# nLab Stein duality

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## Idea

What has come to be called Stein duality establishes an equivalence between a certain category of complex algebras? and a certain category of Stein spaces in a completely analogous manner to the equivalence between commutative rings and affine schemes, and, more closely, C^∞-rings and smooth loci (see also at duality between algebra and geometry).

## Preliminaries

Recall that a Stein manifold is a complex manifold that admits a proper holomorphic immersion into some $C^n$. More generally, a Stein space is a complex analytic space (i.e., a locally ringed space that is locally isomorphic to the vanishing locus of some ideal of holomorphic functions on~$C^n$) whose reduction is a Stein manifold. A Stein spaces is globally finitely presented if it admits a closed embedding in $C^n$ whose defining ideal is globally finitely generated.

## Definition

(See Proposition~1.13 in Pridham.) 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.

(See Theorem~3.23 in Pirkovskii.) The category of Stein spaces of finite embedding dimension is contravariantly equivalent to the category of those finitely generated EFC-algebras defined by closed ideals. The equivalence functor sends a Stein space to its EFC-algebra of global sections.

## 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.