(ectral)i(sequence)t(implies)t(that)t(w)t(e)e(ha)t(v) e(e)d(an)u 125 100 C(equalit)g(y:)e F9 875 99 J(\037)g Fc 52 12 J(E)g Fa 56 12 L(\050)g F8(O)g Fc 98 12 J(Z)g Fe 49 17 L(0)g F9 27 5 J(;)g F8 (M)14 X Fa(\051\051)g(=)23 X F9(\037)t Fc 364 12 J(P)g Fa 55 12 L(\050) g F8(O)g Fc 98 12 J(Z)g Fe 49 17 L(0)g F9 27 5 J(;)g F8(O)14 X Fc 103 12 J(Y)g Fe 53 17 L(0)g Fa 27 5 J(\051)g F9(:)g Fa 1808 149 C(Let)g 747 100 J(0)g F8(!)23 X(F)t Fc 231 12 J(k)g F8 64 12 L(!)g F9(:)t(:)14 X(:) t F8(!)23 X(F)t Fd 392 12 J(1)g F8 60 12 L(!)g(F)t Fd 166 12 J(0)g F8 60 12 L(!)g Fa(0)t 1845 149 C(b)g(e)i(a)z(lo)z(cally)i(free)z (resolution)z(of)z F8(O)z Fc 1159 12 J(Z)g Fe 49 17 L(0)g Fa 56 5 J(o)g (v)e(er)d F8(O)30 X Fc 245 12 J(P)g Fa 55 12 L(.)g(W)41 X(e)-7 X(apply) 29 X(the)t(argumen)t(t)e(of)t(section)t 1564 100 C(2)g(to)p(the)p (\014ltration)p(of)p F8(F)p Fb 802 12 J(\017)g Fa 63 12 L(induced)g(b)p (y)e(the)p(p)p(o)i(w)e(ers)d(of)q F9(I)o Fa(.)8 X(Since)35 X F9(I)26 X Fa(is)y(lo)t(cally)h(generated)t 865 100 C(b)g(y)e(a)35 X(regular)t (sequence,)t(the)v(same)t(argumen)t(t)e(go)t(es)i(through.)t(Ho)59 X(w) d(ev)e(er,)d(there)37 X(are)r H(t)g(w)e(o)d(p)26 X(oin)h(ts)f(whic)r(h) f(are)r(di\013eren)t(t)d(from)t(the)r(case)t(of)r(the)t(Koszul)r (complex.)t(First)35 X(of)26 X(all,)r F8 99 G(F)g Fb 60 12 J(\017)g Fa 63 12 L(will)g(in)s(general)s(not)s(b)s(e)i(a)s(minimal)s(complex)s(lo) s(cally)i(,)-7 X(so)25 X(that)t(it)t(is)t(lo)t(cally)i(a)t(direct)t 123 100 C(sum)g(of)z(a)z(Koszul)z(complex)z(and)z(a)z(trivial)z(\050split)z (exact\051)z(complex.)z(Ho)48 X(w)d(ev)e(er,)d(in)33 X(the)q 99 G(lo)g (cal)i(computation,)26 X(the)t(split)t(exact)t(part)t(is)t(eliminated)t (in)t(the)t(step)t(from)t F9(E)t Fd 2577 30 L(0)g Fa 63 30 J(to)g F9(E) t Fd 166 30 L(1)g Fa 37 30 J(,)g 2843 100 C(so)g(from)z(that)z(p)z(oin) i(t)e(the)z(argumen)z(t)e(go)z(es)i(through)z(as)z(b)z(efore.)i(The)50 X(second)32 X(p)t(oin)i(t)e(is)t H(that)g(in)r(taking)r(the)r(homology) r(at)r F9(E)r Fd 1235 30 L(1)g Fa 37 30 J(,)g(the)s(homology)r(is)r(no) r(longer)r(of)r(\014nite)r(length,)r 1272 99 C(but)g(only)r(supp)s (orted)i(at)s(the)r(maximal)s(ideal)r(of)s F9(R)r Fa(.)h(Ho)45 X(w)e(ev) d(er,)e(it)31 X(is)t(still)s(zero)t(except)s 100 G(for)g(\014nitely)t (man)t(y)e F9(i)t Fa(and)t F9(n)t Fa(and)t(w)t(e)e(can)t(conclude)t (that)t(the)t(Euler)t(c)t(haracteristic)e(is)t H(the)g(same)x(using)x (the)x(additivit)x(y)e(of)x F9(\036)x Fb 1280 12 J(\003)g Fa 73 12 L (and)g(the)x(Euler)x(c)x(haracteristic)e(on)x(Sp)x(ec)i(\050)g F9(R)g Fa(\051.)h 1353 99 C(Th)g(us)e(the)p(argumen)q(t)d(go)q(es)i(through,)q (and)p(w)q(e)d(ha)q(v)d(e)e(the)p(ab)q(o)i(v)e(e)d(equalit)q(y)d 125 100 J(There)g(is)p(one)p(more)p(reduction,)p(whic)p(h)e(reduces)p(to)p (the)p(\014b)p(ers)i(o)p(v)e(er)d(Sp)q(ec\050)h F9(R)h(=)g F1(m)h Fa (\051)f(=)24 X F9 125 99 C(s)g Fa(.)g(Let)65 X F8(M)37 X Fa(b)t(e)i(the) t(sheaf)t(on)t F9(E)t Fa(asso)y(ciated)i(graded)t(ring)t F9(g)t(r)j Fc 2091 12 J(I)g Fa 38 12 L(\050)g F9(B)g Fa(\051)k(considered)t(as)t(a)t 2129 100 C(mo)g(dule)i(o)r(v)d(er)e F9(g)q(r)j Fc 565 12 J(I)g Fa 38 12 L(\050)g F9(A)g Fa(\051.)g(Then,)57 X(since)36 X(for)r(the)s(original)r (ideals)s F1(p)r Fa(and)s F1(q)r Fa(w)s(e)d(had)s(that)r F9 603 100 C(R) g(=)h F1(p)g F8(\012)15 X F9(R)u(=)g F1(q)h Fa(had)25 X(\014nite)v (length,)t F8(M)w Fa(is)t(annihilated)v(b)t(y)f(a)t(p)v(o)h(w)f(er)c(of) w(the)s(maximal)w(ideal)s F1 99 G(m)g Fa(of)t F9(R)t Fa(.)h(Hence)36 X (it)25 X(has)t(a)t(\014nite)t(\014ltration)t(with)t(quotien)t(ts)e F8(M) t Fc 2037 12 J(i)g Fa 53 12 L(whic)g(h)e(are)t(annihilated)t 679 247 C (9)g EOP end %%PageTrailer %======================================================================= %%Page: 10 10 %%BeginPageSetup DVIALWdict begin %%EndPageSetup BOP Fa 1123 5385 A(b)g(y)e F1(m)36 X Fa(.)h(It)62 X(then)38 X(su\016ces)q (to)t(sho)q(w)f(that)q F9(\037)t Fc 1430 12 J(E)g Fa 56 12 L(\050)g F8 (O)g Fc 98 12 J(Z)g Fe 49 17 L(0)g F9 27 5 J(;)g F8(M)14 X Fc 137 12 J (i)g Fa 27 12 L(\051)g(is)37 X(non-negativ)s(e)e(for)s(eac)t(h)d F9(i)t Fa(.)g 1824 100 C(W)g(e)-7 X(can)39 X(compute)s(this)t(Euler)s(c)t (haracteristic)d(b)t(y)d(taking)t(a)s(lo)t(cally)i(free)t(resolution)s H(for)g F8(O)r Fc 203 12 J(Z)g Fe 49 17 L(0)g Fa 64 5 J(and)g(tensoring) r(with)s F8(M)r Fc 840 12 J(i)g Fa 28 12 L(,)g(and,)u(since)t F8(M)s Fc 573 12 J(i)g Fa 65 12 L(is)g(annihilated)r(b)s(y)d F1(m)s Fa(,)f(w)v(e) c(can)t 1822 100 C(tensor)g(\014rst)32 X(with)t F9(R)t(=)g F1(m)h Fa(.) f(Let)50 X F9(s)31 X Fa(denote)v(the)t(closed)v(p)t(oin)j(t)d(of)u(Sp)u (ec\050)i F9(R)g Fa(\051)h(as)t(ab)v(o)h(v)f(e,)c(let)w F9 H(E)g Fc 61 12 J(s)g Fa 64 12 L(=)g(Pro)r(j)k(\050)h F9(g)f(r)k Fc 366 12 J(I)g Fa 38 12 L(\050)g F9(A)g Fa(\051)g F8(\012)21 X Fc 212 12 J(R)g F9 75 12 L (k)g Fa(\051,)j(where)32 X F9(k)s Fa(=)t F9(R)p(=)h F1(m)g Fa(,)g(and)t (let)s F9(Z)s Fb 1121 30 L(0)g Fc 6 51 C(s)g Fa 64 21 L(=)g(Pro)q(j)k (\050)h F9(A=I)f F8(\012)p Fc 518 12 J(R)g F9 75 12 L(k)g Fa(\051)j F9 (:)g Fa(The)s 2588 100 C(ab)g(o)i(v)e(e)d(argumen)s(t)d(sho)s(ws)d(that) s(for)q(eac)s(h)d F9(i)s Fa(w)q(e)f(ha)q(v)f(e)c F9(\037)t Fc 1785 12 J (E)g Fa 56 12 L(\050)g F8(O)g Fc 98 12 J(Z)g Fe 49 17 L(0)g F9 27 5 J(;) g F8(M)14 X Fc 137 12 J(i)g Fa 27 12 L(\051)g(=)27 X F9(\037)t Fc 203 12 J(E)g Ff 49 8 J(s)g Fa 36 20 L(\050)g F8(O)g Fc 98 12 J(Z)g Fe 49 17 L(0)g Ff 4 33 C(s)g F9 36 28 L(;)g F8(M)14 X Fc 137 12 J(i)g Fa 28 12 L (\051.)g 2811 100 C(Hence)g F9 455 100 J(\037)g Fc 52 12 J(E)g Fa 56 12 L(\050)g F8(O)g Fc 98 12 J(Z)g Fe 49 17 L(0)g F9 27 5 J(;)g F8(M)t Fa (\051)g(=)23 X F2 280 79 L(X)g Fc 48 177 J(i)g F9 85 98 L(\037)g Fc 52 12 J(E)g Fa 56 12 L(\050)g F8(O)g Fc 98 12 J(Z)g Fe 49 17 L(0)g F9 27 5 J(;)g F8(M)14 X Fc 137 12 J(i)g Fa 28 12 L(\051)g(=)23 X F2 143 79 L(X) g Fc 48 177 J(i)g F9 86 98 L(\037)g Fc 52 12 J(E)g Ff 49 8 J(s)g Fa 36 20 L(\050)g F8(O)g Fc 98 12 J(Z)g Fe 49 17 L(0)g Ff 4 33 C(s)g F9 36 28 L(;)g F8(M)14 X Fc 137 12 J(i)g Fa 28 12 L(\051)g F9(:)g Fa 2230 226 C (W)g(e)-7 X(recall)33 X(that)t(the)t(dimension)t(of)t F8(M)t Fa(is)t (equal)t(to)t(dim)t(\050)g F9(R)g(=)h F1(p)g Fa(\051)g(+)22 X F9(n)t Fa (\050where)33 X F9(A)t Fa(=)s 125 100 C(Pro)g(j)l(\050)g F9(R)g Fa(\133) h F9(X)g Fd 349 12 J(0)g F9 37 12 L(;)g(:)14 X(:)t(:)t(;)t(X)t Fc 254 12 J(n)g Fa 45 12 L(\135\051.)g(Th)83 X(us)e(to)43 X(pro)t(v)e(e)d(the) u(v)s(anishing)c(and)s(non-negativit)u(y)d(con-)u 685 100 C(jectures)g (it)o(su\016ces)o(to)o(sho)o(w)d(that)o(whenev)o(er)d F8(M)o Fa(is)o(a) o(coheren)o(t)d(sheaf)o(on)o F9(E)o Fc 2659 12 J(s)g Fa 73 12 L(and)g 2732 99 C(dim\050)g F8(M)g Fa(\051)g(+)17 X(dim)u(\050)f F9(Z)h Fb 635 30 L(0)g Fa 23 30 J(\051)g F8(\024)z Fa(dim\050)z F9(R)g Fa(\051)h(+)t F9(n)u Fa(w)26 X(e)f(ha)t(v)f(e)c F9(\037)w Fc 951 12 J(E)g Ff 49 8 J(s) g Fa 36 20 L(\050)g F8(O)g Fc 98 12 J(Z)g Fe 49 17 L(0)g F9 27 5 J(;)g F8(M)14 X Fa(\051)g F8(\025)23 X Fa(0,)t(and)x(that)x(w)x(e)e(ha)x(v)e (e)d 1868 100 C(equalit)g(y)e(when)x(dim\050)y F8(M)g Fa(\051)g(+)o(dim) p(\050)f F9(Z)h Fb 1171 30 L(0)g Fa 23 30 J(\051)g F9(\074)t Fa(dim\050) t F9(R)g Fa(\051)h(+)o F9(n)p Fa(.)f 1069 100 C(T)g(o)-7 X(pro)30 X(v)d (e)e(the)s(p)t(ositivit)i(y)e(conjecture)s(it)t(w)s(ould)e(of)s(course) t(su\016ce)s(to)t(sho)s(w)e(that)s(if)t 125 99 C(dim\050)g F8(M)g Fa (\051)g(+)13 X(dim\050)t F9(Z)g Fb 626 30 L(0)g Fa 23 30 J(\051)g(=)23 X(dim\050)t F9(R)g Fa(\051)h(+)12 X F9(n)v Fa(,)f(the)26 X(Euler)r(c)t (haracteristic)d(is)t(p)r(ositiv)j(e.)d(Ho)37 X(w)d(ev)e(er,)d 649 100 C(this)g(is)26 X(not)t(true)t(in)t(generatl)t(\050w)t(e)e(giv)t(e)e(an) t(example)t(b)t(elo)i(w\051.)e(Ho)36 X(w)e(ev)d(er,)e(assuming)26 X(the) u 99 G(non-negativit)g(y)e(conjecture)p(for)q(a)p(momen)q(t,)d(w)r(e)d (sho)q(w)d(that)q(there)p(is)q(a)p(simple)q(criterion)p 100 G(for)g(p)v (ositivit)i(y)e(.)-7 X F6 183 G(Prop)g(osition)j(1)31 X F7(L)42 X(et)b (notation)n(b)35 X(e)c(as)t(ab)u(ove,)b(and)w(let)s F8(M)v Fd 1958 12 J (0)g F7 73 12 L(b)g(e)c(the)t(sehaf)u(de\014ne)t(d)c(by)t F9 2031 99 C (g)g(r)j Fc 80 12 J(I)g Fa 38 12 L(\050)g F9(B)g Fa(\051)k F8(\012)9 X Fc 205 12 J(R)g F9 64 12 L(k)g F7(c)28 X(onsider)c(e)c(d)c(as)q(a)r(mo) q(dule)c(over)q F9(g)r(r)i Fc 1173 12 J(I)g Fa 38 12 L(\050)g F9(A)g Fa (\051)g F8(\012)9 X Fc 201 12 J(R)g F9 64 12 L(k)g F7(.)j(Assume)36 X (that)26 X(dim)s Fa(\050)g F9(R)g(=)h F1(p)g Fa(\051)g(+)9 X F7 1863 100 C(dim)g Fa(\050)g F9(R)g(=)h F1(q)g Fa(\051)g(=)23 X F7(dim)t Fa (\050)g F9(R)g Fa(\051)h F7(.)g(Then)38 X(the)29 X(p)t(ositivity)c(c)t (onje)c(ctur)c(e)c(holds)t(for)t(the)t(ide)t(als)c F1(p)t F7(and)t F9(q) t F7 H(if)g(and)u(only)t(if)u F9(\037)t Fc 553 12 J(E)g Ff 49 8 J(s)g Fa 36 20 L(\050)g F8(O)g Fc 98 12 J(Z)g Fe 49 17 L(0)g F9 27 5 J(;)g F8 (M)14 X Fd 137 12 J(0)g Fa 37 12 L(\051)g F9(\076)23 X Fa(0)t F7(.)g F6 986 182 C(Pro)g(of.)j Fa(Since)55 X F8(M)34 X Fd 637 12 J(0)g Fa 71 12 L(is)g(a)t(quotien)s(t)e(of)s(the)t(sheaf)s F8(M)t Fa(de\014ned)s(b)t(y) d F9(g)t(r)j Fc 1592 12 J(I)g Fa 38 12 L(\050)g F9(B)g Fa(\051)k(and)t (Euler)s 2338 100 C(c)g(haracteristics)e(are)r(non-negativ)r(e,)e(if)s F9(\037)r Fc 1342 12 J(E)g Ff 49 8 J(s)g Fa 36 20 L(\050)g F8(O)g Fc 98 12 J(Z)g Fe 49 17 L(0)g F9 27 5 J(;)g F8(M)14 X Fd 137 12 J(0)g Fa 37 12 L(\051)g F9(\076)30 X Fa(0,)u(then)w F9(\037)v Fc 500 12 J(E)g Fa 56 12 L(\050)g F8(O)g Fc 98 12 J(Z)g Fe 49 17 L(0)g F9 27 5 J(;)g F8(M)14 X Fa(\051)g F9(\076)30 X Fa(0.)u 2505 99 C(Con)g(v)e(ersely)d(,)a(supp) v(ose)j(that)u F9(\037)w Fc 996 12 J(E)g Ff 49 8 J(s)g Fa 36 20 L(\050) g F8(O)g Fc 98 12 J(Z)g Fe 49 17 L(0)g F9 27 5 J(;)g F8(M)14 X Fd 137 12 J(0)g Fa 37 12 L(\051)g(=)30 X(0.)t(Since)50 X F9(g)31 X(r)k Fc 572 12 J(I)g Fa 38 12 L(\050)g F9(B)g Fa(\051)k(is)u(annihlated)u(b)u(y)e(a) u 2039 100 C(p)g(o)i(w)e(er)d(of)w F1(m)w Fa(,)g(it)x(has)w(a)w (\014ltration)w(with)w(quotien)w(ts)d(whic)w(h)d(are)w(homomorphic)w (images)w H(of)g(direct)x(sums)x(of)x(copies)x(of)x F9(g)x(r)j Fc 1098 12 J(I)g Fa 38 12 L(\050)g F9(B)g Fa(\051)k F8(\012)23 X Fc 219 12 J(R) g F9 78 12 L(k)g Fa(.)j(Again)58 X(using)36 X(non-negativit)r(y)f(,)-8 X(w)38 X(e)d(can)r 1433 99 C(deduce)g(that)29 X(if)t F8(M)t Fc 636 12 J (i)g Fa 57 12 L(is)g(the)t(sheaf)t(de\014ned)t(b)t(y)e(an)t(y)e(of)t (these)t(quotien)t(ts,)e(then)t F8(M)t Fc 1989 12 J(i)g Fa 57 12 L(is)g (a)t 2739 100 C(quotien)g(t)e(of)t(a)t(direct)t(sum)t(of)t(copies)t(of) t F8(M)t Fd 1448 12 J(0)g Fa 37 12 L(,)g(so)t(w)t(e)e(ha)t(v)e(e)d F9 (\037)u Fc 523 12 J(E)g Ff 49 8 J(s)g Fa 36 20 L(\050)g F8(O)g Fc 98 12 J(Z)g Fe 49 17 L(0)g F9 27 5 J(;)g F8(M)14 X Fc 137 12 J(i)g Fa 28 12 L (\051)g(=)25 X(0.)t(Th)41 X(us)d 2432 100 C(the)g(addivitiy)28 X(of)s (the)t(Euler)s(c)t(haracteristic)d(implies)t(that)s F9(\037)t Fc 1980 12 J(E)g Fa 56 12 L(\050)g F8(O)g Fc 98 12 J(Z)g Fe 49 17 L(0)g F9 27 5 J(;)g F8(M)14 X Fa(\051)g(=)23 X(0.)t F4 2210 374 C(4)g(Some)137 X (examples)46 X Fa 182 G(W)g(e)-7 X(giv)33 X(e)d(here)t(three)s(simple)t (examples)s(of)t(the)s(t)t(yp)d(e)j(of)s(bigraded)t(ring)s(whic)t(h)d (migh)t(t)d 99 G(result)g(from)s(this)s(construction.)s(Eac)49 X(h)e(of) 31 X(these)v(examples)t(is)v(obtained)t(b)v(y)c(taking)v(a)t 100 G (birational)g(map)24 X(from)t(a)t(regular)t(sc)t(heme)e(to)t(Sp)t (ec\050)i F9(R)g(=)h F1(q)g Fa(\051,)g(and)u(the)s(last)u(t)s(w)f(o)c (are)v(simple)r 99 G(resolutions)g(of)x(singularities.)w 125 100 J(W)g (e)-7 X(\014rst)38 X(summarize)t(the)t(construction)t(up)t(to)t(this)t (p)t(oin)i(t.)e(W)67 X(e)a(b)37 X(egan)j(with)t(a)v 125 100 C(regular)g (lo)21 X(cal)j(ring)t F9(R)u Fa(and)u(prime)u(ideals)t F1(p)u Fa(and)t F1(q)u Fa(.)f(W)36 X(e)-8 X(then)23 X(to)q(ok)k(a)q(regular)t(subsc)q (heme)f 1391 294 J(10)g EOP end %%PageTrailer %======================================================================= %%Page: 11 11 %%BeginPageSetup DVIALWdict begin %%EndPageSetup BOP F9 1123 5385 A(Z)g Fb 63 30 L(0)g Fa 61 30 J(of)g(Pro)37 X(j)l(\050)g F9(R)g Fa(\133)h F9(X)g Fd 453 12 J(0)g F9 37 12 L(;)g(:)14 X(:)t(:)t(;) t(X)t Fc 254 12 J(n)g Fa 45 12 L(\135\051)g(whic)37 X(h)e(w)t(as)e (generically)t(\014nite)u(o)t(v)e(er)d(Sp)u(ec)i(\050)g F9(R)g(=)h F1(q) g Fa(\051.)g(The)66 X 913 100 C(next)g(step)37 X(w)t(as)d(to)t(replace) t F9(R)t Fa(\133)g F9(X)h Fd 1108 12 J(0)g F9 37 12 L(;)g(:)14 X(:)t(:) t(;)t(X)t Fc 254 12 J(n)g Fa 45 12 L(\135)g(with)37 X(the)t(asso)t (ciated)i(graded)t(ring)t(of)t F9(I)t Fa 1444 100 C(tensored)g(with)n F9(R)o(=)g F1(m)h Fa(=)28 X F9(k)w Fa(.)h(The)50 X(sheaf)30 X F8(O)w Fc 1393 12 J(Y)g Fe 53 17 L(0)g Fa 58 5 J(de\014ned)g(b)u(y)e F9(B)u Fa(=) x F9(R)s(=)h F1(p)g Fa(\133)g F9(X)g Fd 838 12 J(0)g F9 37 12 L(;)g(:) 14 X(:)t(:)t(;)t(X)t Fc 254 12 J(n)g Fa 45 12 L(\135)g(w)31 X(as)e 2678 100 C(then)g(replaced)o(with)p(the)o(sheaf)p F8(M)o Fa(de\014ned)p(b)o (y)e F9(g)o(r)j Fc 1655 12 J(I)g Fa 38 12 L(\050)g F9(B)g Fa(\051,)k (again)p(tensored)o(with)p F9(k)o Fa(.)j(The)y 1693 100 C(assumption)g (of)v(regularit)u(y)e(implies)u(that)v F9(I)u(=I)7 X Fd 1528 30 L(2)g Fa 69 30 J(is)g(lo)33 X(cally)i(free)t(o)s(v)e(er)d F9(A=I)t Fa(;)m (denote)w(its)s 1597 100 C(rank)g F9(r)24 X Fa(.)j(Then)35 X(the)25 X (dimension)s(of)t F8(M)s Fa(is)t(at)s(most)t F9(r)s Fa(,)j(and)s(it)t (is)s(equal)t(to)s F9(r)t Fa(if)u(and)t(only)s(if)t H(w)g(e)e(had)x (dim\050)x F9(R)g(=)h F1(p)g Fa(\051)g(+)n(dim)o(\050)f F9(R)h(=)g F1(q) h Fa(\051)f(=)u(dim\050)t F9(R)g Fa(\051.)h(W)41 X(e)-7 X(note)29 X (that)t(the)t(\014b)t(er)i F9(Z)t Fb 2446 30 L(0)g Fc 6 51 C(s)g Fa 64 21 L(of)g F9(Z)t Fb 159 30 L(0)g Fa 52 30 J(o)g(v)e(er)d 2715 100 C(the) g(maximal)x(ideal)w(of)x F9(R)w Fa(has)y(dimension)w(at)x(most)w (dim\050)x F9(R)g(=)h F1(q)g Fa(\051)g F8(\000)22 X Fa(1,)t(but)34 X (apart)s(from)r 99 G(that)g(w)n(e)e(do)n(not)n(kno)n(w)e(m)n(uc)e(h)d (ab)o(out)h(it.)o(It)w(is)o(the)27 X(pro)v(jectiv)k(e)e(sc)u(heme)e (de\014ned)u(b)u(y)e(the)u 100 G(graded)g(ring)w(\050)w F9(A=I)g Fa (\051)7 X F8(\012)19 X Fc 745 12 J(R)g F9 74 12 L(k)g Fa(,)j(whic)30 X (h)e(is)s(the)u(part)s(of)u(degree)s(zero)u(in)s(the)u(grading)s(in)u (the)s 819 100 C(second)g(comp)r(onen)i(t)e(of)q(the)r(bigraded)q(ring) r(w)q(e)e(are)q(considering.)r 125 99 J(F)g(or)-7 X(the)32 X(\014rst)t (example,)t(let)u F9(R)t Fa(ha)u(v)d(e)e(dimension)t(four,)t(let)u F9 (t;)t(u;)14 X(v)t(;)j(w)t Fa(b)34 X(e)j(a)q(regular)s 125 100 C(system) g(of)o(parameters,)o(and)o(de\014ne)o(the)o(prime)o(ideals)o F1(p)o Fa (and)o F1(q)o Fa(b)o(y)e(letting)o F1(p)o Fa(=)25 X(\050)u F9(t;)f(u)15 X Fa(\051)f H(and)g F1(q)31 X Fa(=)r(\050)q F9(v)h(;)i(w)15 X Fa(\051.) h(In)48 X(this)30 X(case,)v(Sp)u(ec\050)j F9(R)f(=)i F1(q)f Fa(\051)h (is)t(already)v(regular,)t(and)w(w)t(e)f(can)t(simply)v 99 G(tak)g(e)e (the)q(pro)r(jectiv)k(e)e(sc)q(heme)e(Pro)q(j\050)l F9(R)g Fa(\133)h F9 (X)g Fa(\135\051)7 X(=)23 X(Sp)t(ec\050)i F9(R)g Fa(\051.)h 125 100 J(F) g(or)-7 X(a)40 X(sligh)t(tly)e(more)t(complicated)t(example,)t(consider) w(the)t(subsc)t(heme)e(of)t(the)t 125 99 C(pro)g(jectiv)l(e)d(space)n (Pro)n(j\050)k F9(R)h Fa(\133)g F9(X)h(;)h(Y)15 X Fa(\135\051)v (de\014ned)35 X(b)r(y)e(the)r(ideal)t F9(I)r Fa(generated)z(b)r(y)e F9 (X)r(u)7 X F8(\000)22 X F9(Y)u(t)p Fa(.)g 100 G(Then)g F9(Z)29 X Fb 281 30 L(0)g Fa 52 30 J(is)g(the)t(blo)t(w)e(up)t(of)t F9(R)t(=)h F1(q)g Fa (at)t(the)t(p)t(oin)i(t)e(de\014ned)t(b)t(y)e(the)t(maximal)t(ideal,)t (and)u 333 100 C(the)g(\014b)36 X(er)i(o)t(v)e(er)d F9(s)u Fa(is)s(pro) u(jectiv)k(e)e(space)t(of)t(dimension)t(1.)t(One)62 X(could)36 X (de\014ne)t(similar)t 99 G(examples)g(in)28 X(higher)s(dimension.)t 125 100 J(F)g(or)-7 X(a)38 X(third)t(example,)t(let)v F9(R)t Fa(ha)t(v)e(e) d(dimension)u(2,)s(and)w(let)s F9(I)u Fa(b)y(e)k(generated)r(b)v(y)c F9 125 100 C(X)g(u)7 X F8(\000)15 X F9(Y)u(t;)w(Z)s(u)m F8(\000)t F9(X)u (t;)m(X)s Fd 777 31 L(2)g F8 52 31 J(\000)g F9(Y)t(Z)x Fa(.)l(The)37 X (pro)26 X(jectiv)l(e)d(space)u F9(P)s Fa(has)39 X(dimension)25 X(2,)v (and)u(the)v 829 99 C(\014b)g(er)i(o)y(v)d(er)e(the)x(maximal)z(ideal)x (has)z(co)x(dimension)j(one)x(in)z(Pro)x(j\050)l F9(k)g Fa(\133)j F9(X) g(;)i(Y)14 X(;)k(Z)t Fa(\135\051)m(and)30 X(th)t(us)d 100 G(has)g (dimension)v(one.)u(The)49 X(sheaf)31 X(de\014ned)u(b)t(y)e F9(I)t(=I)7 X Fd 1672 30 L(2)g Fa 68 30 J(has)g(rank)32 X(2,)s(but)u F9(I)s Fa(is) 39 X(minimally)31 X 1740 99 C(generated)g(b)q(y)d(three)q(elemen)p(ts.) e 125 100 J(In)g(the)x(ab)x(o)i(v)e(e)d(examples)y(it)w(w)y(as)d(not)y (really)w(necessary)y(to)w(reduce)y(to)w(pro)y(jectiv)k(e)e 125 100 C (space)g(since)v(the)w(original)v(quotien)w(ts)d F9(R)w(=)g F1(q)h Fa(w) u(ere)f(regular.)u(W)55 X(e)-8 X(next)34 X(consider)s(an)t(ex-)s 99 G (ample)g(where)o(the)n(original)o(sc)n(heme)e(is)n(not)o(regular.)n(Let) 41 X F1(m)28 X Fa(b)u(e)i(minimally)u(generated)t 100 G(b)g(y)e F9(t;)x (u)14 X Fa(,)g(and)34 X(let)r F1(q)s Fa(b)r(e)j(the)r(principal)s(ideal) r(generated)s(b)r(y)e F9(t)r Fd 2011 30 L(2)g F8 59 30 J(\000)g F9(u)22 X Fd 134 30 L(3)g Fa 37 30 J(.)g(W)52 X(e)-7 X(can)33 X(resolv)s(e)e 2241 100 C(the)g(singularit)u(y)e(b)u(y)e(letting)u F9(Z)v Fb 1022 31 L (0)g Fa 58 31 J(b)g(e)i(de\014ned)v(b)u(y)e(the)u(ideal)v(in)u F9(R)v Fa(\133)g F9(X)h(;)h(Y)15 X Fa(\135)v(generated)36 X(b)q(y)f F9 1080 99 C(t)g Fd 30 30 L(2)g F8 56 30 J(\000)g F9(u)18 X Fd 131 30 L(3)g F9 37 30 J(;)g(uX)p F8(\000)25 X F9(tY)18 X(;)l(X)o Fd 458 30 L(2)g F8 55 30 J(\000)g F9(uY)t Fd 197 30 L(2)g Fa 38 30 J(.)g(The)37 X(\014b)28 X(er) i F9(Z)t Fb 485 30 L(0)g Fc 6 51 C(s)g Fa 63 21 L(in)g(this)t(case)s(is) t(Pro)s(j\050)l F9(k)g Fa(\133)j F9(X)g(;)i(Y)14 X Fa(\135)x F9(=)g Fa (\050)g F9(X)g Fd 1126 30 L(2)g Fa 37 30 J(\051\051.)g 2582 100 C (Finally)g(,)-7 X(w)42 X(e)e(consider)q(the)r(case)q(where)r F1(q)q Fa (is)r(the)q(determinan)r(tal)d(ideal)r(in)q F9(R)r Fa(of)q 125 99 C (dimension)g(4)s(generated)t(b)s(y)e F9(w)s(u)j F8(\000)27 X F9(t)u Fd 1271 30 L(2)g F9 37 30 J(;)g(w)14 X(v)i F8(\000)31 X F9(tu)p Fa(,)h(and) 44 X F9(tv)r F8(\000)30 X F9(u)r Fd 822 30 L(2)g Fa 37 30 J(.)g(In)78 X (this)41 X(case)u(the)t 2167 100 C(resolution)g(can)26 X(b)s(e)j(found) 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X (section)s(w)t(e)d(sho)t(w)d(that)t(similar)s(results)t(hold)s(in)t(the) s(presen)t(t)d H(situation.)g(W)t(e)-7 X(no)27 X(w)e(let)t F9(C)t Fa (denote)z(the)t(bigraded)t(ring)t(whic)t(h)e(w)t(e)e(previously)t (denoted)t F9 H(g)g(r)j Fc 80 12 J(I)g Fa 38 12 L(\050)g F9(A)g Fa(\051) g F8(\012)i Fc 193 12 J(R)g F9 56 12 L(k)g Fa(,)j(where)n F9(C)19 X Fc 381 12 J(i;j)g Fa 97 12 L(consists)g(of)t(the)t(elemen)t(ts)e(of)t(\050) t F9(I)g Fc 1010 30 L(j)g F9 34 30 J(=I)g Fc 85 30 L(j)g Fd(+1)j Fa 119 30 J(\051)g F8(\012)i F9(k)h Fa(of)w(degree)t F9(i)t Fa(.)g(Th)34 X(us) e 2093 100 C(in)g(our)q(presen)q(t)e(notation,)q F9(E)r Fc 966 12 J(s)g Fa 64 12 L(=)g(Pro)o(j\050)l F9(C)g Fa(\051,)m(where)r(the)q(grading)q (on)q F9(C)q Fa(is)w(that)q(in)q(the)q 1030 100 C(\014rst)g(co)25 X (ordinate.)j(Let)35 X F9(C)26 X Fd 815 12 J(0)g Fa 63 12 L(denote)g(the) s(subring)t F8(\010)s Fc 764 12 J(i)g Fa 28 12 L(\050)g F9(C)g Fc 91 12 J(i;)g Fd(0)g Fa 80 12 L(\051.)g(Let)36 X F9(r)25 X Fa(b)w(e)i(the)u 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(degree)n(\050)n F9(k)g Fc 480 12 J(i)g F9 28 12 L(;)g Fa(1\051)14 X(in) 26 X(the)t(bigrading)t(on)t F9(C)t Fa(.)m(As)36 X(men)26 X(tioned)d(ab) t(o)i(v)e(e,)d(the)u 994 100 C(non-negativit)g(y)e(conjecture)x(will)x (b)x(e)i(reduced)x(to)x(this)x(situation)x(in)x(the)x(next)x(section.)x H(In)g(this)z(case)z(the)z(resolution)z(is)z(the)z(usual)z(Koszul)z (complex)z(on)z F9(T)z Fd 2222 12 J(1)g F9 37 12 L(;)g(:)14 X(:)t(:)t(;) t(T)t Fc 233 12 J(r)g Fa 37 12 L(,)g(and)33 X(the)s 2529 100 C(Hilb)g (ert)i(p)n(olynomial)i(of)n F9(M)n F8(\012)n F9(F)17 X Fc 1050 12 J(i)g Fa 55 12 L(is)g(a)27 X(sum)t(of)t(Hilb)t(ert)i(p)t(olynomials)i(of)t F9 (M)t Fa(with)36 X(shifts)27 X 1105 99 C(in)g(the)41 X(degrees.)u(F)78 X (urthermore,)-7 X(the)45 X(Koszul)p(complex)q(on)p F9(T)q Fd 2115 12 J (1)g F9 37 12 L(;)g(:)14 X(:)t(:)t(;)t(T)t Fc 234 12 J(r)g Fa 78 12 L (is)g(a)41 X(tensor)u 2464 100 C(pro)g(duct)i(of)q(Koszul)q(complexes)q (on)q(the)q(individual)q F9(T)q Fc 1845 12 J(i)g Fa 28 12 L(,)g(and)t(w) q(e)e(can)q(compute)q(the)q 1873 100 C(Hilb)g(ert)i(p)32 X(olynomial)i (of)t(the)s(tensor)t(pro)s(duct)j F9(K)s Fb 1606 12 J(\017)g Fa 38 12 L (\050)g F9(T)g Fd 81 12 J(1)g F9 37 12 L(;)g(:)14 X(:)t(:)t(;)t(T)t Fc 234 12 J(r)g Fa 37 12 L(\051)g F8(\012)21 X F9(M)t Fa(b)41 X(y)d (tensoring)32 X(b)s(y)e 2033 99 C(eac)g(h)e(factor)t F9(K)t Fb 506 12 J (\017)g Fa 38 12 L(\050)g F9(T)g Fc 81 12 J(i)g Fa 27 12 L(\051)g(in)t (turn)t(and)t(k)t(eeping)e(trac)t(k)e(of)t(the)t(result.)t(As)50 X(ab) 32 X(o)i(v)e(e,)d(assume)v 652 100 C(that)g(the)y(dimension)y(of)y F9(M) y Fa(is)46 X(at)37 X(most)t F9(r)t Fa(,)i(and)w(let)s F9(Q)u Fc 1858 30 L(r)g 53 G(M)g Fa 74 23 L(\050)g F9(m;)g(n)14 X Fa(\051)g(b)37 X(e)i (the)t(comp)t(onen)i(t)e 1932 100 C(of)g F9(P)r Fc 155 12 J(M)g Fa 74 12 L(\050)g F9(m;)g(n)14 X Fa(\051)g(of)35 X(degree)t F9(r)t Fa(.)i(Let) 59 X F9(T)34 X Fc 953 12 J(i)g Fa 62 12 L(ha)g(v)e(e)d(degree)v(\050)t F9(k)h(;)i Fa(1\051,)14 X(and)36 X(consider)t(the)r(Hilb)t(ert)h 1244 99 C(p)g(olynomial)i(obtained)28 X(b)s(y)e(tensoring)s(with)t(the)s (complex)t 938 203 J(0)g F8(!)o F9(C)o Fa(\133\050)m F8(\000)g F9(k)g(;) j F8(\000)13 X Fa(1\051\135)h Fc 633 48 L(T)g Ff 39 8 J(i)g F8 48 40 C (!)g F9(C)23 X F8(!)z Fa(0)t F9(:)g Fa 1562 183 C(The)g(Hilb)30 X(ert)j (p)t(olynomial)j(of)t(the)t(resulting)t(complex)t F9(K)t Fb 1879 12 J (\017)g Fa 38 12 L(\050)g F9(T)g Fc 81 12 J(i)g Fa 28 12 L(\051)g F8 (\012)20 X F9(M)t Fa(will)39 X(b)30 X(e)j(giv)t(en)e(b)t(y)e 2026 100 C (the)g(p)37 X(olynomial)i(whose)t(v)s(alue)c(at)s(\050)t F9(m;)g(n)14 X Fa(\051)g(is)37 X F9(P)s Fc 1587 12 J(M)g Fa 74 12 L(\050)g F9(m;)g(n) 14 X Fa(\051)g F8(\000)24 X F9(P)u Fc 389 12 J(M)g Fa 74 12 L(\050)g F9 (m)g F8(\000)t F9(k)u(;)i(n)14 X F8(\000)24 X Fa(1\051.)u(W)64 X(e)-7 X 2124 100 C(compute)g(this)28 X(di\013erence)s(for)t(a)s(monomial)t F9(m) s Fc 1524 31 L(i)g F9 28 31 J(n)g Fc 50 31 L(j)g Fa 62 31 J(and)g (obtain)t F9 1612 182 C(m)g Fc 73 34 L(i)g F9 28 34 J(n)g Fc 50 34 L(j) g F8 53 34 J(\000)g Fa(\050)18 X F9(m)g F8(\000)t F9(k)u Fa(\051)i Fc 367 34 L(i)g Fa 28 34 J(\050)g F9(n)g F8(\000)t Fa(1\051)u Fc 257 34 L (j)g Fa 58 34 J(=)g F9(m)y Fc 161 34 L(i)g F9 28 34 J(n)g Fc 50 34 L(j) g F8 53 34 J(\000)g Fa(\050)t F9(m)g Fc 188 34 L(i)g F8 46 34 J(\000)g F9(ik)t(m)j Fc 231 34 L(i)g Fb(\000)g Fd(1)g Fa 131 34 J(+)g F9(:)t(:)p (:)p Fa(\051\050)g F9(n)g Fc 294 34 L(j)g F8 53 34 J(\000)g F9(j)t(n)l Fc 172 34 L(j)g Fb(\000)j Fd(1)g Fa 138 34 J(+)g F9(:)t(:)p(:)p Fa(\051) g(=)y F9 2375 183 C(m)g Fc 73 34 L(i)g F9 28 34 J(n)g Fc 50 34 L(j)g F8 53 34 J(\000)g F9(m)t Fc 156 34 L(i)g F9 28 34 J(n)g Fc 50 34 L(j)g Fa 53 34 J(+)g F9(ik)t(m)j Fc 231 34 L(i)g Fb(\000)g Fd(1)g F9 113 34 J(n) g Fc 50 34 L(j)g Fa 53 34 J(+)g F9(j)t(m)l Fc 195 34 L(i)g F9 28 34 J(n) g Fc 50 34 L(j)g Fb(\000)j Fd(1)g Fa 138 34 J(+)g F9(:)t(:)p(:)p Fa(=)y F9(ik)y(m)j Fc 439 34 L(i)g Fb(\000)g Fd(1)g F9 112 34 J(n)g Fc 50 34 L (j)g Fa 53 34 J(+)g F9(j)t(m)l Fc 195 34 L(i)g F9 28 34 J(n)g Fc 50 34 L(j)g Fb(\000)j Fd(1)g Fa 138 34 J(+)g F9(:)t(:)p(:)p Fa 2550 149 C (where)g(the)27 X(remaining)t(terms)t(ha)t(v)e(e)d(lo)u(w)d(er)e (degree.)t(Since)37 X(w)27 X(e)e(are)t(concerned)t(with)t(the)t 100 G (comp)g(onen)i(t)e(of)y(highest)z(degree,)y(this)34 X(su\016ces)r(for)s (our)r(computation.)s(W)50 X(e)a(note)31 X(that)w 99 G(w)g(e)e(can)p (express)q(this)p(result)q(b)p(y)e(the)p(form)q(ula)d F9 921 221 J(Q)g Fc 66 36 L(r)g Fb(\000)i Fd(1)g Fc 64 G(K)g Fe 56 8 J(\017)g Fd 35 8 L (\050)g Fc(T)g Ff 65 8 J(i)g Fd 26 8 L(\051)g Fb(\012)g Fc(M)g Fa 174 28 L(=)g F9 98 56 L(@)g(Q)l 37 G 152 4 B Fc 115 67 L(r)g F9 89 143 C(@) g(n)l Fa 154 57 L(+)g F9(k)18 X 139 56 L(@)g(Q)l 37 G U Fc 115 67 L(r)g F9 100 143 C(@)g(m)l 146 57 L(:)g Fa 1796 211 C(Iterating)g(this)25 X (pro)t(cess,)i(where)u(w)t(e)e(let)t F9(T)t Fc 1336 12 J(i)g Fa 52 12 L (ha)g(v)e(e)d(degree)u(\050)s F9(k)h Fc 520 12 J(i)g F9 28 12 L(;)g Fa (1\051)14 X(for)25 X(eac)t(h)e F9(i)t Fa(,)g(w)u(e)d(ha)u(v)d(e)e F9 1295 249 C(\037)g 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Fa(\051\135)k (denotes)32 X(a)s(shift)t(in)s(degrees.)t(Supp)47 X(ose)j(some)31 X F9 (T)u Fc 2401 12 J(i)g Fa 59 12 L(is)g(not)t(in)u F1(p)t Fa(.)g 2460 100 C(Then)g F9(T)24 X Fc 262 12 J(i)g Fa 51 12 L(is)g(a)t(non-zero)s (divisor)t(on)s F9(M)t Fa(,)9 X(and)24 X(w)t(e)d(can)t(tensor)s(with)t (the)s(Koszul)t(complex)s 313 99 C(on)g F9(T)t Fc 161 12 J(i)g Fa 28 12 L(,)g(replacing)t F9(M)t Fa(with)32 X F9(M)24 X(=T)k Fc 872 12 J(i)g F9 28 12 L(M)g Fa(and)33 X(reducing)23 X F9(r)u Fa(b)v(y)f(one.)s(Th)37 X (us)c(the)25 X(result)q(follo)t(ws)c 1089 100 C(b)g(y)e(induction.)35 X (If)59 X(all)35 X F9(T)t Fc 808 12 J(i)g Fa 63 12 L(are)g(in)t F1(p)t Fa(,)g(then)v(its)t(Hilb)t(ert)i(p)t(olynomial)i(is)t(constan)t(t)e (with)t 871 100 C(resp)g(ect)i(to)s F9(n)s Fa(,)g(so)u(w)s(e)e(ha)s(v)e (e)d F9(Q)t Fc 1010 30 L(r)g Fa 37 30 J(\050)g F9(m;)g(n)14 X Fa(\051)g (=)34 X F9(\013m)t Fc 483 30 L(r)g Fa 71 30 J(for)g(some)t F9(\013)t F8 (\025)t Fa(0.)t(Hence)56 X(the)35 X(ab)r(o)j(v)d(e)e 1601 99 C(form)g (ula)e(states)27 X(that)u F9 922 100 J(\037)g Fa(\050)g F9(C)g Fd 143 12 J(0)g F9 37 12 L(;)g(M)14 X Fa(\051)o(=)23 X F9(k)t Fd 313 12 J(1)g F9 37 12 L(k)g Fd 43 12 J(2)g F8 51 12 L(\001)g(\001)14 X(\001)t F9(k)t Fc 154 12 J(r)g Fa 37 12 L(\050)g F9(r)g Fa(!\051)i F9(\013:)g Fa 1737 149 C(If)g(all)38 X(the)t F9(k)t Fc 415 12 J(i)g Fa 65 12 L(are)g (greater)t(than)t(or)t(equal)t(to)t(zero,)t(w)v(e)e(th)t(us)e(ha)t(v)e (e)d F9(\037)u Fa(\050)f F9(C)h Fd 1981 12 J(0)g F9 37 12 L(;)g(M)14 X Fa(\051)o F8(\025)40 X Fa(0.)t 2498 100 C(If)g(all)q(the)q F9(k)q Fc 412 12 J(i)g Fa 64 12 L(are)g(greater)q(than)q(zero)q(and)q F9(M)q Fa (has)z(dimension)q F9(r)q Fa(,)i(then)s F9(\013)q(\076)r Fa(0)s(and)p F9 476 100 C(\037)g Fa(\050)g F9(C)g Fd 143 12 J(0)g F9 37 12 L(;)g(M) 14 X Fa(\051)o F9(\076)23 X Fa(0.)t(This)37 X(pro)28 X(v)d(es)e (\050ii\051)s(and)t(\050iv\051.)s 55 116 C(If)g(all)v(the)v F9(k)v Fc 391 12 J(i)g Fa 58 12 L(are)g(zero,)v(then)w F9(\037)v Fa(\050)g F9(C)g Fd 678 12 J(0)g F9 37 12 L(;)g(M)14 X Fa(\051)o(is)30 X(simply)t(the)t F9(r)t Fa 727 30 L(th)g 113 30 J(deriv)g(ativ)b(e)e(of)t F9(P)t Fc 534 12 J(M)g Fa 74 12 L(,)g(so)u 2737 100 C(it)g(is)s(p)t(ositiv)i(e)e(if)s (and)t(only)s(if)t(the)s(co)t(e\016cien)i(t)e(of)s F9(n)t Fc 1660 30 L (r)g Fa 66 30 J(is)g(p)s(ositiv)j(e.)d(On)43 X(the)29 X(other)u(hand,)t 1726 100 C(this)g(co)q(e\016cien)i(t)e(giv)q(es)e(the)q(length)q(of)q (the)q(mo)q(dule)i F8(\010)q Fc 1732 36 L(n)g Fb(\000)g Fd(1)g Fc 59 G (i)g Fd(=1)g F9 130 23 L(M)g Fc 81 12 J(m;i)g Fa 132 12 L(for)g (su\016cien)q(tly)e(large)q F9(n)q Fa 2075 99 C(up)g(to)s(terms)r(of)s (lo)r(w)e(er)d(degree)s(in)r F9(n)s Fa(,)g(so)s(it)r(cannot)s(b)r(e)j (negativ)r(e.)e 125 199 J(The)g(graded)38 X(ring)t(obtained)t(from)t (the)t(original)t(situation)t(will)t(b)t(e)i(of)t(the)t(form)t 125 100 C(considered)g(here)24 X(when)s F9(I)t Fa(is)z(globally)t(de\014ned)s(b) t(y)d(a)t(regular)s(sequence,)t(and)t(the)t F9(k)s Fc 2686 12 J(i)g Fa 51 12 L(will)g 2737 100 C(then)g(b)34 X(e)i(the)t(degrees)t(of)t(the)t (generators.)t(W)55 X(e)a(giv)33 X(e)f(an)t(example)v(to)t(sho)v(w)d (that)v(the)t 99 G(condition)g(the)q F9(M)r Fa(has)z(dimension)r F9(r)q Fa(do)t(es)i(not)r(su\016ce)q(for)r F9(\037)q Fa(\050)g F9(C)g Fd 2113 12 J(0)g F9 38 12 L(;)g(M)14 X Fa(\051)o(to)30 X(b)u(e)i(p)u(ositiv)i (e.)e 2151 100 C(Let)g F9(R)u Fa(ha)u(v)e(e)d(dimension)v(3)t(and)v(let) t F9(t;)v(u;)13 X(v)v Fa(b)33 X(e)j(a)q(regular)s(system)q(of)s (parameters.)q(Let)47 X F1 H(q)g Fa(b)32 X(e)j(the)t(ideal)u(generated) t(b)u(y)d F9(v)u Fa(,)i(and)v(let)t F9(I)u Fa(b)z(e)j(the)t(ideal)u(of) t F9(R)u Fa(\133)g F9(X)h(;)h(Y)15 X Fa(\135)v(generated)34 X(b)q(y)f F9 99 G(v)g Fa(and)z F9(uX)x F8(\000)q F9(tY)25 X Fa(.)18 X(Then)67 X (the)37 X(\014b)u(er)i(o)u(v)d(er)e(the)t(closed)u(p)t(oin)j(t)d(is)u (pro)t(jectiv)l(e)d(space)u(of)t 100 G(dimension)g(one,)23 X F9(C)u Fd 619 12 J(0)g Fa 60 12 L(=)g F9(k)t Fa(\133)j F9(X)g(;)i(Y)14 X Fa(\135,) x(and)24 X F9(C)s Fa(=)y F9(C)s Fd 793 12 J(0)g Fa 37 12 L(\133)g F9(T) g Fd 72 12 J(1)g F9 37 12 L(;)g(T)14 X Fd 86 12 J(2)g Fa 37 12 L(\135)g (with)23 X F9(k)t Fd 273 12 J(1)g Fa 60 12 L(=)g(0)t(and)t F9(k)t Fd 353 12 J(2)g Fa 60 12 L(=)g(1.)t(Then)35 X 2487 100 C(if)g F9(M)28 X Fa (=)x F9(C)o(=T)k Fd 431 12 J(1)g Fa 37 12 L(,)g F9(M)t Fa(has)36 X (dimension)28 X(2)s(and)t F9(\037)s Fa(\050)h F9(C)f Fd 1083 12 J(0)g F9 37 12 L(;)g(M)14 X Fa(\051)o(=)23 X(0.)t 1463 99 C(Exercise:)g(Pro) 64 X(v)e(e)d(\050without)42 X(using)s(the)t(Serre)s(p)t(ositivit)i(y)e (conjecture\051)s(that)t(the)s 125 100 C(mo)g(dule)i F9(M)28 X Fa(in)36 X(the)28 X(previous)s(paragraph)t(could)s(not)t(arise)s(from)t(a)s (prime)t(ideal)s F1(p)t Fa(suc)s(h)e H(that)g F9(R)t(=)g F1(p)h F8(\012) 17 X F9(R)w(=)f F1(q)i Fa(has)26 X(\014nite)w(length)t(and)w(dim\050)t F9(R)i(=)f F1(p)i Fa(\051)e(+)n(dim)17 X(\050)h F9(R)f(=)i F1(q)f Fa (\051)h(=)y(dim\050)24 X F9(R)f Fa(\051.)i F4 274 G(6)g(Gabb)137 X(er's) k(pro)46 X(of)k(of)t(non-negativit)t(y)c Fa 182 G(In)g(this)36 X (section)t(w)t(e)e(complete)t(Gabb)t(er's)i(pro)t(of)i(of)t(the)t (non-negativit)t(y)e(of)t(in)t(tersec-)e 100 G(tion)g(m)s (ultiplicities.)e(W)59 X(e)-7 X(ha)35 X(v)e(e)d(seen)u(in)s(the)u(last) s(section)u(that)s(if)u F9(g)s(r)k Fc 2320 12 J(I)g Fa 38 12 L(\050)g F9(A)g Fa(\051)g F8(\012)23 X Fc 214 12 J(R)g F9 78 12 L(k)g Fa(is)38 X (a)q 2650 99 C(p)g(olynomial)i(ring)26 X(o)t(v)d(er)e(\050)s F9(A=I)h Fa(\051)m F8(\012)15 X Fc 1065 12 J(R)g F9 69 12 L(k)g Fa(generated)28 X(b)r(y)d(elemen)r(ts)d(of)r(non-negativ)r(e)d(degree,)r 1134 100 C (then)g(non-negativit)q(y)d(follo)q(ws.)d(W)37 X(e)-8 X(sho)26 X(w)c (here)t(that)q(w)t(e)c(can)t(em)q(b)f(ed)h F9(g)t(r)i Fc 2345 12 J(I)g Fa 38 12 L(\050)g F9(A)g Fa(\051)g F8(\012)13 X Fc 204 12 J(R)g F9 67 12 L(k)g Fa(in)27 X(to)e 1263 271 C(14)g EOP end %%PageTrailer %======================================================================= %%Page: 15 15 %%BeginPageSetup DVIALWdict begin %%EndPageSetup BOP Fa 1123 5385 A(a)g(p)34 X(olynomial)i(ring)t(of)t(this)t(t)t(yp)e(e.)i (Let)57 X F9(A)34 X Fd 1414 12 J(0)g Fa 71 12 L(denote)g F9(A=I)t F8 (\012)p Fc 508 12 J(R)g F9 77 12 L(k)g Fa(.)j(Actually)56 X(,)a(w)35 X (e)f(sho)r(w)f 2070 100 C(instead)g(that)q(w)p(e)e(can)p(em)q(b)d(ed)j (the)p(symmetric)q(algebra)p(Sym)q Fc 2034 20 J(A)g Fg 50 8 J(0)g Fa 34 28 L(\050\050)g F9(I)g(=I)7 X Fd 192 30 L(2)g Fa 37 30 J(\051)g F8(\012) 21 X Fc 118 12 J(R)g F9 75 12 L(k)g Fa(\051)j(in)31 X(to)e(a)t 2540 100 C(p)g(olynomial)i(ring)r(b)r(y)e(a)r(lo)r(cally)i(\015at)r(map.)r(Since) 40 X F9(I)29 X(=I)m Fd 1769 30 L(2)g Fa 66 30 J(is)g(lo)t(cally)i(free,) t(the)t(map)t(from)t 1835 100 C(the)g(symmetric)p(algebra)q(to)p(the)q (asso)p(ciated)j(graded)p(algebra)q(de\014nes)p(an)q(isomorphism)p H(of) g(sc)s(hemes,)d(so)s(this)r(su\016ces)s(to)r(pro)s(v)d(e)e(the)r (result.)s(Let)37 X F9(S)28 X Fa(=)t(Sym)o Fc 2189 20 J(A)g Fg 50 8 J(0) g Fa 34 28 L(\050\050)g F9(I)g(=I)7 X Fd 192 30 L(2)g Fa 37 30 J(\051)g F8(\012)18 X Fc 115 12 J(R)g F9 73 12 L(k)g Fa(\051)j F9(:)g Fa 2565 100 C(Let)g F9(E)y Fc 205 12 J(s)g Fa 58 12 L(denote)g(Pro)y(j)k(\050)h F9(g)f(r)k Fc 535 12 J(I)g Fa 38 12 L(\050)g F9(A)g Fa(\051)g F8(\012)8 X Fc 199 12 J(R)g F9 63 12 L(k)g Fa(\051)j(=)23 X(Pro)t(j)l(\050Sym)g Fc 541 20 J(A)g Fg 50 8 J(0)g Fa 36 28 L(\050\050)g F9(I)g(=I)7 X Fd 192 30 L(2)g Fa 37 30 J(\051)g F8(\012)u Fc 105 12 J(R)g F9 63 12 L(k)g Fa(\051\051)j(as)22 X(ab)u(o)i(v)e(e.)d(Let)36 X F9 2247 100 C(W)g Fa(=) v(Pro)26 X(j)l(\050)g F9(A)g Fd 462 12 J(0)g Fa 37 12 L(\133)g F9(T)g Fd 72 12 J(1)g F9 37 12 L(;)g(:)14 X(:)t(:)t(;)t(T)t Fc 234 12 J(r)g Fe 33 17 L(0)g Fa 26 5 J(\135\051)g(for)30 X F9(T)s Fc 264 12 J(i)g Fa 57 12 L(of)g(degree)t(\050)s F9(k)h Fc 433 12 J(i)g F9 28 12 L(;)g Fa (1\051)14 X(for)30 X(some)s(in)t(teger)d F9(r)t Fb 795 30 L(0)g Fa 24 30 J(.)g(Supp)43 X(ose)i 2502 100 C(that)g F9(f)31 X Fa(is)40 X(a)31 X (map)t(from)t F9(S)t Fa(in)y(to)d(the)u(p)s(olynomial)j(ring)s F9(A)u Fd 1885 12 J(0)g Fa 37 12 L(\133)g F9(T)g Fd 72 12 J(1)g F9 37 12 L(;)g (:)14 X(:)t(:)t(;)t(T)t Fc 233 12 J(r)g Fe 33 17 L(0)g Fa 26 5 J(\135)g (suc)31 X(h)e(that)t(the)t 2323 99 C(map)g F9(\036)q Fa(induced)r(b)q(y) e F9(f)q Fa(from)38 X F9(W)28 X Fa(to)40 X F9(E)28 X Fc 1244 12 J(s)g Fa 64 12 L(is)g(\015at)t(of)u(relativ)t(e)e(dimension)t F9(r)u Fb 1057 30 L(0)g F8 42 30 J(\000)g F9(r)19 X Fa(,)i(where)29 X F9(r)s Fa 2407 100 C(is)g(the)s(rank)r(of)s F9(I)r(=I)7 X Fd 642 30 L(2)g Fa 37 30 J(.) g(Then)37 X(w)28 X(e)d(ha)t(v)d(e)e(a)s(comm)t(utativ)d(e)e(diagram)s F9 314 211 J(S)g Fc 185 51 L(\036)g F8 47 51 C(\000)g(!)-14 X F9(A)83 X Fd 279 12 J(0)g Fa 37 12 L(\133)g F9(T)g Fd 72 12 J(1)g F9 37 12 L(;)g (:)14 X(:)t(:)t(;)t(T)t Fc 234 12 J(r)g Fe 33 17 L(0)g Fa 26 5 J(\135)g F8 692 100 C(&)g(#)349 X F9 403 100 J(A)g Fd 62 12 J(0)g Fa 1622 167 C (whic)g(h)e(induces)27 X(a)u(comm)t(utativ)e(e)d(diagram)u(of)t(sc)u (hemes)d F9 1176 206 J(E)g Fc 61 12 J(s)g 166 63 L(\036)g F8 47 51 C (\040)g(\000)-14 X F9(W)84 X F8 25 100 J(-)g(")134 X F9 192 100 J(Z)g Fb 63 30 L(0)g Fc 6 51 C(s)g F9 77 135 L(:)g Fa 1707 293 C(Let)g F8(M) 21 X Fa(b)s(e)j(a)s(sheaf)t(on)s F9(E)t Fc 804 12 J(s)g Fa 56 12 L (de\014ned)g(b)t(y)d(a)t(bigraded)s(mo)t(dule)i F9(M)t Fa(.)9 X(Since) 35 X(w)20 X(e)e(are)t(assuming)u 860 100 C(that)g F9(\036)28 X Fa(is)s (\015at,)t(w)s(e)e(ha)s(v)e(e)d(an)t(isomorphism)s F8 678 192 J(T)g F9 (or)21 X Fb 145 45 L(O)g Ff 53 8 J(E)g 42 8 J(s)g Fc 97 52 C(i)g Fa 137 23 L(\050)g F8(O)g Fc 98 12 J(Z)g Fe 49 17 L(0)g Ff 4 33 C(s)g F9 36 28 L(;)g F8(M)14 X Fa(\051)g F8 192 22 L(\030)g Fa 26 G(=)g F8 88 4 L(T)g F9(or)21 X Fb 145 37 L(O)g Ff 53 8 J(W)g Fc 55 52 C(i)g Fa 124 23 L (\050)g F8(O)g Fc 98 12 J(Z)g Fe 49 17 L(0)g Ff 4 33 C(s)g F9 36 28 L(;) g(\036)14 X Fb 87 34 L(\003)g Fa 38 34 J(\050)g F8(M)g Fa(\051\051)g 1988 183 C(for)g(all)28 X F9(i)s Fa(.)h(Th)36 X(us)f(w)26 X(e)f(ha)t(v) f(e)c(an)w(equalit)t(y)f(of)t(Euler)w(c)t(haracteristics)f F9 808 182 J (\037)g Fc 52 12 J(E)g Ff 49 8 J(s)g Fa 36 20 L(\050)g F8(O)g Fc 98 12 J(Z)g Fe 49 17 L(0)g Ff 4 33 C(s)g F9 36 28 L(;)g F8(M)14 X Fa(\051)g(=) 23 X F9(\037)t Fc 332 12 J(W)g Fa 76 12 L(\050)g F8(O)g Fc 98 12 J(Z)g Fe 49 17 L(0)g Ff 4 33 C(s)g F9 36 28 L(;)g(\036)14 X Fb 86 34 L(\003)g Fa 38 34 J(\050)g F8(M)g Fa(\051\051)g F9(:)g Fa 1710 183 C(Th)g(us)e (if)28 X(w)t(e)e(can)t(\014nd)t(suc)t(h)e(a)t(map)t F9(f)t Fa(suc)37 X (h)e(that)28 X(all)t(the)t F9(k)t Fc 1919 12 J(i)g Fa 56 12 L(are)g (non-negativ)t(e,)e(the)t 2100 100 C(conjecture)g(will)u(follo)v(w.)d(W) 42 X(e)-7 X(no)29 X(w)e(sho)t(w)e(that)t(suc)u(h)d(an)u(em)t(b)e(edding) i(exists.)u(Gabb)41 X(er's)j 99 G(pro)g(of)i(uses)28 X(the)s(fact)t (that)s(the)t(dual)s(of)t F9(I)s(=I)7 X Fd 1431 30 L(2)g Fa 64 30 J(o)g (v)e(er)d F9(s)28 X Fa(is)s(generated)t(b)s(y)e(global)s(sections;)t 1495 100 C(w)g(e)e(de\014ne)z(this)35 X(map)s(directly)t(without)s (dualizing.)t(A)57 X(t)e(this)34 X(p)u(oin)i(t)e(w)t(e)e(assume)t(that) u F9 H(R)g Fa(is)p(rami\014ed.)o(Although)41 X(this)29 X(is)t(an)t(un)t (usual)e(assumption,)t(it)u(is)t(p)t(ossible)i(to)t(reduce)t 99 G(to)g (the)z(rami\014ed)y(case)z(b)y(y)e(a)y(\014nite)z(\015at)y(extension)z (of)y F9(R)z Fa(,)g(for)37 X(example)q(b)r(y)d(adjoining)r 100 G(a)g (square)v(ro)u(ot)j(of)u F9(p)v Fa(,)g(where)x F9(R)v Fa(has)v(mixed)v (c)u(haracteristic)e F9(p)u Fa(.)h(Let)69 X F9(t)40 X Fd 2372 12 J(1)g F9 37 12 L(;)g(:)14 X(:)t(:)t(;)t(t)t Fc 214 12 J(d)g Fa 77 12 L(b)g(e) i(a)39 X 2700 99 C(minimal)g(set)p(of)p(generators)p(of)p(the)p(maximal) p(ideal)p F1(m)p Fa(of)p F9(R)p Fa(.)h(Since)59 X F9(R)36 X Fa(is)s (rami\014ed,)t F9 100 G(R)g(=)h F1(m)g Fd 170 30 L(2)g Fa 65 30 J(is)g (isomorphic)28 X(to)s(a)t(p)s(olynomial)j(ring)s(in)t F9(t)s Fd 1402 12 J(1)g F9 37 12 L(;)g(:)14 X(:)t(:)t(;)t(t)t Fc 214 12 J(d)g Fa 66 12 L (mo)g(dulo)i(the)28 X(square)s(of)t(the)s 1954 143 C(ideal)g(generated) s(b)t(y)d F9(t)t Fd 722 12 J(1)g F9 37 12 L(;)g(:)14 X(:)t(:)t(;)t(t)t Fc 215 12 J(d)g Fa 39 12 L(.)g(Th)37 X(us)e(for)27 X(eac)u(h)d F9(i)u Fa(,)f(the)v(partial)s(deriv)v(ativ)a(e)f 1464 19 L 107 4 B F9 29 37 L (@)g 29 113 C(@)g(t)l Fc 79 12 J(i)g Fa 63 69 L(de\014nes)g 2619 145 C (a)g(map)s(from)r F9(R)s(=)g F1(m)h Fd 615 30 L(2)g Fa 37 30 J(\133)g F9(X)g Fd 92 12 J(0)g F9 37 12 L(;)g(:)14 X(:)t(:)t(;)t(X)t Fc 253 12 J (n)g Fa 45 12 L(\135)g(to)26 X F9(R)s(=)h F1(m)g Fa(\133)g F9(X)g Fd 410 12 J(0)g F9 37 12 L(;)g(:)14 X(:)t(:)t(;)t(X)t Fc 253 12 J(n)g Fa 45 12 L(\135.)g(By)36 X(taking)26 X(the)s(comp)t(osition)i 1824 100 C (with)g(the)u(map)t(from)u F9(I)t Fa(to)34 X F9(R)26 X(=)h F1(m)g Fd 1050 30 L(2)g Fa 37 30 J(\133)g F9(X)g Fd 92 12 J(0)g F9 37 12 L(;)g(:) 14 X(:)t(:)t(;)t(X)t Fc 253 12 J(n)g Fa 45 12 L(\135)g(induced)27 X(b)s (y)e(the)s(inclusion)t(of)s F9(I)t Fa(in)z(to)e F9(A)s Fa 1514 99 C(and) g(with)z(the)z(map)z(from)z F9(R)z(=)h F1(m)g Fa(\133)g F9(X)g Fd 1163 12 J(0)g F9 37 12 L(;)g(:)14 X(:)t(:)t(;)t(X)t Fc 254 12 J(n)g Fa 45 12 L(\135)g(=)32 X F9(A)u F8(\012)22 X Fc 301 12 J(R)g F9 77 12 L(k)g Fa (to)36 X(\050)q F9(A=I)g Fa(\051)7 X F8(\012)22 X Fc 484 12 J(R)g F9 77 12 L(k)g Fa(=)35 X F9(A)q Fd 237 12 J(0)g Fa 37 12 L(,)g(w)t(e)d 1321 335 C(15)g EOP end %%PageTrailer %======================================================================= %%Page: 16 16 %%BeginPageSetup DVIALWdict begin %%EndPageSetup BOP Fa 1123 5385 A(obtain)g(a)23 X(map)s(from)t F9(I)s Fa(to)30 X F9(A)22 X Fd 914 12 J(0)g Fa 37 12 L(.)g(Since)35 X(for)23 X(all)s F9(a)t Fa(in)s F9(A)t Fa(and)s F9(i)t Fa(in)s F9(A)t Fa(the)s(partial)t(deriv)s(ativ)c (es)d 951 100 C(satisfy)g F9 1071 87 J(@)g Fa(\050)l F9(ai)g 37 G 186 4 B Fa 154 37 L(\051)g F9 114 113 C(@)g(t)l Fc 79 12 J(i)g Fa 100 69 L(=) g F9(a)t 156 56 L(@)g 14 37 C 107 4 B 63 37 L(i)g 63 113 C(@)g(t)l Fc 79 12 J(i)g Fa 56 69 L(+)g F9(i)o 129 56 L(@)g 7 37 C U 56 37 L(a)g 56 113 C(@)g(t)l Fc 79 12 J(i)g Fa 1768 172 C(and)g F9(A)s Fd 218 12 J(0)g Fa 59 12 L(is)g(annihilated)s(b)s(y)e F9(I)s Fa(,)m(w)u(e)d(can)t 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X(cal)i(ring)t(at)s F9(p)t Fa(in)s F9(P)t Fa(,)11 X(whic)25 X(h)d(w) s(e)d(denote)s F9 655 100 C(A)g Fc 62 12 J(p)g Fa 38 12 L(,)g(is)y (isomorphic)y(to)y F9(R)y Fa(\133)g F9(u)h Fd 802 12 J(1)g F9 37 12 L(;) g(:)14 X(:)t(:)t(;)t(u)t Fc 233 12 J(n)g Fa 45 12 L(\135)g F1 23 12 J(m) g Ff 63 8 J(p)g Fa 39 20 L(,)g(where,)30 X(after)u(a)s(c)u(hange)d(of)u (co)s(ordinates,)j(w)t(e)e 1342 99 C(ma)g(y)e(assume)w(that)x F9(u)w Fd 714 12 J(1)g F9 37 12 L(;)g(:)14 X(:)t(:)t(;)t(u)t Fc 232 12 J(n)g Fa 79 12 L(together)g(with)34 X(a)s(set)t(of)s(generators)t(of)s F1(m)t Fc 1411 12 J(R)g Fa 87 12 L(generate)g F1 2560 100 C(m)g Fc 64 12 J(p)g Fa 38 12 L(.)g(Since)55 X F9(Z)34 X Fb 364 30 L(0)g Fa 57 30 J(is)g (regular,)t F9(I)u Fa(is)41 X(generated)33 X(lo)v(cally)h(b)v(y)c(part) v(of)t(a)v(regular)t(system)v(of)t 523 99 C(parameters)g F9(i)21 X Fd 453 12 J(1)g F9 37 12 L(;)g(:)14 X(:)t(:)t(;)t(i)t Fc 214 12 J(r)g Fa 37 12 L(.)g(F)35 X(urthermore,)-7 X(the)22 X(quotien)s(t)d F9(I)s(=I)m Fd 1140 30 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