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I just added two "sections needing references" templates. The section on the relation to topological cohomology theories just requires a book. The section on the history makes several different claims and probably requires several references. For some of the statements regarding algebraic number
888:
Could someone (preferably someone more knowledgeable than me!) please rewrite the opening paragraph? As it stands it's not a good intro to the article. For example, "generalization to non-abelian coefficients" makes no sense in a context where no coefficients, abelian or otherwise, have been
2492:{\displaystyle \left(d^{n+1}\varphi \right)(g_{1},\ldots ,g_{n+1})=g_{1}{\tilde {\varphi }}(g_{0},g_{2},\dots ,g_{n+1})+\sum _{i=1}^{n}(-1)^{i}{\tilde {\varphi }}\left(g_{0},g_{1},\ldots ,g_{i-1},g_{i}g_{i+1},\ldots ,g_{n+1}\right)+(-1)^{n+1}{\tilde {\varphi }}(g_{0},g_{1},\ldots ,g_{n})}
168:
That's not true. Knowing the set of 1-cocycles and the set of 1-coboundaries isn't enough. One needs an equivalence relation on 1-cocycles, and the H^1 is the equiv classes. In the non-abelian case the equiv classes can be of different sizes so you need to know the full equiv reln.
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I just added a sentence saying that the article deals only with finite groups for now (it may be in an awkward place...) It would be good to say something about at least profinite group cohomology, but to do it justice one would have to add quite a bit.
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1657:{\displaystyle \left(d^{n+1}\varphi \right)(g_{1},\ldots ,g_{n+1})=g_{1}\varphi (g_{2},\dots ,g_{n+1})+\sum _{i=1}^{n}(-1)^{i}\varphi \left(g_{1},\ldots ,g_{i-1},g_{i}g_{i+1},\ldots ,g_{n+1}\right)+(-1)^{n+1}\varphi (g_{1},\ldots ,g_{n})}
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The relation between central extensions and the second cohomology group are mentioned briefly. Most
Knowledge articles, including this one, are incomplete; please feel free to add more material on this subtopic.
904:
I am even less knowledgeable than you, but I support your request. I believe that a reader who does not already know what group cohomology is, is unlikely to be able to figure it out from the opening paragraph.
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Does anyone know why the last equation of the "cochain complexes" section is so small? As this is where we're defining H^n it should be bigger, if anything. I looked at the code and it seems normal to me.
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2802:{\displaystyle H^{k}(C_{m},\mathbb {Z} )={\begin{cases}\mathbb {Z} /m\mathbb {Z} &k{\text{ even}},n\geq 2\\0&k{\text{ odd}},n\geq 1\end{cases}}}
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But the "n"s on the right side of the equation do not occur on the left, and the "m" on the left side does not occur on the right.
1020:, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.
611:{\displaystyle \delta \phi _{2}(g_{1},g_{2},g_{3})=\phi _{2}(g_{2},g_{3})-\phi _{2}(g_{1},g_{3})+\phi _{2}(g_{2},g_{3})\,\!}
227:
The article is, however, very confusing in its present form. The definition in terms of the Ext functor seems to produce
162:
A passing comment from someone with no account: as I write (18th Nov 09) it says, in the non-abelian cohomology section:
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1886:{\displaystyle {\tilde {\varphi }}:\{g_{0}\}\times G^{n}\to M:(g_{0},g_{1},...g_{n})\mapsto \varphi (g_{1},...g_{n})}
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165:"The first cohomology of G with coefficents in A is defined as above using 1-cocycles and 1-coboundaries."
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Needs a discussion of restriction, inflation, transfer, corestriction. Needs a section on applications.
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
409:{\displaystyle \phi _{n}(gg_{1},gg_{2},\ldots ,gg_{n})=g\phi _{n}(g_{1},g_{2},\ldots ,g_{n})\,\!}
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with its more complicated-appearing coboundary formula (which is correct as written though).
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I think the g0 in front of the first term and the g_i in the second term are probably mistakes.
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The character for R in RP^\infty does not display properly. I would suggest a switch to tex.
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The only reference link, Turkelli, Szilágyi, Lukács: Cohomology of Groups, is a broken link.
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Examples (e.g. the cohomology ring of an elementary abelian p-group, with coefficients in
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In the column "Formal construction", is there a "g_n+1" missing in the definition of d^n?
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Last edited at 22:34, 16 December 2008 (UTC). Substituted at 02:09, 5 May 2016 (UTC)
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I'm a professional mathematician and I cold not get anything out of this article.
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Add computations of free products, weibel pg 170, give group cohomology of SL_2(Z)
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The definitions of coboundaries in the
Cochain Complexes section are incomplete
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I rewrote the first part of the lead to be more accessible, see if it helps. --
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But this looks a bit ugly. Could somebody formulate this a bit more nicely?
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Re the previous comment: the definition of the coboundary is correct for the
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No mention here of the use of low-dimensional cohomology groups to classify
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757:{\displaystyle \alpha _{2}(g_{1},g_{2})=\phi _{3}(1,g_{1},g_{1}g_{2})\,\!}
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is missing. What is it? It would be needed for applying the definitions
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theory, the current references (Serre and Milne) are probably enough.
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I have now added a bit linking homogeneous and inhomogeneous chains
223:
cochains seen in applications such as projective representations.
2589:{\displaystyle d^{1}(g_{1})=g_{1}\varphi (g_{0})-\varphi (g_{0})}
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I'd love to rewrite it, but I am not an expert on Ext and Tor.
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I hope someone knowledgeable about this subject can fix this.
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The graded product (cup product, in the topological context)
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Suggestions (I may add these myself if I feel energetic):
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and so on. The description with the explicit formula for
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Add computations of semi-direct products from Totaro -
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universal central extensions and long exact sequences
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1268:{\displaystyle H^{n}(G,M)=Z^{n}(G,M)/B^{n}(G,M).}
204:The formula for d^n does look a little strange.
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1010:The comment(s) below were originally left at
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1159:{\displaystyle Z^{n}(G,M)=\ker(d^{n+1})}
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2076:. Then the general formula would be
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95:This article is within the scope of
646:cochains defined as, for example,
38:It is of interest to the following
1026:Needs references and proper lead.
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2861:Mid-priority mathematics articles
2638:group for riemann surfaces pg 205
1728:, we should regard the functions
1018:several discussions in past years
115:Knowledge:WikiProject Mathematics
2069:{\displaystyle G^{0}:=\{g_{0}\}}
118:Template:WikiProject Mathematics
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830:02:31, 3 September 2007 (UTC)
779:13:36, 23 February 2012 (UTC)
416:and with coboundary operator
188:13:24, 18 November 2009 (UTC)
109:and see a list of open tasks.
2856:C-Class mathematics articles
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2826:07:48, 6 December 2020 (UTC)
2654:06:33, 23 October 2020 (UTC)
2615:05:27, 29 January 2019 (UTC)
1666:has no sense in the case of
2010:is the identity element of
1063:section, the definition of
850:Sections needing references
635:{\displaystyle \delta \,\!}
231:co chains, i.e. functions
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795:15:27, 19 March 2013 (UTC)
257:{\displaystyle G^{n}\to M}
213:10:04, 16 March 2007 (UTC)
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835:Finite groups and further
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141:project's priority scale
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2003:{\displaystyle g_{0}}
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1976:{\displaystyle n=0}
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889:mentioned. TIA :-)
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2282:
2274:
2270:
2266:
2263:
2260:
2255:
2250:
2247:
2244:
2240:
2236:
2233:
2228:
2225:
2222:
2218:
2214:
2211:
2208:
2203:
2199:
2195:
2190:
2186:
2182:
2176:
2173:
2165:
2161:
2157:
2154:
2149:
2146:
2143:
2139:
2135:
2132:
2129:
2124:
2120:
2116:
2112:
2108:
2103:
2100:
2097:
2093:
2088:
2065:
2060:
2056:
2052:
2049:
2044:
2040:
2019:
1997:
1993:
1972:
1969:
1966:
1946:
1943:
1937:
1934:
1909:
1906:
1903:
1882:
1877:
1873:
1869:
1866:
1863:
1860:
1855:
1851:
1847:
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1825:
1822:
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1811:
1807:
1803:
1798:
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1772:
1768:
1765:
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1756:
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1694:
1681:
1678:
1675:
1653:
1648:
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1637:
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1629:
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1621:
1618:
1613:
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1603:
1599:
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1580:
1577:
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1555:
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1535:
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1526:
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1497:
1492:
1488:
1483:
1479:
1475:
1472:
1469:
1464:
1459:
1456:
1453:
1449:
1445:
1442:
1437:
1434:
1431:
1427:
1423:
1420:
1417:
1412:
1408:
1404:
1401:
1396:
1392:
1388:
1385:
1380:
1377:
1374:
1370:
1366:
1363:
1360:
1355:
1351:
1347:
1343:
1339:
1334:
1331:
1328:
1324:
1319:
1293:
1290:
1287:
1276:
1275:
1264:
1261:
1258:
1255:
1252:
1249:
1244:
1240:
1235:
1231:
1228:
1225:
1222:
1219:
1214:
1210:
1206:
1203:
1200:
1197:
1194:
1191:
1186:
1182:
1167:
1166:
1155:
1150:
1147:
1144:
1140:
1136:
1133:
1130:
1127:
1124:
1121:
1118:
1115:
1112:
1107:
1103:
1077:
1073:
1056:
1053:
1047:
1046:
1033:
1007:
1004:
1003:
1002:
963:
960:
959:
958:
957:
956:
939:
938:
927:134.157.88.219
917:
885:
882:
866:
863:
851:
848:
836:
833:
824:
823:
816:
813:
806:
798:
749:
744:
740:
734:
730:
726:
721:
717:
713:
710:
707:
702:
698:
694:
691:
686:
682:
678:
673:
669:
665:
660:
656:
627:
603:
598:
594:
590:
585:
581:
577:
572:
568:
564:
561:
556:
552:
548:
543:
539:
535:
530:
526:
522:
519:
514:
510:
506:
501:
497:
493:
488:
484:
480:
477:
472:
468:
464:
459:
455:
451:
446:
442:
438:
433:
429:
425:
401:
396:
392:
388:
385:
382:
377:
373:
369:
364:
360:
356:
351:
347:
343:
340:
337:
332:
328:
324:
321:
318:
315:
310:
306:
302:
299:
294:
290:
286:
283:
278:
274:
253:
250:
245:
241:
226:
217:
216:
215:
202:
196:
180:155.198.192.80
171:
159:
156:
153:
152:
149:
148:
145:
144:
133:
127:
126:
124:
107:the discussion
94:
93:
77:
65:
64:
56:
44:
43:
37:
26:
13:
10:
9:
6:
4:
3:
2:
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2862:
2859:
2857:
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2853:
2851:
2842:
2838:
2834:
2830:
2829:
2828:
2827:
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2814:
2810:
2809:
2789:
2786:
2783:
2780:
2772:
2767:
2760:
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2754:
2751:
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2625:
2624:
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2604:
2597:
2578:
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2564:
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2110:
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2098:
2095:
2091:
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2054:
2047:
2042:
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2017:
1995:
1991:
1970:
1967:
1964:
1944:
1941:
1932:
1907:
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625:
596:
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583:
579:
570:
566:
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541:
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214:
211:
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206:
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2601:— Preceding
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137:Mid-priority
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62:Mid‑priority
40:WikiProjects
992:Mark viking
946:Mark viking
229:homogeneous
174:—Preceding
112:Mathematics
103:mathematics
59:Mathematics
2850:Categories
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803:Owen Jones
787:Mike Stone
771:Mike Stone
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264:obeying
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923:unsigned
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176:unsigned
158:Untitled
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2646:Wundzer
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907:Maproom
891:Educres
139:on the
30:C-class
2030:, and
1922:, and
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36:scale.
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1957:for
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980:talk
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775:talk
197:no.
184:talk
2777:odd
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131:Mid
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42::
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