arXiv subjects
Publications and source records attributed to Stephanie Ziegenhagen.
A multicomplex, also known as a twisted chain complex, has an associated spectral sequence via a filtration of its total complex. We give explicit formulas for all the differentials in this spectral sequence.
In recent work, Hess and Shipley defined a theory of topological coHochschild homology (coTHH) for coalgebras. In this paper we develop computational tools to study this new theory. In particular, we prove a Hochschild-Kostant-Rosenberg type theorem in the cofree case for differential graded coalgebras. We also develop a coBökstedt spectral sequence to compute the homology of coTHH for coalgebra spectra. We use a coalgebra structure on this spectral sequence to produce several computations.
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 Pirashvili's 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 Kähler differentials.
The first author proved in a previous paper that the n-fold bar construction for commutative algebras can be generalized to E_n-algebras, and that one can calculate E_n-homology with trivial coefficients via this iterated bar construction. We extend this result to E_n-homology and E_n-cohomology of a commutative algebra A with coefficients in a symmetric A-bimodule.
Building on work of Livernet and Richter, we prove that E_n-homology and E_n-cohomology of a commutative algebra with coefficients in a symmetric bimodule can be interpreted as functor homology and cohomology. Furthermore we show that the associated Yoneda algebra is trivial.
The notion of a derived A-infinity algebra arose in the work of Sagave as a natural generalisation of the classical A-infinity algebra, relevant to the case where one works over a commutative ring rather than a field. We develop some of the basic operadic theory of derived A-infinity algebras, building on work of Livernet-Roitzheim-Whitehouse. In particular, we study the coalgebras over the Koszul dual cooperad of the operad dAs, and provide a simple description of these. We study representations of derived A-infinity algebras and explain how these are a two-sided version of Sagave's modules over derived A-infinity algebras. We also give a new explicit example of a derived A-infinity algebra.