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# Homological Properties of Comodules Over MU*(MU) and BP*(BP)

Peter S. Landweber
American Journal of Mathematics
Vol. 98, No. 3 (Autumn, 1976), pp. 591-610
DOI: 10.2307/2373808
Stable URL: http://www.jstor.org/stable/2373808
Page Count: 20
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## Abstract

Let M U denote the category of comodules over MU*(MU) which are finitely presented as MU*-modules. We give a criterion for an MU*-module G to have the property that the functor M ∣→ M$\bigotimes_{MU_\ast} G$ is exact on the category M U. Similarly, let B P denote the category of comodules over BP*(BP) which are finitely presented as BP*-modules; again we have a criterion for a functor M ∣→ M $\bigotimes_{BP_\ast}G$ to be exact on B P. Applications include the theorem of Conner and Floyd that complex cobordism determines K-theory, as well as mostly algebraic proofs of the results of Johnson and Wilson on the homological dimension of BP*(X), including the theorem of Wilson that for a finite complex the natural homomorphism $BP_\ast(X) \rightarrow BP \langle n \rangle_\ast (X)$ is epic in almost all dimensions, which is the starting point for their study of the homological properties of the homology theories in the BP-tower.

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