Let E be a homotopy commutative ring spectrum, and suppose the ring of cooperations E#E is flat over E# . We wish to address the following question: given a commutative E# -algebra A in E#E-comodules, is there an E# -ring spectrum X with E#X = A as comodule algebras? We will formulate this as a moduli problem, and give a way -- suggested by work of Dwyer, Kan, and Stover -- of dissecting the resulting moduli space as a tower with layers governed by appropriate Andre-Quillen cohomology groups. A special case is A = E#E itself. The final section applies this to discuss the Lubin-Tate or Morava spectra En .

Notion of an oriented cohomology pretheory on algebraic varieties is introduced and a Riemann-Roch theorem for ring morphisms between oriented pretheories is proved. An explicit formula for the Todd genus related to a ring morphism is given.

In this document we make good on all the assertions we made in the previous paper “Moduli spaces of commutative ring spectra ” [20] wherein we laid out a theory a moduli spaces and problems for the existence and uniqueness of E∞-ring spectra. In that paper, we discussed the the Hopkins-Miller theorem on the Lubin-Tate or Morava spectra En; in particular, we showed how to prove that the moduli space of all E ∞ ring spectra X so that (En)∗X ∼ = (En)∗En as commutative (En) ∗ algebras had the homotopy type of BG, where G was an appropriate variant of the Morava stabilizer group. This is but one point of view on these results, and the reader should also consult [3], [38], and [41], among others. A point worth reiterating is that the moduli problems here begin with algebra: we have a homology theory E ∗ and a commutative ring A in E∗E comodules and we wish to discuss the homotopy type of the space T M(A) of all E∞-ring spectra so that E∗X ∼ = A. We do not, a priori, assume that T M(A) is non-empty, or even that there is a spectrum X so that E∗X ∼ = A as comodules.

An algebraic version of a theorem due to Quillen is proved. More precisely, for a ground field k we consider the motivic stable homotopy category SH(k) of P 1-spectra, equipped with the symmetric monoidal structure described in [PPR1]. The algebraic cobordism P 1-spectrum MGL is considered as a commutative monoid equipped with a canonical orientation th MGL ∈ MGL 2,1 (Th(O(−1))). For a commutative monoid E in the category SH(k) it is proved that assignment ϕ ↦ → ϕ(th MGL) identifies the set of monoid homomorphisms ϕ: MGL → E in the motivic stable homotopy category SH(k) with the set of all orientations of E. The result was stated originally in a slightly different form by G. Vezzosi in [Ve]. 1 Oriented commutative ring spectra We refer to [PPR1, Appendix] for the basic terminology, notation, constructions, definitions, results. For the convenience of the reader we recall the basic definitions. Let S be a Noetherian scheme of finite Krull dimension. One may think of S being the spectrum of

Abstract. We prove the Arnold conjecture for closed symplectic manifolds with π2(M) = 0 and cat M = dim M. Furthermore, we prove an analog of the Lusternik– Schnirelmann theorem for functions with “generalized hyperbolicity ” property.

We outline an obstruction theory for deciding when a homotopy commutative and associative ring spectrum is actually an E∞ ring spectrum. The obstruction groups are André-Quillen cohomology groups of an algebra over an E∞ operad. The same cohomology theory is part of a spectral sequence for computing the homotopy type of mapping spaces between E∞ ring spectrum. The obstruction theory arises out of techniques of Dwyer, Kan, and Stover, and the main application here is to prove an analog of a theorem of Haynes Miller and the second author: the Lubin-Tate spectra En are E∞ and the space of E∞ self-maps has weakly contractible components.

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Let C be a closed model category. Following work of Dwyer, Kan, and Stover, we develop a technique for building simplicial resolutions of objects in C which are free, or homotopically free, in a precise sense. Specifically, we specify in advance a set A of small, co brant, co-H objects in C and the simplicial resolutions X will have the property that at each level n, the object Xn is homotopy equivalent to a coproduct of objects of A. These Stover-type resolutions are modeled on the free resolutions developed by Quillen, and are meant to be a replacement for projective resolutions in this context. In the end we develop some computational tools for spectra and structured ring spectra.

Quillen’s algebraic K-theory is reconstructed via Voevodsky’s algebraic cobordism. More precisely, for a ground field k the algebraic cobordism P 1-spectrum MGL of Voevodsky is considered as a commutative P 1-ring spectrum. Setting MGL i = ⊕p−2q=i MGL p,q we regard the bigraded theory MGL p,q as just a graded theory. There is a unique ring morphism φ: MGL 0 (k) → Z which sends the class [X]MGL of a smooth projective k-variety X to the Euler characteristic χ(X,OX) of the structure sheaf OX. Our main result states that there is a canonical grade preserving isomorphism of ring cohomology theories

We examine the foundations of simplicial algebras in spectra over a simplicial operad. We are led to simplicial operads and simplicial algebras over simplicial operads because certain operads which are notoriously hard to work with { mainly the E1 operad { can be simplicially resolved by simpler pieces. Our main goals are to a build spectral sequence for computing spaces of maps between structured ring spectra, and to develop a Dwyer-Kan-Stover style obstruction theory for deciding when a spectrum actually can be a structured ring spectrum. In this paper we work out some of the foundations of the homotopy theory of simplicial ring spectra over a simplicial operad. This is not a gratuitous act of generalization. Simplicial objects in any category are a standard mechanism for building the resolutions necessary for computations; this is how simplicial spectra arise. The simplicial operads arise as an answer to an immediate practical problem. If T is an E1 operad over the linear isometrie...