We derive anétale descent formula for topological Hochschild homology and prove a HKR theorem for smooth S-algebras.Key words: spectra with additional structure,étale descent, smooth S-algebras MCS: 55P42 IntroductionOne of the main results for computing the Hochschild homology of smooth discrete algebras is the Hochschild-Kostant-Rosenberg (HKR) theorem (e.g. see Chapter 3 of [9]), which states that for a smooth algebra k → A, the Hochschild homology coincides with differential forms:In fact this result is often used not only to compute the Hochschild homology, but also the other way around: in order to generalize some results to non-smooth (or even non-commutative) algebras one replaces the differential forms by Hochschild homology. Other applications include a comparison theorem between cyclic and de Rham homology theories.One of our objectives is to develop a topological analogue of the HKR theorem in the framework provided in [5], or more precisely, in the category of commutative of S-algebras. Recall that S-algebras are equivalent to the more traditional notion of E ∞ -ring spectra, and are a generalization to stable homotopy theory of the algebraic notion of a commutative ring. In this context, the topological André-Quillen homology of a commutative S-algebra A is the natural replacement of the module of differentials Ω 1 A|k , as it is evident from the definition of TAQ. The definitions of TAQ, as well as THH, in our context are recalled in Section 2, and we refer to [1] and Chapter IX of [5] for detailed discussion of these notions. Noting that the orbits of the n ′ th smash powers of the suspension of T AQ, (ΣT AQ(A)) ∧An /Σ n , are analogous to symmetric powers in the graded context, and therefore correspond to taking exterior powers (and thus are the analogues of the higher order modules of differentials), we state our main theorem. Theorem 1.1. (HKR) For a connective smooth S-algebra A, the natural (derivative) map T HH(A) → ΣT AQ(A) has a section in the category of A-modules which induces an equivalence of A-algebras: P A ΣT AQ(A) ≃ → T HH(A),where P is the symmetric algebra triple.

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Abstract. Using functional equations, we define functors that generalize standard examples from calculus of one variable. Examples of such functors are discussed, and their Taylor towers are computed. We also show that these functors factor through objects enriched over the homology of little n-cubes operads and discuss the relationship between functors defined via functional equations and operads. In addition, we compute the differentials of the forgetful functor from the category of n-Poisson algebras in terms of the homology of configuration spaces.