The classical Hochschild--Kostant--Rosenberg (HKR) theorem computes the Hochschild homology and cohomology of smooth commutative algebras. In this paper, we generalise this result to other kinds of algebraic structures. Our main insight is that produ
cing HKR isomorphisms for other types of algebras is directly related to computing quasi-free resolutions in the category of left modules over an operad; we establish that an HKR-type result follows as soon as this resolution is diagonally pure. As examples we obtain a permutative and a pre-Lie HKR theorem for smooth commutative and smooth brace algebras, respectively. We also prove an HKR theorem for operads obtained from a filtered distributive law, which recovers, in particular, all the aspects of the classical HKR theorem. Finally, we show that this property is Koszul dual to the operadic PBW property defined by V. Dotsenko and the second author (1804.06485).
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 producin
g 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 $p$.
Aguiar and Mahajan introduced a cohomology theory for the twisted coalgebras of Joyal, with particular interest in the computation of their second cohomology group, which gives rise to their deformations. We use the Koszul duality theory between twis
ted algebras and coalgebras on the twisted coalgebra that gives rise to their cohomology theory to give a new alternative description of it which, in particular, allows for its effective computation. We compute it completely in various examples, including those proposed by Aguiar and Mahajan, and obtain structural results: in particular, we study its multiplicative structure and provide a Kunneth formula.
We define derived Poincare--Birkhoff--Witt maps of dg operads or derived PBW maps, for short, which extend the definition of PBW maps between operads of V.~Dotsenko and the second author in 1804.06485, with the purpose of studying the universal envel
oping algebra of dg Lie algebras as a functor on the homotopy category. Our main result shows that the map from the homotopy Lie operad to the homotopy associative operad is derived PBW, which gives us an amenable description of the homology of the universal envelope of an $L_infty$-algebra in the sense of Lada--Markl. We deduce from this several known results involving universal envelopes of $L_infty$-algebras of V. Baranovsky and J. Moreno-Fernandez, and extend D. Quillens classical quasi-isomorphism $mathcal C longrightarrow BU$ from dg Lie algebras to $L_infty$-algebras; this confirms a conjecture of J. Moreno-Fernandez.
We show how to compute the Tamarkin-Tsygan calculus of an associative algebra by providing, for a given cofibrant replacement of it, a `small $mathsf{Calc}_infty$-model of its calculus, which we make somewhat explicit at the level of $mathsf{Calc}$-a
lgebras. To do this, we prove that the operad $mathsf{Calc}$ is inhomogeneous Koszul; to our best knowledge, this result is new. We illustrate our technique by carrying out some computations for two monomial associative algebra using the cofibrant replacement obtained by the author in 1804.01435.
Using combinatorics of chains going back to works of Anick, Green, Happel and Zacharia, we give, for any monomial algebra $A$, an explicit description of its minimal model. This also provides us with formulas for a canonical $A_infty$-structure on th
e Ext-algebra of the trivial $A$-module. We do this by exploiting the combinatorics of chains going back to works of Anick, Green, Happel and Zacharia, and the algebraic discrete Morse theory of Jollenbeck, Welker and Skoldberg. We then show how this result can be used to obtain models for algebras with a chosen Grobner basis, and briefly outline how to compute some classical homological invariants with it.
