Morava K-theory
{{Short description|Cohomology theory}}
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Morava K-theory is one of a collection of cohomology theories introduced in algebraic topology by Jack Morava in unpublished preprints in the early 1970s.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 in the sense of homotopy theory. {{harvtxt|Johnson|Wilson|1975}} published the first account of the theories.
Details
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
:Fp[vn,vn−1]
where vn has degree 2(pn − 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 isomorphisms for arbitrary pairs of spaces: that is, for X and Y CW complexes, we have
:
- They are "fields" in the category of ring spectra. In other words every module spectrum over K(n) is free, i.e. a wedge of suspensions of K(n).
- They are complex oriented (at least after being periodified by taking the wedge sum of (pn − 1) shifted copies), and the formal group they define has height n.
- Every finite p-local spectrum X has the property that K(n)∗(X) = 0 if and only if n is less than a certain number N, called the type of the spectrum X. By a theorem of Devinatz–Hopkins–Smith, every thick subcategory of the category of finite p-local spectra is the subcategory of type-n spectra for some n.
See also
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= 55−75|doi=10.1007/BF01214408 }}
- Hovey-Strickland, "[https://web.archive.org/web/20210118104753/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 Studies|volume=128|year=1992}}
- {{citation|mr=1133896
|last=Würgler|first= Urs
|chapter=Morava K-theories: a survey|title= Algebraic topology Poznan 1989|pages= 111–138
|series=Lecture Notes in Math.|volume= 1474|publisher= Springer|location= Berlin|year= 1991
|doi=10.1007/BFb0084741|isbn=978-3-540-54098-4}}