%!PS-Adobe-2.0 %%Creator: dvips 5.58 Copyright 1986, 1994 Radical Eye Software %%Title: ln42.dvi %%CreationDate: Mon Jun 26 16:21:02 2000 %%Pages: 6 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: DVIPS16 -a ln42 %DVIPSParameters: dpi=300, compressed, comments removed %DVIPSSource: TeX output 2000.06.26:1531 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B 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Fm(ma)f Fq(:=)g Fm(am)p Fq(.)27 b(W)l(e)19 b(denote)f(the)g(set)g(of)h (deriv)m(ations)g(from)e Fm(A)h Fq(to)h Fm(M)24 b Fq(b)o(y)0 928 y(Der)k(\()p Fm(A;)8 b(M)d Fq(\))255 935 y Fk(c)272 928 y Fq(.)75 1015 y Fn(Prop)r(osition)17 b(4.2.2.)24 b Fq(1.)42 b Fi(L)n(et)24 b Fm(A)g Fi(b)n(e)h(a)63 b(-algebr)n(a.)43 b(Then)25 b(the)g(functor)f Fq(Der)k(\()p Fm(A;)8 b Fi(-)p Fq(\))26 b(:)0 1073 y Fm(A)p Fi(-)p Fn(Mo)r(d)p Fi(-)p Fm(A)14 b Fj(\000)-29 b(!)13 b Fn(V)-5 b(ec)18 b Fi(is)f(r)n(epr)n (esentable)i(by)e(the)h Fq(mo)q(dule)d(of)i(di\013eren)o(tials)f(\012) 1430 1080 y Fk(A)1459 1073 y Fi(.)75 1131 y Fq(2.)35 b Fi(L)n(et)21 b Fm(A)g Fi(b)n(e)h(a)g(c)n(ommutative)61 b(-algebr)n(a.)36 b(Then)22 b(the)g(functor)g Fq(Der)28 b(\()p Fm(A;)8 b Fi(-)p Fq(\))1558 1138 y Fk(c)1597 1131 y Fq(:)21 b Fm(A)p Fi(-)p Fn(Mo)r(d)0 1189 y Fj(\000)-29 b(!)14 b Fn(V)-5 b(ec)17 b Fi(is)g(r)n(epr)n(esentable)i(by)e(the)h Fq(mo)q(dule)d(of)i(comm)o(utativ)n(e)c(di\013eren)o(tials)k(\012)1486 1171 y Fk(c)1486 1202 y(A)1514 1189 y Fi(.)75 1277 y Fd(Pr)o(oof.)i Fq(1.)32 b(Represen)o(t)19 b Fm(A)g Fq(as)h(a)g(quotien) o(t)f(of)h(a)g(free)58 b(-algebra)20 b Fm(A)f Fq(:=)59 b Fj(h)p Fm(X)1582 1284 y Fk(i)1596 1277 y Fj(j)p Fm(i)19 b Fj(2)h Fm(J)5 b Fj(i)p Fm(=I)0 1335 y Fq(where)19 b Fm(B)j Fq(=)58 b Fj(h)p Fm(X)358 1342 y Fk(i)372 1335 y Fj(j)p Fm(i)19 b Fj(2)g Fm(J)5 b Fj(i)20 b Fq(is)f(the)g(free)g (algebra)g(with)h(generators)g Fm(X)1348 1342 y Fk(i)1362 1335 y Fq(.)31 b(W)l(e)19 b(\014rst)h(pro)o(v)o(e)e(the)0 1393 y(theorem)d(for)h(free)g(algebras.)75 1451 y(a\))g(A)g(represen)o (ting)f(mo)q(dule)h(for)g(Der)28 b(\()p Fm(B)s(;)8 b Fq(-)o(\))17 b(is)f(\(\012)1050 1458 y Fk(B)1080 1451 y Fm(;)8 b(d)14 b Fq(:)g Fm(B)i Fj(\000)-30 b(!)13 b Fq(\012)1329 1458 y Fk(B)1360 1451 y Fq(\))j(with)588 1530 y(\012)623 1537 y Fk(B)667 1530 y Fq(:=)d Fm(B)h Fj(\012)d Fm(F)c Fq(\()p Fm(dX)956 1537 y Fk(i)970 1530 y Fj(j)p Fm(i)13 b Fj(2)h Fm(J)5 b Fq(\))11 b Fj(\012)g Fm(B)0 1608 y Fq(where)18 b Fm(F)7 b Fq(\()p Fm(dX)266 1615 y Fk(i)280 1608 y Fj(j)p Fm(i)16 b Fj(2)i Fm(J)5 b Fq(\))17 b(is)h(the)g(free)56 b(-mo)q(dule)17 b(on)i(the)f(set)g(of)g (formal)f(sym)o(b)q(ols)g Fj(f)p Fm(dX)1631 1615 y Fk(i)1646 1608 y Fj(j)p Fm(i)f Fj(2)i Fm(J)5 b Fj(g)0 1666 y Fq(as)17 b(a)g(basis.)75 1725 y(W)l(e)k(ha)o(v)o(e)f(to)h(sho)o(w)h(that)f(for)h (ev)o(ery)d(deriv)m(ation)i Fm(D)j Fq(:)e Fm(B)i Fj(\000)-30 b(!)22 b Fm(M)k Fq(there)21 b(exists)f(a)i(unique)0 1783 y(homomorphisms)14 b Fm(')f Fq(:)h(\012)472 1790 y Fk(B)516 1783 y Fj(\000)-30 b(!)14 b Fm(M)21 b Fq(of)c Fm(B)s Fq(-)p Fm(B)s Fq(-bimo)q(dules)d(suc)o(h)i(that)h(the)f(diagram)747 1863 y Fm(B)194 b Fq(\012)1013 1870 y Fk(B)p 801 1850 163 2 v 922 1849 a Fc(-)873 1840 y Fk(d)828 1982 y(D)801 1925 y Fc(@)842 1966 y(@)884 2008 y(@)925 2049 y(@)935 2059 y(@)-42 b(R)984 2110 y Fm(M)p 1010 2059 2 176 v 1010 2059 a Fc(?)1030 1975 y Fk(')0 2172 y Fq(comm)o(utes.)24 b(The)18 b(mo)q(dule)e(\012)564 2179 y Fk(B)613 2172 y Fq(is)i(a)g Fm(B)s Fq(-)p 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Fq(1.)21 b(T)l(o)14 b(see)f(that)h Fm(d)g Fq(is)g(a)g(deriv)m(ation)f(it)g(su\016ces)h(to)g (sho)o(w)g(this)g(on)g(the)f(basis)h(elemen)o(ts:)199 2419 y Fm(d)p Fq(\()p Fm(X)283 2426 y Fg(1)312 2419 y Fm(:)8 b(:)g(:)g(X)418 2426 y Fk(k)439 2419 y Fm(X)479 2426 y Fk(k)q Fg(+1)555 2419 y Fm(:)g(:)g(:)f(X)660 2426 y Fk(n)684 2419 y Fq(\))439 2484 y(=)491 2446 y Ff(P)544 2459 y Fk(k)544 2498 y(j)r Fg(=1)615 2484 y Fm(X)655 2491 y Fg(1)684 2484 y Fm(:)h(:)g(:)f(X)789 2491 y Fk(j)r Fe(\000)p Fg(1)864 2484 y Fj(\012)k Fm(dX)979 2491 y Fk(j)1009 2484 y Fj(\012)g Fm(X)1099 2491 y Fk(j)r Fg(+1)1171 2484 y Fm(:)d(:)g(:)g(X)1277 2491 y Fk(k)1299 2484 y Fm(X)1339 2491 y Fk(k)q Fg(+1)1414 2484 y Fm(:)g(:)g(:)f(X)1519 2491 y Fk(n)456 2545 y Fq(+)502 2508 y Ff(P)554 2521 y Fk(n)554 2560 y(j)r Fg(=)p Fk(k)q Fg(+1)673 2545 y Fm(X)713 2552 y Fg(1)741 2545 y Fm(:)h(:)g(:)g(X)847 2552 y Fk(k)869 2545 y Fm(X)909 2552 y Fk(k)q Fg(+1)984 2545 y Fm(:)g(:)g(:)f(X)1089 2552 y Fk(j)r Fe(\000)p Fg(1)1164 2545 y Fj(\012)k Fm(dX)1279 2552 y Fk(j)1309 2545 y Fj(\012)g Fm(X)1399 2552 y Fk(j)r Fg(+1)1471 2545 y Fm(:)d(:)g(:)g(X)1577 2552 y Fk(n)439 2607 y Fq(=)14 b Fm(d)p Fq(\()p Fm(X)575 2614 y Fg(1)604 2607 y Fm(:)8 b(:)g(:)g(X)710 2614 y Fk(k)731 2607 y Fq(\))p Fm(X)790 2614 y Fk(k)q Fg(+1)866 2607 y Fm(:)g(:)g(:)f(X)971 2614 y Fk(n)1006 2607 y Fq(+)k Fm(X)1095 2614 y Fg(1)1124 2607 y Fm(:)d(:)g(:)f(X)1229 2614 y Fk(k)1251 2607 y Fm(d)p Fq(\()p Fm(X)1335 2614 y Fk(k)q Fg(+1)1410 2607 y Fm(:)h(:)g(:)g(X)1516 2614 y Fk(n)1540 2607 y Fq(\))p eop %%Page: 132 3 132 2 bop 0 118 a Fo(132)534 b(4.)17 b(THE)f(INFINITESIMAL)g(THEOR)m(Y) 0 213 y Fq(No)o(w)i(let)g Fm(D)h Fq(:)e Fm(B)i Fj(\000)-30 b(!)17 b Fm(M)24 b Fq(b)q(e)18 b(a)h(deriv)m(ation.)27 b(De\014ne)18 b Fm(')g Fq(b)o(y)g Fm(')p Fq(\(1)13 b Fj(\012)f Fm(dX)1321 220 y Fk(i)1349 213 y Fj(\012)g Fq(1\))17 b(:=)g Fm(D)q Fq(\()p Fm(X)1629 220 y Fk(i)1645 213 y Fq(\).)27 b(This)0 271 y(map)14 b(ob)o(viously)g(extends)g(to)h (a)g(homomorphism)d(of)j 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Fq(let)f Fm( )i Fq(:)f(\012)867 566 y Fk(B)911 559 y Fj(\000)-30 b(!)14 b Fm(M)19 b Fq(b)q(e)c(a)f(bimo)q(dule)f (homomorphism)e(suc)o(h)0 617 y(that)18 b Fm( )r(d)f Fq(=)f Fm(D)q Fq(.)27 b(Then)18 b Fm( )r Fq(\(1)12 b Fj(\012)g Fm(dX)653 624 y Fk(i)680 617 y Fj(\012)g Fq(1\))17 b(=)f Fm( )r(d)p Fq(\()p Fm(X)963 624 y Fk(i)978 617 y Fq(\))g(=)h Fm(D)q Fq(\()p Fm(X)1168 624 y Fk(i)1183 617 y Fq(\))f(=)h Fm(')p Fq(\(1)12 b Fj(\012)g Fm(dX)1476 624 y Fk(i)1503 617 y Fj(\012)g Fq(1\).)27 b(Since)17 b Fm( )0 675 y Fq(and)g Fm(')f Fq(are)g Fm(B)s Fq(-)p Fm(B)s Fq(-bimo)q(dules)f(homomorphisms)f(this)i(extends)f(to)i Fm( )e Fq(=)f Fm(')p Fq(.)75 733 y(b\))19 b(No)o(w)g(let)f Fm(A)g Fq(:=)57 b Fj(h)p Fm(X)550 740 y Fk(i)564 733 y Fj(j)p Fm(i)18 b Fj(2)h Fm(J)5 b Fj(i)p Fm(=I)23 b Fq(b)q(e)c(an)g(arbitrary)g(algebra)h(with)e Fm(B)j Fq(=)58 b Fj(h)p Fm(X)1634 740 y Fk(i)1648 733 y Fj(j)p Fm(i)18 b Fj(2)h Fm(J)5 b Fj(i)0 792 y Fq(free.)20 b(De\014ne)418 857 y(\012)453 864 y Fk(A)496 857 y Fq(:=)13 b(\012)596 864 y Fk(B)627 857 y Fm(=)p Fq(\()p Fm(I)t Fq(\012)731 864 y Fk(B)772 857 y Fq(+)e(\012)856 864 y Fk(B)887 857 y Fm(I)j Fq(+)d Fm(B)s(d)1037 864 y Fk(B)1067 857 y Fq(\()p Fm(I)t Fq(\))g(+)g Fm(d)1216 864 y Fk(B)1246 857 y Fq(\()p Fm(I)t Fq(\))p Fm(B)s Fq(\))p Fm(:)0 932 y Fq(W)l(e)g(\014rst)h(sho)o (w)g(that)g Fm(I)t Fq(\012)453 939 y Fk(B)485 932 y Fq(+)r(\012)560 939 y Fk(B)590 932 y Fm(I)5 b Fq(+)r Fm(B)s(d)722 939 y Fk(B)752 932 y Fq(\()p Fm(I)t Fq(\))r(+)r Fm(d)883 939 y Fk(B)912 932 y Fq(\()p Fm(I)t Fq(\))p Fm(B)13 b Fq(is)e(a)h Fm(B)s Fq(-)p Fm(B)s Fq(-subbimo)q(dule.)19 b(Since)10 b(\012)1679 939 y Fk(B)1721 932 y Fq(and)0 990 y Fm(I)21 b Fq(are)d Fm(B)s Fq(-)p Fm(B)s Fq(-bimo)q(dules)e(the)i (terms)e Fm(I)t Fq(\012)755 997 y Fk(B)803 990 y Fq(and)i(\012)934 997 y Fk(B)965 990 y Fm(I)j Fq(are)d(bimo)q(dules.)24 b(F)l(urthermore)16 b(w)o(e)i(ha)o(v)o(e)0 1048 y Fm(bd)46 1055 y Fk(B)76 1048 y Fq(\()p Fm(i)p Fq(\))p Fm(b)152 1030 y Fe(0)177 1048 y Fq(=)c Fm(bd)275 1055 y Fk(B)305 1048 y Fq(\()p Fm(ib)362 1030 y Fe(0)373 1048 y Fq(\))c Fj(\000)f Fm(bid)513 1055 y Fk(B)543 1048 y Fq(\()p Fm(b)583 1030 y Fe(0)594 1048 y Fq(\))14 b Fj(2)g Fm(B)s(d)739 1055 y Fk(B)769 1048 y Fq(\()p Fm(I)t Fq(\))9 b(+)g Fm(I)t Fq(\012)950 1055 y Fk(B)996 1048 y Fq(hence)15 b Fm(I)t Fq(\012)1192 1055 y Fk(B)1231 1048 y Fq(+)9 b(\012)1313 1055 y Fk(B)1344 1048 y Fm(I)k Fq(+)c Fm(B)s(d)1491 1055 y Fk(B)1521 1048 y Fq(\()p Fm(I)t Fq(\))g(+)g Fm(d)1666 1055 y Fk(B)1697 1048 y Fq(\()p Fm(I)t Fq(\))p Fm(B)0 1106 y Fq(is)16 b(a)h(bimo)q(dule.)75 1164 y(No)o(w)g Fm(I)t Fq(\012)248 1171 y Fk(B)296 1164 y Fq(and)h(\012)427 1171 y Fk(B)458 1164 y Fm(I)j Fq(are)c(subbimo)q(dules)g(of)h Fm(I)t Fq(\012)1008 1171 y Fk(B)1050 1164 y Fq(+)12 b(\012)1135 1171 y Fk(B)1165 1164 y Fm(I)k Fq(+)c Fm(B)s(d)1318 1171 y Fk(B)1348 1164 y Fq(\()p Fm(I)t Fq(\))f(+)h Fm(d)1498 1171 y Fk(B)1529 1164 y Fq(\()p Fm(I)t Fq(\))p Fm(B)s Fq(.)24 b(Hence)0 1222 y Fm(A)13 b Fq(=)h Fm(B)s(=I)20 b Fq(acts)c(on)h(b)q(oth)g(sides)f(on)h(\012)711 1229 y Fk(A)756 1222 y Fq(so)g(that)g(\012)957 1229 y Fk(A)1002 1222 y Fq(b)q(ecomes)e(an)h Fm(A)p Fq(-)p Fm(A)p Fq(-bimo)q(dule.)75 1280 y(Let)k Fm(\027)j Fq(:)c(\012)281 1287 y Fk(B)332 1280 y Fj(\000)-30 b(!)19 b Fq(\012)445 1287 y Fk(A)494 1280 y Fq(and)h(also)h Fm(\027)i Fq(:)c Fm(B)k Fj(\000)-30 b(!)19 b Fm(A)h Fq(b)q(e)g(the)f(residue)h(homomorphisms.)29 b(Since)0 1338 y Fm(\027)s(d)52 1345 y Fk(B)83 1338 y Fq(\()p Fm(i)p Fq(\))13 b Fj(2)h Fm(\027)s(d)250 1345 y Fk(B)281 1338 y Fq(\()p Fm(I)t Fq(\))f(=)h(0)g Fj(\022)g Fq(\012)536 1345 y Fk(A)579 1338 y Fq(w)o(e)f(get)h(a)g(unique)f (factorization)h(map)f Fm(d)1333 1345 y Fk(A)1375 1338 y Fq(:)h Fm(A)f Fj(\000)-30 b(!)13 b Fq(\012)1560 1345 y Fk(A)1603 1338 y Fq(suc)o(h)h(that)784 1629 y Fm(A)145 b Fq(\012)1001 1636 y Fk(A)p 835 1616 117 2 v 910 1615 a Fc(-)872 1648 y Fk(d)890 1654 y Fb(A)783 1434 y Fm(B)g Fq(\012)1000 1441 y Fk(B)p 837 1421 114 2 v 909 1420 a Fc(-)871 1405 y Fk(d)889 1411 y Fb(B)p 802 1581 2 127 v 803 1581 a Fc(?)763 1525 y Fk(\027)p 997 1581 V 997 1581 a Fc(?)1017 1525 y Fk(\027)0 1732 y Fq(comm)o(utes.)18 b(Since)e Fm(d)401 1739 y Fk(A)429 1732 y Fq(\()p 448 1691 21 2 v Fm(b)p Fq(\))e(=)p 554 1689 115 2 v 14 w Fm(d)579 1739 y Fk(B)609 1732 y Fq(\()p Fm(b)p Fq(\))i(it)g(is)g(clear) g(that)g Fm(d)1029 1739 y Fk(A)1074 1732 y Fq(is)g(a)h(deriv)m(ation.) 75 1790 y(Let)12 b Fm(D)k Fq(:)d Fm(A)h Fj(\000)-30 b(!)13 b Fm(M)18 b Fq(b)q(e)13 b(a)g(deriv)m(ation.)19 b(The)13 b Fm(A)p Fq(-)p Fm(A)p Fq(-bimo)q(dule)e Fm(M)18 b Fq(is)12 b(also)h(a)g Fm(B)s Fq(-)p Fm(B)s Fq(-bimo)q(dule)e(b)o(y)0 1848 y Fm(bm)i Fq(=)p 129 1808 21 2 v 14 w Fm(bm)p Fq(.)21 b(F)l(urthermore)14 b Fm(D)q(\027)k Fq(:)13 b Fm(B)k Fj(\000)-31 b(!)14 b Fm(A)g Fj(\000)-31 b(!)14 b Fm(M)22 b Fq(is)15 b(again)j(a)e(deriv)m(ation.)21 b(Let)16 b Fm(')1524 1855 y Fk(B)1568 1848 y Fq(:)e(\012)1631 1855 y Fk(B)1675 1848 y Fj(\000)-30 b(!)14 b Fm(M)0 1907 y Fq(b)q(e)22 b(the)g(unique)f(factorization)h(map)f(for)h(the)g Fm(B)s Fq(-deriv)m(ation)g Fm(D)q(\027)s Fq(.)39 b(Consider)23 b(the)e(follo)o(wing)0 1965 y(diagram)747 2037 y Fm(B)169 b Fq(\012)988 2044 y Fk(B)p 801 2024 138 2 v 897 2023 a Fc(-)847 2008 y Fk(d)865 2014 y Fb(B)748 2232 y Fm(A)145 b Fq(\012)965 2239 y Fk(A)p 799 2219 117 2 v 874 2218 a Fc(-)836 2203 y Fk(d)854 2209 y Fb(A)p 766 2184 2 127 v 767 2184 a Fc(?)p 961 2184 V 153 w(?)p 961 2379 V 962 2379 a(?)918 2324 y Fk( )p 1010 2379 2 322 v 1010 2379 a Fc(?)1030 2222 y Fk(')963 2430 y Fm(M)804 2327 y Fk(D)801 2294 y Fc(@)842 2336 y(@)884 2377 y(@)886 2379 y(@)-42 b(R)0 2489 y Fq(W)l(e)21 b(w)o(an)o(t)h(to)f(construct)h Fm( )h Fq(suc)o(h)e(that)h(the)f(diagram)g(comm)o(utes.)34 b(Let)22 b Fm(i!)i Fj(2)f Fm(I)t Fq(\012)1608 2496 y Fk(B)1638 2489 y Fq(.)37 b(Then)0 2547 y Fm(')p Fq(\()p Fm(i!)r Fq(\))16 b(=)p 189 2508 17 2 v 16 w Fm(i')p Fq(\()p Fm(!)r Fq(\))g(=)g(0)i(and)h(similarly)c Fm(')p Fq(\()p Fm(!)r(i)p Fq(\))g(=)i(0.)26 b(Let)17 b Fm(bd)1105 2554 y Fk(B)1136 2547 y Fq(\()p Fm(i)p Fq(\))f Fj(2)g Fm(B)s(d)1321 2554 y Fk(B)1351 2547 y Fq(\()p Fm(I)t Fq(\))h(then)h Fm(')p Fq(\()p Fm(bd)1642 2554 y Fk(B)1672 2547 y Fq(\()p Fm(i)p Fq(\)\))e(=)p 0 2567 21 2 v 0 2608 a Fm(b'd)78 2615 y Fk(B)108 2608 y Fq(\()p Fm(i)p Fq(\))e(=)p 229 2567 V 14 w Fm(b)o(D)q Fq(\()p 309 2569 17 2 v Fm(i)q Fq(\))g(=)f(0)j(and)f(similarly)e Fm(')p Fq(\()p Fm(d)819 2615 y Fk(B)849 2608 y Fq(\()p Fm(i)p Fq(\))p Fm(b)p Fq(\))g(=)h(0.)21 b(Hence)14 b Fm(')h Fq(v)m(anishes)h(on)f Fm(I)t Fq(\012)1577 2615 y Fk(B)1616 2608 y Fq(+)8 b(\012)1697 2615 y Fk(B)1728 2608 y Fm(I)k Fq(+)p eop %%Page: 133 4 133 3 bop 748 118 a Fo(2.)17 b(DERIV)l(A)m(TIONS)677 b(133)0 213 y Fm(B)s(d)65 220 y Fk(B)95 213 y Fq(\()p Fm(I)t Fq(\))6 b(+)g Fm(d)234 220 y Fk(B)265 213 y Fq(\()p Fm(I)t Fq(\))p Fm(B)16 b Fq(and)e(th)o(us)g(factorizes)g(through)h(a)f (unique)f(map)h Fm( )h Fq(:)e(\012)1385 220 y Fk(A)1428 213 y Fj(\000)-30 b(!)13 b Fm(M)5 b Fq(.)21 b(Ob)o(viously)0 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790 y Fk(i)1743 783 y Fq(\))p Fj(g)p Fq(.)75 842 y(3.)34 b(Let)21 b Fm(f)26 b Fj(2)21 b Fm(B)j Fq(=)60 b Fj(h)p Fm(X)561 849 y Fk(i)576 842 y Fj(i)p Fq(.)34 b(Let)21 b Fm(B)775 824 y Fk(op)832 842 y Fq(b)q(e)f(the)h(algebra)g(opp)q(osite)g(to)g Fm(B)i Fq(\(with)d(opp)q(osite)0 900 y(m)o(ultiplic)o(ation\).)e(Then)d(\012) 511 907 y Fk(B)556 900 y Fq(=)e Fm(B)e Fj(\012)d Fm(F)f Fq(\()p Fm(dX)825 907 y Fk(i)839 900 y Fq(\))h Fj(\012)g Fm(B)17 b Fq(is)e(the)f(free)g Fm(B)d Fj(\012)d Fm(B)1324 882 y Fk(op)1375 900 y Fq(left)14 b(mo)q(dule)f(o)o(v)o(er)h(the)0 958 y(free)h(generating)i(set)f Fj(f)p Fm(d)p Fq(\()p Fm(X)517 965 y Fk(i)532 958 y Fq(\))p Fj(g)p Fq(.)21 b(Hence)15 b Fm(d)p Fq(\()p Fm(f)5 b Fq(\))17 b(has)g(a)f(unique)g (represen)o(tation)675 1082 y Fm(d)p Fq(\()p Fm(f)5 b Fq(\))15 b(=)834 1035 y Ff(X)864 1140 y Fk(i)931 1048 y Fm(@)s(f)p 919 1071 84 2 v 919 1116 a(@)s(X)988 1123 y Fk(i)1007 1082 y Fm(d)p Fq(\()p Fm(X)1091 1089 y Fk(i)1106 1082 y Fq(\))0 1223 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Fk(i)862 550 y Fm(d)887 557 y Fk(A)916 550 y Fq(\()p 935 510 55 2 v Fm(X)975 557 y Fk(i)989 550 y Fq(\))d(=)g(0)p Fm(:)0 666 y Fq(These)f(are)g(the)g(de\014ning)g (relations)h(for)f(the)g Fm(A)p Fq(-)p Fm(A)p Fq(-bimo)q(dule)e(\012) 1176 673 y Fk(A)1218 666 y Fq(with)i(the)g(generators)h Fm(d)1665 673 y Fk(A)1694 666 y Fq(\()p 1713 626 V Fm(X)1753 673 y Fk(i)1767 666 y Fq(\).)75 756 y(F)l(or)k(motiv)m(ation)e(of)i (the)g(quan)o(tum)e(group)j(case)f(w)o(e)f(consider)h(an)g(a\016ne)g (algebraic)f(group)0 814 y Fm(G)e Fq(with)g(represen)o(ting)f(comm)o (utativ)n(e)e(Hopf)j(algebra)g Fm(A)p Fq(.)20 b(Recall)14 b(that)h(Hom)o(\()p Fm(A;)8 b(R)p Fq(\))15 b(is)f(an)i(alge-)0 873 y(bra)23 b(with)f(the)g(con)o(v)o(olution)g(m)o(ultiplic)o(ation)e (for)j(ev)o(ery)e Fm(R)k Fj(2)64 b Fq(-)p Fn(cAlg)22 b Fq(and)h(that)g Fm(G)p Fq(\()p Fm(R)p Fq(\))i(=)39 931 y(-)p Fn(cAlg)p Fq(\()p Fm(A;)8 b(R)p Fq(\))29 b Fj(\022)g Fq(Hom)o(\()p Fm(A;)8 b(R)p Fq(\))25 b(is)g(a)h(subgroup)h 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Fj(\003)p Fm(S)14 b Fj(\003)e Fm(T)22 b Fq(=)15 b Fm(T)7 b Fq(,)16 b(so)i(that)g Fm(T)23 b Fq(is)0 1388 y(in)16 b(the)g(image)f(of)h(\010.)p 1765 1388 2 33 v 1767 1357 30 2 v 1767 1388 V 1796 1388 2 33 v 75 1475 a Fn(Prop)r(osition)h(4.2.7.)24 b Fi(L)n(et)18 b Fm(d)g Fj(2)f Fq(Hom)o(\()p Fm(H)q(;)47 b Fq(\))19 b Fi(and)g Fq(\010\()p Fm(d)p Fq(\))f(=)f Fm(D)h Fj(2)g Fq(Hom)n(\()p Fm(H)q(;)8 b(H)t Fq(\))20 b Fi(b)n(e)f(given.)0 1533 y(The)f(fol)r(lowing)h(ar)n(e)e(e)n(quivalent:)75 1601 y Fq(1.)j Fm(d)15 b Fq(:)e Fm(H)18 b Fj(\000)-29 b(!)332 1608 y Fk(")41 b(")425 1601 y Fi(is)17 b(a)g(derivation.)75 1660 y Fq(2.)j Fm(D)c Fq(:)d Fm(H)19 b Fj(\000)-29 b(!)348 1667 y Fk(H)382 1660 y Fm(H)422 1667 y Fk(H)473 1660 y Fi(is)18 b(a)f(\(left)h(tr)n(anslation)g(invariant\))g(derivation.)0 1728 y(In)g(p)n(articular)g Fq(\010)g Fi(induc)n(es)h(an)g(isomorphism) e(b)n(etwe)n(en)i(the)g(set)g(of)f(derivations)h Fm(d)c Fq(:)g Fm(H)20 b Fj(\000)-30 b(!)1725 1735 y Fk(")41 b(")0 1786 y Fi(and)18 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Ff(P)61 1989 y Fm(a)87 1997 y Fg(\(1\))134 1989 y Fm(b)155 1997 y Fg(\(1\))202 1989 y Fm(d)p Fq(\()p Fm(a)272 1997 y Fg(\(2\))319 1989 y Fq(\))p Fm(")p Fq(\()p Fm(b)401 1997 y Fg(\(2\))447 1989 y Fq(\))j(=)g Fm(aD)q Fq(\()p Fm(b)p Fq(\))13 b(+)g Fm(D)q Fq(\()p Fm(a)p Fq(\))p Fm(b:)19 b Fq(Con)o(v)o(ersely)f(assume)g(that)i Fm(D)q Fq(\()p Fm(ab)p Fq(\))f(=)g Fm(aD)q Fq(\()p Fm(b)p Fq(\))13 b(+)0 2048 y Fm(D)q Fq(\()p Fm(a)p Fq(\))p Fm(b)p Fq(.)21 b(Then)c Fm(d)p Fq(\()p Fm(ab)p Fq(\))c(=)h Fm("D)q Fq(\()p Fm(ab)p Fq(\))g(=)g Fm(")p Fq(\()p Fm(a)p Fq(\))p Fm("D)q Fq(\()p Fm(b)p Fq(\))c(+)h Fm("D)q Fq(\()p Fm(a)p Fq(\))p Fm(")p Fq(\()p Fm(b)p Fq(\))j(=)g Fm(")p Fq(\()p Fm(a)p Fq(\))p Fm(d)p Fq(\()p Fm(b)p Fq(\))c(+)h Fm(d)p Fq(\()p Fm(a)p Fq(\))p Fm(")p Fq(\()p Fm(b)p Fq(\).)p 1765 2048 V 1767 2016 30 2 v 1767 2048 V 1796 2048 2 33 v eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF