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nLab: semifree dga

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    • CommentRowNumber1.
    • CommentAuthorJulianK
    • CommentTimeMar 6th 2025
    I think the article on semifree dgas doesn't disambiguate two concepts of the same name.

    It starts with one saying that a semifree dga is one that is the tensor algebra of a graded vector space (possibly it should be generalised to graded quiver here). Such an algebra can have all possible degrees, but the degree zero part is a tensor algebra.

    But then it continues mentioning Roiter's theorem. This in contrast is about dgas whose underlying algebra is isomorphic to the tensor algebra of a bimodule (in degree 1) over its degree zero part. Evidently such a dga is concentrated in nonnegative degrees, but the degree 0 part can have an arbitrary algebra.
    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMar 6th 2025

    The tensor algebra in question is free over the given ground ring — or “ground algebra” if you wish — which appears in degree 0. I have now adjusted the wording of the first paragraph to make this clearer (announced there).

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