# nLab EFC-algebra

Contents

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## 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 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.

Last revised on October 18, 2021 at 11:51:50. See the history of this page for a list of all contributions to it.