%!PS-Adobe-3.0 %%Title: dvialw ourpaper2.dvi %%CreationDate: Mon Jul 12 15:03:32 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 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X(morphism)n(Pro)o(j\050)k Fk(A=I)h Fn(\051) m Fh(!)p Fn(Sp)p(ec)i Fk(R)14 X(=)g F2(p)h Fn(is)36 X(generically)u (\014nite;)t(that)y(is,)t(the)w 106 100 J(extension)g(of)28 X(function) s(\014elds)t(de\014ned)s(b)t(y)d(this)t(morphism)s(is)t(\014nite.)s 83 166 C(W)g(e)-7 X(remark)19 X(that)t(the)t(existence)t(of)t(the)t(ideal) t Fk(I)t Fn(with)26 X(the)19 X(ab)t(o)i(v)e(e)d(prop)u(erties)h (requires)u 125 100 C(that)g(the)w(regular)w(lo)w(cal)i(ring)w Fk(R)w Fn(is)w(essen)w(tially)d(of)w(\014nite)w(t)w(yp)d(e)j(o)v(v)e(er)d(a)w (\014eld)w(or)w(a)w(complete)w H(discrete)g(v)28 X(aluation)b(ring,)t (but)t(that)t(the)t(m)t(ultiplicit)e(y)d(conjectures)u(can)s(b)u(e)h (reduced)u(to)s H(this)g(case.)o(W)35 X(e)-7 X(refer)23 X(to)t (Berthelot)t(\1332\135)t(and)t(Ho)t(c)i(hster)e(\1336\135)t(for)t (details)t(of)t(this)t(reduction.)t H(Throughout)g(this)38 X(section)t (w)t(e)d(assume)t(that)t(these)t(prop)t(erties)i(hold)t(and)t(that)t (this)t H(construction)g(can)28 X(b)s(e)j(carried)s(out.)t(W)37 X(e)-7 X(let)28 X Fk(X)s Fn(denote)35 X(Pro)27 X(j\050)l Fk(A=I)g Fn(\051.)7 X 125 100 J(The)g(\014rst)32 X(step)s(in)t(Gabb)s(er's)j(pro)s(of)j(of)s (nonnegativit)t(y)d(reduces)t(the)s(computation)t 125 99 C(of)g(in)r (tersection)e(m)r(ultiplicities)e(to)r(a)s(computation)r(on)s(asso)r (ciated)j(graded)r(rings.)s(Let)44 X Fk 100 G(g)g(r)j Fl 80 12 J(I)g Fn 38 12 L(\050)g Fk(A)g Fn(\051)g(b)23 X(e)i(the)t(asso)t(ciated)i(graded) t(ring)t(of)t Fk(I)t Fn(;)7 X(since)24 X Fk(I)t Fn(is)y(a)t(graded)r (ideal)t(of)r(the)t(graded)r 118 100 C(ring)g Fk(A)33 X Fn(,)g Fk(g)u(r) j Fl 376 12 J(I)g Fn 38 12 L(\050)g Fk(A)g Fn(\051)g(is)t(a)t(bigraded) t(ring.)t(One)53 X(grading,)33 X(whic)v(h)d(w)u(e)d(sometimes)u(refer)s (to)u 414 99 C(as)g(the)r(\014rst)r(grading,)r(is)r(induced)r(b)r(y)e (the)r(grading)r(on)r Fk(A)r Fn(,)g(while)s(the)q(second)s(grading)q 100 G(is)g(determined)v(b)u(y)e(the)u(p)v(o)i(w)e(ers)d(of)v Fk(I)u Fn (.)7 X(T)57 X(o)a(the)34 X(ring)u Fk(g)t(r)k Fl 1849 12 J(I)g Fn 38 12 L(\050)g Fk(A)g Fn(\051)g(w)u(e)d(asso)u(ciate)i(a)u(sc)t(heme)e Fk 1887 100 C(Y)g Fn(as)53 X(follo)35 X(ws.)e(The)58 X(comp)35 X(onen)i(t) e(of)t Fk(g)t(r)j Fl 1330 12 J(I)g Fn 38 12 L(\050)g Fk(A)g Fn(\051)g (of)t(degree)t(0)t(in)t(the)t(second)t(grading)t(is)t Fk 1368 99 C(A=I) g Fn(,)7 X(whic)30 X(h)e(de\014nes)t(the)t(pro)t(jectiv)l(e)d(sc)u(heme) d Fk(X)u Fn(=)v(Pro)r(j\050)k Fk(A=I)h Fn(\051.)m(F)44 X(or)-7 X(eac)30 X(h)e(elemen)t(t)e Fk(x)t Fn 100 G(of)g(degree)n(one)n(in)n Fk(A=I)n Fn (there)u(is)n(an)n(op)n(en)i(a\016ne)n(subset)n(of)n(Pro)n(j)l(\050)g Fk(A=I)g Fn(\051)7 X(with)24 X(asso)t(ciated)i 99 G(ring)g(consisting)z (of)z(the)z(elemen)z(ts)e(of)z(degree)z(zero)z(in)z(the)z(lo)z (calization)i(\050)z Fk(A=I)g Fn(\051)7 X Fl 2579 12 J(x)g Fn 42 12 L(.) g(Eac)44 X(h)d 2621 100 C(suc)g(h)e Fk(x)30 X Fn(de\014nes)v(an)t (elemen)v(t)c(of)v Fk(g)t(r)k Fl 1152 12 J(I)g Fn 38 12 L(\050)g Fk(A)g 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Fk(I)t Fn(is)44 X 99 G(lo)g(cally)i(generated)35 X(b)s(y)e(a)s (regular)t(sequence,)s(the)v(asso)s(ciated)j(graded)s(ring)t Fk(g)s(r)k Fl 2610 12 J(I)g Fn 38 12 L(\050)g Fk(A)g Fn(\051)g(is)t 2648 100 C(lo) g(cally)i(isomorphic)24 X(to)s(the)t(symmetric)s(algebra)t(of)s Fk(I)t (=I)m Fo 1818 30 L(2)g Fn 60 30 J(o)g(v)e(er)d Fk(A=I)t Fn(.)m(Th)36 X (us)d(if)24 X(w)s(e)e(de\014ne)s 1878 99 C(a)g(sc)38 X(heme)e(Pro)t(j)l (\050Sym)g Fl 727 20 J(A=I)g Fn 122 20 L(\050)g Fk(I)g(=I)7 X Fo 160 30 L(2)g Fn 37 30 J(\051\051)g(asso)38 X(ciated)j(to)t(the)u(bigraded)t (ring)u(Sym)t Fl 1468 20 J(A=I)g Fn 121 20 L(\050)g Fk(I)g(=I)7 X Fo 160 30 L(2)g Fn 37 30 J(\051)g 2832 100 C(using)g(a)23 X(similar)s (de\014nition,)t(the)t(map)t(induced)s(on)t(sc)s(hemes)e(b)s(y)e(the)s (natural)t(surjection)s H(from)g(Sym)y Fl 356 20 J(A=I)g Fn 121 20 L (\050)g Fk(I)g(=I)7 X Fo 160 30 L(2)g Fn 37 30 J(\051)g(to)28 X Fk(g)s (r)k Fl 242 12 J(I)g Fn 38 12 L(\050)g Fk(A)g Fn(\051)g(is)t(an)s (isomorphism.)t 829 99 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J(\051)g(to)33 X Fk(g)t(r)j Fo 260 10 J(~)g Fl 8 15 C(I)g Fn 38 25 L(\050)g Fk(A=)g F2(p)g Fk(A)g Fn (\051.)g(Let)54 X Fk(K)33 X Fn(denote)z(the)t(k)t(ernel)e 1612 99 C(of) g(this)r(map,)r(and)s(let)r Fh(K)r Fn(denote)s(the)r(asso)r(ciated)i (sheaf)r(of)r(ideals)r(in)r Fh(O)r Fl 2387 12 J(Y)g Fn 57 12 L(,)g (where)s Fh(O)r Fl 365 12 J(Y)g Fn 2809 88 C(denotes)g(the)z(structure) z(sheaf)z(of)z Fk(Y)z Fn(.)18 X(Then)71 X Fh(K)38 X Fn(de\014nes)w(a)t (closed)v(subsc)t(heme)f(of)t Fk(Y)v Fn 100 G(whic)g(h)e(is)p(the)q(pro) p(jectiv)l(e)d(sc)q(heme)d(de\014ned)q(b)p(y)e Fk(g)p(r)j Fo 1669 10 J (~)g Fl 8 15 C(I)g Fn 38 25 L(\050)g Fk(A=)g F2(p)g Fk(A)g Fn(\051.)g (The)57 X(dimension)34 X(of)u(this)t 1699 99 C(subsc)g(heme)e(is)23 X (equal)t(to)t(the)t(dimension)t(of)t(Pro)t(j\050)l Fk(A=)g F2(p)g Fk(A) g Fn(\051,)g(whic)u(h)e(is)t Fk(n)t Fn(+)9 X(dim\050)t Fk(R)g(=)h F2(p) g Fn(\051.)g(Let)35 X Fh 100 G(I)g Fn(denote)x(the)s(sheaf)q(of)s (ideals)q(in)s Fh(O)q Fl 1226 12 J(Y)g Fn 91 12 L(de\014ned)g(b)r(y)e (the)r(ideal)r Fk(I)r(=I)7 X Fo 892 30 L(2)g Fh 59 30 J(\010)g Fk(I)22 X Fo 130 30 L(2)g Fk 37 30 J(=I)g Fo 85 30 L(3)g Fh 59 30 J(\010)g(\001) t(\001)14 X(\001)t Fn(of)33 X Fk 2579 99 C(g)g(r)j Fl 80 12 J(I)g Fn 38 12 L(\050)g Fk(A)g Fn(\051.)g 7 100 J(W)g(e)-7 X(wish)27 X(to)t (de\014ne)t(an)t(Euler)t(c)t(haracteristic)e Fk(\037)t Fl 1570 12 J(Y)g Fn 57 12 L(\050)g Fh(O)g Fl 98 12 J(Y)g Fk 57 12 L(=)g Fh(I)g Fk(;)m Fh (O)14 X Fl 196 12 J(Y)g Fk 57 12 L(=)g Fh(K)g Fn(\051.)h(W)37 X(e)-7 X (\014rst)27 X(recall)t 749 280 C(3)g EOP end %%PageTrailer %======================================================================= %%Page: 4 4 %%BeginPageSetup DVIALWdict begin %%EndPageSetup BOP Fn 1123 5385 A(the)g(construction)27 X(of)t(T)t(or)-7 X Fi 839 37 L(O)g Fm 53 8 J(Y)g Fl 54 52 C(i)g Fn 107 23 L(\050)g Fh(O)g Fl 98 12 J(Y)g Fk 57 12 L(=)g Fh(I)g Fk(;)m Fh(O)14 X Fl 196 12 J(Y)g Fk 57 12 L(=)g Fh(K)g Fn(\051.)h(The)37 X(sheaf)27 X(T)t(or)-7 X Fi 705 37 L(O)g Fm 53 8 J(Y)g Fl 52 52 C(i)g Fn 106 23 L(\050)g Fh(O)g Fl 98 12 J(Y)g Fk 57 12 L(=)g Fh(I)g Fk(;)m Fh(O)14 X Fl 196 12 J(Y)g Fk 57 12 L(=)g Fh(K)g Fn(\051)h(can)27 X 2573 100 C(b)g(e)i(computed)41 X(using)r(a)t(lo)r (cally)k(free)r(resolution)t(in)r(either)t(v)r(ariable)d(and)r (tensoring)t H(with)g(the)33 X(other,)t(or)u(b)t(y)e(taking)t(the)t (asso)t(ciated)i(shea)t(v)e(es)d(of)u(the)s(lo)u(cal)h(computations)u H (of)g(T)v(or.)-7 X(The)58 X(lo)34 X(cally)j(free)t(resolutions)u(are)t (\014nite)u(since)t(the)u(graded)t(ring)u Fk(g)t(r)k Fl 2610 12 J(I)g Fn 38 12 L(\050)g Fk(A)g Fn(\051)g(is)u 2648 100 C(lo)g(cally)i(a)q(p)q (olynomial)i(ring)q(o)q(v)e(er)d(a)r(regular)p(ring,)r(whic)q(h)f(follo) p(ws)f(from)p(the)r(fact)p(that)r H(Pro)g(j)l(\050)g Fk(A=I)g Fn(\051)7 X(is)30 X(a)t(regular)t(sc)t(heme.)e(Eac)44 X(h)e(T)30 X(or)-7 X Fi 1505 37 L(O)g Fm 53 8 J(Y)g Fl 55 52 C(i)g Fn 107 23 L(\050)g Fh(O)g Fl 98 12 J(Y)g Fk 57 12 L(=)g Fh(I)g Fk(;)m Fh(O)14 X Fl 196 12 J(Y)g Fk 57 12 L(=)g Fh(K)g 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Fh(O)g Fl 98 12 J(Z)g Fk 53 12 L(=)g Fh(I)g Fl 87 12 J(F)g Fk 56 12 L(;) g Fh(O)14 X Fl 103 12 J(Z)g Fk 53 12 L(=)g Fh(K)g Fl 105 12 J(F)g Fn 56 12 L(\051)g(and)38 X Fk(\037)s Fl 294 12 J(W)g Fn 76 12 L(\050)g Fh(O)g Fl 98 12 J(W)g Fk 76 12 L(=)g Fh(I)g Fl 87 12 J(G)g Fk 54 12 L(;)g Fh(O) 14 X Fl 103 12 J(W)g Fk 76 12 L(=)g Fh(K)g Fl 105 12 J(G)g Fn 55 12 L (\051)g(de\014ned)38 X(b)s(y)e(the)s(same)t(pro)s(cess)j 1740 99 C(whic) g(h)e(w)o(e)e(outlined)o(ab)p(o)i(v)e(e.)d(These)54 X(Euler)33 X(c)u (haracteristics)d(are)u(simpler)t(to)u(compute)t 100 G(than)g(those)w (on)w Fk(Y)w Fn(,)18 X(since)39 X(the)p(ideals)r Fk(I)p Fl 1318 12 J(F) g Fn 91 12 L(and)g Fk(I)q Fl 206 12 J(G)g Fn 92 12 L(are)g(generated)q (b)q(y)e(v)q(ariables)b(in)q(a)q 1707 100 C(p)g(olynomial)i(ring,)23 X (so)t(that)t(the)s(T)t(ors)-7 X(in)23 X(the)s(\014rst)t(step)s(can)t(b) s(e)j(computed)s(using)t(Koszul)s 99 G(complexes.)g(W)43 X(e)-7 X(note) 30 X(that)s(in)t(the)s(case)t(of)s Fk(F)t Fn(the)41 X(degrees)30 X(of)s (the)t Fk(T)s Fl 2288 12 J(i)g Fn 57 12 L(m)g(ust)e(b)s(e)j(tak)s(en)e 2345 100 C(in)g(to)e(accoun)q(t)e(in)q(computing)r(Euler)q(c)r (haracteristics.)d(W)38 X(e)-8 X(then)29 X(ha)q(v)f(e)c F7 183 G (Theorem)g(1.1)w(\050Gabb)w(er\051)i F6(With)42 X(notation)29 X(as)u(ab) t(ove,)c(we)t(have)u 98 166 J(1.)g Fn(\133)42 X Fk(R)g Fn(\050)h Fk(X)g Fn(\051)7 X(:)25 X Fk(Q)t Fn(\050)g Fk(R)g(=)h F2(q)g Fn(\051\135)g Fh (\001)n Fk(\037)n Fl 823 12 J(R)g Fn 54 12 L(\050)g Fk(R)g(=)h F2(p)g Fk(;)g(R)14 X(=)g F2(q)h Fn(\051)f(=)26 X Fk(\037)r Fl 561 12 J(Y)g Fn 57 12 L(\050)g Fh(O)g Fl 98 12 J(Y)g Fk 57 12 L(=)g Fh(I)g Fk(;)m Fh(O) 14 X Fl 196 12 J(Y)g Fk 57 12 L(=)g Fh(K)g Fn(\051)h F6(,)g(wher)31 X(e) c Fk(R)t Fn(\050)g Fk(X)h Fn(\051)m F6(is)t(the)t 1793 99 C(function)g (\014eld)y(of)x Fk(X)y Fn(=)40 X(Pro)n(j)k(\050)h Fk(A=I)f Fn(\051)8 X F6(and)34 X Fk(Q)w Fn(\050)f Fk(R)h(=)g F2(q)h Fn(\051)f F6(is)w(the)t (\014eld)w(of)t(fr)w(actions)a(of)w Fk 100 G(R)g(=)h F2(q)g F6(.)g 110 166 C(2.)g Fk(\037)42 X Fl 162 12 J(Y)g Fn 57 12 L(\050)g Fh(O)g Fl 98 12 J(Y)g Fk 57 12 L(=)g Fh(I)g Fk(;)m Fh(O)14 X Fl 196 12 J(Y)g Fk 57 12 L(=)g Fh(K)g Fn(\051)h Fk(\076)23 X Fn(0)t F6(if)30 X(and)s(only)t (if)s Fk 117 183 J(\037)g Fl 52 12 J(Y)g Fj 53 17 L(0)g Fn 27 5 J (\050\050)g Fh(O)g Fl 130 12 J(Y)g Fk 57 12 L(=)g Fh(I)g Fn(\051)m Fh (\012)19 X Fl 208 12 J(R)g Fk 73 12 L(k)g(;)j Fn(\050)n Fh(O)h Fl 182 12 J(Y)g Fk 57 12 L(=)g Fh(K)g Fn(\051)h Fh(\012)t Fl 221 12 J(R)g Fk 73 12 L(k)g Fn(\051)j Fk(\076)x Fn(0)x Fk(;)g F6 1767 182 C(wher)g(e)c (we)29 X(put)u Fk(Y)t Fi 568 30 L(0)g Fn 47 30 J(=)g(Pro)n(j\050)l Fk(g) g(r)j Fl 361 12 J(I)g Fn 38 12 L(\050)g Fk(A)g Fn(\051)g Fh(\012)18 X Fl 209 12 J(R)g Fk 73 12 L(k)g Fn(\051)j F6(.)g 1406 166 C(3.)g(We)42 X (have)29 X(the)u(e)t(qualities)c Fk 467 183 J(\037)g Fl 52 12 J(Y)g Fj 53 17 L(0)g Fn 27 5 J(\050)g Fh(O)g Fl 98 12 J(Y)g Fk 57 12 L(=)g Fh(I) g(\012)p Fl 177 12 J(R)g Fk 73 12 L(k)g(;)j Fh(O)13 X Fl 149 12 J(Y)g Fk 57 12 L(=)g Fh(K)g(\012)20 X Fl 190 12 J(R)g Fk 73 12 L(k)g Fn(\051) j(=)w Fk(\037)w Fl 240 12 J(Z)g Fn 53 12 L(\050)g Fh(O)g Fl 98 12 J(Z)g Fk 53 12 L(=)g Fh(I)g Fl 87 12 J(F)g Fk 56 12 L(;)g Fh(O)n Fl 103 12 J (Z)g Fk 53 12 L(=)g Fh(K)g Fl 105 12 J(F)g Fn 55 12 L(\051)g 1401 183 C (=)g Fk(\037)w Fl 140 12 J(W)g Fn 76 12 L(\050)g Fh(O)g Fl 98 12 J(W)g Fk 76 12 L(=)g Fh(I)g Fl 87 12 J(G)g Fk 54 12 L(;)g Fh(O)n Fl 103 12 J (W)g Fk 76 12 L(=)g Fh(K)g Fl 105 12 J(G)g Fn 56 12 L(\051)g Fk(:)g Fn 533 477 C(5)g EOP end %%PageTrailer %======================================================================= %%Page: 6 6 %%BeginPageSetup DVIALWdict begin %%EndPageSetup BOP F6 1123 5385 A(Pr)g(o)c(of.)c Fn(F)90 X(or)-7 X(the)21 X(pro)s(of)j(of) s(ev)t(erything)d(except)t(statemen)s(t)e(2)s(w)t(e)d(refer)t(to)s(the) t(references)s 100 G(Berthelot)g(\1332\135,)38 X(Ho)x(c)h(hster)f (\1336\135,)s(or)y(Rob)s(erts)k(\13315)s(\135)h(cited)s(ab)v(o)h(v)f(e.) c(Statemen)71 X(t)c(2)40 X(is)q(not)t H(pro)g(v)e(en)d(explicitly)24 X (there.)s(Let)36 X Fk(r)23 X Fn(b)w(e)i(the)u(co)t(dimension)j(of)t Fk (X)u Fn(=)z(Pro)u(j\050)k Fk(A=I)h Fn(\051)m(in)u(Pro)t(j\050)l Fk(A)g Fn(\051.)g H(Then)g(Gabb)q(er)i(sho)q(ws)d(that)q Fk(\037)p Fl 955 12 J (Y)g Fn 57 12 L(\050)g Fh(O)g Fl 98 12 J(Y)g Fk 57 12 L(=)g Fh(I)g Fk(;) m Fh(M)14 X Fn(\051)g(is)z(nonnegativ)y(e)e(for)y(mo)z(dules)i(of)z (dimension)y 1167 100 C(at)g(most)23 X Fk(r)u Fn(suc)v(h)e(that)t(the)u (supp)t(ort)j(of)t Fh(M)u Fn(lies)t(o)u(v)d(er)e(the)t(maximal)u(ideal) t(of)u Fk(R)t Fn(,)h(and)u(that)u H(it)g(v)30 X(anishes)c(for)t(mo)t (dules)i(of)t(dimension)t(less)t(than)t Fk(r)t Fn(.)i(Hence)45 X(w)30 X (e)e(ha)t(v)e(e)d(that)u(whether)s Fk H(\037)g Fl 52 12 J(Y)g Fn 57 12 L(\050)g Fh(O)g Fl 98 12 J(Y)g Fk 57 12 L(=)g Fh(I)g Fk(;)m Fh(O)14 X Fl 196 12 J(Y)g Fk 57 12 L(=)g Fh(K)g Fn(\051)h Fk(\076)31 X Fn(0)t(or) v(not)u(dep)v(ends)i(only)v(on)u(the)v(supp)u(ort)j(of)u Fh(O)v Fl 1861 12 J(Y)g Fk 57 12 L(=)g Fh(K)g Fn(\050and,)w(in)w 2435 100 C(fact,)g (only)y(on)w(the)w(prime)w(ideals)w(of)w(dimension)w Fk(r)w Fn(in)y(the) w(supp)w(ort\051.)i(Since)56 X(a)34 X(p)t(o)i(w)e(er)d 99 G(of)g F2(m)v Fn(annihilates)v Fh(O)v Fl 698 12 J(Y)g Fk 57 12 L(=)g Fh(K)g Fn(,)h (the)y(supp)v(ort)j(of)v Fh(O)v Fl 803 12 J(Y)g Fk 57 12 L(=)g Fh(K)g Fn(is)w(the)w(same)u(as)w(the)u(supp)w(ort)i(of)w 1615 100 C(\050)g Fh (O)g Fl 98 12 J(Y)g Fk 57 12 L(=)g Fh(K)g Fn(\051)h Fh(\012)19 X Fl 221 12 J(R)g Fk 73 12 L(k)g Fn(.)j(This)37 X(pro)28 X(v)d(es)e(the)s(second) t(statemen)s(t.)e F7(q.e.d.)843 X Fn 324 149 C(F)g(rom)-7 X(the)29 X (theorem)s(ab)t(o)i(v)e(e,)d(it)u(su\016ces)s(to)t(test)s(the)t(p)s (ositivit)j(y)d(of)t(the)s(Euler)t(c)s(har-)e 125 100 C(acteristic)g Fk (\037)t Fl 410 12 J(W)g Fn 76 12 L(\050)g Fh(O)g Fl 98 12 J(W)g Fk 76 12 L(=)g Fh(I)g Fl 87 12 J(G)g Fk 54 12 L(;)g Fh(O)14 X Fl 103 12 J(W)g Fk 76 12 L(=)g Fh(K)g Fl 105 12 J(G)g Fn 55 12 L(\051.)g(W)40 X(e)a (note)28 X(that)v Fk(k)t Fn(\133)j Fk(S)g Fl 726 12 J(i)g Fk 28 12 L(;) g(U)14 X Fl 94 12 J(j)g(k)j Fn 72 12 L(\135)g(is)29 X(a)t(subring)t(of) t Fk(G)t Fn(.)g(In)40 X 2060 100 C(what)g(follo)30 X(ws,)d(if)u Fk(J)s Fn(is)38 X(an)29 X(ideal)u(of)t(one)u(of)t(the)u(graded)t(rings)u(w)t(e) e(are)t(considering,)u(w)u(e)e 265 32 J 55 4 B 265 67 C(denote)g Fk(J)p Fn(the)y(ideal)p(of)q(elemen)p(ts)e Fk(a)p Fn(suc)q(h)d(that)q(\050)p Fk(X)g Fo 1642 12 J(0)g Fk 37 12 L(;)g(:)14 X(:)t(:)t(;)t(X)t Fl 254 12 J(n)g Fn 45 12 L(\051)g Fl 32 30 L(k)g Fk 41 30 J(a)g Fh(\022)23 X Fk(J) t Fn(for)33 X(some)26 X(in)s(teger)e Fk 2051 100 C(k)g Fn(.)j(The)37 X (aim)28 X(of)s(this)t(section)s(is)t(to)s(pro)t(v)d(e)e(the)s(next)t (theorem:)s F7 183 G(Theorem)g(1.2)x F6(With)41 X(notation)27 X(as)r(ab) t(ove,)b(the)u(fol)r(lowing)l(statements)r(ar)t(e)b(e)t(quivalent:)b 98 166 J(1.)g Fk(\037)42 X Fl 162 12 J(R)g Fn 54 12 L(\050)g Fk(R)g(=)h F2 (p)g Fk(;)g(R)14 X(=)g F2(q)h Fn(\051)f Fk(\076)24 X Fn(0)r F6(.)h 106 99 C 127 4 B 110 67 C(2.)g Fk(K)42 X Fl 181 12 J(G)g Fh 73 12 L(\\)g Fk (k)18 X Fn(\133)j Fk(S)g Fo 193 12 J(1)g Fk 37 12 L(;)g(:)p(:)p(:)p(;)p (S)p Fl 236 12 J(d)g Fk 39 12 L(;)g(U)p Fo 94 12 J(00)g Fk 70 12 L(;)g (:)p(:)p(:)p(U)p Fl 205 12 J(nn)g Fn 86 12 L(\135)g(=)y(0)y F6(.)g Fn 1187 182 C(In)g(the)28 X(remainder)s(of)t(this)s(section,)t(w)s(e)e(giv) s(e)e(a)s(pro)t(of)i(of)t(Theorem)s(1.2.)t 100 G(W)g(e)-7 X(recall)30 X (that)t(w)t(e)e(ha)t(v)e(e)d(sho)u(wn)d(that)u(it)s(su\016ces)u(to)s (sho)u(w)d(that)u(the)s(second)u(con-)s 125 100 C(dition)g(of)r(the)r (theorem)r(is)r(equiv)r(alen)c(t)d(to)s(the)q(fact)s(that)q Fk(\037)s Fl 1922 12 J(W)g Fn 76 12 L(\050)g Fh(O)g Fl 98 12 J(W)g Fk 76 12 L(=)g Fh(I)g Fl 87 12 J(G)g Fk 57 12 L(;)g Fh(O)14 X Fl 103 12 J(W)g Fk 76 12 L(=)g Fh(K)g Fl 105 12 J(G)g Fn 55 12 L(\051)g Fk(\076)24 X Fn(0.)t 2655 99 C(W)g(e)-7 X(next)28 X(examine)s(the)t(computation)s(of)t(this) s(Euler)t(c)s(haracteristic)e(more)s(closely)t(.)-7 X 125 100 J(Let)g Fk(C)31 X Fn(denote)y(the)t(ring)s Fk(k)t Fn(\133)i Fk(S)h Fl 960 12 J (i)g Fk 27 12 L(;)g(U)14 X Fl 94 12 J(j)g(k)j Fn 72 12 L(\135.)g(W)46 X (e)-7 X(then)31 X(ha)s(v)e(e)d(a)t(comm)s(utativ)e(e)d(diagram)t(of)s 1278 99 C(inclusions)g(of)q(rings:)p Fk 1241 95 J(G)g Fh(\040)83 X Fk(B) t Fh 12 99 J(")g(")274 X Fk 12 100 C(C)g Fh(\040)89 X Fk(k)y Fn 1241 145 C(and)g Fk(G)20 X Fn(=)w Fk(C)w Fh(\012)9 X Fl 463 12 J(k)g Fk 44 12 L(B)g Fn(.)k(These)34 X(maps)20 X(induce)t(a)t(comm)t(utativ)e(e)d (diagram)u(of)s(maps)u(of)s(sc)u(hemes)d 505 178 J(Pro)g(j)l(\050)g Fk (G)g Fn(\051)g Fh(!)87 X Fn(Pro)u(j)k(\050)h Fk(B)f Fn(\051)l Fh 124 99 J(#)g(#)508 X Fn 128 100 C(Sp)g(ec\050)i Fk(C)g Fn(\051)m Fh(!)83 X Fn (Sp)t(ec\050)i Fk(k)g Fn(\051)j Fk(:)g Fn 1008 179 C(In)g(this)30 X (diagram)t(the)t(v)t(ertical)e(arro)t(ws)e(are)t(pro)t(jectiv)l(e)d (morphisms.)u(F)43 X(or)a(a)29 X(sc)v(heme)d Fk(T)v Fn 100 G(w)g(e)e (let)n Fk(K)n Fo 304 12 J(0)g Fn 37 12 L(\050)g Fk(T)g Fn(\051)12 X (denote)22 X(the)v(Grothendiec)t(k)f(group)t(of)v(coheren)t(t)f(shea)t (v)f(es)c(on)w Fk(T)s Fn(.)13 X(W)34 X(e)a(then)22 X 341 99 C(ha)g(v)e (e)d(a)z(corresp)y(onding)j(diagram)y(of)z(Grothendiec)y(k)e(groups)y Fk 839 177 J(K)g Fo 71 12 J(0)g Fn 37 12 L(\050Pro)g(j)l(\050)g Fk(G)g Fn(\051\051)g Fh(!)87 X Fk(K)u Fo 684 12 J(0)g Fn 37 12 L(\050Pro)g(j)l (\050)g Fk(B)g Fn(\051\051)k Fh 618 100 C(#)g(#)680 X Fk 215 100 C(K)g Fo 71 12 J(0)g Fn 37 12 L(\050Sp)g(ec\050)i Fk(C)g Fn(\051\051)m Fh(!) 83 X Fk(K)t Fo 683 12 J(0)g Fn 37 12 L(\050Sp)g(ec\050)i Fk(k)g Fn (\051\051)j Fk(:)g Fn 1095 99 L(\0501\051)g 1347 419 C(6)g EOP end %%PageTrailer %======================================================================= %%Page: 7 7 %%BeginPageSetup DVIALWdict begin %%EndPageSetup BOP Fn 1123 5385 A(The)g(v)23 X(ertical)e(maps)t(are)t(obtained)t(b)t(y)e (taking)t(alternating)t(sums)t(of)t(sheaf)t(cohomology)t(.)-7 X 100 G (The)g(top)19 X(horizon)t(tal)d(map)t(is)t(obtained)t(b)t(y)d(taking)t (a)t(coheren)t(t)d(sheaf)t Fh(M)t Fn(to)t(the)t(alternat-)t H(ing)g (sums)29 X(of)t(T)t(or)-7 X Fi 573 37 L(O)g Fm 53 8 J(W)g Fl 53 52 C(i) g Fn 121 23 L(\050)g Fh(O)g Fl 98 12 J(W)g Fk 76 12 L(=)g Fh(I)g Fl 87 12 J(G)g Fk 56 12 L(;)g Fh(M)14 X Fn(\051,)g(while)29 X(the)t(b)t(ottom) i(horizon)t(tal)e(map)t(is)t(obtained)t 1011 100 C(b)g(y)e(taking)u(a)w Fk(C)u Fn(-mo)7 X(dule)h Fk(M)32 X Fn(to)39 X(the)32 X(alternating)r (sum)t(of)r(T)t(or)-8 X Fl 2043 36 L(C)g 2 57 C(i)g Fn 56 21 L(\050)g Fk(C)g(=)k Fn(\050)g Fk(S)g Fl 220 12 J(i)g Fk 28 12 L(;)g(U)14 X Fl 94 12 J(j)g(k)j Fn 72 12 L(\051)g Fk(;)g(M)t Fn(\051.)o(T)46 X(o)-7 X 2511 100 C(see)g(that)21 X(the)t(diagram)t(comm)t(utes,)e(let)u Fk(K)u Fi 1347 12 J(\017)g Fn 59 12 L(b)g(e)i(the)t(Koszul)t(complex)t(on)t(the)t Fk(d)t Fn(+)l(\050)m Fk(n)f Fn(+)m(1\051)l Fo 1422 30 L(2)g Fn 2828 130 C(elemen)g(ts)e Fk(S)29 X Fl 392 12 J(i)g Fk 28 12 L(;)g(U)14 X Fl 94 12 J(j)g(k)j Fn 101 12 L(o)g(v)e(er)d Fk(C)30 X Fn(,)l(and)u(let)r Fk(C) t Fi 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X Fl 103 12 J(W)g Fk 76 12 L(=)g Fh(K)g Fl 105 12 J(G)g Fn 55 12 L(\051)g(=)23 X Fk(\015)t(\014)l Fn(\050)k Fh(O)g Fl 340 12 J(W)g Fk 76 12 L(=)g Fh(K)g Fl 105 12 J(G)g Fn 56 12 L(\051)g Fk(:)g Fn 2085 149 C (W)g(e)-7 X(recall)28 X(that)s(the)t(sheaf)s Fh(O)t Fl 966 12 J(W)g Fk 75 12 L(=)g Fh(K)g Fl 105 12 J(G)g Fn 84 12 L(has)g(supp)t(ort)i(of)t (dimension)s Fk(r)t Fn(=)q Fk(d)o Fn(+)19 X(\050)s Fk(n)g Fn(+)s(1\051) t Fo 1493 30 L(2)g Fn 37 30 J(.)g 2635 100 C(It)g(follo)z(ws)e(from)z (results)26 X(of)s(Serre)t(\13316)s(\135)g(that)s(for)t(an)s(y)e (\014nitely)s(generated)t Fk(C)s Fn(-mo)m(dule)i Fk 125 100 C(M)g Fn(,) 9 X(if)26 X(w)s(e)e(iden)s(tify)e Fk(K)s Fo 702 12 J(0)g Fn 37 12 L (\050Sp)g(ec\050)i Fk(k)g Fn(\051\051)j(with)s F4(Z)s Fn(,)g(then)t Fk (\015)s Fn(\050\133)l Fk(M)g Fn(\135\051)9 X(is)25 X(equal)t(to)t(the)t (rank)t(of)t Fk(M)t Fn(as)34 X 739 100 C(a)g Fk(C)n Fn(-mo)m(dule.)i(Th) w(us)e(w)27 X(e)e(ha)t(v)e(e)d Fk 432 182 J(\015)g(\014)l Fn(\050)k Fh (O)g Fl 197 12 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g(elemen)t(ts)d(whic)t(h)d(are)t(annihilated)s(b)t(y)d(a)t(p)s(o)j(w)d (er)e(of)s(\050)t Fk(X)g Fo 1868 12 J(0)g Fk 37 12 L(;)g(:)14 X(:)t(:)t (;)t(X)t Fl 254 12 J(n)g Fn 45 12 L(\051)g(mo)28 X(dulo)i Fk(J)t Fn(.)8 X F7 2204 178 C(Theorem)g(2.1)32 X F6(L)41 X(et)c Fk(P)x F6(b)57 X(e)b (a)46 X(bigr)r(ade)d(d)b(prime)t(ide)r(al)d(in)r Fk(B)t Fn(\133)j Fk(S) h Fo 2102 12 J(1)g Fk 37 12 L(;)g(:)14 X(:)t(:)t(;)t(S)t Fl 236 12 J(d) g Fk 39 12 L(;)g(T)t Fo 86 12 J(0)g Fk 37 12 L(;)g(:)t(:)t(:)t(;)t(T)t Fl 234 12 J(n)g Fn 45 12 L(\135)g F6(.)g 2816 100 C(Supp)g(ose)c(that) 29 X Fn 1411 247 J(8)g EOP end %%PageTrailer %======================================================================= %%Page: 9 9 %%BeginPageSetup DVIALWdict begin %%EndPageSetup BOP F6 1196 5385 A(\050i\051)g Fk(k)42 X Fn(\133)i Fk(S)h Fo 255 12 J(1)g Fk 37 12 L(;)g(:)14 X(:)t(:)t(;)t(S)t Fl 236 12 J(d)g Fn 39 12 L(\135)g Fh(\\)x Fk(P)y Fn(=)34 X(0)24 X F6(.)f 432 94 C 249 4 B 161 72 C (\050ii\051)g Fk(')42 X Fn(\050)g 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L(0)g Fk 23 30 J(=P)g Fi 107 30 L(0)g Fh 65 8 J(\030)g Fn 26 G(=)g Fk 106 4 L(F)g(=P)7 X Fn(,)x(it)42 X(is)q(clear)q(that)q(dim\050)q Fk(F) g(=P)7 X Fn(\051)x Fh(\024)42 X Fk(d)s Fn(+)27 X Fk(n)r Fn(if)40 X(and) r(only)t(if)r 775 100 C(dim\050)g Fk(F)g Fi 235 30 L(0)g Fk 23 30 J(=P) g Fi 107 30 L(0)g Fn 23 30 J(\051)g Fh(\024)26 X Fk(d)t Fn(+)n Fk(n)19 X Fn(.)h(W)41 X(e)a(th)28 X(us)f(only)t(ha)w(v)c(e)f(to)t(c)w(hec)c(k)f (that)t(the)w(h)t(yp)f(otheses)h(are)w(also)t 388 100 C(equiv)g(alen)b (t.)e 125 99 J(T)g(o)-7 X(c)28 X(hec)d(k)e(the)s(\014rst)t(h)s(yp)e (othesis,)i(w)t(e)d(note)t(that)s(there)t(is)s(a)t(comm)s(utativ)e(e)d (diagram)t Fk 835 177 J(k)g Fn(\133)j Fk(S)g Fo 120 12 J(1)g Fk 37 12 L (;)g(:)14 X(:)t(:)t(;)t(S)t Fl 236 12 J(d)g Fn 39 12 L(\135)g Fh(!)83 X Fk(F)t Fi 336 30 L(0)g Fk 23 30 J(=P)g Fi 107 30 L(0)g Fh 361 130 C(&)g (#)171 X Fk 192 100 J(F)g(=P)7 X 234 99 L(:)g Fn 1923 278 C(Since)g(the) 22 X(\014rst)u(h)t(yp)e(othesis)i(is)u(satis\014ed)t(for)u Fk(P)t Fn(if) 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g(ectiv)i(ely)e(,)-7 X(w)28 X(e)e(ha)t(v)e(e)d(an)u(isomorphism)s Fk(G) u Fi 1451 30 L(0)g Fk 23 30 J(=Q)g Fi 108 30 L(0)g Fh 47 8 J(\030)g Fn 26 G(=)g Fk 88 4 L(G=Q)g Fn(.)g 2671 100 C(W)g(e)-7 X(pro)34 X(v)d(e)e (this)s(fact)t(directly;)s(let)w Fk(\013)s Fn(b)u(e)h(the)u(surjectiv)r (e)f(map)r(from)u Fk(G)r Fi 2369 30 L(0)g Fn 57 30 J(to)g Fk(G)t Fn(.)g (Since)55 X 2426 100 C(the)g(map)35 X(induced)t(b)t(y)e Fk(\013)t Fn (from)t Fk(G)t Fi 1138 30 L(0)g Fk 23 30 J(=Q)g Fi 108 30 L(0)g Fn 58 30 J(to)g Fk(G=Q)t Fn(is)t(clearly)t(surjectiv)t(e,)e(it)u(su\016ces)u (to)s 1327 100 C(sho)g(w)e(that)28 X(it)t(is)t(injectiv)t(e.)e(Let)38 X Fk(c)28 X Fi 1102 30 L(0)g Fn 51 30 J(b)g(e)i(an)t(elemen)t(t)e(of)t Fk (G)t Fi 696 30 L(0)g Fn 51 30 J(suc)g(h)e(that)t Fk(\013)t Fn(\050)g Fk (c)g Fi 489 30 L(0)g Fn 23 30 J(\051)g Fh(2)p Fk(Q)o Fn(.)h(Since)37 X Fk 2412 100 C(Q)g Fn(is)n(generated)30 X(b)u(y)d Fk(')u Fn(\050)g Fk(P) g Fn(\051,)12 X(this)31 X(means)t(that)s(w)t(e)d(can)t(write)s Fk(\013) t Fn(\050)g Fk(c)g Fi 2144 30 L(0)g Fn 23 30 J(\051)g(=)q Fq 153 62 L(P) g Fk 101 62 J(c)g Fl 36 12 J(i)g Fk 28 12 L(')g Fn(\050)g Fk(p)g Fl 128 12 J(i)g Fn 28 12 L(\051)g(with)t Fk 2641 100 C(c)g Fl 36 12 J(i)g Fh 53 12 L(2)g Fk(G)n Fn(and)r Fk(p)r Fl 379 12 J(i)g Fh 53 12 L(2)g Fk(P) n Fn(.)12 X(W)41 X(e)-7 X(can)29 X(lift)t(these)t(elemen)t(ts)e(to)t (elemen)t(ts)e Fk(c)t Fi 1675 30 L(0)g Fl 52 G(i)g Fn 57 22 L(and)g Fk (p)t Fi 205 30 L(0)g Fl 52 G(i)g Fn 57 22 L(in)g Fk(G)t Fi 163 30 L(0)g Fn 52 30 J(and)g Fk 2730 100 C(P)g Fi 65 30 L(0)g Fn 23 30 J(,)g(and)v (then)t Fk(d)v Fi 452 30 L(0)g Fn 50 30 J(=)g Fq 92 62 L(P)g Fk 101 62 J(')g Fi 54 30 L(0)g Fn 23 30 J(\050)g Fk(p)g Fi 74 30 L(0)g Fl 52 G(i) g Fn 28 22 L(\051)g Fk(c)g Fi 68 30 L(0)g Fl 52 G(i)g Fn 58 22 L(is)g (in)u Fk(Q)u Fi 251 30 L(0)g Fn 53 30 J(and)g Fk(\013)u Fn(\050)g Fk(c) g Fi 285 30 L(0)g Fh 43 30 J(\000)g Fk(d)20 X Fi 128 30 L(0)g Fn 23 30 J(\051)g(=)27 X(0.)t(No)44 X(w)d(the)31 X(k)r(ernel)f(of)r Fk(\013)t Fn 1871 100 C(is)g(generated)t(b)t(y)e(the)t(k)t(ernel)e(of)t(the)t(map)t (from)t Fk(B)t Fi 1682 30 L(0)g Fn 54 30 J(to)g Fk(B)t Fn(,)k(and)u (this)t(k)t(ernel)e(is)t(clearly)t 1736 99 C(con)g(tained)e(in)q Fk(P)r Fi 541 30 L(0)g Fn 51 30 J(and)g(hence)r(the)q(ideal)r(it)q(generates)r (in)q Fk(G)r Fi 1350 30 L(0)g Fn 52 30 J(is)g(con)r(tained)d(in)r Fk(Q) q Fi 625 30 L(0)g Fn 23 30 J(.)g(This)39 X 2642 100 C(completes)g(the) 28 X(pro)s(of)j(that)s(w)t(e)d(can)t(reduce)s(to)t(the)s(case)t(of)s Fk (B)t Fi 2044 30 L(0)g Fn 23 30 J(,)g Fk(F)t Fi 116 30 L(0)g Fn 23 30 J (,)g Fk(G)t Fi 116 30 L(0)g Fn 23 30 J(,)g(and)t Fk(P)s Fi 278 30 L(0)g Fn 23 30 J(.)g 2521 100 C(W)g(e)-7 X(no)32 X(w)e(c)t(hange)e(notation,) t(replace)u Fk(B)t Fn(,)k Fk(F)u Fn(,)12 X Fk(G)33 X Fn(,)g Fk(P)t Fn(,) 12 X(and)33 X Fk(Q)s Fn(with)s Fk(B)s Fi 2261 31 L(0)g Fn 23 31 J(,)g Fk(F)t Fi 121 31 L(0)g Fn 23 31 J(,)g Fk(G)t Fi 121 31 L(0)g Fn 23 31 J (,)g Fk(P)t Fi 121 31 L(0)g Fn 23 31 J(,)g 2841 99 C(and)g Fk(Q)o Fi 228 30 L(0)g Fn 51 30 J(and)g(assume)o(that)n Fk(X)o Fo 698 12 J(0)g Fk 37 12 L(;)g(:)14 X(:)t(:)t(X)t Fl 217 12 J(n)g Fn 73 12 L(are)g (algebraically)28 X(indep)s(enden)j(t)d(o)t(v)d(er)e Fk(k)s Fn(.)j 1179 100 C(W)g(e)-7 X(next)29 X(sho)t(w)e(that)t(it)t(su\016ces)t(to)t(pro)t (v)e(e)d(that)u(under)s(the)u(giv)s(en)f(h)s(yp)f(otheses)h(the)v 125 100 C(heigh)g(t)e(of)r Fk(P)s Fn(is)39 X(at)28 X(least)s(equal)t(to)s Fk(n)t Fn(+)18 X(1.)u(Since)36 X(w)29 X(e)c(ha)t(v)c(e)f 483 182 J(h)g (t\050)e Fk(P)g Fn(\051)12 X(+)z(dim)19 X(\050)f Fk(F)h(=P)m Fn(\051)12 X(=)23 X(dim\050)t Fk(F)g Fn(\051)12 X(=)23 X(dim\050)t Fk(B)g Fn(\051) k(+)p Fk(n)o Fn(+)p Fk(d)o Fn(+)p(1)o Fk(;)g Fn 483 183 C(w)g(e)e(ha)x (v)e(e)d(that)y(h)x(t)e(\050)g Fk(P)g Fn(\051)12 X Fh(\025)23 X Fn (dim\050)t Fk(B)g Fn(\051)k(+)p(1)o(if)y(and)x(only)y(if)x 637 182 J (dim\050)g Fk(F)g(=P)7 X Fn(\051)x Fh(\024)24 X Fn(dim\050)r Fk(F)h Fn (\051)11 X Fh(\000)19 X Fn(dim)s(\050)g Fk(B)g Fn(\051)k Fh(\000)t Fn(1) s(=)y Fk(n)w Fn(+)u Fk(d:)r Fn 637 183 C(Th)g(us)e(it)28 X(su\016ces)u (to)t(sho)u(w)d(that)u(the)t(heigh)u(t)d(of)u Fk(P)t Fn(is)40 X(at)28 X (least)u(the)t(dimension)u(of)t Fk(B)u Fn(\050or)x 100 G(of)g(Pro)t(j)l (\050)g Fk(B)g Fn(\051\051)k(plus)t(1,)t(whic)t(h,)e(no)t(w)e(that)t(w) t(e)e(ha)t(v)e(e)d(reduced)u(to)s(the)u(case)s(in)u(whic)s(h)f Fk(B)s Fn(is)y 99 G(the)g(p)t(olynomial)i(ring)t Fk(k)s Fn(\133)j Fk(X)g Fo 883 12 J(0)g Fk 37 12 L(;)g(:)14 X(:)t(:)t(;)t(X)t Fl 253 12 J(n)g Fn 45 12 L(\135,)g(is)28 X Fk(n)s Fn(+)19 X(1.)s 1093 100 C(W)g(e)-7 X (next)34 X(lo)s(calize)j(at)s(the)t(m)s(ultiplicativ)e(ely)d(closed)t (set)s Fk(S)t Fn(consisting)x(of)t(nonzero)s 125 100 C(elemen)g(ts)e(of) 27 X Fk(k)t Fn(\133)j Fk(S)g Fo 553 12 J(1)g Fk 37 12 L(;)g(:)14 X(:)t (:)t(;)t(S)t Fl 236 12 J(d)g Fn 39 12 L(\135)g(and)27 X(reduce)t(the)t (problem)t(to)t(a)t(question)t(on)t(graded)t(rings)t 865 99 C(o)g(v)e (er)d(a)20 X(\014eld.)r(Let)35 X Fk(L)18 X Fn(denote)v(the)t(quotien)v (t)d(\014eld)v(of)t Fk(k)v Fn(\133)i Fk(S)h Fo 1746 12 J(1)g Fk 37 12 L (;)g(:)p(:)p(:)p(;)p(S)p Fl 236 12 J(d)g Fn 39 12 L(\135.)g(Hyp)34 X (othesis)i(\050i\051)20 X(of)r(the)t 2058 100 C(theorem)g(implies)28 X (that)u Fk(P)t Fn(de\014nes)40 X(a)28 X(prime)u(ideal)t(of)u(the)t(lo)u (calization,)i(and)u(its)t(heigh)u(t)d 99 G(will)g(remain)21 X(the)t (same.)t(F)34 X(urthermore,)a(the)21 X(second)u(h)s(yp)e(othesis)i (implies)t(that)t(there)t(is)t(a)t 100 G(non-zero)g(elemen)u(t)d(in)u Fk(L)u Fn(\133)g Fk(U)g Fo 861 12 J(00)g Fk 70 12 L(;)g(:)14 X(:)t(:)t (;)t(U)t Fl 242 12 J(nn)g Fn 86 12 L(\135)g(suc)22 X(h)d(that)t(some)t (p)t(o)i(w)e(er)d(of)t(the)t(ideal)t(generated)t 1259 100 C(b)g(y)e Fk (X)u Fo 180 12 J(0)g Fk 37 12 L(;)g(:)14 X(:)t(:)t(;)t(X)t Fl 254 12 J (n)g Fn 68 12 L(annihilates)g(this)23 X(elemen)u(t)d(mo)u(dulo)i(the)u (image)t(of)u Fk(P)t Fn(after)35 X(lo)23 X(calization)j 539 99 C(at)g Fk(S)r Fn(.)l(Changing)35 X(notation,)21 X(w)v(e)d(use)u Fk(F)t(;)j(G;) 13 X(P)v Fn(and)32 X Fk(Q)22 X Fn(to)s(denote)t(the)s(lo)t(calizations) i(of)t(the)s 100 G(original)g(rings)w Fk(F)v Fn(and)37 X Fk(G)24 X Fn (and)u(ideals)t Fk(P)u Fn(and)36 X Fk(Q)25 X Fn(at)s Fk(S)t Fn(.)k(In) 37 X(the)23 X(lo)w(calization)h(w)w(e)c(consider)w H(the)g(grading)y (induced)y(b)y(y)e(the)y(grading)y(in)y(the)y(\014rst)y(comp)y(onen)i (t;)e(that)y(is,)y(elemen)y(ts)e(of)y Fk 99 G(L)g Fn(ha)30 X(v)d(e)e (degree)s(zero,)t(the)t(degrees)t(of)s(the)t Fk(X)s Fl 1491 12 J(i)g Fn 57 12 L(and)g Fk(T)t Fl 213 12 J(i)g Fn 57 12 L(are)g(1,)t(and)t(the)t (degrees)s(of)t Fk(U)s Fl 990 12 J(ij)g Fn 2808 88 C(are)g(zero.)r(It) 37 X(su\016ces)28 X(to)s(pro)t(v)d(e)e(the)s(follo)t(wing)d(lemma.)t F7 183 G(Lemma)g(2.3)x F6(L)41 X(et)c Fk(L)22 X F6(b)u(e)b(a)u(\014eld,)t (and)v(let)t Fk(P)u F6(b)33 X(e)d(a)21 X(gr)w(ade)a(d)e(prime)s(ide)x (al)-7 X(of)25 X(the)o(p)t(olynomial)-7 X 99 G(ring)g Fk(F)39 X Fn(=)51 X Fk(L)40 X Fn(\133)g Fk(X)g Fo 542 12 J(0)g Fk 37 12 L(;)g(:)14 X(:)t (:)t(X)t Fl 217 12 J(n)g Fk 45 12 L(;)g(T)t Fo 86 12 J(0)g Fk 37 12 L(;) g(:)t(:)t(:)t(;)t(T)t Fl 234 12 J(n)g Fn 45 12 L(\135)g F6(.)g(L)66 X (et)b Fk(Q)40 X F6(b)r(e)d(the)r(ide)t(al)b(gener)t(ate)b(d)d(by)r Fk(') t Fn(\050)f Fk(P)h Fn(\051)11 X F6(in)39 X Fk 1243 100 C(G)g Fn(=)23 X Fk(L)t Fn(\133)g Fk(X)g Fo 325 12 J(0)g Fk 37 12 L(;)g(:)14 X(:)t(:)t(;) t(X)t Fl 254 12 J(n)g Fk 45 12 L(;)g(U)t Fo 94 12 J(00)g Fk 70 12 L(;)g (:)t(:)t(:)t(;)t(U)t Fl 242 12 J(nn)g Fn 86 12 L(\135)g F6(,)g(and)30 X 480 115 C 66 4 B 719 115 L(let)g Fk 719 182 C(Q)g Fn(=)23 X Fh(f)t Fk(c) g Fh(2)t Fk(G)t Fh(j9)g Fk(k)g F6(with)w Fn(\050)t Fk(X)g Fo 832 12 J(0) g Fk 37 12 L(;)g(:)14 X(:)t(:)t(;)t(X)t Fl 254 12 J(n)g Fn 45 12 L(\051) g Fl 32 34 L(k)g Fk 41 34 J(c)g Fh(\022)23 X Fk(Q)t Fh(g)g Fk(:)g 1827 116 C U F6 87 67 C(If)g Fk(Q)30 X Fh(\\)18 X Fk(L)u Fn(\133)f Fk(U)h Fo 381 12 J(00)g Fk 70 12 L(;)g(:)p(:)p(:)p(;)p(U)p Fl 242 12 J(nn)g Fn 86 12 L(\135)g Fh(6)y Fn(=)g(0)y F6(,)g(then)30 X(the)s(height)t(of)s Fk(P) t F6(is)41 X(at)30 X(le)s(ast)c Fk(n)s Fn(+)19 X(1)s F6(.)g Fn 612 363 J(10)g EOP end %%PageTrailer %======================================================================= %%Page: 11 11 %%BeginPageSetup DVIALWdict begin %%EndPageSetup BOP Fn 1248 5385 A(If)g Fk(P)34 X Fn(con)45 X(tains)e(\050)33 X Fk(X)h Fo 621 12 J(0)g Fk 37 12 L(;)g(:)14 X(:)t(:)t(;)t(X)t Fl 254 12 J(n)g Fn 45 12 L(\051,)g(then)35 X(the)s(heigh)r(t)e(of)r Fk(P)s Fn(is)45 X (clearly)34 X(at)s(least)t Fk(n)s Fn(+)23 X(1.)s 1082 100 C(Therefore,) g(w)34 X(e)e(ma)s(y)e(assume)s Fk(P)s Fh(6\023)43 X Fn(\050)32 X Fk(X)g Fo 1306 12 J(0)g Fk 37 12 L(;)g(:)14 X(:)t(:)t(;)t(X)t Fl 254 12 J(n)g Fn 45 12 L(\051.)g(Since)52 X(the)33 X(ideal)t Fk(P)t Fn(is)44 X (graded,)33 X(it)u 1642 100 C(su\016ces)g(to)o(lo)p(calize)i(at)p(one)o (of)p(the)o Fk(X)p Fl 1241 12 J(i)g Fn 28 12 L(,)g(whic)o(h)e(w)o(e)e (can)o(tak)p(e)d(to)p(b)o(e)j Fk(X)o Fo 1030 12 J(0)g Fn 37 12 L(,)g (tak)o(e)e(the)o(part)p 2336 100 C(of)g(degree)26 X(zero,)s(and)u(sho)s (w)e(that)s(the)t(heigh)s(t)e(of)s(the)t(resulting)s(prime)t(is)s(at)t (least)s Fk(n)t Fn(+)14 X(1.)u H(After)g(lo)38 X(calization,)i(the)w 1100 67 L 66 4 B 242 67 C(ideals)g Fk(Q)t Fn(and)t Fk(Q)t Fn(are)t (equal,)t 1138 67 L U 112 67 C(so)g Fk(Q)t Fn(will)t(b)t(e)i(generated) t(b)t(y)e Fk 1884 100 C(')g Fn(\050)g Fk(P)g Fn(\051.)12 X(Denote)48 X Fk(X)32 X Fl 613 12 J(i)g Fk 28 12 L(=X)g Fo 111 12 J(0)g Fn 69 12 L(b) g(y)e Fk(a)s Fl 164 12 J(i)g Fn 59 12 L(and)g(denote)t Fk(T)s Fl 486 12 J(i)g Fk 27 12 L(=X)g Fo 111 12 J(0)g Fn 69 12 L(b)g(y)e Fk(t)s Fl 149 12 J(i)g Fn 27 12 L(.)g(Then)49 X(the)31 X(part)u(of)t(degree)u 1913 100 C(zero)g(of)n(the)n(lo)n(calization)i(of)n Fk(F)n Fn(is)z(the)n(p)n (olynomial)i(ring)n Fk(L)n Fn(\133)g Fk(a)g Fo 1967 12 J(1)g Fk 37 12 L (;)g(:)14 X(:)t(:)t(;)t(a)t Fl 228 12 J(n)g Fk 45 12 L(;)g(t)t Fo 67 12 J(0)g Fk 37 12 L(;)g(:)t(:)t(:)t(;)t(t)t Fl 215 12 J(n)g Fn 45 12 L (\135.)g(The)36 X 2641 100 C(part)g(of)o(degree)n(0)o(of)n(the)o(lo)n (calization)j(of)n Fk(G)o Fn(is)n Fk(L)o Fn(\133)g Fk(a)g Fo 1707 12 J (1)g Fk 37 12 L(;)g(:)14 X(:)t(:)t(;)t(a)t Fl 228 12 J(n)g Fk 45 12 L(;) g(U)t Fo 94 12 J(00)g Fk 70 12 L(;)g(:)t(:)t(:)t(;)t(U)t Fl 242 12 J(nn) g Fn 86 12 L(\135,)g(and)31 X(the)t 2509 99 C(lo)g(calization)i(of)y (the)x(map)y(de\014ned)x(b)y(y)d Fk(')y Fn(sends)x Fk(a)y Fl 1677 12 J (i)g Fn 63 12 L(to)g(itself)y(and)x(sends)y Fk(t)x Fl 748 12 J(i)g Fn 63 12 L(to)g Fk(U)y Fl 166 12 J(i)g Fo(0)g Fn 84 12 L(+)g Fk 2801 100 C (a)g Fo 44 12 J(1)g Fk 37 12 L(U)g Fl 57 12 J(i)g Fo(1)g Fn 79 12 L(+)g Fh(\001)18 X(\001)p(\001)p Fn(+)t Fk(a)u Fl 326 12 J(n)g Fk 45 12 L(U)g Fl 57 12 J(in)g Fn 69 12 L(.)g(W)37 X(e)-7 X(iden)28 X(tify)d Fk(t)t Fl 535 12 J(i)g Fn 55 12 L(with)g(its)t(image)s(in)t Fk(G)s Fn(,)h(and)s(w) t(e)d(no)t(w)d(ha)t(v)d(e)e Fk 422 149 C(U)g Fl 57 12 J(i)g Fo(0)g Fn 84 12 L(=)g Fk(t)o Fl 118 12 J(i)g Fh 46 12 L(\000)g Fk(a)18 X Fo 127 12 J(1)g Fk 37 12 L(U)g Fl 57 12 J(i)g Fo(1)g Fh 79 12 L(\000)g(\001)t 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X Fk(n)v Fn(,)f(w)w(e)c(consider)v(the)t(total)v (degree)t(of)v Fk(N)t Fn(in)33 X(the)22 X(v)v(ariables)a Fk 125 100 C(U) g Fl 57 12 J(i)g Fo(1)g Fk 61 12 L(;)g(:)14 X(:)t(:)t(;)t(U)t Fl 242 12 J(in)g Fn 69 12 L(.)g(By)36 X(our)25 X(construction)t(and)t(c)t(hoice)e (of)t Fk(N)t Fn(,)9 X(this)26 X(degree)s(is)s Fk(k)s Fl 1963 12 J(i)g Fo(1)g Fn 74 12 L(+)g Fh(\001)13 X(\001)u(\001)u Fn(+)t Fk(k)t Fl 308 12 J(in)g Fn 69 12 L(.)g 2843 99 C(On)g(the)25 X(other)t(hand,)t(the)t (degree)t(of)t(an)t(y)d(monomial)t(in)t(the)t(expansion)t(of)t Fk(M)t Fi 2539 30 L(0)g Fn 48 30 J(in)g(these)t 2587 100 C(v)g(ariables)b(is)u (at)t(least)u(equal)t(to)u Fk(n)t Fl 1081 12 J(i)g Fo(1)g Fn 75 12 L(+) g Fh(\001)14 X(\001)t(\001)t Fn(+)t Fk(n)t Fl 318 12 J(in)g Fn 69 12 L (.)g(Hence)36 X(if)26 X(the)s(monomial)t Fk(N)s Fn(o)35 X(ccurs)i(in)26 X 1543 100 C(the)g(expansion,)v(w)u(e)e(m)u(ust)e(ha)u(v)e(e)d Fk 843 182 J(n)g Fl 50 12 J(i)g Fo(1)g Fn 79 12 L(+)g Fh(\001)18 X(\001)p(\001) p Fn(+)t Fk(n)u Fl 332 12 J(in)g Fh 92 12 L(\024)g Fk(k)y Fl 131 12 J(i) g Fo(1)g Fn 79 12 L(+)g Fh(\001)t(\001)p(\001)p Fn(+)t Fk(k)u Fl 325 12 J(in)g Fk 69 12 L(:)g Fn(\0503\051)737 X 1875 183 C(If)g(an)28 X(y)d(of) t(the)s(inequalities)t(\0502\051)s(or)t(\0503\051)s(w)t(ere)d(strict,)t (w)s(e)e(w)s(ould)e(then)s(ha)t(v)d(e)e(that)s Fl 475 145 J(m)g Fq 31 25 C(X)g Fl 7 177 J(i)g Fo(=0)g Fn 113 98 L(\050)g Fk(n)g Fl 82 12 J(i) g Fo(0)g Fn 79 12 L(+)g Fk(n)18 X Fl 133 12 J(i;i)g Fo(+1)g Fn 155 12 L (\051)g Fh(\000)t Fl 222 104 L(n)g Fq 40 25 C(X)g Fl 49 177 C(i)g Fo(=) g Fl(m)g Fo(+1)g Fn 219 98 L(\050)g Fk(n)g Fl 82 12 J(i)g Fo(1)g Fn 79 12 L(+)g Fk(n)t Fl 133 12 J(i)g Fo(2)g Fn 79 12 L(+)g Fh(\001)t(\001)p (\001)p Fn(+)t Fk(n)u Fl 332 12 J(in)g Fn 69 12 L(\051)g 2264 262 C(w)g (as)e(strictly)27 X(greater)u(than)t Fl 605 141 J(m)g Fq 31 25 C(X)g Fl 7 177 J(i)g Fo(=0)g Fn 113 98 L(\050)g Fk(k)g Fl 75 12 J(i)g Fo(0)g Fn 79 12 L(+)g Fk(k)18 X Fl 126 12 J(i;i)g Fo(+1)g Fn 155 12 L(\051)g Fh (\000)t Fl 222 104 L(n)g Fq 40 25 C(X)g Fl 49 177 C(i)g Fo(=)g Fl(m)g Fo(+1)g Fn 219 98 L(\050)g Fk(k)g Fl 75 12 J(i)g Fo(1)g Fn 79 12 L(+)g Fk(k)t Fl 126 12 J(i)g Fo(2)g Fn 79 12 L(+)g Fh(\001)t(\001)p(\001)p Fn (+)t Fk(k)u Fl 325 12 J(in)g Fn 69 12 L(\051)g Fk(:)g Fn 2234 262 C(Ho) g(w)e(ev)d(er,)e(the)39 X(monomial)r(w)r(as)e(c)r(hosen)e(to)r(maximize) r(this)r(quan)r(tit)e(y)d(.)a(Th)64 X(us)e(all)37 X(the)t 100 G (inequalities)g(\0502\051)28 X(and)s(\0503\051)t(are)s(equalities.)t 1391 264 J(12)g EOP end %%PageTrailer %======================================================================= %%Page: 13 13 %%BeginPageSetup DVIALWdict begin %%EndPageSetup BOP Fn 1248 5385 A(W)g(e)-7 X(no)29 X(w)d(compute)t(the)s(con)t(tribution)d (of)t(the)s(monomial)t Fk(M)s Fi 1988 30 L(0)g Fn 52 30 J(to)g(the)t(co) s(e\016cien)j(t)d(of)t Fk 2165 100 C(N)g Fn(.)9 X(Let)37 X Fk(i)28 X Fn (b)s(e)j(b)s(et)j(w)d(een)e(0)s(and)t Fk(m)s Fn(.)h(Then,)36 X(as)29 X (sho)q(wn)f(ab)q(o)k(v)c(e,)f(w)q(e)f(ha)q(v)f(e)c Fk 955 182 J(n)g Fl 50 12 J(i)g Fo(0)g Fn 79 12 L(+)g Fk(n)18 X Fl 133 12 J(i;i)g Fo(+1)g Fn 178 12 L(=)g Fk(k)y Fl 131 12 J(i)g Fo(0)g Fn 79 12 L(+)g Fk(k)t Fl 126 12 J(i;i)g Fo(+1)g Fk 155 12 L(:)g Fn 1886 182 C(T)g(o)-7 X(\014nd) 28 X(the)s(co)t(e\016cien)i(t)e(of)s Fk(N)t Fn(,)9 X(w)28 X(e)d(need)t (to)s(tak)t(e)d(the)t(part)s(of)t(the)s(expansion)t(of)s 256 182 J(\050) g Fk(t)g Fl 62 12 J(i)g Fh 46 12 L(\000)g Fk(a)18 X Fo 127 12 J(1)g Fk 37 12 L(U)g Fl 57 12 J(i)g Fo(1)g Fh 79 12 L(\000)g(\001)t(\001)p(\001) p(\000)t Fk(a)u Fl 326 12 J(i)g Fo(+1)g Fk 112 12 L(U)g Fl 57 12 J(i;i) g Fo(+1)g Fh 172 12 L(\000)g(\001)t(\001)p(\001)p(\000)t Fk(a)u Fl 326 12 J(n)g Fk 45 12 L(U)g Fl 57 12 J(in)g Fn 69 12 L(\051)g Fl 32 34 L(n) g Fm 41 8 J(i)g Fp(0)g Fk 59 26 J(U)g Fl 66 37 L(n)g Fm 41 8 J(i)g Fp(1) g Fl 50 52 C(i)g Fo(1)g Fh 123 23 L(\001)g(\001)p(\001)p Fk(U)p Fl 175 45 L(n)g Fm 41 8 J(i;i)g Fp(+1)g Fl 50 60 C(i;i)g Fo(+1)g Fh 206 23 L (\001)g(\001)p(\001)p Fn 2512 182 C(for)g(whic)36 X(h)d(the)t(exp)s (onen)j(t)d(of)t Fk(U)s Fl 1058 12 J(i;i)g Fo(+1)g Fn 188 12 L(is)g Fk (n)t Fl 142 12 J(i)g Fo(0)g Fn 84 12 L(+)g Fk(n)24 X Fl 139 12 J(i;i)g Fo(+1)g Fn 155 12 L(.)g(Ev)61 X(ery)d(term)36 X(will)s(ha)t(v)d(e)e(exp) s(o-)j 1766 100 C(nen)g(t)e(strictly)t(lo)v(w)c(er)f(than)s Fk(n)w Fl 970 12 J(i)g Fo(0)g Fn 85 12 L(+)g Fk(n)25 X Fl 139 12 J(i;i)g Fo(+1)g Fn 190 12 L(except)g(one,)37 X(and)v(that)t(is)t(the)t(one)t(coming)t 1384 100 C(from)g(\050)28 X Fh(\000)g Fk(a)g Fl 338 12 J(i)g Fo(+1)g Fk 112 12 L(U)g Fl 57 12 J(i;i)g Fo(+1)g Fn 155 12 L(\051)g Fl 32 30 L(n)g Fm 41 8 J(i)g Fp(0)g Fn 59 22 J(.)g(Hence)38 X(the)28 X(only)t(terms)t (whic)t(h)e(end)t(up)t(con)t(tributing)e(to)t(the)t 794 100 C(co)g (e\016cien)i(t)e(of)s Fk(N)t Fn(are)36 X(obtained)28 X(b)s(y)e (substituting)s(\050)t Fh(\000)g Fk(a)g Fl 1785 12 J(i)g Fo(+1)g Fk 112 12 L(U)g Fl 57 12 J(i;i)g Fo(+1)g Fn 155 12 L(\051)g(for)t Fk(U)s Fl 244 12 J(i)g Fo(0)g Fn 60 12 L(.)g 2288 100 C(W)g(e)-7 X(next)28 X 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C(\014cien)g(t)e(is)q(nonzero)q(and)q(do)q(es)i(not)q (in)q(v)e(olv)d(e)e Fk(t)q Fo 1464 12 J(0)g Fk 37 12 L(;)g(:)14 X(:)t(:) t(;)t(t)t Fl 215 12 J(m)g Fn 93 12 L(or)g Fk(a)30 X Fl 148 12 J(m)g Fo (+2)g Fk 147 12 L(;)g(:)14 X(:)t(:)t(;)t(a)t Fl 229 12 J(n)g Fn 45 12 L (.)g(This)44 X(pro)30 X(v)e(es)d 2378 99 C(Lemma)g(2.3.)r F8 274 G(3)g (P)134 X(ositivit)d(y)c(and)45 X(Sym)t(b)c(olic)k(P)t(o)c(w)d(ers)c Fn 182 G(F)g(or)-7 X(a)26 X(prime)u(ideal)t F2(q)u Fn(of)t(a)u(comm)t (utativ)e(e)d(ring)u Fk(R)t Fn(,)h F2(q)t Fo 1694 30 L(\050)g Fl(k)g Fo (\051)h Fn 116 30 J(=)g F2(q)q Fl 128 30 L(k)g Fk 41 30 J(R)g F1 63 12 J(q)g Fh 54 12 L(\\)g Fk(R)16 X Fn(is)27 X(called)s(the)t Fk(k)s Fn(-th) j F6 2096 100 C(symb)g(olic)c(p)w(ower)c Fn(of)u F2(q)u Fn(.)g(The)38 X Fk(k)27 X Fn(-th)j(sym)t(b)f(olic)h(p)v(o)h(w)f(er)c(can)w(also)s(b)w(e) g(de\014ned)w(as)s(the)w(set)s(of)w Fk H(r)g Fh(2)r Fk(R)p Fn(suc)u(h)e (that)t(there)u(exists)t Fk(s)u Fh(62)p F2(p)p Fn(with)u Fk(sr)t Fh(2)s F2(p)o Fl 1662 30 L(k)g Fn 41 30 J(.)g 1578 99 C(In)g(this)u(section)t (w)u(e)d(discuss)u(the)t(follo)u(wing)d(conjecture:)u F7 125 182 C (Conjecture)g(3.1)y Fn(Let)41 X(\050)24 X Fk(R)f(;)i F2(m)12 X Fn(\051) i(b)21 X(e)k(a)t(regular)x(lo)t(cal)k(ring.)t(Let)37 X F2(p)21 X Fn(and) x F2(q)t Fn(b)x(e)g(prime)x(ideals)t 100 G(of)g Fk(R)28 X Fn(that)t (satisfy)t 695 59 L 184 4 B Fh 69 E(p)g F2 69 58 J(p)g Fn(+)18 X F2(q)u Fn 206 1 J(=)g F2(m)y Fn(and)28 X(h)s(t)e Fl 418 12 J(R)g F2 67 12 L(p) g Fn(+)18 X(h)u(t)d Fl 220 12 J(R)g F2 67 12 L(q)g Fn(=)y Fk(d)y Fn(.)g (Then)37 X F2 524 182 C(p)g Fh(\\)18 X F2(q)u Fo 174 35 L(\050)g Fl(k)g Fo(\051)h Fh 116 35 J(\022)g F2(m)y Fl 152 35 L(k)g Fo(+1)h Fn 1591 216 C(for)g(an)28 X(y)d Fk(k)t(\076)q Fn(0.)p 1391 247 J(13)g EOP end %%PageTrailer %======================================================================= %%Page: 14 14 %%BeginPageSetup DVIALWdict begin %%EndPageSetup BOP Fn 1248 5385 A(W)g(e)-7 X(sho)40 X(w)e(b)t(elo)j(w)d(that)u(in)t(the)u 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(we)z(have)y F2(p)z Fh(\\)n F2(q)o Fo 703 30 L(\050)g Fl(k)g Fo(\051)h Fh 116 30 J(\022)g Fk(m)s Fl 161 30 L(k)g Fo(+1)h F6 155 30 J(for)g(any) z(p)y(ositive)c(inte)y(ger)c Fk(k)y F6(.)j Fn 1010 156 C(If)g(\050)u Fk (R)g(;)h F2(m)13 X Fn(\051)h(is)24 X(equic)v(haracteristic)d(in)v (Theorem)t(3.2,)v(then)u(the)v(third)t(assumption)v(for)t F2 125 100 C (p)g Fn(and)w F2(q)w Fn(follo)w(ws)d(from)w(the)w(\014rst)w(and)w (second)w(ones)w(b)w(y)d(the)w(p)w(ositivit)i(y)e(theorem)v(due)x(to)v H(Serre)g(\13316)x(\135.)g(Therefore,)37 X(if)28 X(\050)s Fk(R)h(;)g F2 (m)14 X Fn(\051)g(is)28 X(equic)s(haracteristic,)e(Conjecture)s(3.1)t (is)s(true.)t 125 100 J(By)g(Theorem)u(3.2,)u(Serre's)v(p)u(ositivit)i (y)e(conjecture)u(implies)u(Conjecture)u(3.1)u(in)u(the)u 125 100 C (case)g(where)t Fk(R)s Fn(is)u(rami\014ed.)s 125 99 J(Since)g(the)x (connection)y(b)x(et)j(w)d(een)e(the)x(p)y(ositivit)i(y)e(conjecture)x (and)y(Conjecture)x(3.1)y 125 100 C(dep)g(ends)i(on)v(the)v 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(2)g Fn 37 30 J(.)g(If)36 X Fk(R)23 X Fn(con)v(tains)d(a)u(\014eld)t(of) u(c)t(haracteristic)e(0)t(\050resp.)u Fk(p)t(\076)u Fn(0\051,)s 733 99 C(then)g F2(q)36 X Fo 238 30 L(\050)g Fl(k)g Fo(\051)h Fh 130 30 J(\022) g F2(m)u Fl 166 30 L(k)g Fo(+1)h Fn 160 30 J(for)g(an)t(y)e Fk(k)t(\076) x Fn(0)u(\050resp.)t(0)t Fk(\074)u(k)u(\074)x(p)u Fn(\051)g(b)t(y)e (Prop)t(osition)i(3.4)t 694 100 C(b)g(elo)i(w.)e 128 G(On)g(the)29 X (other)t(hand,)t(w)u(e)d(no)u(w)d(presen)u(t)d(an)u(example)s(in)u(p)s (ositiv)j(e)d(c)u(haracteristic)d 99 G(in)g(whic)s(h)d F2(q)s Fh(\022)n F2(m)n Fo 551 30 L(2)g Fn 65 30 J(but)g F2(q)s Fo 193 30 L(\050)g Fl(p) g Fo(\051)g Fh 113 30 J(6\022)g F2(m)n Fl 151 30 L(p)g Fo(+1)g Fn 122 30 J(.)g 1195 128 C(Let)g Fk(E)o Fn(b)t(e)i(a)o(3-dimensional)o(regular) o(lo)o(cal)i(ring)o(with)o(regular)o(system)o(of)o(param-)o 99 G(eters) g Fk(x)s Fn(,)g Fk(y)s Fn(,)j Fk(z)s Fn(,)j(and)s(let)r 685 156 J(\010) g(:)51 X Fk(R)s Fn(=)24 X Fk(E)s Fn(\133\133)l Fk(S;)g(T)14 X(;)7 X(U;) 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J(i)g Fn 28 12 L(,)g Fk(\014)u Fi 96 30 L(0)g Fn 46 30 J(=)g Fk(@)v(\014)l(=@)k(s)l Fl 317 12 J(i)g Fn 48 12 L(and)g (\050)t Fk(\013\014)g Fn(\051)k Fi 323 30 L(0)g Fn 46 30 J(=)g Fk(@)v Fn(\050)l Fk(\013\014)g Fn(\051)k Fk(=@)g(s)l Fl 435 12 J(i)g Fn 28 12 L(.)g(Since)35 X(\050)20 X Fk(\013\014)h Fn(\051)j Fi 436 30 L(0)g Fn 46 30 J(=)g Fk(\013)w Fi 141 30 L(0)g Fk 24 30 J(\014)g Fn(+)9 X Fk (\013\014)k Fi 229 30 L(0)g Fn 24 30 J(,)g 2843 100 C(w)g(e)e(ha)27 X(v) e(e)d Fk 988 100 J(\014)g Fo 51 34 L(2)g Fk 38 34 J(\013)g Fi 53 34 L(0) g Fn 47 34 J(=)g Fh(\000)p Fk(\014)g Fi 204 34 L(0)g Fk 24 34 J (\013\014)g Fn(+)p Fk(\014)18 X Fn(\050)l Fk(\013\014)f Fn(\051)l Fi 425 34 L(0)g Fk 23 34 J(:)g Fn 1728 149 C(On)g(the)30 X(other)t(hand,)t (since)u Fk(\013\014)t Fh(2)u F2(q)q Fl 1200 30 L(k)g Fo(+1)h Fn 143 9 J(^)g Fk 18 21 C(R)g Fn(=)r Fk(I)q Fl 226 30 L(k)g Fo(+1)h Fn 124 30 J (,)g(w)u(e)d(ha)u(v)d(e)e(\050)t Fk(\013\014)g Fn(\051)k Fi 541 30 L(0) g Fh 50 30 J(2)g Fk(I)q Fl 125 30 L(k)g Fn 40 30 J(.)g(There-)44 X 2556 100 C(fore,)g(w)33 X(e)e(ha)s(v)e(e)d Fk(\014)t Fo 568 30 L(2)g Fk 38 30 J(\013)g Fi 53 30 L(0)g Fh 54 30 J(2)g Fk(I)r Fl 129 30 L(k)g Fn 40 30 J(.)g(Hence,)50 X Fk(\013)34 X Fi 401 30 L(0)g Fh 54 30 J(2)g Fk(I)q Fl 129 30 L(k)g Fn 71 30 J(:)g Fo 37 12 J(^)g Fl 14 15 C(R)g Fk 86 27 L (\014)g Fo 51 30 L(2)g Fn 38 30 J(.)g(By)50 X(\0504\051,)33 X(w)t(e)e (obtain)s Fk(\013)s Fi 813 30 L(0)g Fh 54 30 J(2)g F2(m)r Fl 150 30 L(l) g Fn 43 9 J(^)g Fk 18 21 C(R)g Fn(,)h 2777 100 C(whic)g(h)e(con)n (tradicts)e(\0505\051.)n F7(q.e.d.)1844 X Fn 125 149 J(It)g(follo)23 X (ws)e(immediately)t(from)t(Prop)t(osition)i(3.4)t(that)t(if)t Fk(R)t Fh (\033)u F4(Q)t Fn(and)t F2(q)t Fn(is)t(con)t(tained)e 125 100 C(in)g F2 (m)z Fo 162 30 L(2)g Fn 37 30 J(,)g(then)z F2(p)y Fh(\\)q F2(q)o Fo 418 30 L(\050)g Fl(k)g Fo(\051)h Fh 117 30 J(\022)g F2(m)v Fl 154 30 L(k)g Fo(+1)h Fn 125 30 J(.)g(The)40 X(general)28 X(case)u(in)t(whic)u(h)d Fk (R)u Fh(\033)q F4(Q)q 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