We discuss the Adams Spectral Sequence for R-modules based on commutative localized regular quotient ring spectra over a commutative S -algebra R in the sense of Elmendorf, Kriz, Mandell, May and Strickland. The formulation of this spectral sequence is similar to the classical case and the calculation of its E 2 -term involves the cohomology of certain 'brave new Hopf algebroids' E R * E . In working out the details we resurrect Adams' original approach to Universal Coefficient Spectral Sequences for modules over an R ring spectrum.We show that the Adams Spectral Sequence for S R based on a commutative localized regular quotient R ring spectrum E = R/I[X −1 ] converges to the homotopy of the E -nilpotent completionWe also show that when the generating regular sequence of I * is finite,the Bousfield localization of S R with respect to E -theory. The spectral sequence here collapses at its E 2 -term but it does not have a vanishing line because of the presence of polynomial generators of positive cohomological degree. Thus only one of Bousfield's two standard convergence criteria applies here even though we have this equivalence. The details involve the construction of an I -adic towerwhose homotopy limit is L R E S R . We describe some examples for the motivating case R = M U .

This paper builds a general framework in which to study cohomology theories of strongly homotopy algebras, namely A 1 -, C 1 -and L 1 -algebras. This framework is based on noncommutative geometry as expounded by Connes and Kontsevich. The developed machinery is then used to establish a general form of Hodge decomposition of Hochschild and cyclic cohomology of C 1 -algebras. This generalises and puts in a conceptual framework previous work by Loday and Gerstenhaber-Schack. 13D03, 13D10; 46L87

We determine the $$L_\infty $$ L ∞ -algebra that controls deformations of a relative Rota–Baxter Lie algebra and show that it is an extension of the dg Lie algebra controlling deformations of the underlying $$\mathsf {Lie}\mathsf {Rep}$$ Lie Rep  pair by the dg Lie algebra controlling deformations of the relative Rota–Baxter operator. Consequently, we define the cohomology of relative Rota–Baxter Lie algebras and relate it to their infinitesimal deformations. A large class of relative Rota–Baxter Lie algebras is obtained from triangular Lie bialgebras and we construct a map between the corresponding deformation complexes. Next, the notion of a homotopy relative Rota–Baxter Lie algebra is introduced. We show that a class of homotopy relative Rota–Baxter Lie algebras is intimately related to pre-Lie$$_\infty $$ ∞ -algebras.

We give a definition of a derivation of an A ∞ ring spectrum and relate this notion to topological Hochschild cohomology. Strict multiplicative structure is introduced into Postnikov towers and generalized Adams towers of A ∞-ring spectra. An obstruction theory for lifting multiplicative maps is constructed. The developed techniques are then applied to show that a broad class of complex-oriented spectra admit structures of M U-algebras where M U is the complex cobordism spectrum. Various computations of topological derivations and topological Hochschild cohomology are made.

We construct an explicit minimal model for an algebra over the cobar-construction of a differential graded operad. The structure maps of this minimal model are expressed in terms of sums over decorated trees. We introduce the appropriate notion of a homotopy equivalence of operadic algebras and show that our minimal model is homotopy equivalent to the original algebra. All this generalizes and gives a conceptual explanation of well-known results for A ∞algebras. Furthermore, we show that these results carry over to the case of algebras over modular operads; the sums over trees get replaced by sums over general Feynman graphs. As a by-product of our work we prove gauge-independence of Kontsevich's 'dual construction' producing graph cohomology classes from contractible differential graded Frobenius algebras.