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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
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