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For certain motivic spectra, we construct a square of spectral sequences relating the effective slice spectral sequence and the motivic Adams spectral sequence. We show the square can be constructed for connective algebraic K-theory, motivic Morava K-theory, and truncated motivic Brown-Peterson spectra. In these cases, we show that the $mathbb{R}$-motivic effective slice spectral sequence is completely determined by the $rho$-Bockstein spectral sequence. Using results of Heard, we also obtain applications to the Hill-Hopkins-Ravenel slice spectral sequences for connective Real K-theory, Real Morava K-theory, and truncated Real Brown-Peterson spectra.
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We completely compute the slice spectral sequence of the $C_4$-spectrum $BP^{((C_4))}langle 2 rangle$. After periodization and $K(4)$-localization, this spectrum is equivalent to a height-4 Lubin-Tate theory $E_4$ with $C_4$-action induced from the Goerss-Hopkins-Miller theorem. In particular, our computation shows that $E_4^{hC_{12}}$ is 384-periodic.
Operadic tangent cohomology generalizes the existing theories of Harrison cohomology, Chevalley--Eilenberg cohomology and Hochschild cohomology. These are usually non-trivial to compute. We complement the existing computational techniques by producing a spectral sequence that converges to the operadic tangent cohomology of a fixed algebra. Our main technical tool is that of filtrations arising from towers of cofibrations of algebras, which play the same role cell attaching maps and skeletal filtrations do for topological spaces. As an application, we consider the rational Adams--Hilton construction on topological spaces, where our spectral sequence gives rise to a seemingly new and completely algebraic description of the Serre spectral sequence, which we also show is multiplicative and converges to the Chas--Sullivan loop product. Finally, we consider relative Sullivan--de Rham models of a fibration $p$, where our spectral sequence converges to the rational homotopy groups of the identity component of the space of self-fiber homotopy equivalences of $p$.
In this note, we use Curtiss algorithm and the Lambda algebra to compute the algebraic Atiyah-Hirzebruch spectral sequence of the suspension spectrum of $mathbb{R}P^infty$ with the aid of a computer, which gives us its Adams $E_2$-page in the range of $t<72$. We also compute the transfer map on the Adams $E_2$-pages. These data are used in our computations of the stable homotopy groups of $mathbb{R}P^infty$ in [6] and of the stable homotopy groups of spheres in [7].
The aim of this paper is to construct and examine three candidates for local-to-global spectral sequences for the cohomology of diagrams of algebras with directed indexing. In each case, the $E^2$ -terms can be viewed as a type of local cohomology relative to a map or an object in the diagram.
In the world of chain complexes E_n-algebras are the analogues of based n-fold loop spaces in the category of topological spaces. Fresse showed that operadic E_n-homology of an E_n-algebra computes the homology of an n-fold algebraic delooping. The aim of this paper is to construct two spectral sequences for calculating these homology groups and to treat some concrete classes of examples such as Hochschild cochains, graded polynomial algebras and chains on iterated loop spaces. In characteristic zero we gain an identification of the summands in Pirashvilis Hodge decomposition of higher order Hochschild homology in terms of derived functors of indecomposables of Gerstenhaber algebras and as the homology of exterior and symmetric powers of derived Kahler differentials.
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