# nLab semifree dga

### Context

and

rational homotopy theory

## Rational spaces

#### Homological algebra

homological algebra

and

nonabelian homological algebra

diagram chasing

# Definition

A differential graded algebra is semifree (or semi-free) if the underlying graded algebra is free: if after forgetting the differential, it is isomorphic as a graded algebra to a (polynomial) tensor algebra of some (super)graded vector space.

A differential graded-commutative algebra is semifree (or semi-free) if the underlying graded-commutative algebra is free: if after forgetting the differential, it is isomorphic as a graded-commutative algebra to a Grassmann algebra of some graded vector space .

# Roiter’s theorem

Roiter’s theorem

• A. V. Roiter, Matrix problems and representations of BOCS’s; in Lec. Notes. Math. 831, 288–324 (1980)

says: semi-free differential graded algebras are in bijective correspondence with corings with a grouplike element:

to an $A$-coring $\left(C,\Delta ,A\right)$ with a grouplike element $g$ associate its Amitsur complex? with underlying graded module ${T}_{A}\left({\Omega }^{1}A\right)={\oplus }_{n=0}^{\infty }\left({\Omega }^{1}A{\right)}^{{\otimes }_{A}n}$ where ${\Omega }^{1}=\mathrm{ker}\phantom{\rule{thinmathspace}{0ex}}ϵ$ and differential linearly extending the formulas $da=ga-ag$ for $a\in A$ and

$dc=g\otimes c+\left(-1{\right)}^{n}c\otimes g+\sum _{i=1}^{n}\left(-1{\right)}^{i}{c}_{1}\otimes \dots \otimes {c}_{i-1}\otimes \Delta \left({c}_{i}\right)\otimes {c}_{i+1}\otimes \dots \otimes {c}_{n}$d c = g\otimes c + (-1)^n c\otimes g +\sum_{i=1}^n (-1)^i c_1\otimes\ldots\otimes c_{i-1}\otimes\Delta(c_i)\otimes c_{i+1}\otimes\ldots\otimes c_n

for $c={c}_{1}{\otimes }_{A}\dots {\otimes }_{A}{c}_{n}\in \left(\mathrm{ker}\phantom{\rule{thinmathspace}{0ex}}ϵ{\right)}^{{\otimes }_{A}n}$;

conversely, to a semi-free dga ${\Omega }^{•}A$ one associates the $A$-coring $Ag\oplus {\Omega }^{1}A$ where $g$ isa new group-like indeterminate; this is by definition a direct sum of left $A$-modules with a right $A$-module structure given by

$\left(ag+\omega \right)a\prime :=aa\prime g+ada\prime +\omega a\prime .$(a g +\omega)a' := a a' g + a d a'+\omega a'.

In other words, we want the commutator $\left[g,a\prime \right]=d\omega \prime$. We obtain an $A$-bimodule. The coproduct on $\mathrm{Ag}\oplus {\Omega }^{1}A$ is $\Delta \left(ag\right)=ag\otimes g$ and $\Delta \left(\omega \right)=g\otimes \omega +\omega \otimes g-d\omega$. The two operations are mutual inverses (see lectures by Brzezinski or the arxiv version math/0608170).

Moreover flat connections for a semi-free dga are in $1$-$1$ correspondence with the comodules over the corresponding coring with a group-like element.

# Relation to Lie $\infty$-algebroids

One can identify semifree differential graded algebras in non-negative degree with Chevalley–Eilenberg algebras of (degreewise finite dimensional) Lie infinity-algebroids

At least when the algebra in degree $0$ is of the form ${C}^{\infty }\left(X\right)$ for some space $X$, which then is the space of objects of the Lie infinity-algebroid. But if it is a more general algebra in degree $0$ one can think of a suitably generalized ${L}_{\infty }$-algebroid, for instance with a noncommutative space of objects. This generalizes the step from Lie algebroids to Lie–Rinehart pairs.

# Terminology

Sometimes semi-free DGAs are called quasi-free, but this is in collision with the terminology about formal smoothness of noncommutative algebras, i.e. quasi-free algebras in the sense of Cuntz and Quillen (and with extensions to homological smootheness of dg-algebras by Kontsevich).

Revised on March 25, 2011 00:35:45 by Urs Schreiber (87.212.203.135)