%!PS-Adobe-3.0 %%Title: dvialw nntadams.dvi %%CreationDate: Mon Jul 12 15:02:15 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 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Fp(.)i (By)38 X(Prop)27 X(osition)j(2.1,)t(w)u(e)d(ha)u(v)d(e)e Fm 583 178 J (\025)g Fn 48 34 L(k)g Fp 41 34 J(\050)g Fd(E)g Fm(:)g Fp(\051)g(=)p (\050)p Fj(\000)g Fp(1\051)g Fn 423 34 L(k)g Fk(\000)h Fq(1)g Fp 126 34 J(\133)g Fd(E)g Fm(:)g Fj(\012)18 X(O)u Fn 269 12 J(X)g Fp 63 12 L(\050) g Fj(\000)g Fp(\050)g Fm(k)g Fj(\000)w Fp(1\051)u Fm(D)f Fp(\051\135)j 1678 178 C(for)g Fm(k)39 X Fj(\025)y Fp(1.)w(In)70 X(particular,)39 X(w) v(e)e(ha)s(v)e(e)d Fm(\040)t Fq 1396 31 L(1)g Fp 37 31 J(\050)g Fd(E)g Fm(:)g Fp(\051)g(=)v Fd(E)w Fm(:)f Fp(=)w Fm(\025)v Fq 561 31 L(1)g Fp 38 31 J(\050)g Fd(E)g Fm(:)g Fp(\051.)g(Supp)70 X(ose)i Fm(k)39 X Fj (\025)y Fp(2.)w 2032 99 C(Assertion)g(\050c\051)q(follo)p(ws)e(easily)p (from)q(\050b\051,)p(equation)s(\0502\051)p(of)q(section)p(2,)q(and)r (the)q(fact,)p 100 G(pro)g(v)e(en)d(in)28 X(the)s(same)t(w)s(a)e(y)d(as) t(\050b\051,)s(that)t(\133)s Fd(E)h Fm(:)f Fj(\012)19 X Fd(E)s Fm(:)g Fp(\135)g(=)x(\133)x Fd(E)g Fm(:)g Fp(\135)g Fj(\000)s Fp(\133)t Fd(E)f Fm(:)h Fj(\012)r(O)u Fn 2238 12 J(X)g Fp 63 12 L(\050)g Fj(\000)g Fm(D) g Fp(\051\135.)i Fh(q.e.d.)89 X Fp 2176 147 C(W)g(e)-7 X(no)25 X(w)e (pro)t(v)e(e)d(Theorem)u(3.1)s(for)u(elemen)s(tary)f(complexes.)s(W)37 X(e)-8 X(ma)26 X(y)d(assume)t(that)r Fm 125 100 C(X)g Fp(is)34 X(an)n (in)27 X(tegral)e(sc)t(heme,)e(and)t(w)u(e)d(ha)u(v)d(e)e(only)t(to)u (pro)t(v)e(e)d(that)u 792 178 J(c)g(h)e Fn 80 12 J(i)g Fp 28 12 L(\050) g Fm(\040)g Fn 89 35 L(k)g Fp 41 35 J(\050)g Fd(F)g Fm(:)g Fp(\051)g (\051)g Fj(\\)18 X Fp(\133)u Fm(X)f Fp(\135)8 X(=)22 X Fm(k)v Fn 540 35 L(i)g Fp 41 35 J(c)g(h)e Fn 81 12 J(i)g Fp 27 12 L(\050)g Fd(F)g Fm(:)g Fp(\051)g Fj(\\)p Fp(\133)q Fm(X)f Fp(\135)8 X 1719 177 C(for)g Fm(k)28 X Fj(\025)q Fp(1)p(and)s Fm(i)t Fj(\025)o Fp(0.)o 125 100 J(First,)g (assume)t(that)s Fm(F)t Fq 746 12 J(0)g Fp 65 12 L(is)g(a)t(line)s (bundle)t(and)s Fm(F)t Fn 796 12 J(t)g Fp 52 12 L(=)g(0)o(for)t Fm(t)s Fj(6)p Fp(=)f(0.)p(In)36 X(this)28 X(case)s(the)t 1784 99 C(lo)g (calized)i(Chern)s(c)r(haracter)e(of)r Fd(F)s Fm(:)f Fp(is)s(the)r (Chern)s(c)r(haracter)e(of)r Fm(F)s Fq 2119 12 J(0)g Fp 64 12 L(as)g(a) r(sheaf,)s(and)r(\050b)s(y)d 2183 100 C(Prop)g(osition)i(18.1)t (\050a\051)s(in)t(\1333\135)s(or)t(\13314\135,)s(section)t(11.4\051,)s (w)t(e)d(ha)t(v)d(e)e 891 217 J(c)g(h)e Fn 81 12 J(i)g Fp 27 12 L(\050) g Fd(F)g Fm(:)g Fp(\051)g(=)o(c)o(h)e Fn 330 12 J(i)g Fp 27 12 L(\050)g Fm(F)g Fq 85 12 J(0)g Fp 38 12 L(\051)g(=)o 153 19 L 52 4 B 5 37 L(1)g Fm 5 113 C(i)g Fp(!)g Fm 62 57 L(c)g Fq 36 12 J(1)g Fp 37 12 L(\050)g Fm(F)g Fq 85 12 J(0)g Fp 37 12 L(\051)g Fn 32 35 L(i)g Fm 28 35 J(;)g Fp 1949 205 C(where)g Fm(c)42 X Fq 291 12 J(1)g Fp 37 12 L(\050)g Fm(F) g Fq 85 12 J(0)g Fp 37 12 L(\051)g(is)t(the)t(\014rst)t(Chern)t(class)t (of)t(the)t(line)t(bundle)t Fm(F)t Fq 1766 12 J(0)g Fp 37 12 L(.)g (Therefore,)81 X(b)46 X(y)e 2253 100 C(Lemma)g(3.2)28 X(\050a\051,)s(w) t(e)d(ha)t(v)d(e)e 459 217 J(c)g(h)e Fn 81 12 J(i)g Fp 28 12 L(\050)g Fm(\040)g Fn 89 35 L(k)g Fp 41 35 J(\050)g Fd(F)g Fm(:)g Fp(\051)g(\051) g(=)o(c)o(h)e Fn 362 12 J(i)g Fp 27 12 L(\050)g Fm(F)g Fk 97 36 L(\012) g Fn(k)g Fq 12 58 C(0)g Fp 106 22 L(\051)g(=)o 153 20 L U 5 37 L(1)g Fm 5 113 C(i)g Fp(!)g Fj 62 56 L(f)g Fm(k)g Fj(\001)21 X Fm(c)r Fq 183 12 J(1)g Fp 37 12 L(\050)g Fm(F)g Fq 85 12 J(0)g Fp 37 12 L(\051)g Fj(g)g Fn 74 35 L(i)g Fp 51 35 J(=)g Fm(k)v Fn 134 35 L(i)g Fp 41 35 J(c)g(h)e Fn 81 12 J(i)g Fp 27 12 L(\050)g Fd(F)g Fm(:)g Fp(\051)g Fm(:)g Fp 832 270 C(7)g EOP end %%PageTrailer %======================================================================= %%Page: 8 8 %%BeginPageSetup DVIALWdict begin %%EndPageSetup BOP Fp 1248 5385 A(Next,)g(w)25 X(e)d(supp)t(ose)h(that)t Fm(Y)r Fj(6)42 X Fp(=)g Fm(X)23 X Fp(and)30 X(that)n Fd(F)n Fm(:)g Fp(is)n(an)n(elemen)n (tary)e(complex)n(concen-)n 125 100 C(trated)g(in)q(degrees)q(0)q(and)q (1.)q(Then,)37 X(there)27 X(exist)t(a)t(line)t(bundle)t Fj(L)t Fp(and)t (an)t(elemen)t(tary)e H(complex)g Fd(E)v Fm(:)g 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e(the)t(Dutta)u(m)t(ultiplicit)e(y)d(of)u Fd(F)t Fm(:)h Fp(is)t(p)u (ositiv)i(e.)e Fh(q.e.d.)1013 X 232 G(Remark)g(4.5)32 X Fp(Compare)41 X (equation)28 X(\0501\051)s(in)t(Section)s(1)t(with)s(\0503\051)t(in)s (Lemma)t(4.4.)s 125 100 J(F)g(or)-7 X(a)24 X(No)t(etherian)i(lo)t(cal)i (ring)t Fm(A)t Fp(con)t(taining)e(a)t(\014eld)t(of)t(c)t(haracteristic) e Fm(p)t(\076)s Fp(0)s(and)t(a)t 125 100 C(p)g(erfect)i Fm(A)y Fp (-complex)g Fd(F)y Fm(:)g Fp(,)g Fm(F)y Fn 884 30 L(e)g Fp 35 30 J(\050) g Fd(F)g Fm(:)g Fp(\051)g(denotes)y(a)y(p)y(erfect)i Fm(A)y Fp(-complex) g(de\014ned)y(b)y(y)e(matrices)y 919 100 C(whose)g(en)x(tries)d(are)x (the)w Fm(p)x Fn 836 30 L(e)g Fp 36 30 J(-th)g(p)x(o)i(w)e(ers)d(of)x (those)w(of)x(the)w(complex)x Fd(F)w Fm(:)h Fp(.)f 747 100 C(Then,)g(b) 37 X(y)e(a)r(remark)r(follo)r(wing)e(Prop)r(osition)i(4.13)r(in)r (Gillet-Soul)r(\023)e(e)-40 X(\1334\135,)36 X(w)t(e)f(ha)r(v)f(e)c 125 100 C(\050)g Fm(\040)g Fn 89 30 L(p)g Fp 38 30 J(\051)g Fn 32 30 L(e)g Fp 36 30 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(of)s(Dutta)u(\1332\135)s(on)u(the)s(p)u(ositivit)h(y)f 100 G(of)g(in)y (tersection)d(m)y(ultiplicities.)d Fh 182 G(Corollary)g(5.1)32 X Fg(L) 41 X(et)c Fp(\050)27 X Fm(A;)g(m)14 X Fp(\051)g Fg(b)27 X(e)c(a)t (homomorphic)u(image)t(of)u(a)t(r)u(e)b(gular)d(lo)s(c)d(al)b(ring)u (with)t Fm 100 G(d)g Fp(=)p(dim)p Fm(A)14 X Fg(.)g(Assume)35 X(that)22 X Fm(\034)s Fn 929 12 J(A)g Fp 54 12 L(\050)g Fm(A)g Fp(\051)g Fj(2)u Fp(A)u Fn 289 12 J(d)g Fp 39 12 L(Sp)g(ec)i Fm(A)14 X F4 246 20 J(Q)g Fg 49 20 L(.)g(L)35 X(et)c Fm(M)21 X Fg(and)31 X Fm(N)21 X Fg(b)31 X(e) b(\014nitely)22 X(gener)s(ate)c(d)c Fm 1606 100 C(A)g Fg(-mo)g(dules)c (such)29 X(that)u Fj 125 166 J(\017)g Fp(p)42 X(d)i Fn 177 20 J(A)g Fm 68 20 L(M)g(\074)32 X Fj(1)23 X Fg(,)g Fp(p)30 X(d)i Fn 433 20 J(A)g Fm 68 20 L(N)g(\074)v Fj(1)23 X Fg(,)g Fj 746 166 C(\017)g Fp(0)42 X Fm (\074)23 X(`)t Fn 271 12 J(A)g Fp 54 12 L(\050)g Fm(M)g Fj(\012)y Fn 205 12 J(A)g Fm 73 12 L(N)g Fp(\051)9 X Fm(\074)23 X Fj(1)t Fg(,)g Fj 603 166 C(\017)g Fp(dim)42 X Fm(M)13 X Fp(+)28 X(dim)18 X Fm(N)p Fp(=) 32 X(dim)23 X Fm(A)14 X Fg(,)g Fj H(\017)g Fp(depth)42 X Fm(M)13 X Fp(=) 33 X(dim)22 X Fm(M)15 X Fg(,)8 X Fj H(\017)g Fg(for)42 X(any)23 X(prime) v(ide)t(al)d Fm(Q)t Fg(in)v(the)t(supp)v(ort)b(of)v Fm(N)t Fg(with)34 X Fp(dim\050)23 X Fm(A=Q)h Fp(\051)f(=)u(dim\050)s Fm(N)h Fp(\051)8 X Fg (,)h Fm(A=Q)24 X Fg 83 100 J(c)g(ontains)c(a)y(\014eld,)z 208 166 C (wher)g(e)c Fp(p)31 X(d)j Fn 330 20 J(A)g Fm 68 20 L(M)g Fg(\050r)40 X (esp.)c Fp(p)31 X(d)j Fn 451 20 J(A)g Fm 69 20 L(N)g Fg(\051)9 X (denotes)31 X(the)t(pr)t(oje)c(ctive)c(dimension)t(of)t Fm(M)t Fg(\050r) 41 X(esp.)b Fm(N)32 X Fg(\051.)8 X 918 99 C(Then)g Fr 887 21 J(X)g Fn 48 177 J(i)g Fp 72 98 L(\050)g Fj(\000)g Fp(1\051)g Fn 171 34 L(i)g Fm 28 34 J(`)g Fn 35 12 J(A)g Fp 54 12 L(\050T)g(or)-7 X Fn 159 36 L(A)g 56 G(i)g Fp 53 20 L(\050)g Fm(M)g(;)k(N)14 X Fp(\051\051)o Fm(\076)23 X Fp(0)t Fm(:)g Fg 1507 255 C(Pr)g(o)c(of.)c Fp(Put)107 X Fm(X)38 X Fp(=) 48 X(Sp)40 X(ec)j Fm(A)13 X Fp(,)h Fm(Y)40 X Fp(=)60 X(Supp)40 X Fm(M) 15 X Fp(and)46 X Fm(Z)n Fp(=)s(Supp)p Fm(N)13 X Fp(.)q(Then,)68 X(w)42 X(e)d(ha)q(v)d(e)e Fm 100 G(Y)g Fj(\\)p Fm(Z)19 X Fp(=)32 X(Sp)25 X(ec) j Fm(A=m)13 X Fp(\050as)30 X(a)s(set\051)t(and)s(dim)t Fm(Y)13 X Fp(+) 39 X(dim)18 X Fm(Z)q Fp(=)31 X Fm(d)o Fp(b)r(y)e(our)r(assumption.)s (Put)41 X Fm 99 G(s)g Fp(=)23 X(dim)t Fm(Y)14 X Fp(and)46 X Fm(t)28 X Fp(=)o(dim)o Fm(Z)14 X Fp(.)m 1391 349 J(14)g EOP end %%PageTrailer %======================================================================= %%Page: 15 15 %%BeginPageSetup DVIALWdict begin %%EndPageSetup BOP Fp 1248 5385 A(Let)g Fd(F)25 X Fm(:)g Fp(and)t Fd(G)t Fm(:)g Fp(b)t(e)i (minimal)t Fm(A)t Fp(-free)g(resolutions)t(of)t Fm(M)t Fp(and)34 X Fm(N) 25 X Fp(resp)34 X(ectiv)i(ely)e(.)-7 X(Then)36 X Fd 125 100 C(F)g Fm(:) g Fp(and)28 X Fd(G)u Fm(:)f Fp(are)u(p)t(erfect)j Fm(A)t Fp(-complexes) g(with)t(supp)u(orts)i Fm(Y)u Fp(and)46 X Fm(Z)29 X Fp(.)l(Then,)40 X(b) 28 X(y)e(the)t(lo)u(cal)i 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