### Continuous Morava K-Theory and the Geometry of the In-Adic Tower.

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We continue our programme of extending the Roman-Rota umbral calculus to the setting of delta operators over a graded ring ${E}_{*}$ with a view to applications in algebraic topology and the theory of formal group laws. We concentrate on the situation where ${E}_{*}$ is free of additive torsion, in which context the central issues are number- theoretic questions of divisibility. We study polynomial algebras which admit the action of two delta operators linked by an invertible power series, and make related constructions...

We examine the behaviour of a complex oriented cohomology theory ${G}^{*}(-)$ on ${D}_{p}\left(X\right)$, the ${C}_{p}$-extended power of a space $X$, seeking a description of ${G}^{*}({D}_{p}\left(X\right))$ in terms of the cohomology ${G}^{*}\left(X\right)$. We give descriptions for the particular cases of Morava $K$-theory $K\left(n\right)$ for any space $X$ and for complex cobordism $MU$, the Brown-Peterson theories BP and any Landweber exact theory for a wide class of spaces.

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