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Ft(\))0 4745 y(or)32 b(in)g(other)h(w)m(ord)g(can)g(b)s(e)g(factorized)f(through)g (the)h(residue)g(class)g(map)799 4902 y Fp(\027)i Ft(:)27 b Fq(K)18 b Ft([)p Fp(x)1090 4917 y Ff(1)1136 4902 y Fp(;)f(:)g(:)g(:)32 b(;)17 b(x)1426 4917 y Fk(n)1473 4902 y Ft(])28 b Fo(\000)-60 b(!)28 b Fq(K)17 b Ft([)p Fp(x)1827 4917 y Ff(1)1872 4902 y Fp(;)g(:)g(:)g(:)33 b(;)17 b(x)2163 4917 y Fk(n)2210 4902 y Ft(])p Fp(=)p Ft(\()p Fp(p)2373 4917 y Ff(1)2412 4902 y Fp(;)g(:)g(:)g(:)32 b(;)17 b(p)2696 4917 y Fk(m)2763 4902 y Ft(\))0 5059 y(This)33 b(induces)g(a)f(bijection)179 5216 y(Mor)355 5231 y Fj(K)5 b Ft(-)g Fb(cAlg)604 5216 y Ft(\()p Fq(K)18 b Ft([)p Fp(x)796 5231 y Ff(1)842 5216 y Fp(;)f(:)g(:)g(:)32 b(;)17 b(x)1132 5231 y Fk(n)1179 5216 y Ft(])p Fp(=)p Ft(\()p Fp(p)1342 5231 y Ff(1)1381 5216 y Fp(;)g(:)g(:)g(:)33 b(;)17 b(p)1666 5231 y Fk(m)1732 5216 y Ft(\))p Fp(;)g(A)p Ft(\))27 b Fo(3)h Fp(f)39 b Fo(7!)27 b Ft(\()p Fp(f)11 b Ft(\()p Fp(x)2450 5231 y Ff(1)2489 5216 y Ft(\))p Fp(;)17 b(:)g(:)g(:)33 b(;)17 b(f)11 b Ft(\()p Fp(x)2915 5231 y Fk(n)2962 5216 y Ft(\)\))27 b Fo(2)h(X)15 b Ft(\()p Fp(A)p Ft(\))p Fp(:)p eop %%Page: 7 7 7 6 bop 542 236 a Fl(1.)34 b(THE)g(PRINCIPLES)f(OF)g(COMMUT)-6 b(A)g(TIVE)33 b(ALGEBRAIC)g(GEOMETR)-6 b(Y)478 b(7)0 425 y Ft(No)m(w)33 b(it)f(is)g(easy)h(to)g(see)g(that)g(this)f (bijection)f(is)h(a)h(natural)e(isomorphism)f(\(in)i Fp(A)p Ft(\).)p 3529 425 4 66 v 3533 363 59 4 v 3533 425 V 3591 425 4 66 v 149 638 a(If)26 b(no)g(p)s(olynomials)d(are)j (giv)m(en)g(for)g(the)g(ab)s(o)m(v)m(e)h(construction,)g(then)g(the)f (functor)h(under)f(this)0 755 y(construction)i(is)f(the)i(a\016ne)f (space)h Fq(A)1388 719 y Fk(n)1469 755 y Ft(of)e(dimension)g Fp(n)p Ft(.)42 b(By)28 b(giving)f(p)s(olynomials)d(the)29 b(functor)0 871 y Fo(X)43 b Ft(b)s(ecomes)28 b(a)g(subfunctor)h(of)e Fq(A)1227 835 y Fk(n)1280 871 y Ft(,)i(b)s(ecause)h(it)d(de\014nes)i (subsets)h Fo(X)15 b Ft(\()p Fp(A)p Ft(\))28 b Fo(\022)g Fq(A)2868 835 y Fk(n)2921 871 y Ft(\()p Fp(A)p Ft(\))f(=)h Fp(A)3274 835 y Fk(n)3321 871 y Ft(.)42 b(Both)0 987 y(functors)26 b(are)g(represen)m(table)h(functors.)42 b(The)27 b(em)m(b)s(edding)e(is)h(induced)g(b)m(y)h(the)f(homomorphism) 0 1103 y(of)32 b(algebras)g Fp(\027)i Ft(:)28 b Fq(K)17 b Ft([)q Fp(x)782 1118 y Ff(1)827 1103 y Fp(;)g(:)g(:)g(:)33 b(;)17 b(x)1118 1118 y Fk(n)1165 1103 y Ft(])28 b Fo(\000)-60 b(!)27 b Fq(K)18 b Ft([)p Fp(x)1518 1118 y Ff(1)1564 1103 y Fp(;)f(:)g(:)g(:)32 b(;)17 b(x)1854 1118 y Fk(n)1901 1103 y Ft(])p Fp(=)p Ft(\()p Fp(p)2064 1118 y Ff(1)2104 1103 y Fp(;)g(:)g(:)g(:)32 b(;)17 b(p)2388 1118 y Fk(m)2454 1103 y Ft(\).)149 1300 y Fr(Problem)36 b(1.1.1.)199 b Ft(1.)41 b(Determine)36 b(the)h(a\016ne)g(algebra)e(of)h(the)h(functor) f(\\unit)g(circle")267 1416 y Fp(S)333 1380 y Ff(1)405 1416 y Ft(in)31 b Fq(A)584 1380 y Ff(2)630 1416 y Ft(.)149 1532 y(2.)42 b(Determine)31 b(the)i(a\016ne)g(algebra)f(of)g(the)h (functor)g(\\unit)e(sphere")j Fp(S)2803 1496 y Fk(n)p Fm(\000)p Ff(1)2973 1532 y Ft(in)e Fq(A)3152 1496 y Fk(n)3205 1532 y Ft(.)149 1648 y(3.)42 b(Let)33 b Fo(X)47 b Ft(denote)33 b(the)g(plane)f(curv)m(e)i Fp(y)d Ft(=)c Fp(x)1799 1612 y Ff(2)1839 1648 y Ft(.)44 b(Then)33 b Fo(X)47 b Ft(is)32 b(isomorphic)f(to)h(the)h(a\016ne)g(line.)149 1765 y(4.)42 b(Let)35 b Fo(Y)43 b Ft(denote)36 b(the)f(plane)g(curv)m(e)i Fp(xy)e Ft(=)d(1.)51 b(Then)36 b Fo(Y)43 b Ft(is)35 b(not)g(isomorphic) e(to)i(the)g(a\016ne)267 1881 y(line.)58 b(\(Hin)m(t:)53 b(An)38 b(isomorphism)d Fq(K)18 b Ft([)p Fp(x;)f(x)1806 1845 y Fm(\000)p Ff(1)1907 1881 y Ft(])36 b Fo(\000)-60 b(!)37 b Fq(K)17 b Ft([)p Fp(y)t Ft(])43 b(sends)d Fp(x)e Ft(to)f(a)h(p)s(olynomial)c Fp(p)p Ft(\()p Fp(y)t Ft(\))267 1997 y(whic)m(h)43 b(m)m(ust)f(b)s(e)g(in)m(v)m(ertible.)72 b(Consider)43 b(the)f(highest)h(co)s(e\016cien)m(t)g(of)e Fp(p)p Ft(\()p Fp(y)t Ft(\))g(and)i(sho)m(w)267 2113 y(that)32 b Fp(p)p Ft(\()p Fp(y)t Ft(\))27 b Fo(2)h Fq(K)17 b Ft(.)50 b(But)33 b(that)f(means)h(that)f(the)h(map)f(cannot)g(b)s(e)h (bijectiv)m(e.\))149 2230 y(5.)42 b(Let)47 b Fq(K)77 b Ft(=)53 b Fq(C)73 b Ft(b)s(e)48 b(the)g(\014eld)f(of)g(complex)g(n)m (um)m(b)s(ers.)89 b(Sho)m(w)49 b(that)e(the)h(unit)f(functor)267 2346 y Fp(U)d Ft(:)34 b Fq(K)17 b Ft(-)6 b Fr(cAlg)32 b Fo(\000)-60 b(!)34 b Fr(Set)i Ft(in)f(Lemma)f(2.3.7)i(is)g(naturally) e(isomorphic)g(to)i(the)g(unit)g(circle)267 2462 y(functor)f Fp(S)674 2426 y Ff(1)713 2462 y Ft(.)51 b(\(Hin)m(t:)d(There)36 b(is)e(an)h(algebra)f(isomorphism)e(b)s(et)m(w)m(een)38 b(the)d(represen)m(ting)267 2578 y(algebras)d Fq(K)17 b Ft([)p Fp(e)q(;)g(e)880 2542 y Fm(\000)p Ff(1)980 2578 y Ft(])32 b(and)h Fq(K)18 b Ft([)p Fp(c;)f(s)p Ft(])p Fp(=)p Ft(\()p Fp(c)1616 2542 y Ff(2)1683 2578 y Ft(+)22 b Fp(s)1827 2542 y Ff(2)1888 2578 y Fo(\000)h Ft(1\).\))149 2695 y(6.)267 2658 y Fm(\003)337 2695 y Ft(Let)30 b Fq(K)54 b Ft(b)s(e)31 b(an)f(algebraically)d(closed)k(\014eld.)43 b(Let)30 b Fp(p)g Ft(b)s(e)h(an)f(irreducible)g(square)h(p)s(olyno-)267 2811 y(mial)d(in)i Fq(K)17 b Ft([)p Fp(x)q(;)g(y)t Ft(].)48 b(Let)31 b Fo(Z)39 b Ft(b)s(e)31 b(the)g(conic)f(section)h(de\014ned)i (b)m(y)e Fp(p)g Ft(with)f(the)i(a\016ne)f(algebra)267 2927 y Fq(K)17 b Ft([)p Fp(x)q(;)g(y)t Ft(])p Fp(=)p Ft(\()p Fp(p)p Ft(\).)79 b(Sho)m(w)44 b(that)f Fo(Z)51 b Ft(is)42 b(naturally)g(isomorphic)f(either)i(to)g Fo(X)57 b Ft(or)43 b(to)f Fo(Y)51 b Ft(from)267 3043 y(parts)33 b(3.)43 b(resp.)h(4.)149 3240 y Fr(Remark)37 b(1.1.8.)50 b Ft(A\016ne)37 b(algebras)f(of)h(a\016ne)g(sc)m(hemes)i(are)e (\014nitely)f(generated)h(comm)m(u-)0 3356 y(tativ)m(e)32 b(algebras)f(and)h(an)m(y)g(suc)m(h)i(algebra)c(is)i(an)g(a\016ne)g (algebra)f(of)g(some)h(a\016ne)g(sc)m(heme,)i(since)0 3472 y Fp(A)101 3444 y Fo(\030)102 3476 y Ft(=)206 3472 y Fq(K)17 b Ft([)q Fp(x)360 3487 y Ff(1)405 3472 y Fp(;)g(:)g(:)g(:)33 b(;)17 b(x)696 3487 y Fk(n)743 3472 y Ft(])p Fp(=)p Ft(\()p Fp(p)906 3487 y Ff(1)945 3472 y Fp(;)g(:)g(:)g(:)33 b(;)17 b(p)1230 3487 y Fk(m)1296 3472 y Ft(\))32 b(\(Hilb)s(ert)f(basis)i (theorem\).)149 3588 y(The)i(p)s(olynomials)c Fp(p)944 3603 y Ff(1)983 3588 y Fp(;)17 b(:)g(:)g(:)32 b(;)17 b(p)1267 3603 y Fk(m)1367 3588 y Ft(are)34 b(not)f(uniquely)h (determined)f(b)m(y)i(the)f(a\016ne)g(algebra)e(of)0 3705 y(an)e(a\016ne)h(sc)m(heme.)44 b(Not)31 b(ev)m(en)h(the)e(ideal)f (generated)j(b)m(y)f(the)g(p)s(olynomials)c(in)j(the)h(p)s(olynomial)0 3821 y(ring)e Fq(K)17 b Ft([)p Fp(x)352 3836 y Ff(1)397 3821 y Fp(;)g(:)g(:)g(:)33 b(;)17 b(x)688 3836 y Fk(n)735 3821 y Ft(])30 b(is)f(uniquely)h(determined.)43 b(Also)29 b(the)i(n)m(um)m(b)s(er)f(of)f(v)-5 b(ariables)29 b Fp(x)3127 3836 y Ff(1)3167 3821 y Fp(;)17 b(:)g(:)g(:)32 b(;)17 b(x)3457 3836 y Fk(n)3534 3821 y Ft(is)0 3937 y(not)32 b(uniquely)h(determined.)149 4053 y(The)e Fq(K)18 b Ft(-p)s(oin)m(ts)35 b(\()p Fp(\013)848 4068 y Ff(1)887 4053 y Fp(;)17 b(:)g(:)g(:)33 b(;)17 b(\013)1185 4068 y Fk(n)1231 4053 y Ft(\))28 b Fo(2)g(X)15 b Ft(\()p Fq(K)i Ft(\))36 b(of)29 b(an)h(a\016ne)g(sc)m (heme)i Fo(X)44 b Ft(\(with)29 b(co)s(e\016cien)m(ts)i(in)f(the)0 4169 y(base)39 b(\014eld)g Fq(K)17 b Ft(\))45 b(are)39 b(called)e Fn(r)-5 b(ational)40 b(p)-5 b(oints)p Ft(.)62 b(They)40 b(do)e(not)h(su\016ce)h(to)f(completely)e(describ)s(e)0 4286 y(the)c(a\016ne)g(sc)m(heme.)149 4402 y(Let)49 b(for)e(example)h Fq(K)78 b Ft(=)54 b Fq(R)59 b Ft(the)49 b(set)g(of)e(rational)f(n)m(um) m(b)s(ers.)91 b(If)48 b Fo(X)63 b Ft(and)48 b Fo(Y)56 b Ft(are)48 b(a\016ne)0 4518 y(sc)m(hemes)28 b(with)f(a\016ne)g (algebras)f Fo(O)s Ft(\()p Fo(X)15 b Ft(\))28 b(:=)f Fq(K)18 b Ft([)p Fp(x;)f(y)t Ft(])p Fp(=)p Ft(\()p Fp(x)2032 4482 y Ff(2)2087 4518 y Ft(+)10 b Fp(y)2225 4482 y Ff(2)2274 4518 y Ft(+)g(1\))26 b(and)h Fo(O)s Ft(\()p Fo(Y)8 b Ft(\))28 b(:=)f Fq(K)18 b Ft([)p Fp(x)p Ft(])p Fp(=)p Ft(\()p Fp(x)3372 4482 y Ff(2)3427 4518 y Ft(+)10 b(1\))0 4634 y(then)45 b(b)s(oth)f(sc)m(hemes)j(ha)m(v)m(e)f(no)e(rational)e(p) s(oin)m(ts.)79 b(The)45 b(sc)m(heme)h Fo(Y)8 b Ft(,)48 b(ho)m(w)m(ev)m(er,)i(has)45 b(exactly)0 4751 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Fp(B)5 b Ft(\).)94 b(Since)50 b(suc)m(h)g(a)f(homomorphism)e(is)i(completely)f (describ)s(ed)i(b)m(y)h(the)f(image)d(of)i Fp(x)h Ft(w)m(e)0 4169 y(get)40 b Fp(\036)p Ft(\()p Fp(a)p Ft(\)\()p Fp(B)5 b Ft(\)\()p Fp(p)p Ft(\)\()p Fp(x)p Ft(\))41 b(=)g Fp(p)p Ft(\()p Fp(a)p Ft(\).)67 b(T)-8 b(o)41 b(compare)f(the)h(algebra)e (structures)j(of)e Fp(A)g Ft(and)h(Nat\()p Fo(X)15 b Fp(;)i Fq(A)k Ft(\))0 4286 y(let)45 b Fp(a;)17 b(a)300 4250 y Fm(0)374 4286 y Fo(2)51 b Fp(A)p Ft(.)83 b(W)-8 b(e)46 b(ha)m(v)m(e)h Fp(\036)p Ft(\()p Fp(a)p Ft(\)\()p Fp(B)5 b Ft(\)\()p Fp(p)p Ft(\)\()p Fp(x)p Ft(\))50 b(=)h Fp(p)p Ft(\()p Fp(a)p Ft(\))45 b(and)h Fp(\036)p Ft(\()p Fp(a)2437 4250 y Fm(0)2461 4286 y Ft(\)\()p Fp(B)5 b Ft(\)\()p Fp(p)p Ft(\)\()p Fp(x)p Ft(\))50 b(=)g Fp(p)p Ft(\()p Fp(a)3224 4250 y Fm(0)3247 4286 y Ft(\),)g(hence)0 4402 y Fp(\036)p Ft(\()p Fp(a)26 b Ft(+)h Fp(a)327 4366 y Fm(0)350 4402 y Ft(\)\()p Fp(B)5 b Ft(\)\()p Fp(p)p Ft(\)\()p Fp(x)p Ft(\))39 b(=)f Fp(p)p Ft(\()p Fp(a)27 b Ft(+)f 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Ft(\)\()p Fp(B)5 b Ft(\)\()p Fp(p)p Ft(\)\()p Fp(x)p Ft(\))36 b(=)g Fp(p)p Ft(\()p Fp(aa)1882 4598 y Fm(0)1905 4634 y Ft(\))g(=)g Fp(p)p Ft(\()p Fp(a)p Ft(\))p Fp(p)p Ft(\()p Fp(a)2405 4598 y Fm(0)2428 4634 y Ft(\))g(=)g(\()p Fp(\036)p Ft(\()p Fp(a)p Ft(\))26 b Fo(\001)f Fp(\036)p Ft(\()p Fp(a)3063 4598 y Fm(0)3086 4634 y Ft(\)\)\()p Fp(B)5 b Ft(\)\()p Fp(p)p Ft(\)\()p Fp(x)p Ft(\),)0 4751 y(and)42 b(th)m(us)i Fp(\036)p Ft(\()p Fp(a)28 b Ft(+)h Fp(a)755 4714 y Fm(0)778 4751 y Ft(\))44 b(=)g Fp(\036)p Ft(\()p Fp(a)p Ft(\))29 b(+)f Fp(\036)p Ft(\()p Fp(a)1445 4714 y Fm(0)1469 4751 y Ft(\))42 b(and)g Fp(\036)p Ft(\()p Fp(aa)1946 4714 y Fm(0)1969 4751 y Ft(\))i(=)g Fp(\036)p Ft(\()p Fp(a)p Ft(\))29 b Fo(\001)f Fp(\036)p Ft(\()p Fp(a)2588 4714 y Fm(0)2611 4751 y Ft(\).)73 b(Hence)43 b(addition)e(and)0 4867 y(m)m(ultiplication)26 b(in)j(Nat\()p Fo(X)15 b Fp(;)i Fq(A)k Ft(\))37 b(are)30 b(de\014ned)i(b)m(y)f(the)g(addition)e(and)h(the)h(m)m(ultiplication)26 b(of)k(the)0 4983 y(v)-5 b(alues)32 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Fp(;)g(:)g(:)g(:)33 b(;)17 b(x)2076 1178 y Fk(r)2114 1163 y Ft(\))28 b Fo(2)g Fq(K)17 b Ft([)p Fp(x)2428 1178 y Ff(1)2473 1163 y Fp(;)g(:)g(:)g(:)33 b(;)17 b(x)2764 1178 y Fk(r)2802 1163 y Ft(])p Fp(;)0 1320 y Fn(such)35 b(that)639 1436 y Fp(\036)p Ft(\()p Fp(A)p Ft(\)\()p Fp(a)935 1451 y Ff(1)974 1436 y Fp(;)17 b(:)g(:)g(:)32 b(;)17 b(a)1260 1451 y Fk(r)1298 1436 y Ft(\))28 b(=)f(\()p Fp(p)1554 1451 y Ff(1)1594 1436 y Ft(\()p Fp(a)1683 1451 y Ff(1)1722 1436 y Fp(;)17 b(:)g(:)g(:)33 b(;)17 b(a)2009 1451 y Fk(r)2047 1436 y Ft(\))p Fp(;)g(:)g(:)g(:)32 b(;)17 b(p)2369 1451 y Fk(s)2405 1436 y Ft(\()p Fp(a)2494 1451 y Ff(1)2534 1436 y Fp(;)g(:)g(:)g(:)32 b(;)17 b(a)2820 1451 y Fk(r)2858 1436 y Ft(\)\))p Fp(;)0 1573 y Fn(for)42 b(al)5 b(l)42 b Fp(A)f Fo(2)g Fq(K)35 b Fn(-)5 b Fr(A\013)53 b Fn(and)41 b(al)5 b(l)42 b Ft(\()p Fp(a)1295 1588 y Ff(1)1335 1573 y Fp(;)17 b(:)g(:)g(:)32 b(;)17 b(a)1621 1588 y Fk(r)1659 1573 y Ft(\))41 b Fo(2)g(X)15 b Ft(\()p Fp(A)p Ft(\))p Fn(,)44 b(i.e.)d(the)i(morphisms)d(b)-5 b(etwe)g(en)42 b(a\016ne)0 1689 y(algebr)-5 b(aic)34 b(varieties)g(ar)-5 b(e)35 b(of)f(p)-5 b(olynomial)34 b(typ)-5 b(e.)149 1864 y Fc(Pr)n(oof.)42 b Ft(Let)47 b Fo(O)s Ft(\()p Fo(X)15 b Ft(\))51 b(=)f Fq(K)18 b Ft([)p Fp(x)1298 1879 y Ff(1)1344 1864 y Fp(;)f(:)g(:)g(:)32 b(;)17 b(x)1634 1879 y Fk(r)1672 1864 y Ft(])p Fp(=I)54 b Ft(and)47 b Fo(O)s Ft(\()p Fo(Y)8 b Ft(\))50 b(=)h Fq(K)18 b Ft([)p Fp(y)2606 1879 y Ff(1)2651 1864 y Fp(;)f(:)g(:)g(:)32 b(;)17 b(y)2934 1879 y Fk(s)2970 1864 y Ft(])p Fp(=J)9 b Ft(.)85 b(F)-8 b(or)45 b Fp(A)52 b Fo(2)0 1980 y Fq(K)17 b Ft(-)6 b Fr(Alg)42 b Ft(and)h(\()p Fp(a)614 1995 y Ff(1)654 1980 y Fp(;)17 b(:)g(:)g(:)32 b(;)17 b(a)940 1995 y Fk(r)978 1980 y Ft(\))46 b Fo(2)g(X)15 b Ft(\()p Fp(A)p Ft(\))42 b(let)h Fp(f)56 b Ft(:)46 b Fq(K)17 b Ft([)q Fp(x)1934 1995 y Ff(1)1979 1980 y Fp(;)g(:)g(:)g(:)33 b(;)17 b(x)2270 1995 y Fk(r)2308 1980 y Ft(])p Fp(=I)53 b Fo(\000)-60 b(!)46 b Fp(A)d Ft(with)g Fp(f)11 b Ft(\()p Fp(x)3144 1995 y Fk(i)3172 1980 y Ft(\))46 b(=)f Fp(a)3428 1995 y Fk(i)3500 1980 y Ft(b)s(e)0 2096 y(the)36 b(homomorphism)d (obtained)i(from)g Fo(X)15 b Ft(\()p Fp(A)p Ft(\))1768 2068 y Fo(\030)1769 2100 y Ft(=)1879 2096 y Fq(K)i Ft(-)6 b Fr(Alg)n Ft(\()p Fq(K)18 b Ft([)p Fp(x)2352 2111 y Ff(1)2398 2096 y Fp(;)f(:)g(:)g(:)32 b(;)17 b(x)2688 2111 y Fk(r)2726 2096 y Ft(])p Fp(=I)8 b(;)17 b(A)p Ft(\).)52 b(The)37 b(natural)0 2212 y(transformation)20 b Fp(\036)h Ft(is)g(giv)m(en)h(b)m(y)h(comp)s(osition)c(with)i(a)h(homomorphism)d Fp(g)31 b Ft(:)d Fq(K)17 b Ft([)p Fp(y)2962 2227 y Ff(1)3007 2212 y Fp(;)g(:)g(:)g(:)33 b(;)17 b(y)3291 2227 y Fk(s)3327 2212 y Ft(])p Fp(=J)36 b Fo(\000)-60 b(!)0 2328 y Fq(K)17 b Ft([)q Fp(x)154 2343 y Ff(1)199 2328 y Fp(;)g(:)g(:)g(:)33 b(;)17 b(x)490 2343 y Fk(r)528 2328 y Ft(])p Fp(=I)40 b Ft(hence)34 b(w)m(e)g(get)187 2485 y Fp(\036)p Ft(\()p Fp(A)p Ft(\))28 b(:)g Fq(K)17 b Ft(-)6 b Fr(cAlg)n Ft(\()p Fq(K)18 b Ft([)p Fp(x)1000 2500 y Ff(1)1046 2485 y Fp(;)f(:)g(:)g(:)32 b(;)17 b(x)1336 2500 y Fk(r)1374 2485 y Ft(])p Fp(=I)8 b(;)17 b(A)p Ft(\))27 b Fo(3)h Fp(f)39 b Fo(7!)27 b Fp(f)11 b(g)31 b Fo(2)d Fq(K)17 b Ft(-)6 b Fr(cAlg)o Ft(\()p Fq(K)17 b Ft([)p Fp(y)2738 2500 y Ff(1)2783 2485 y Fp(;)g(:)g(:)g(:)32 b(;)17 b(y)3066 2500 y Fk(s)3102 2485 y Ft(])p Fp(=J)n(;)g(A)p Ft(\))p Fp(:)0 2642 y Ft(Since)33 b Fp(g)j Ft(is)c(describ)s(ed)h(b)m (y)g Fp(g)t Ft(\()p Fp(y)1139 2657 y Fk(i)1166 2642 y Ft(\))28 b(=)g Fp(p)1385 2657 y Fk(i)1413 2642 y Ft(\()p Fp(x)1506 2657 y Ff(1)1546 2642 y Fp(;)17 b(:)g(:)g(:)32 b(;)17 b(x)1836 2657 y Fk(r)1874 2642 y Ft(\))28 b Fo(2)g Fq(K)17 b Ft([)p Fp(x)2188 2657 y Ff(1)2233 2642 y Fp(;)g(:)g(:)g(:)33 b(;)17 b(x)2524 2657 y Fk(r)2562 2642 y Ft(])32 b(w)m(e)i(get)872 2798 y Fp(\036)p Ft(\()p Fp(A)p Ft(\)\()p Fp(a)1168 2813 y Ff(1)1207 2798 y Fp(;)17 b(:)g(:)g(:)33 b(;)17 b(a)1494 2813 y Fk(s)1530 2798 y Ft(\))28 b(=)f(\()p Fp(f)11 b(g)t Ft(\()p Fp(y)1933 2813 y Ff(1)1971 2798 y Ft(\))p Fp(;)17 b(:)g(:)g(:)33 b(;)17 b(f)11 b(g)t Ft(\()p Fp(y)2441 2813 y Fk(s)2476 2798 y Ft(\)\))872 2914 y(=)27 b(\()p Fp(f)11 b Ft(\()p Fp(p)1159 2929 y Ff(1)1198 2914 y Ft(\()p Fp(x)1291 2929 y Ff(1)1331 2914 y Fp(;)17 b(:)g(:)g(:)33 b(;)17 b(x)1622 2929 y Fk(r)1660 2914 y Ft(\)\))p Fp(;)g(:)g(:)g(:)32 b(;)17 b(f)11 b Ft(\()p Fp(p)2117 2929 y Fk(s)2153 2914 y Ft(\()p Fp(x)2246 2929 y Ff(1)2286 2914 y Fp(;)17 b(:)g(:)g(:)32 b(;)17 b(x)2576 2929 y Fk(r)2614 2914 y Ft(\)\)\))872 3030 y(=)27 b(\()p Fp(p)1062 3045 y Ff(1)1102 3030 y Ft(\()p Fp(a)1191 3045 y Ff(1)1230 3030 y Fp(;)17 b(:)g(:)g(:)33 b(;)17 b(a)1517 3045 y Fk(r)1555 3030 y Ft(\))p Fp(;)g(:)g(:)g(:)32 b(;)17 b(p)1877 3045 y Fk(s)1913 3030 y Ft(\()p Fp(a)2002 3045 y Ff(1)2042 3030 y Fp(;)g(:)g(:)g(:)32 b(;)17 b(a)2328 3045 y Fk(r)2366 3030 y Ft(\)\))p Fp(:)p 3529 3180 4 66 v 3533 3117 59 4 v 3533 3180 V 3591 3180 4 66 v 149 3346 a Ft(An)33 b(analogous)f(statemen)m(t)h(holds)f(for)g(geometric)f (spaces.)149 3520 y Fr(Example)37 b(1.1.14.)49 b Ft(The)31 b(isomorphism)c(b)s(et)m(w)m(een)32 b(the)e(a\016ne)g(line)e(\(1.1.2\)) h(and)h(the)g(para-)0 3636 y(b)s(ola)35 b(is)h(giv)m(en)h(b)m(y)g(the)g (isomorphism)d Fp(f)46 b Ft(:)34 b Fq(K)18 b Ft([)p Fp(x;)f(y)t Ft(])p Fp(=)p Ft(\()p Fp(y)33 b Fo(\000)25 b Fp(x)2221 3600 y Ff(2)2261 3636 y Ft(\))34 b Fo(\000)-60 b(!)35 b Fq(K)18 b Ft([)p Fp(z)t Ft(],)44 b Fp(f)11 b Ft(\()p Fp(x)p Ft(\))35 b(=)f Fp(z)t Ft(,)k Fp(f)11 b Ft(\()p Fp(y)t Ft(\))33 b(=)i Fp(z)3560 3600 y Ff(2)0 3753 y Ft(that)j(has)g(the)h(in)m(v)m(erse)h(function)d Fp(f)1345 3716 y Fm(\000)p Ff(1)1439 3753 y Ft(\()p Fp(z)t Ft(\))h(=)f Fp(x)p Ft(.)61 b(On)38 b(the)h(a\016ne)f(sc)m(hemes)i Fq(A)22 b Ft(,)46 b(the)38 b(a\016ne)h(line,)0 3869 y(and)47 b Fq(P)p Ft(,)52 b(the)47 b(parab)s(ola,)i(the)e(induced)g(map)f(is)h Fp(f)62 b Ft(:)52 b Fq(A)22 b Ft(\()p Fp(A)p Ft(\))58 b Fo(3)52 b Fp(a)g Fo(7!)g Ft(\()p Fp(a;)17 b(a)2895 3833 y Ff(2)2934 3869 y Ft(\))52 b Fo(2)g Fq(P)p Ft(\()p Fp(A)p Ft(\))d(resp.)0 3985 y Fp(f)59 3949 y Fm(\000)p Ff(1)181 3985 y Ft(:)27 b Fq(P)p Ft(\()p Fp(A)p Ft(\))j Fo(3)e Ft(\()p Fp(a;)17 b(b)p Ft(\))28 b Fo(7!)f Fp(a)h Fo(2)g Fq(A)22 b Ft(\()p Fp(A)p Ft(\).)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF