1178:
Early in the course I formed a one-step projective resolution of a module, and remarked that if the kernel was projective in one resolution it was projective in all. I added that, although the statement was so simple and straightforward, it would be a while before we proved it. Steve
Schanuel spoke
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855:{\displaystyle {\begin{aligned}\ker \pi &=\{(0,q):(0,q)\in X\}\\&=\{(0,q):\phi '(q)=0\}\\&\cong \ker \phi '\cong K'.\end{aligned}}}
1029:
871:
1236:
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up and told me and the class that it was quite easy, and thereupon sketched what has come to be known as "Schanuel's lemma."
1266:
40:. It is useful in defining the Heller operator in the stable category, and in giving elementary descriptions of
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1231:. Chicago Lectures in Mathematics (2nd ed.). University Of Chicago Press. pp. 165–168.
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63:
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37:
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33:
21:
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403:
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25:
1023:, and the same argument as above shows that there is another short exact sequence
17:
441:
115:
135:
1071:{\displaystyle 0\rightarrow K\rightarrow X\rightarrow P'\rightarrow 0,}
910:{\displaystyle 0\rightarrow K'\rightarrow X\rightarrow P\rightarrow 0.}
338:{\displaystyle X=\{(p,q)\in P\oplus P':\phi (p)=\phi '(q)\}.}
36:, allows one to compare how far modules depart from being
400:
is defined as the projection of the first coordinate of
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We may conclude that there is a short exact sequence
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1146:The above argument may also be generalized to
78: → 0 and 0 →
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1190:
981:. Similarly, we can write another map
1173:in Autumn of 1958. Kaplansky writes:
1118:. Combining the two equivalences for
66:with identity. If 0 →
7:
236:{\displaystyle \phi '\colon P'\to M}
1016:{\displaystyle \pi '\colon X\to P'}
568:{\displaystyle \phi (p)=\phi '(q)}
194:{\displaystyle \phi \colon P\to M}
14:
1111:{\displaystyle X\cong P'\oplus K}
974:{\displaystyle X\cong K'\oplus P}
373:{\displaystyle \pi \colon X\to P}
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1:
1200:Lectures on Modules and Rings
940:is projective this sequence
55:is the following statement:
20:, especially in the area of
1161:discovered the argument in
647:{\displaystyle \pi (p,q)=p}
1288:
1138:gives the desired result.
606:{\displaystyle (p,q)\in X}
162:{\displaystyle P\oplus P'}
90: → 0 are
469:is surjective, for any
519:{\displaystyle q\in P'}
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488:{\displaystyle p\in P}
463:
462:{\displaystyle \phi '}
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1171:University of Chicago
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134:Define the following
106:are projective, then
92:short exact sequences
1148:long exact sequences
1142:Long exact sequences
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671:{\displaystyle \pi }
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393:{\displaystyle \pi }
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1267:Homological algebra
1214:pgs. 165–167.
1167:homological algebra
654:. Now examine the
86: →
82: →
74: →
70: →
1198:Lam, T.Y. (1999).
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42:dimension shifting
1225:Kaplansky, Irving
1131:{\displaystyle X}
933:{\displaystyle P}
495:, one may find a
433:{\displaystyle P}
413:{\displaystyle X}
1279:
1251:
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1229:Fields and Rings
1221:
1215:
1213:
1195:
1163:Irving Kaplansky
1159:Stephen Schanuel
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53:Schanuel's lemma
34:Stephen Schanuel
30:Schanuel's lemma
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1169:course at the
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584:
575:. This gives
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49:
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32:, named after
13:
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4:
3:
2:
1284:
1273:
1272:Module theory
1270:
1268:
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1264:
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1248:
1244:
1240:
1238:0-226-42451-0
1234:
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1209:0-387-98428-3
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992:
989:
968:
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927:
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738:
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681:
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665:
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632:
629:
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559:
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117:
113:
109:
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101:
98:-modules and
97:
93:
89:
85:
81:
77:
73:
69:
65:
61:
56:
54:
47:
45:
43:
39:
35:
31:
27:
26:module theory
23:
19:
1228:
1219:
1202:. Springer.
1199:
1193:
1177:
1157:
1145:
1080:
919:
864:
347:
133:
123:
119:
111:
107:
103:
99:
95:
87:
83:
79:
75:
71:
67:
59:
57:
52:
51:
29:
15:
658:of the map
18:mathematics
1261:Categories
1247:1001.16500
526:such that
442:surjective
116:isomorphic
38:projective
1103:⊕
1092:≅
1060:→
1049:→
1043:→
1037:→
1003:→
997::
990:π
966:⊕
955:≅
902:→
896:→
890:→
879:→
835:≅
828:ϕ
824:
818:≅
786:ϕ
745:∈
699:π
696:
666:π
621:π
598:∈
550:ϕ
534:ϕ
506:∈
480:∈
453:ϕ
444:. Since
388:π
365:→
359::
356:π
314:ϕ
298:ϕ
284:⊕
278:∈
228:→
217::
210:ϕ
186:→
180::
177:ϕ
149:⊕
136:submodule
48:Statement
24:known as
1227:(1972).
1099:′
1056:′
1010:′
993:′
962:′
886:′
842:′
831:′
789:′
553:′
513:′
456:′
380:, where
348:The map
317:′
291:′
224:′
213:′
169:, where
156:′
122:⊕
110:⊕
1154:Origins
1081:and so
22:algebra
1245:
1235:
1206:
942:splits
920:Since
656:kernel
1185:Notes
944:, so
613:with
440:, is
420:into
130:Proof
62:be a
1233:ISBN
1204:ISBN
201:and
102:and
64:ring
58:Let
1243:Zbl
1165:'s
821:ker
693:ker
138:of
118:to
114:is
94:of
16:In
1263::
1241:.
1150:.
905:0.
678::
243::
126:.
120:K′
112:P′
104:P′
84:P′
80:K′
44:.
28:,
1249:.
1212:.
1126:X
1106:K
1096:P
1089:X
1066:,
1063:0
1053:P
1046:X
1040:K
1034:0
1007:P
1000:X
969:P
959:K
952:X
928:P
899:P
893:X
883:K
876:0
846:.
839:K
808:}
805:0
802:=
799:)
796:q
793:(
782::
779:)
776:q
773:,
770:0
767:(
764:{
761:=
751:}
748:X
742:)
739:q
736:,
733:0
730:(
727::
724:)
721:q
718:,
715:0
712:(
709:{
706:=
642:p
639:=
636:)
633:q
630:,
627:p
624:(
601:X
595:)
592:q
589:,
586:p
583:(
563:)
560:q
557:(
546:=
543:)
540:p
537:(
510:P
503:q
483:P
477:p
428:P
408:X
368:P
362:X
333:.
330:}
327:)
324:q
321:(
310:=
307:)
304:p
301:(
295::
288:P
281:P
275:)
272:q
269:,
266:p
263:(
260:{
257:=
254:X
231:M
221:P
189:M
183:P
153:P
146:P
124:P
108:K
100:P
96:R
88:M
76:M
72:P
68:K
60:R
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