Marcinkiewicz interpolation theorem: Difference between revisions

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In [[stable homotopy theory]], a branch of [[mathematics]], '''Morava K-theory''' is one of a collection of [[cohomology theories]] introduced in [[algebraic topology]] by [[Jack Morava]] in unpublished preprints in the early 1970s. For every [[prime number]] ''p'' (which is suppressed in the notation), it consists of theories ''K''(''n'') for each nonnegative integer ''n'', each a [[ring spectrum (homotopy theory)|ring spectrum]] in the sense of [[homotopy theory]].   {{harvtxt|Johnson|Wilson|1975}} published the first account of the theories.
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The theory ''K''(0) agrees with [[singular homology]] with rational coefficients, whereas ''K''(1) is a summand of mod-''p'' [[complex K-theory]]. The theory ''K''(''n'') has coefficient ring
 
:'''F'''<sub>''p''</sub>[''v''<sub>''n''</sub>,''v''<sub>''n''</sub><sup>&minus;1</sup>]
 
where ''v''<sub>''n''</sub> has degree 2(''p''<sup>''n''</sup>&nbsp;&minus;&nbsp;1). In particular, Morava K-theory is periodic with this period, in much the same way that complex K-theory has period 2.
 
These theories have several remarkable properties.
* They have [[Künneth theorem|Künneth isomorphisms]] for arbitrary pairs of spaces: that is, for ''X'' and ''Y'' CW complexes, we have
:<math>K(n)_*(X \times Y) \cong K(n)_*(X) \otimes_{K(n)_*} K(n)_*(Y).</math>
* They are "fields" in the [[Category (mathematics)|category]] of [[ring spectrum (homotopy theory)|ring spectra]]. In other words every [[module spectrum]] over ''K''(''n'') is free, i.e. a [[wedge sum|wedge]] of [[suspension (topology)|suspensions]] of ''K''(''n'').
* They are [[complex cobordism|complex oriented]] (at least after being periodified by taking the wedge sum of (''p''<sup>''n''</sup>&nbsp;&minus;&nbsp;1) shifted copies), and the [[formal group]] they define has [[Formal_group#The_height_of_a_formal_group_law|height]] ''n''.
* Every finite ''p''-local [[spectrum (homotopy theory)|spectrum]] ''X'' has the property that ''K''(''n'')<sub>∗</sub>(''X'') = 0 if and only if ''n'' is less than a certain number ''N'', called the [[Type (homotopy theory)|type]] of the spectrum ''X''. By a theorem of Devinatz&ndash;[[Michael J. Hopkins|Hopkins]]&ndash;Smith, every [[thick subcategory]] of the [[category (mathematics)|category]] of finite ''p''-local spectra is the subcategory of type-''n'' spectra for some ''n''.
 
==References==
*{{citation|mr=0377856
|last=Johnson|first= David Copeland|last2= Wilson|first2= W. Stephen
|title=BP operations and Morava's extraordinary K-theories.
|journal=Math. Z. |volume=144 |year=1975|issue= 1|pages= 55&minus;75|doi=10.1007/BF01214408 }}
*Hovey-Strickland, "[http://hopf.math.purdue.edu/cgi-bin/generate?/Hovey-Strickland/kn Morava K-theory and localisation]"
*{{citation|mr=1192553
|last=Ravenel|first=Douglas C.
|title=Nilpotence and periodicity in stable homotopy theory
|publisher=Princeton University Press|series=Annals of Mathematics Sudies|volume=128|year=1992}}
*{{citation|mr=1133896
|last=Würgler|first= Urs
|chapter=Morava K-theories: a survey|title= Algebraic topology Poznan 1989|pages= 111&nbsp;138
|series=Lecture Notes in Math.|volume= 1474|publisher= Springer|publication-place= Berlin|year= 1991
|doi=10.1007/BFb0084741|isbn=978-3-540-54098-4}}
 
[[Category:Algebraic topology]]
[[Category:Cohomology theories]]

Latest revision as of 05:30, 9 December 2014

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