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1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeOct 18th 2021
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexCreated: ## 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__. ## Related concepts * [[C^∞-ring]] * [[Stein space]] * [[duality between geometry and algebra]] ## 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. <a href="https://ncatlab.org/nlab/revision/EFC-algebra/1">v1</a>, <a href="https://ncatlab.org/nlab/show/EFC-algebra">current</a>

   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.

   Related concepts

   • C^∞-ring

   • Stein space

   • duality between geometry and algebra

   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

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeOct 18th 2021
   • (edited Oct 18th 2021)
   • PermaLink
   Author: Urs
   Format: MarkdownItexInteresting, 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. <a href="https://ncatlab.org/nlab/revision/diff/EFC-algebra/2">diff</a>, <a href="https://ncatlab.org/nlab/revision/EFC-algebra/2">v2</a>, <a href="https://ncatlab.org/nlab/show/EFC-algebra">current</a>

   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