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• CommentRowNumber1.
• CommentAuthorDmitri Pavlov
• CommentTimeOct 18th 2021

Created:

## Definition

An entire functional calculus algebra is a product-preserving functor

$CartHolo \to Set,$

where $CartHolo$ 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.

## 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.

• 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.