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The congruence criterion for power operations in Morava $$E$$-theory. (English) Zbl 1193.55010
In this paper a monad $${\mathbb T}$$ is constructed on the category of graded $$\pi_*E$$-modules, where $$E$$ is the cohomology theory associated to the universal deformations of a height $$n$$ formal group $$G_0$$ over a perfect field $$k$$ of characteristic $$p$$, with the property that the homotopy of a $$K(n)$$-local commutative algebra is an algebra over $${\mathbb T}$$. All $${\mathbb T}$$-algebras are modules for a certain associative ring $$\Gamma$$ (an analogue of the May-Dyer-Lashof algebra), and one of the main results of this paper gives a congruence criterion for determining which $$p$$-torsion free $$\Gamma$$-algebras admit the structure of $${\mathbb T}$$-algebras. This generalizes work of C. Wilkerson [Commun. Algebra 10, 311–328 (1982; Zbl 0492.55004)]. The second main result gives an explanation of this congruence criterion in terms of formal groups.

##### MSC:
 55S25 $$K$$-theory operations and generalized cohomology operations in algebraic topology 55S12 Dyer-Lashof operations 14L05 Formal groups, $$p$$-divisible groups
##### Keywords:
power operation; Morava E-theory
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