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quasi-free algebra

Contents

Idea

An associative algebra (over a field kk) is quasi-free if dually regarded as a noncommutative scheme it is formally smooth.

Definition

Given an associative algebra AA let ΩA\Omega A be its universal differential envelope.

An associative unital kk-algebra AA is quasi-free (or formally smooth) if one of the following equivalent conditions is satisfied

  • Given an extension 0NEqB00\to N\to E\stackrel{q}\to B\to 0 of algebras where the ideal NN is nilpotent and a:ABa:A\to B an algebra map. Then there exists a homomorphism a:AEa' : A \to E such that qa=aq \circ a' = a.

  • AA has cohomological dimension 1\leq 1 with respect to Hochschild cohomology;

  • Ω 1A\Omega^1 A is a projective AA-bimodule;

  • the universal Hochschild 2-cocycle c:AAΩ 2Ac : A\otimes A\to\Omega^2 A, c:abdadbc : a\otimes b\mapsto d a d b is a coboundary, i.e. c=bϕc = b\phi for some ϕ:AΩ 2A\phi:A\to\Omega^2 A satisfying the cocycle condition ϕ(ab)=aϕ(b)+ϕ(a)bdadb\phi(a b)=a\phi(b)+\phi(a)b-d a d b;

  • there exists a “right connection” :Ω 1AΩ 2A\nabla:\Omega^1 A\to \Omega^2 A i.e. a kk-linear map satisfying (aw)=a(w)\nabla(a w)=a\nabla(w) and (wa)=(w)a+wda\nabla (w a) = \nabla(w)a+w d a where wΩ 1Aw\in\Omega^1 A and aAa\in A.

This is due to (CuntzQuillen).

Properties

For AA an associative algebra, the object SpecA[Alg k,Set]Spec A \in [Alg_k, Set] is formally smooth with respect to the standard infinitesimal cohesive structure over non-commutative algebras (see there for details) precisely if it is quasi-free.

Notice that the characterization via nilpotent extensions is similar to the definition of commutative formally smooth algebras as in EGAIV4 17.1.1. However most commutative formally smooth algebras are not formally smooth in the associative noncommutative sense.

Examples

Path algebras of quivers and free algebras are some of the (few classes of) examples.

References

  • J. Cuntz and Daniel Quillen, Algebra extensions and nonsingularity , J.Amer. Math. Soc. 8 (1995), 251-289.
Revised on April 12, 2011 15:46:29 by Urs Schreiber (131.211.238.95)