nLab
semifree dga

Context

Differential-graded objects

Homological algebra

homological algebra

and

nonabelian homological algebra

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

Contents

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 AA-coring (C,Δ,A)(C,\Delta, A) with a grouplike element gg associate its Amitsur complex with underlying graded module T A(Ω 1A)= n=0 (Ω 1A) AnT_A(\Omega^1 A)=\oplus_{n=0}^\infty (\Omega^1 A)^{\otimes_A n} where Ω 1=kerϵ\Omega^1=ker\,\epsilon and differential linearly extending the formulas da=gaagd a = g a - a g for aAa\in A and

dc=gc+(1) ncg+ i=1 n(1) ic 1c i1Δ(c i)c i+1c 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 A Ac n(kerϵ) Anc=c_1\otimes_A\ldots\otimes_A c_n\in (ker\,\epsilon)^{\otimes_A n};

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

(ag+ω)a:=aag+ada+ωa. (a g +\omega)a' := a a' g + a d a'+\omega a'.

In other words, we want the commutator [g,a]=dω[g,a']=d\omega'. We obtain an AA-bimodule. The coproduct on AgΩ 1AAg\oplus\Omega^1 A is Δ(ag)=agg\Delta(a g)=a g\otimes g and Δ(ω)=gω+ωgdω\Delta(\omega)= 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 11-11 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 00 is of the form C (X)C^\infty(X) for some space XX, which then is the space of objects of the Lie infinity-algebroid. But if it is a more general algebra in degree 00 one can think of a suitably generalized L 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)