Is the complex cobordism spectrum, $MU$, a finite spectrum?
If yes, what other examples of finite spectrums there are? Is the Eilenberg-MacLane spectrum finite?
What about the connective $K$-theory $ku$?
2026-03-25 07:48:41.1774424921
Is the complex cobordism spectrum, $MU$, a finite spectrum?
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No, none of these are finite spectra. An easy way to tell is by taking homology: any finite spectrum has nontrivial homology in only finitely many degrees, but $MU$, $H\mathbb{Z}$, and $ku$ all have nontrivial homology in infinitely many degrees.
Typically, finite spectra arise not as particularly natural cohomology theories, but as suspension spectra of spaces. Indeed, a spectrum is finite iff it is a shift of the suspension spectrum of a finite CW-complex.