On the indecomposable elements of the bar construction

Abstract

An explicit formula for a canonical splitting s : Q B ( E ⋅ ) → B ( E ⋅ ) s:Q\mathcal {B}({\mathcal {E}^ \cdot }) \to \mathcal {B}({\mathcal {E}^ \cdot }) of the projection B ( E ⋅ ) → Q B ( E ⋅ ) \mathcal {B}({\mathcal {E}^ \cdot }) \to Q\mathcal {B}({\mathcal {E}^ \cdot }) of the bar construction on a commutative d.g. algebra E ⋅ {\mathcal {E}^\cdot } onto its indecomposables is given. We prove that s s induces a d.g. algebra isomorphism Λ ( Q B ( E ⋅ ) ) → B ( E ⋅ ) \Lambda (Q\mathcal {B}({\mathcal {E}^\cdot })) \to \mathcal {B}({\mathcal {E}^\cdot }) and that H ( Q B ( E ⋅ ) ) H(Q\mathcal {B}({\mathcal {E}^\cdot })) is isomorphic with Q H ( B ( E ⋅ ) ) QH(\mathcal {B}({\mathcal {E}^\cdot })) .