nLab Stein duality

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 $\mathbf{C}^n$. 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 $\mathbf{C}^n$) whose reduction is a Stein manifold. A Stein space is globally finitely presented if it admits a closed embedding in $\mathbf{C}^n$ 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 (holomorphically finitely generated algebras in Pirkovskii’s terminology of Definition 3.16). The equivalence functor sends a Stein space to its EFC-algebra of global sections.

Related concepts

• Stein space

• EFC-algebra

• C^∞-ring

• duality between geometry and algebra

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.