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induces an isomorphism of [[profinite group|prorings]]


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Here, the induced map has as [[Domain of a function|domain]] the [[completion (ring theory)|completion]] of the ''G''-equivariant K-theory of ''X'' with respect to ''I'', where ''I'' denotes the [[augmentation ideal]] of the [[representation ring]] of ''G''.


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In the special case of ''X'' a point, the theorem specializes to give an isomorphism <math>K^*(BG)\cong R(G)\hat{_I}</math> between the K-theory of the [[classifying space]] of ''G'' and the completion of the representation ring.
 
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The theorem can be interpreted as giving a comparison between the geometrical process of completing a ''G''-space by making the [[group action|action]] free and the algebraic process of completing with respect to an ideal.
<ref name=Greenlees>{{cite conference
| author = Greenlees, J.P.C.
| year = 1996
| title = An introduction to equivariant K-theory.
| conference = Equivariant homotopy and cohomology theory
| booktitle = CBMS Regional Conference Series
| volume = 91
| pages = 143–152
| publisher = Published for the Conference Board of the Mathematical Sciences,
    Washington, DC
| url =
| conferenceurl =
}}</ref>


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The theorem was first proved for [[finite groups]] by [[Michael Atiyah]] in 1961,
 
<ref name=Atiyah1961>{{cite journal
  <li>[http://crossgate.site90.com/forum.php?mod=viewthread&tid=91897 http://crossgate.site90.com/forum.php?mod=viewthread&tid=91897]</li>
| last = Atiyah|first= M.F.|authorlink=Michael Atiyah
 
| year = 1961
  <li>[http://www.nuright.com/bbs/forum.php?mod=viewthread&tid=1086995 http://www.nuright.com/bbs/forum.php?mod=viewthread&tid=1086995]</li>
| title = Characters and cohomology of finite groups
 
| volume = 9
  <li>[http://www.expo-2015milan.com/activity/p/76717/ http://www.expo-2015milan.com/activity/p/76717/]</li>
| issue = 1
 
| pages = 23–64
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| url = http://www.springerlink.com/index/3228267279648852.pdf
 
| accessdate = 2008-06-19
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and a proof of the general case was published by Atiyah together with [[Graeme Segal]] in 1969.
</ul>
<ref name=Atiyah1969>{{cite journal
| last1 = Atiyah|first1= M.F.|author1-link=Michael Atiyah
| last2 = Segal|first2= G.B.|author2-link=Graeme Segal
| year = 1969
| title = Equivariant K-theory and completion
| journal = [[Journal of Differential Geometry]]
| volume = 3
| pages = 1–18
| url = http://intlpress.com/JDG/archive/pdf/1969/3-1&2-1.pdf
| accessdate = 2008-06-19
}}</ref>
Different proofs have since appeared generalizing the theorem to completion with respect to families of subgroups.
<ref name=Jackowski1985>{{cite journal
| last = Jackowski | first= S.
| year = 1985
| title = Families of subgroups and completion
| journal = J. Pure Appl. Algebra
| volume = 37
| issue = 2
| pages = 167–179
| doi = 10.1016/0022-4049(85)90094-5
}}</ref>
<ref name=Adams1988>{{cite journal
| last1 = Adams | first1= J.F.
| last2 = Haeberly|first2= J.P.|last3= Jackowski|first3= S.; |last4=May|first4= J.P.|author4-link=J. Peter May
| year = 1988
| title = A generalization of the Atiyah-Segal Completion Theorem
| journal = Topology
| volume = 27
| issue = 1
| pages = 1–6
| doi = 10.1016/0040-9383(88)90002-X
}}</ref>


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*[[Segal conjecture]]


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{{DEFAULTSORT:Atiyah-Segal completion theorem}}
[[Category:K-theory]]
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Revision as of 20:34, 20 January 2014

The Atiyah–Segal completion theorem is a theorem in mathematics about equivariant K-theory in homotopy theory. Let G be a compact Lie group and let X be a G-CW-complex. The theorem then states that the projection map

π:X×EGX

induces an isomorphism of prorings

π*:KG*(X)I^KG*(X×EG).

Here, the induced map has as domain the completion of the G-equivariant K-theory of X with respect to I, where I denotes the augmentation ideal of the representation ring of G.

In the special case of X a point, the theorem specializes to give an isomorphism K*(BG)R(G)I^ between the K-theory of the classifying space of G and the completion of the representation ring.

The theorem can be interpreted as giving a comparison between the geometrical process of completing a G-space by making the action free and the algebraic process of completing with respect to an ideal. [1]

The theorem was first proved for finite groups by Michael Atiyah in 1961, [2] and a proof of the general case was published by Atiyah together with Graeme Segal in 1969. [3] Different proofs have since appeared generalizing the theorem to completion with respect to families of subgroups. [4] [5]

See also

References

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    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang