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    • CommentRowNumber1.
    • CommentAuthorDmitri Pavlov
    • CommentTimeOct 18th 2021

    Created:

    Definition

    An entire functional calculus algebra is a product-preserving functor

    CartHoloSet,CartHolo \to Set,

    where CartHoloCartHolo is the category of finite-dimensional complex vector spaces and holomorphic maps.

    This is in complete analogy to C^∞-rings, and EFC-algebras are applicable in similar contexts.

    Properties

    The category of globally finitely presented Stein spaces is contravariantly equivalent to the category of finitely presented EFC-algebras. The equivalence functor sends a Stein space to its EFC-algebra of global sections.

    The category of Stein spaces of finite embedding dimension is contravariantly equivalent to the category of finitely generated EFC-algebras. The equivalence functor sends a Stein space to its EFC-algebra of global sections.

    These statements can thus be rightfully known as Stein duality.

    Related concepts

    References

    • Alexei~Yu.~Pirkovskii, Holomorphically finitely generated algebras. Journal of Noncommutative Geometry 9 (2015), 215–264. arXiv:1304.1991, doi:10.4171/JNCG/192.

    • J.~P.~Pridham, A differential graded model for derived analytic geometry. Advances in Mathematics 360 (2020), 106922. arXiv:1805.08538v1, doi:10.1016/j.aim.2019.106922.

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeOct 18th 2021
    • (edited Oct 18th 2021)

    Interesting, wasn’t aware of this. But could you give me a feeling for the scope of “finitely presented Stein spaces”?

    \,

    I have added more hyperlinks, also a hyperlink back from Stein space and from C-infinity ring. It would be good to state the equivalence there, too.

    In fact, it would be great to have a page for Stein duality! to which both entries could point.

    diff, v2, current

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